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0ee56c2ef05b58cf | 1861: Quantum
Explain xkcd: It's 'cause you're dumb.
Revision as of 14:50, 11 July 2017 by (Talk)
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If you draw a diagonal line from lower left to upper right, that's the ICP 'Miracles' axis.
Title text: If you draw a diagonal line from lower left to upper right, that's the ICP 'Miracles' axis.
[edit] Explanation
Ambox notice.png This explanation may be incomplete or incorrect: Initial explanation of the idea behind the comic, still needs more detail. General relativity not mentioned yet. Seems it is listed as needing as much math as QM but gives less philosophical arguments...?
The comic depicts a relationship between how philosophically exciting the questions in a field of study are, versus how many years are required to understand the answers. For example, special relativity poses very intriguing philosophical questions, such as "can the temporal ordering of spatially separated events depend on the observer?", or "can time run at different rates for differerent observers?". But it doesn't take a lot of mathematical knowledge to understand the answers - that when objects move very close to the speed of light, time slows down and their lengths contract: the key Lorentz transformations ultimately involve little more than high-school algebra. Hence, Special Relativity is very high up on the y-axis but not very far on the x-axis. Basic physics is not very philosophically interesting but also not very complicated. Fluid dynamics, as captured by the Navier–Stokes equations is very complicated, but it's concerned with a very specific topic - how water or other fluids flow around - so it doesn't lead to big philosophical questions.
The "danger zone" in the top right of the chart is when a field of study is wide-ranging enough to pose broad philosophical questions, and also so complicated that most people can't answer those questions. Quantum mechanics deals with some very strange concepts that readily lend themselves to philosophical questions, such as the idea that merely observing something can change it, or the idea that something can be both a wave and a particle at the same time. However, the explanation for those phenomena is a very complicated piece of math, notably the Schrödinger equation, which means that most people don't have accurate answers to those questions. Randall suggests that this is the reason why so many people have "weird ideas" about quantum mechanics.
1240: Quantum Mechanics also discusses weird ideas that people have about quantum mechanics.
General relativity also presupposes considerable mathematical sophistication to understand the Einstein field equations. However, the main contribution of GR – the explanation of gravity in terms of a curved spacetime – does not seem to induce a lot of philosophical novelty beyond that already seen in special relativity, possibly with the exception of black holes.
The title text references the Insane Clown Posse (ICP) song "Miracles", made memetic by the lyric "Fucking magnets, how do they work?" An axis is the direction on a graph in which some quantity is increasing or decreasing. So things that are far along the "miracle" axis are presumably more miraculous. As you move from bottom-left to top-right on the graph, items become both more philosophically interesting and harder to understand. It would be fair to describe something that's hard to understand and raises big philosophical questions as a "miracle". The ICP "Miracles" axis would also intersect the topic "magnets" infamously mentioned in the song.
[edit] Transcript
[A chart with the Y-axis titled "How Philosophically Exciting the Questions Are to a Novice Student" and the X-axis titled "How Many Years of Math are Needed to Understand the Answers". The upper-right portion of the chart is labeled "Danger Zone". The following topics are charted as follows:
Basic Physics: low excitement, low prerequisites
Fluid Dynamics: low excitement, high prerequisites
Magnets: medium excitement, medium prerequisites
General Relativity: medium excitement, high prerequisites (on the border to the "Danger Zone")
Special Relativity: high excitement, low prerequisites
Quantum Mechanics: high excitement, high prerequisites (in the "Danger Zone")]
[Caption below the panel:]
Why so many people have weird ideas about Quantum Mechanics
The final paragraph probably should note that Magnets are directly on the ICP "Miracles" axis. JamesCurran (talk) 18:34, 10 July 2017 (UTC)
And now I have to listen to "Miracles" again. Thanks explainxkcd. OldCorps (talk) 19:03, 10 July 2017 (UTC)
Unless Randall includes Quantum Field Theory in Quantum Mechanics (which is unusual), General Relativity certainly must be on the right of QM, but on the chart they are almost same level, why? All physics students learn QM, but only small minority take GR course, because mathematically it's much more demanding.
If you look closely, General Relativity is slightly to the right of Quantum Mechanics. 20:33, 10 July 2017 (UTC)
_I'M_ extremely intrigued by Special Relativity being depicted as requiring not much more math than Basic Physics (the only thing I've studied on this chart - I'm not counting magnets as all I know are the grade school basics), but as being vastly more exciting (I enjoyed the physics courses I took, as far as I remember). :) NiceGuy1 (talk) 04:46, 11 July 2017 (UTC)
It's interesting that special relativity is to the left of magnets when you can explain magnetism as a consequence of special relativity, from each charged particle's frame of reference, it's experiencing an electrostatic attraction or repulsion due to length contraction or an altered electric current due to time dilation. 05:11, 11 July 2017 (UTC)
That's way more complicated than special relativity, at least to me.--TheSandromatic (talk) 07:55, 11 July 2017 (UTC)
The thin with magnets is that they are like lasers; they are easy to get used to, but hard to understand the math behind. 07:19, 6 November 2017 (UTC)
He forgot entropy. Maybe around where Special Relativity is? 22:22, 11 July 2017 (UTC)
The Maxwell equations are more complicated than the Lorenz equations. That is why Magnets are to the right of special relativity. 08:33, 11 July 2017 (UTC)
Now I'm listening to "Highway To The Danger Zone". Thanks, upper-right corner! 13:03, 11 July 2017 (UTC)
Every idea anyone has about quantum mechanics is weird. That includes those who can do the math for basic field theory (I have) and beyond. There are no non-weird mental models that fit what the math describes, and experiments validate. 15:02, 12 July 2017 (UTC)
The explanation mentions a couple of philosophical questions, but I'm not sure that a novice to the field would even understand the question. I just can't imagine a room full of people getting excited if you said "Lets explore whether the temporal ordering of spatially separated events depend on the observer." Pudder (talk) 08:06, 11 August 2017 (UTC)
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6262a29902a02083 | Friday, 15 January 2016
Quantum finance
Quantum finance
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Quantum finance is an interdisciplinary research field, applying theories and methods developed by quantum physicists and economists in order to solve problems in finance. It is a branch of econophysics.
Background on instrument pricing[edit]
Finance theory is heavily based on financial instrument pricing such as stock option pricing. Many of the problems facing the finance community have no known analytical solution. As a result, numerical methods and computer simulations for solving these problems have proliferated. This research area is known as computational finance. Many computational finance problems have a high degree of computational complexity and are slow to converge to a solution on classical computers. In particular, when it comes to option pricing, there is additional complexity resulting from the need to respond to quickly changing markets. For example, in order to take advantage of inaccurately priced stock options, the computation must complete before the next change in the almost continuously changing stock market. As a result, the finance community is always looking for ways to overcome the resulting performance issues that arise when pricing options. This has led to research that applies alternative computing techniques to finance.
Background on quantum finance[edit]
One of these alternatives is quantum computing. Just as physics models have evolved from classical to quantum, so has computing. Quantum computers have been shown to outperform classical computers when it comes to simulating quantum mechanics[1] as well as for several other algorithms such as Shor's algorithm for factorization and Grover's algorithm for quantum search, making them an attractive area to research for solving computational finance problems.
Quantum continuous model[edit]
Most quantum option pricing research typically focuses on the quantization of the classical Black–Scholes–Merton equation from the perspective of continuous equations like the Schrödinger equation. Haven [2] builds on the work of Chen[3] and others, but considers the market from the perspective of the Schrödinger equation. The key message in Haven's work is that the Black–Scholes–Merton equation is really a special case of the Schrödinger equation where markets are assumed to be efficient. The Schrödinger-based equation that Haven derives has a parameter ħ (not to be confused with the complex conjugate of h) that represents the amount of arbitrage that is present in the market resulting from a variety of sources including non-infinitely fast price changes, non-infinitely fast information dissemination and unequal wealth among traders. Haven argues that by setting this value appropriately, a more accurate option price can be derived, because in reality, markets are not truly efficient.
This is one of the reasons why it is possible that a quantum option pricing model could be more accurate than a classical one. Baaquie [4] has published many papers on quantum finance and even written a book[5] that brings many of them together. Core to Baaquie's research and others like Matacz [6] are Feynman's path integrals.
Baaquie applies path integrals to several exotic options and presents analytical results comparing his results to the results of Black–Scholes–Merton equation showing that they are very similar. Piotrowski et al.[7] take a different approach by changing the Black–Scholes–Merton assumption regarding the behavior of the stock underlying the option. Instead of assuming it follows a Wiener-Bachelier process,[8] they assume that it follows an Ornstein-Uhlenbeck process.[9] With this new assumption in place, they derive a quantum finance model as well as a European call option formula.
Other models such as Hull-White[10] and Cox-Ingersoll-Ross[11] have successfully used the same approach in the classical setting with interest rate derivatives. Khrennikov[12] builds on the work of Haven and others and further bolsters the idea that the market efficiency assumption made by the Black–Scholes–Merton equation may not be appropriate. To support this idea, Khrennikov builds on a framework of contextual probabilities using agents as a way of overcoming criticism of applying quantum theory to finance. Accardi and Boukas[13] again quantize the Black–Scholes–Merton equation, but in this case, they also consider the underlying stock to have both Brownian and Poisson processes.
Quantum binomial model[edit]
Chen published a paper in 2001,[3] where he presents a quantum binomial options pricing model or simply abbreviated as the quantum binomial model. Metaphorically speaking, Chen's quantum binomial options pricing model (referred to hereafter as the quantum binomial model) is to existing quantum finance models what the Cox-Ross-Rubinstein classical binomial options pricing model was to the Black–Scholes–Merton model: a discretized and simpler version of the same result. These simplifications make the respective theories not only easier to analyze but also easier to implement on a computer.
Multi-step quantum binomial model[edit]
In the multi-step model the quantum pricing formula is:
which is the equivalent of the Cox-Ross-Rubinstein binomial options pricing model formula as follows:
This shows that assuming stocks behave according to Maxwell-Boltzmann classical statistics, the quantum binomial model does indeed collapse to the classical binomial model.
Quantum volatility is as follows as per Meyer:[14]
Bose-Einstein assumption[edit]
Maxwell-Boltzmann statistics can be replaced by the quantum Bose-Einstein statistics resulting in the following option price formula:
The Bose-Einstein equation will produce option prices that will differ from those produced by the Cox-Ross-Rubinstein option pricing formula in certain circumstances. This is because the stock is being treated like a quantum boson particle instead of a classical particle.
1. Jump up ^ B. Boghosian (1998). "Simulating quantum mechanics on a quantum computer". Physica D.
2. Jump up ^ Haven, Emmanuel (2002). "A discussion on embedding the Black–Scholes option pricing model in a quantum physics setting". Physica A. Bibcode:2002PhyA..304..507H. doi:10.1016/S0378-4371(01)00568-4.
3. ^ Jump up to: a b Zeqian Chen (2004). "Quantum Theory for the Binomial Model in Finance Theory". Journal of Systems Science and Complexity. arXiv:quant-ph/0112156.
4. Jump up ^ Baaquie, Belal E.; Coriano, Claudio; Srikant, Marakani (2002). "Quantum Mechanics, Path Integrals and Option Pricing: Reducing the Complexity of Finance". ArXiv Condensed Matter e-prints: 8191. arXiv:cond-mat/0208191. Bibcode:2002cond.mat..8191B.
5. Jump up ^ Baaquie, Belal (2004). Quantum Finance: Path Integrals and Hamiltonians for Options and Interest Rates. Cambridge University Press. p. 332. ISBN 978-0-521-84045-3.
6. Jump up ^ "Path dependent option pricing, The path integral partial averaging method". Journal of Computational Finance. 2002. arXiv:cond-mat/0005319v1.
7. Jump up ^ Piotrowski, Edward W.; Schroeder, Małgorzata; Zambrzycka, Anna (2006). "Quantum extension of European option pricing based on the Ornstein Uhlenbeck process". Physica A (Physica A Statistical Mechanics and its Applications) 368: 176. arXiv:quant-ph/0510121. Bibcode:2006PhyA..368..176P. doi:10.1016/j.physa.2005.12.021.
8. Jump up ^ Hull, John (2006). Options, futures, and other derivatives. Upper Saddle River, N.J: Pearson/Prentice Hall. ISBN 0-13-149908-4.
9. Jump up ^ "On the Theory of {B}rownian Motion". The Journal of Political Economy. 1930.
10. Jump up ^ "The pricing of options on interest rate caps and floors using the Hull-White model". Advanced Strategies in Financial Risk Management. 1990.
11. Jump up ^ "A theory of the term structure of interest rates". Physica A. 1985.
12. Jump up ^ Khrennikov, Andrei (2007). "Classical and quantum randomness and the financial market" 0704. ArXiv e-prints: 2865. arXiv:0704.2865. Bibcode:2007arXiv0704.2865K.
13. Jump up ^ Accardi, Luigi; Boukas, Andreas. "The Quantum Black-Scholes Equation". arXiv:0706.1300v1.
14. Jump up ^ Keith Meyer (2009). Extending and simulating the quantum binomial options pricing model. The University of Manitoba.
External links[edit]
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21a531e3713ec93d | Our Universe-Infinite and Eternal: Its Physics, Nature, and Cosmology PDF (Adobe DRM) download by Barry Bruce
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The field equations of Einstein's General Relativity are solved for an infinite universe with uniform density. One of the three solutions, the Infinite Universe of Einstein and Newton, fits all the data for the Hubble diagram better than the Big Bang. Next, using general relativity and the physics that evolved from Newton, the force of gravity between two massive point particles is found. Utilizing this force and the Infinite Universe of Einstein and Newton model, the net force of gravity on a point particle in arbitrary motion, due the uniform mass distribution of the universe, is calculated by integration. This net force of gravity is found to be equal to the Force of Inertia. These calculations explain Newton's First Law, Newton's Second Law, and the equivalence of inertial and gravitational mass. The middle of the book deals with the development of quantum mechanics. Here it is shown that hidden within the classical mechanics of particles there is the phase of a wave, associated with a particle, that moves at the speed of a de Broglie wave. The form of the phase of the wave is developed. Making use of the form of the phase, the Hamilton-Jacobi equation for a particle is setup to be solved using an integrating factor. The resulting equation is manipulated directly into the form of the Schrödinger equation. This development requires that the particle Hamilton-Jacobi equation has a solution whenever the Schrödinger equation has a solution and vice versa. The classical wave function is then shown to have exactly the same mathematical properties as the quantum mechanical wave function, including the fact that the absolute value squared of the classical wave function has the mathematical properties of a probability density. However, the interpretation that this is a probability density for the particle is shown not to hold. Lastly, the missing matter problem is resolved by showing that the dynamics and the mass of a spiral galaxy are better and more naturally explained by using ordinary physics with ordinary interacting matter than they are by postulating and using exotic weakly interacting dark matter.
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393aebca8f01f0c1 | GPGPU with WebGL: solving Laplace’s equation
This is the first post in what will hopefully be a series of posts exploring how to use WebGL to do GPGPU (General-purpose computing on graphics processing units). In this installment we will solve a partial differential equation using WebGL, the Laplace’s equation more specifically.
Discretizing the Laplace’s equation
The Laplace’s equation, \nabla^2 \phi = 0, is one of the most ubiquitous partial differential equations in physics. It appears in lot of areas, including electrostatics, heat conduction and fluid flow.
To get a numerical solution of a differential equation, the first step is to replace the continuous domain by a lattice and the differential operators with their discrete versions. In our case, we just have to replace the Laplacian by its discrete version:
\displaystyle \nabla^2 \phi(x) = 0 \rightarrow \frac{1}{h^2}\left(\phi_{i-1\,j} + \phi_{i+1\,j} + \phi_{i\,j-1} + \phi_{i\,j+1} - 4\phi_{i\,j}\right) = 0,
where h is the grid size.
If we apply this equation at all internal points of the lattice (the external points must retain fixed values if we use Dirichlet boundary conditions) we get a big system of linear equations whose solution will give a numerical approximation to a solution of the Laplace’s equation. Of the various methods to solve big linear systems, the Jacobi relaxation method seems the best fit to shaders, because it applies the same expression at every lattice point and doesn’t have dependencies between computations. Applying this method to our linear system, we get the following expression for the iteration:
\displaystyle \phi_{i\,j}^{(k+1)} = \frac{1}{4}\left(\phi_{i-1\,j}^{(k)} + \phi_{i+1\,j}^{(k)} + \phi_{i\,j-1}^{(k)} + \phi_{i\,j+1}^{(k)}\right),
where k is a step index.
Solving the discretized problem using WebGL shaders
If we use a texture to represent the domain and a fragment shader to do the Jacobi relaxation steps, the shader will follow this general pseudocode:
1. Check if this fragment is a boundary point. If it’s one, return the previous value of this point.
2. Get the four nearest neighbors’ values.
3. Return the average of their values.
To flesh out this pseudocode, we need to define a specific representation for the discretized domain. Taking into account that the currently available WebGL versions don’t support floating point textures, we can use 32 bits RGBA fragments and do the following mapping:
R: Higher byte of \phi.
G: Lower byte of \phi.
B: Unused.
A: 1 if it’s a boundary value, 0 otherwise.
Most of the code is straightforward, but doing the multiprecision arithmetic is tricky, as the quantities we are working with behave as floating point numbers in the shaders but are stored as integers. More specifically, the color numbers in the normal range, [0.0, 1.0], are multiplied by 255 and rounded to the nearest byte value when stored at the target texture.
My first idea was to start by reconstructing the floating point numbers for each input value, do the required operations with the floating numbers and convert the floating point numbers to color components that can be reliably stored (without losing precision). This gives us the following pseudocode for the iteration shader:
// wc is the color to the "west", ec is the color to the "east", ...
float w_val = wc.r + wc.g / 255.0;
float e_val = ec.r + ec.g / 255.0;
// ...
float val = (w_val + e_val + n_val + s_val) / 4.0;
float hi = val - mod(val, 1.0 / 255.0);
float lo = (val - hi) * 255.0;
fragmentColor = vec4(hi, lo, 0.0, 0.0);
The reason why we multiply by 255 in place of 256 is that we need val_lo to keep track of the part of val that will be lost when we store it as a color component. As each byte value of a discrete color component will be associated with a range of size 1/255 in its continuous counterpart, we need to use the “low byte” to store the position of the continuous component within that range.
Simplifying the code to avoid redundant operations, we get:
float val = (wc.r + ec.r + nc.r + sc.r) / 4.0 +
(wc.g + ec.g + nc.g + sc.g) / (4.0 * 255.0);
float lo = (val - hi) * 255.0;
The result of running the full code, implemented in GLSL, is:
Solving the Laplace's equation using a 32x32 grid. Click the picture to see the live solving process (if your browser supports WebGL).
As can be seen, it has quite low resolution but converges fast. But if we just crank up the number of points, the convergence gets slower:
Incompletely converged solution in a 512x512 grid. Click the picture to see a live version.
How can we reconcile these approaches?
The basic idea behind multigrid methods is to apply the relaxation method on a hierarchy of increasingly finer discretizations of the problem, using in each step the coarse solution obtained in the previous grid as the “starting guess”. In this mode, the long wavelength parts of the solution (those that converge slowly in the finer grids) are obtained in the first coarse iterations, and the last iterations just add the finer parts of the solution (those that converge relatively easily in the finer grids).
The implementation is quite straightforward, giving us fast convergence and high resolution at the same time:
Multigrid solution using grids from 8x8 to 512x512. Click the picture to see the live version.
It’s quite viable to use WebGL to do at least basic GPGPU tasks, though it is, in a certain sense, a step backward in time, as there is no CUDA, floating point textures or any feature that helps when working with non-graphic problems: you are on your own. But with the growing presence of WebGL support in modern browsers, it’s an interesting way of partially accessing the enormous computational power present in modern video cards from any JS application, without requiring the installation of a native application.
In the next posts we will explore other kinds of problem-solving where WebGL can provide a great performance boost.
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5 thoughts on “GPGPU with WebGL: solving Laplace’s equation
1. Evgeny says:
Very nice application. There are floating point textures in the nightly Chrome (for about 2 months)
There is “The Energy2D Simulator” open source Java based project
with very nice turbulent flows (3-5 applets). They used implicit scheme and relaxation. You could move in this directions too :)
• mchouza says:
You can see a more complex example of the same techniques in this (not very accurate and still unfinished) simulation of the two slits experiment with the Schrödinger equation:
In my next posts I will probably transition to floating point textures for this kind of simulations, as working with the combination of integer textures and floating point values in the shaders is quite painful :-D
Thanks for your comment and your very interesting website!
2. [...] This is very cool indeed — GPGPU with WebGL: solving Laplace’s equation [...]
3. [...] In a previous post we solved Laplace’s Equation using WebGL. We will see how to implement the Lattice Boltzmann algorithm using WebGL shaders in the next post, but this post has a preview of the solution: Click on the image to go to the demo. New obstacles can be created by dragging the mouse over the simulation area. [...]
4. [...] method is introduced with WebGL demos in this blog. Demidov wrote something about Multigrid recently. Real-Time Gradient-Domain Painting is an [...]
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5b2e95051ff5819c | Kyungmo Ahn, Sun-Kyung Kim, Se-Hyun Cheon
This paper presents the occurrence probability of freak waves based on the analysis of extensive wave data collected during ARSLOE project. It is suggested to use the probability distribution of extreme waves heights as a possible means of defining the freak wave criteria instead of conventional definition which is the wave height greater than the twice of the significant wave height. Analysis of wave data provided such finding as 1) threshold tolerance of 0.2 m is recommended for the discrimination of the false wave height due to noise, 2) no supportive evidence on the linear relationship between the occurrence probability of freak waves and the kurtosis of surface elevation 3) nonlinear wave-wave interactions is not thh primary cause of the generation of freak waves 4) the occurrence of freak waves does not depend on the wave period 5) probability density function of extreme waves can be used to predict the occurrence probability of freak waves. Three different distribution functions of extreme wave height by Rayleigh, Ahn, and Mori were compared for the analysis of freak waves.
freak wave; occurrence probability; extreme wave; significant wave height
Ahn, K. 2002. Probability distribution of extreme wave heights in finite water depth, Proceedings of 29 th International Conference on Coastal Engineering, ASCE, 614-625.
Henderson, K.L, D.H. Peregrine and J.W. Dold. 1999. Unsteady water wave modulations: fully nonlinear solutions and comparison with the nonlinear Schrödinger equation. Wave Motion 29, 241-361.http://dx.doi.org/10.1016/S0165-2125(98)00045-6
Janssen, P.A.E.M. 2003. Nonlinear four-wave interactions and freak waves, Journal of Physical Oceanography 33, 863-884.http://dx.doi.org/10.1175/1520-0485(2003)33<863:NFIAFW>2.0.CO;2
Kharif, C. and E. Pelinovsky 2003. Physical mechanisms of the rogue wave phenomenon. European Journal of Mechanics – B/Fluids 22 (6), 603-634.http://dx.doi.org/10.1016/j.euromechflu.2003.09.002
Liu, P.C., H.S. Chen, H.J. Doong, C.C. Kao and Y.J. Hsu 2009. Freque waves during Typhoon Krosa, Annales Geophysicae, 27, 2633-2642.http://dx.doi.org/10.5194/angeo-27-2633-2009
Mori, N. 2004. Occurrence probability of a freak wave in a nonlinear wave field, Ocean Engineering, 31, 165-175.http://dx.doi.org/10.1016/S0029-8018(03)00119-7
Mori, N. and P.A.E.M Janssen 2006. On kurtosis and occurrence probability of freak waves. Journal of Physical Oceanography 36, 1471-1483http://dx.doi.org/10.1175/JPO2922.1
Mori, N., P. Liu, and Y. Yasuda 2002. Analysis of freak wave measurements in the Sea of Japan, Ocean Engineering, 29 (11), 1399-1414.http://dx.doi.org/10.1016/S0029-8018(01)00073-7
Olagnon, M. and Athanassoulis, G. (Eds.) 2000. Rogue waves. IFRMER, France.
Ochi, M. 1998. Ocean waves: The stochastic approach, Cambridge University Presshttp://dx.doi.org/10.1017/CBO9780511529559
Stansell, P. 2005. Distributions of extreme wave crest and trough heights measured in the North Sea. Ocean Engineering 32, 1015-1036.http://dx.doi.org/10.1016/j.oceaneng.2004.10.016
Yasuda, T. and N. Mori 1997. Occurrence properties of giant freak waves in sea area around Japan.
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61319d087324422e | Eigenvalues and eigenvectors
From Wikipedia, the free encyclopedia
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In this shear mapping the red arrow changes direction but the blue arrow does not. The blue arrow is an eigenvector of this shear mapping, and since its length is unchanged its eigenvalue is 1.
An eigenvector of a square matrix A is a non-zero vector v that, when the matrix is multiplied by v, yields a constant multiple of v, the multiplier being commonly denoted by \lambda. That is:
A v = \lambda v
(Because this equation uses post-multiplication by v, it describes a right eigenvector.)
The number \lambda is called the eigenvalue of A corresponding to v.[1]
In analytic geometry, for example, a three-element vector may be seen as an arrow in three-dimensional space starting at the origin. In that case, an eigenvector v is an arrow whose direction is either preserved or exactly reversed after multiplication by A. The corresponding eigenvalue determines how the length of the arrow is changed by the operation, and whether its direction is reversed or not, determined by whether the eigenvalue is negative or positive.
In abstract linear algebra, these concepts are naturally extended to more general situations, where the set of real scalar factors is replaced by any field of scalars (such as algebraic or complex numbers); the set of Cartesian vectors \mathbb{R}^n is replaced by any vector space (such as the continuous functions, the polynomials or the trigonometric series), and matrix multiplication is replaced by any linear operator that maps vectors to vectors (such as the derivative from calculus). In such cases, the "vector" in "eigenvector" may be replaced by a more specific term, such as "eigenfunction", "eigenmode", "eigenface", or "eigenstate". Thus, for example, the exponential function f(x) = a^x is an eigenfunction of the derivative operator " {}' ", with eigenvalue \lambda = \ln a, since its derivative is f'(x) = (\ln a)a^x = \lambda f(x).
The set of all eigenvectors of a matrix (or linear operator), each paired with its corresponding eigenvalue, is called the eigensystem of that matrix.[2] Any multiple of an eigenvector is also an eigenvector, with the same eigenvalue. An eigenspace of a matrix A is the set of all eigenvectors with the same eigenvalue, together with the zero vector.[1] An eigenbasis for A is any basis for the set of all vectors that consists of linearly independent eigenvectors of A. Not every matrix has an eigenbasis, but every symmetric matrix does.
The terms characteristic vector, characteristic value, and characteristic space are also used for these concepts. The prefix eigen- is adopted from the German word eigen for "self-" or "unique to", "peculiar to", or "belonging to."
Eigenvalues and eigenvectors have many applications in both pure and applied mathematics. They are used in matrix factorization, in quantum mechanics, and in many other areas.
Eigenvectors and eigenvalues of a real matrix[edit]
In many contexts, a vector can be assumed to be a list of real numbers (called elements), written vertically with brackets around the entire list, such as the vectors u and v below. Two vectors are said to be scalar multiples of each other (also called parallel or collinear) if they have the same number of elements, and if every element of one vector is obtained by multiplying each corresponding element in the other vector by the same number (known as a scaling factor, or a scalar). For example, the vectors
u = \begin{bmatrix}1\\3\\4\end{bmatrix}\quad\quad\quad and \quad\quad\quad v = \begin{bmatrix}-20\\-60\\-80\end{bmatrix}
are scalar multiples of each other, because each element of v is −20 times the corresponding element of u.
A vector with three elements, like u or v above, may represent a point in three-dimensional space, relative to some Cartesian coordinate system. It helps to think of such a vector as the tip of an arrow whose tail is at the origin of the coordinate system. In this case, the condition "u is parallel to v" means that the two arrows lie on the same straight line, and may differ only in length and direction along that line.
If we multiply any square matrix A with n rows and n columns by such a vector v, the result will be another vector w = A v , also with n rows and one column. That is,
\begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} \quad\quad is mapped to
\begin{bmatrix} w_1 \\ w_2 \\ \vdots \\ w_n \end{bmatrix} \;=\;
\begin{bmatrix} A_{1,1} & A_{1,2} & \ldots & A_{1,n} \\
A_{2,1} & A_{2,2} & \ldots & A_{2,n} \\
A_{n,1} & A_{n,2} & \ldots & A_{n,n} \\
where, for each index i,
w_i = A_{i,1} v_1 + A_{i,2} v_2 + \cdots + A_{i,n} v_n = \sum_{j = 1}^{n} A_{i,j} v_j
In general, if v is not all zeros, the vectors v and A v will not be parallel. When they are parallel (that is, when there is some real number \lambda such that A v = \lambda v) we say that v is an eigenvector of A. In that case, the scale factor \lambda is said to be the eigenvalue corresponding to that eigenvector.
In particular, multiplication by a 3×3 matrix A may change both the direction and the magnitude of an arrow v in three-dimensional space. However, if v is an eigenvector of A with eigenvalue \lambda, the operation may only change its length, and either keep its direction or flip it (make the arrow point in the exact opposite direction). Specifically, the length of the arrow will increase if |\lambda| > 1, remain the same if |\lambda| = 1, and decrease it if |\lambda|< 1. Moreover, the direction will be precisely the same if \lambda > 0, and flipped if \lambda < 0. If \lambda = 0, then the length of the arrow becomes zero.
An example[edit]
The transformation matrix \bigl[ \begin{smallmatrix} 2 & 1\\ 1 & 2 \end{smallmatrix} \bigr] preserves the direction of vectors parallel to \bigl[ \begin{smallmatrix} 1 \\ 1 \end{smallmatrix} \bigr] (in blue) and \bigl[ \begin{smallmatrix} 1 \\ -1 \end{smallmatrix} \bigr] (in violet). The points that lie on the line through the origin, parallel to an eigenvector, remain on the line after the transformation. The vectors in red are not eigenvectors, therefore their direction is altered by the transformation. See also: An extended version, showing all four quadrants.
For the transformation matrix
A = \begin{bmatrix} 3 & 1\\1 & 3 \end{bmatrix},
the vector
v = \begin{bmatrix} 4 \\ -4 \end{bmatrix}
is an eigenvector with eigenvalue 2. Indeed,
A v = \begin{bmatrix} 3 & 1\\1 & 3 \end{bmatrix} \begin{bmatrix} 4 \\ -4 \end{bmatrix} = \begin{bmatrix} 3 \cdot 4 + 1 \cdot (-4) \\ 1 \cdot 4 + 3 \cdot (-4) \end{bmatrix}
= \begin{bmatrix} 8 \\ -8 \end{bmatrix} = 2 \cdot \begin{bmatrix} 4 \\ -4 \end{bmatrix}.
On the other hand the vector
is not an eigenvector, since
\begin{bmatrix} 3 & 1\\1 & 3 \end{bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 3 \cdot 0 + 1 \cdot 1 \\ 1 \cdot 0 + 3 \cdot 1 \end{bmatrix} = \begin{bmatrix} 4 \\ 4 \end{bmatrix},
and this vector is not a multiple of the original vector v.
Another example[edit]
For the matrix
A= \begin{bmatrix} 1 & 2 & 0\\0 & 2 & 0\\ 0 & 0 & 3\end{bmatrix},
we have
A \begin{bmatrix} 1\\0\\0 \end{bmatrix} = \begin{bmatrix} 1\\0\\0 \end{bmatrix} = 1 \cdot \begin{bmatrix} 1\\0\\0 \end{bmatrix},\quad\quad
A \begin{bmatrix} 0\\0\\1 \end{bmatrix} = \begin{bmatrix} 0\\0\\3 \end{bmatrix} = 3 \cdot \begin{bmatrix} 0\\0\\1 \end{bmatrix}.\quad\quad
Therefore, the vectors [1,0,0]^\mathsf{T} and [0,0,1]^\mathsf{T} are eigenvectors of A corresponding to the eigenvalues 1 and 3 respectively. (Here the symbol {}^\mathsf{T} indicates matrix transposition, in this case turning the row vectors into column vectors.)
Trivial cases[edit]
The identity matrix I (whose general element I_{i j} is 1 if i = j, and 0 otherwise) maps every vector to itself. Therefore, every vector is an eigenvector of I, with eigenvalue 1.
More generally, if A is a diagonal matrix (with A_{i j} = 0 whenever i \neq j), and v is a vector parallel to axis i (that is, v_i \neq 0, and v_j = 0 if j \neq i), then A v = \lambda v where \lambda = A_{i i}. That is, the eigenvalues of a diagonal matrix are the elements of its main diagonal. This is trivially the case of any 1 ×1 matrix.
General definition[edit]
The concept of eigenvectors and eigenvalues extends naturally to abstract linear transformations on abstract vector spaces. Namely, let V be any vector space over some field K of scalars, and let T be a linear transformation mapping V into V. We say that a non-zero vector v of V is an eigenvector of T if (and only if) there is a scalar \lambda in K such that
T(v)=\lambda v.
This equation is called the eigenvalue equation for T, and the scalar \lambda is the eigenvalue of T corresponding to the eigenvector v. Note that T(v) means the result of applying the operator T to the vector v, while \lambda v means the product of the scalar \lambda by v.[3]
The matrix-specific definition is a special case of this abstract definition. Namely, the vector space V is the set of all column vectors of a certain size n×1, and T is the linear transformation that consists in multiplying a vector by the given n\times n matrix A.
Some authors allow v to be the zero vector in the definition of eigenvector.[4] This is reasonable as long as we define eigenvalues and eigenvectors carefully: If we would like the zero vector to be an eigenvector, then we must first define an eigenvalue of T as a scalar \lambda in K such that there is a nonzero vector v in V with T(v) = \lambda v . We then define an eigenvector to be a vector v in V such that there is an eigenvalue \lambda in K with T(v) = \lambda v . This way, we ensure that it is not the case that every scalar is an eigenvalue corresponding to the zero vector.
Eigenspace and spectrum[edit]
If v is an eigenvector of T, with eigenvalue \lambda, then any scalar multiple \alpha v of v with nonzero \alpha is also an eigenvector with eigenvalue \lambda, since T(\alpha v) = \alpha T(v) = \alpha(\lambda v) = \lambda(\alpha v). Moreover, if u and v are eigenvectors with the same eigenvalue \lambda, then u+v is also an eigenvector with the same eigenvalue \lambda. Therefore, the set of all eigenvectors with the same eigenvalue \lambda, together with the zero vector, is a linear subspace of V, called the eigenspace of T associated to \lambda.[5][6] If that subspace has dimension 1, it is sometimes called an eigenline.[7]
The geometric multiplicity \gamma_T(\lambda) of an eigenvalue \lambda is the dimension of the eigenspace associated to \lambda, i.e. number of linearly independent eigenvectors with that eigenvalue.
The eigenspaces of T always form a direct sum (and as a consequence any family of eigenvectors for different eigenvalues is always linearly independent). Therefore the sum of the dimensions of the eigenspaces cannot exceed the dimension n of the space on which T operates, and in particular there cannot be more than n distinct eigenvalues.[8]
The set of eigenvalues of T is sometimes called the spectrum of T.
An eigenbasis for a linear operator T that operates on a vector space V is a basis for V that consists entirely of eigenvectors of T (possibly with different eigenvalues). Such a basis exists precisely if the direct sum of the eigenspaces equals the whole space, in which case one can take the union of bases chosen in each of the eigenspaces as eigenbasis. The matrix of T in a given basis is diagonal precisely when that basis is an eigenbasis for T, and for this reason T is called diagonalizable if it admits an eigenbasis.
Generalizations to infinite-dimensional spaces[edit]
The definition of eigenvalue of a linear transformation T remains valid even if the underlying space V is an infinite dimensional Hilbert or Banach space. Namely, a scalar \lambda is an eigenvalue if and only if there is some nonzero vector v such that T(v) = \lambda v.
A widely used class of linear operators acting on infinite dimensional spaces are the differential operators on function spaces. Let D be a linear differential operator in on the space \mathbf{C^\infty} of infinitely differentiable real functions of a real argument t. The eigenvalue equation for D is the differential equation
D f = \lambda f
The functions that satisfy this equation are commonly called eigenfunctions. For the derivative operator d/dt, an eigenfunction is a function that, when differentiated, yields a constant times the original function. If \lambda is zero, the generic solution is a constant function. If \lambda is non-zero, the solution is an exponential function
f(t) = Ae^{\lambda t}.\
Eigenfunctions are an essential tool in the solution of differential equations and many other applied and theoretical fields. For instance, the exponential functions are eigenfunctions of any shift invariant linear operator. This fact is the basis of powerful Fourier transform methods for solving all sorts of problems.
Spectral theory[edit]
If \lambda is an eigenvalue of T, then the operator T-\lambda I is not one-to-one, and therefore its inverse (T-\lambda I)^{-1} is not defined. The converse is true for finite-dimensional vector spaces, but not for infinite-dimensional ones. In general, the operator T - \lambda I may not have an inverse, even if \lambda is not an eigenvalue.
For this reason, in functional analysis one defines the spectrum of a linear operator T as the set of all scalars \lambda for which the operator T-\lambda I has no bounded inverse. Thus the spectrum of an operator always contains all its eigenvalues, but is not limited to them.
Associative algebras and representation theory[edit]
More algebraically, rather than generalizing the vector space to an infinite dimensional space, one can generalize the algebraic object that is acting on the space, replacing a single operator acting on a vector space with an algebra representation – an associative algebra acting on a module. The study of such actions is the field of representation theory.
A closer analog of eigenvalues is given by the representation-theoretical concept of weight, with the analogs of eigenvectors and eigenspaces being weight vectors and weight spaces.
Eigenvalues and eigenvectors of matrices[edit]
Characteristic polynomial[edit]
The eigenvalue equation for a matrix A is
A v - \lambda v = 0,
which is equivalent to
(A-\lambda I)v = 0,
where I is the n\times n identity matrix. It is a fundamental result of linear algebra that an equation M v = 0 has a non-zero solution v if, and only if, the determinant \det(M) of the matrix M is zero. It follows that the eigenvalues of A are precisely the real numbers \lambda that satisfy the equation
\det(A-\lambda I) = 0
The left-hand side of this equation can be seen (using Leibniz' rule for the determinant) to be a polynomial function of the variable \lambda. The degree of this polynomial is n, the order of the matrix. Its coefficients depend on the entries of A, except that its term of degree n is always (-1)^n\lambda^n. This polynomial is called the characteristic polynomial of A; and the above equation is called the characteristic equation (or, less often, the secular equation) of A.
For example, let A be the matrix
A =
2 & 0 & 0 \\
0 & 3 & 4 \\
0 & 4 & 9
The characteristic polynomial of A is
\det (A-\lambda I) \;=\; \det \left(\begin{bmatrix}
2 & 0 & 0 \\
0 & 3 & 4 \\
0 & 4 & 9
\end{bmatrix} - \lambda
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{bmatrix}\right) \;=\;
\det \begin{bmatrix}
2 - \lambda & 0 & 0 \\
0 & 3 - \lambda & 4 \\
0 & 4 & 9 - \lambda
which is
(2 - \lambda) \bigl[ (3 - \lambda) (9 - \lambda) - 16 \bigr] = -\lambda^3 + 14\lambda^2 - 35\lambda + 22
The roots of this polynomial are 2, 1, and 11. Indeed these are the only three eigenvalues of A, corresponding to the eigenvectors [1,0,0]', [0,2,-1]', and [0,1,2]' (or any non-zero multiples thereof).
In the real domain[edit]
Since the eigenvalues are roots of the characteristic polynomial, an n\times n matrix has at most n eigenvalues. If the matrix has real entries, the coefficients of the characteristic polynomial are all real; but it may have fewer than n real roots, or no real roots at all.
For example, consider the cyclic permutation matrix
This matrix shifts the coordinates of the vector up by one position, and moves the first coordinate to the bottom. Its characteristic polynomial is 1 - \lambda^3 which has one real root \lambda_1 = 1. Any vector with three equal non-zero elements is an eigenvector for this eigenvalue. For example,
A \begin{bmatrix} 5\\5\\5 \end{bmatrix} =
1 \cdot \begin{bmatrix} 5\\5\\5 \end{bmatrix}
In the complex domain[edit]
The fundamental theorem of algebra implies that the characteristic polynomial of an n\times n matrix A, being a polynomial of degree n, has exactly n complex roots. More precisely, it can be factored into the product of n linear terms,
\det(A-\lambda I) = (\lambda_1 - \lambda )(\lambda_2 - \lambda)\cdots(\lambda_n - \lambda)
where each \lambda_i is a complex number. The numbers \lambda_1, \lambda_2, ... \lambda_n, (which may not be all distinct) are roots of the polynomial, and are precisely the eigenvalues of A.
Even if the entries of A are all real numbers, the eigenvalues may still have non-zero imaginary parts (and the elements of the corresponding eigenvectors will therefore also have non-zero imaginary parts). Also, the eigenvalues may be irrational numbers even if all the entries of A are rational numbers, or all are integers. However, if the entries of A are algebraic numbers (which include the rationals), the eigenvalues will be (complex) algebraic numbers too.
The non-real roots of a real polynomial with real coefficients can be grouped into pairs of complex conjugate values, namely with the two members of each pair having the same real part and imaginary parts that differ only in sign. If the degree is odd, then by the intermediate value theorem at least one of the roots will be real. Therefore, any real matrix with odd order will have at least one real eigenvalue; whereas a real matrix with even order may have no real eigenvalues.
In the example of the 3×3 cyclic permutation matrix A, above, the characteristic polynomial 1 - \lambda^3 has two additional non-real roots, namely
\lambda_2 = -1/2 + \mathbf{i}\sqrt{3}/2\quad\quad and \quad\quad\lambda_3 = \lambda_2^* = -1/2 - \mathbf{i}\sqrt{3}/2,
where \mathbf{i}= \sqrt{-1} is the imaginary unit. Note that \lambda_2\lambda_3 = 1, \lambda_2^2 = \lambda_3, and \lambda_3^2 = \lambda_2. Then
A \begin{bmatrix} 1 \\ \lambda_2 \\ \lambda_3 \end{bmatrix} =
\begin{bmatrix} \lambda_2\\ \lambda_3 \\1 \end{bmatrix} =
A \begin{bmatrix} 1 \\ \lambda_3 \\ \lambda_2 \end{bmatrix} =
\begin{bmatrix} \lambda_3 \\ \lambda_2 \\ 1 \end{bmatrix} =
\lambda_3 \cdot \begin{bmatrix} 1 \\ \lambda_3 \\ \lambda_2 \end{bmatrix}
Therefore, the vectors [1,\lambda_2,\lambda_3]' and [1,\lambda_3,\lambda_2]' are eigenvectors of A, with eigenvalues \lambda_2, and \lambda_3, respectively.
Algebraic multiplicities[edit]
Let \lambda_i be an eigenvalue of an n\times n matrix A. The algebraic multiplicity \mu_A(\lambda_i) of \lambda_i is its multiplicity as a root of the characteristic polynomial, that is, the largest integer k such that (\lambda - \lambda_i)^k divides evenly that polynomial.
Like the geometric multiplicity \gamma_A(\lambda_i), the algebraic multiplicity is an integer between 1 and n; and the sum \boldsymbol{\mu}_A of \mu_A(\lambda_i) over all distinct eigenvalues also cannot exceed n. If complex eigenvalues are considered, \boldsymbol{\mu}_A is exactly n.
It can be proved that the geometric multiplicity \gamma_A(\lambda_i) of an eigenvalue never exceeds its algebraic multiplicity \mu_A(\lambda_i). Therefore, \boldsymbol{\gamma}_A is at most \boldsymbol{\mu}_A.
If \gamma_A(\lambda_i) = \mu_A(\lambda_i), then \lambda_i is said to be a semisimple eigenvalue.
For the matrix: A= \begin{bmatrix}
2 & 0 & 0 & 0 \\
1 & 2 & 0 & 0 \\
0 & 1 & 3 & 0 \\
0 & 0 & 1 & 3
the characteristic polynomial of A is \det (A-\lambda I) \;=\;
\det \begin{bmatrix}
2- \lambda & 0 & 0 & 0 \\
1 & 2- \lambda & 0 & 0 \\
0 & 1 & 3- \lambda & 0 \\
0 & 0 & 1 & 3- \lambda
\end{bmatrix}= (2 - \lambda)^2 (3 - \lambda)^2 ,
being the product of the diagonal with a lower triangular matrix.
The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The algebraic multiplicity of each eigenvalue is 2; in other words they are both double roots. On the other hand, the geometric multiplicity of the eigenvalue 2 is only 1, because its eigenspace is spanned by the vector [0,1,-1,1], and is therefore 1 dimensional. Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by [0,0,0,1]. Hence, the total algebraic multiplicity of A, denoted \mu_A, is 4, which is the most it could be for a 4 by 4 matrix. The geometric multiplicity \gamma_A is 2, which is the smallest it could be for a matrix which has two distinct eigenvalues.
Diagonalization and eigendecomposition[edit]
If the sum \boldsymbol{\gamma}_A of the geometric multiplicities of all eigenvalues is exactly n, then A has a set of n linearly independent eigenvectors. Let Q be a square matrix whose columns are those eigenvectors, in any order. Then we will have A Q = Q\Lambda , where \Lambda is the diagonal matrix such that \Lambda_{i i} is the eigenvalue associated to column i of Q. Since the columns of Q are linearly independent, the matrix Q is invertible. Premultiplying both sides by Q^{-1} we get Q^{-1}A Q = \Lambda. By definition, therefore, the matrix A is diagonalizable.
Conversely, if A is diagonalizable, let Q be a non-singular square matrix such that Q^{-1} A Q is some diagonal matrix D. Multiplying both sides on the left by Q we get A Q = Q D . Therefore each column of Q must be an eigenvector of A, whose eigenvalue is the corresponding element on the diagonal of D. Since the columns of Q must be linearly independent, it follows that \boldsymbol{\gamma}_A = n. Thus \boldsymbol{\gamma}_A is equal to n if and only if A is diagonalizable.
If A is diagonalizable, the space of all n-element vectors can be decomposed into the direct sum of the eigenspaces of A. This decomposition is called the eigendecomposition of A, and it is the preserved under change of coordinates.
A matrix that is not diagonalizable is said to be defective. For defective matrices, the notion of eigenvector can be generalized to generalized eigenvectors, and that of diagonal matrix to a Jordan form matrix. Over an algebraically closed field, any matrix A has a Jordan form and therefore admits a basis of generalized eigenvectors, and a decomposition into generalized eigenspaces
Further properties[edit]
Let A be an arbitrary n\times n matrix of complex numbers with eigenvalues \lambda_1, \lambda_2, ... \lambda_n. (Here it is understood that an eigenvalue with algebraic multiplicity \mu occurs \mu times in this list.) Then
• The trace of A, defined as the sum of its diagonal elements, is also the sum of all eigenvalues:
\operatorname{tr}(A) = \sum_{i=1}^n A_{i i} = \sum_{i=1}^n \lambda_i = \lambda_1+ \lambda_2 +\cdots+ \lambda_n.
• The determinant of A is the product of all eigenvalues:
\operatorname{det}(A) = \prod_{i=1}^n \lambda_i=\lambda_1\lambda_2\cdots\lambda_n.
• The eigenvalues of the kth power of A, i.e. the eigenvalues of A^k, for any positive integer k, are \lambda_1^k,\lambda_2^k,\dots,\lambda_n^k
• The matrix A is invertible if and only if all the eigenvalues \lambda_i are nonzero.
• If A is invertible, then the eigenvalues of A^{-1} are 1/\lambda_1,1/\lambda_2,\dots,1/\lambda_n
• If A is equal to its conjugate transpose A^* (in other words, if A is Hermitian), then every eigenvalue is real. The same is true of any a symmetric real matrix. If A is also positive-definite, positive-semidefinite, negative-definite, or negative-semidefinite every eigenvalue is positive, non-negative, negative, or non-positive respectively.
• Every eigenvalue of a unitary matrix has absolute value |\lambda|=1.
Left and right eigenvectors[edit]
The use of matrices with a single column (rather than a single row) to represent vectors is traditional in many disciplines. For that reason, the word "eigenvector" almost always means a right eigenvector, namely a column vector that must be placed to the right of the matrix A in the defining equation
A v = \lambda v.
There may be also single-row vectors that are unchanged when they occur on the left side of a product with a square matrix A; that is, which satisfy the equation
u A = \lambda u
Any such row vector u is called a left eigenvector of A.
The left eigenvectors of A are transposes of the right eigenvectors of the transposed matrix A^\mathsf{T}, since their defining equation is equivalent to
A^\mathsf{T} u^\mathsf{T} = \lambda u^\mathsf{T}
It follows that, if A is Hermitian, its left and right eigenvectors are complex conjugates. In particular if A is a real symmetric matrix, they are the same except for transposition.
Computing the eigenvalues[edit]
The eigenvalues of a matrix A can be determined by finding the roots of the characteristic polynomial. Explicit algebraic formulas for the roots of a polynomial exist only if the degree n is 4 or less. According to the Abel–Ruffini theorem there is no general, explicit and exact algebraic formula for the roots of a polynomial with degree 5 or more.
It turns out that any polynomial with degree n is the characteristic polynomial of some companion matrix of order n. Therefore, for matrices of order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula, and must therefore be computed by approximate numerical methods.
Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the advent of the QR algorithm in 1961. [9] Combining the Householder transformation with the LU decomposition results in an algorithm with better convergence than the QR algorithm.[citation needed] For large Hermitian sparse matrices, the Lanczos algorithm is one example of an efficient iterative method to compute eigenvalues and eigenvectors, among several other possibilities.[9]
Computing the eigenvectors[edit]
Once the (exact) value of an eigenvalue is known, the corresponding eigenvectors can be found by finding non-zero solutions of the eigenvalue equation, that becomes a system of linear equations with known coefficients. For example, once it is known that 6 is an eigenvalue of the matrix
A = \begin{bmatrix} 4 & 1\\6 & 3 \end{bmatrix}
we can find its eigenvectors by solving the equation A v = 6 v, that is
\begin{bmatrix} 4 & 1\\6 & 3 \end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = 6 \cdot \begin{bmatrix}x\\y\end{bmatrix}
This matrix equation is equivalent to two linear equations
\left\{\begin{matrix} 4x + {\ }y &{}= 6x\\6x + 3y &{}=6 y\end{matrix}\right.
\quad\quad\quad that is
\left\{\begin{matrix} -2x+ {\ }y &{}=0\\+6x-3y &{}=0\end{matrix}\right.
Both equations reduce to the single linear equation y=2x. Therefore, any vector of the form [a,2a]', for any non-zero real number a, is an eigenvector of A with eigenvalue \lambda = 6.
The matrix A above has another eigenvalue \lambda=1. A similar calculation shows that the corresponding eigenvectors are the non-zero solutions of 3x+y=0, that is, any vector of the form [b,-3b]', for any non-zero real number b.
In the 18th century Euler studied the rotational motion of a rigid body and discovered the importance of the principal axes. Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.[10] In the early 19th century, Cauchy saw how their work could be used to classify the quadric surfaces, and generalized it to arbitrary dimensions.[11] Cauchy also coined the term racine caractéristique (characteristic root) for what is now called eigenvalue; his term survives in characteristic equation.[12]
Fourier used the work of Laplace and Lagrange to solve the heat equation by separation of variables in his famous 1822 book Théorie analytique de la chaleur.[13] Sturm developed Fourier's ideas further and brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that real symmetric matrices have real eigenvalues.[11] This was extended by Hermite in 1855 to what are now called Hermitian matrices.[12] Around the same time, Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit circle,[11] and Clebsch found the corresponding result for skew-symmetric matrices.[12] Finally, Weierstrass clarified an important aspect in the stability theory started by Laplace by realizing that defective matrices can cause instability.[11]
In the meantime, Liouville studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called Sturm–Liouville theory.[14] Schwarz studied the first eigenvalue of Laplace's equation on general domains towards the end of the 19th century, while Poincaré studied Poisson's equation a few years later.[15]
At the start of the 20th century, Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices.[16] He was the first to use the German word eigen to denote eigenvalues and eigenvectors in 1904, though he may have been following a related usage by Helmholtz. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is standard today.[17]
The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when Von Mises published the power method. One of the most popular methods today, the QR algorithm, was proposed independently by John G.F. Francis[18] and Vera Kublanovskaya[19] in 1961.[20]
Eigenvalues of geometric transformations[edit]
scaling unequal scaling rotation horizontal shear hyperbolic rotation
illustration Equal scaling (homothety) Vertical shrink () and horizontal stretch () of a unit square. Rotation by 50 degrees
Horizontal shear mapping
matrix \begin{bmatrix}k & 0\\0 & k\end{bmatrix}
\begin{bmatrix}c & -s \\ s & c\end{bmatrix}
\begin{bmatrix} c & s \\ s & c \end{bmatrix}
c=\cosh \varphi
s=\sinh \varphi
\ (\lambda - k)^2 (\lambda - k_1)(\lambda - k_2) \lambda^2 - 2c\lambda + 1 \ (\lambda - 1)^2 \lambda^2 - 2c\lambda + 1
eigenvalues \lambda_i \lambda_1 = \lambda_2 = k \lambda_1 = k_1
\lambda_2 = k_2
\lambda_1 = e^{\mathbf{i}\theta}=c+s\mathbf{i}
\lambda_2 = e^{-\mathbf{i}\theta}=c-s\mathbf{i}
\lambda_1 = \lambda_2 = 1 \lambda_1 = e^\varphi
\lambda_2 = e^{-\varphi},
algebraic multipl.
\mu_1 = 2 \mu_1 = 1
\mu_2 = 1
\mu_1 = 1
\mu_2 = 1
\mu_1 = 2 \mu_1 = 1
\mu_2 = 1
geometric multipl.
\gamma_i = \gamma(\lambda_i)
\gamma_1 = 2 \gamma_1 = 1
\gamma_2 = 1
\gamma_1 = 1
\gamma_2 = 1
\gamma_1 = 1 \gamma_1 = 1
\gamma_2 = 1
eigenvectors All non-zero vectors u_1 = \begin{bmatrix}1\\0\end{bmatrix}
u_2 = \begin{bmatrix}0\\1\end{bmatrix}
u_1 = \begin{bmatrix}{\ }1\\-\mathbf{i}\end{bmatrix}
u_2 = \begin{bmatrix}{\ }1\\ +\mathbf{i}\end{bmatrix}
u_1 = \begin{bmatrix}1\\0\end{bmatrix} u_1 = \begin{bmatrix}{\ }1\\{\ }1\end{bmatrix}
u_2 = \begin{bmatrix}{\ }1\\-1\end{bmatrix}.
Note that the characteristic equation for a rotation is a quadratic equation with discriminant D = -4(\sin\theta)^2, which is a negative number whenever \theta is not an integer multiple of 180°. Therefore, except for these special cases, the two eigenvalues are complex numbers, \cos\theta \pm \mathbf{i}\sin\theta; and all eigenvectors have non-real entries. Indeed, except for those special cases, a rotation changes the direction of every nonzero vector in the plane.
Schrödinger equation[edit]
H\psi_E = E\psi_E \,
Molecular orbitals[edit]
Geology and glaciology[edit]
The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space. The three eigenvectors are ordered v_1, v_2, v_3 by their eigenvalues E_1 \geq E_2 \geq E_3;[24] v_1 then is the primary orientation/dip of clast, v_2 is the secondary and v_3 is the tertiary, in terms of strength. The clast orientation is defined as the direction of the eigenvector, on a compass rose of 360°. Dip is measured as the eigenvalue, the modulus of the tensor: this is valued from 0° (no dip) to 90° (vertical). The relative values of E_1, E_2, and E_3 are dictated by the nature of the sediment's fabric. If E_1 = E_2 = E_3, the fabric is said to be isotropic. If E_1 = E_2 > E_3, the fabric is said to be planar. If E_1 > E_2 > E_3, the fabric is said to be linear.[25]
Principal components analysis[edit]
PCA of the multivariate Gaussian distribution centered at (1,3) with a standard deviation of 3 in roughly the (0.878,0.478) direction and of 1 in the orthogonal direction. The vectors shown are unit eigenvectors of the (symmetric, positive-semidefinite) covariance matrix scaled by the square root of the corresponding eigenvalue. (Just as in the one-dimensional case, the square root is taken because the standard deviation is more readily visualized than the variance.
Principal component analysis is used to study large data sets, such as those encountered in data mining, chemical research, psychology, and in marketing. PCA is popular especially in psychology, in the field of psychometrics. In Q methodology, the eigenvalues of the correlation matrix determine the Q-methodologist's judgment of practical significance (which differs from the statistical significance of hypothesis testing; cf. criteria for determining the number of factors). More generally, principal component analysis can be used as a method of factor analysis in structural equation modeling.
Vibration analysis[edit]
1st lateral bending (See vibration for more types of vibration)
Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom. The eigenvalues are used to determine the natural frequencies (or eigenfrequencies) of vibration, and the eigenvectors determine the shapes of these vibrational modes. In particular, undamped vibration is governed by
m\ddot x + kx = 0
m\ddot x = -k x
that is, acceleration is proportional to position (i.e., we expect x to be sinusoidal in time).
In n dimensions, m becomes a mass matrix and k a stiffness matrix. Admissible solutions are then a linear combination of solutions to the generalized eigenvalue problem
-k x = \omega^2 m x
where \omega^2 is the eigenvalue and \omega is the angular frequency. Note that the principal vibration modes are different from the principal compliance modes, which are the eigenvectors of k alone. Furthermore, damped vibration, governed by
m\ddot x + c \dot x + kx = 0
leads to what is called a so-called quadratic eigenvalue problem,
(\omega^2 m + \omega c + k)x = 0.
This can be reduced to a generalized eigenvalue problem by clever use of algebra at the cost of solving a larger system.
Eigenfaces as examples of eigenvectors
In image processing, processed images of faces can be seen as vectors whose components are the brightnesses of each pixel.[26] The dimension of this vector space is the number of pixels. The eigenvectors of the covariance matrix associated with a large set of normalized pictures of faces are called eigenfaces; this is an example of principal components analysis. They are very useful for expressing any face image as a linear combination of some of them. In the facial recognition branch of biometrics, eigenfaces provide a means of applying data compression to faces for identification purposes. Research related to eigen vision systems determining hand gestures has also been made.
Tensor of moment of inertia[edit]
Stress tensor[edit]
Eigenvalues of a graph[edit]
In spectral graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix A, or (increasingly) of the graph's Laplacian matrix (see also Discrete Laplace operator), which is either T - A (sometimes called the combinatorial Laplacian) or I - T^{-1/2}A T^{-1/2} (sometimes called the normalized Laplacian), where T is a diagonal matrix with T_{i i} equal to the degree of vertex v_i, and in T^{-1/2}, the ith diagonal entry is \sqrt{\operatorname{deg}(v_i)}. The kth principal eigenvector of a graph is defined as either the eigenvector corresponding to the kth largest or kth smallest eigenvalue of the Laplacian. The first principal eigenvector of the graph is also referred to merely as the principal eigenvector.
Basic reproduction number[edit]
See Basic reproduction number
The basic reproduction number (R_0) is a fundamental number in the study of how infectious diseases spread. If one infectious person is put into a population of completely susceptible people, then R_0 is the average number of people that one typical infectious person will infect. The generation time of an infection is the time, t_G, from one person becoming infected to the next person becoming infected. In a heterogenous population, the next generation matrix defines how many people in the population will become infected after time t_G has passed. R_0 is then the largest eigenvalue of the next generation matrix.[27][28]
See also[edit]
1. ^ a b Wolfram Research, Inc. (2010) Eigenvector. Accessed on 2010-01-29.
2. ^ William H. Press, Saul A. Teukolsky, William T. Vetterling, Brian P. Flannery (2007), Numerical Recipes: The Art of Scientific Computing, Chapter 11: Eigensystems., pages=563–597. Third edition, Cambridge University Press. ISBN 9780521880688
3. ^ See Korn & Korn 2000, Section 14.3.5a; Friedberg, Insel & Spence 1989, p. 217
5. ^ Shilov 1977, p. 109
6. ^ Lemma for the eigenspace
7. ^ Schaum's Easy Outline of Linear Algebra, p. 111
10. ^ See Hawkins 1975, §2
11. ^ a b c d See Hawkins 1975, §3
12. ^ a b c See Kline 1972, pp. 807–808
13. ^ See Kline 1972, p. 673
14. ^ See Kline 1972, pp. 715–716
15. ^ See Kline 1972, pp. 706–707
16. ^ See Kline 1972, p. 1063
17. ^ See Aldrich 2006
18. ^ Francis, J. G. F. (1961), "The QR Transformation, I (part 1)", The Computer Journal 4 (3): 265–271, doi:10.1093/comjnl/4.3.265 and Francis, J. G. F. (1962), "The QR Transformation, II (part 2)", The Computer Journal 4 (4): 332–345, doi:10.1093/comjnl/4.4.332
19. ^ Kublanovskaya, Vera N. (1961), "On some algorithms for the solution of the complete eigenvalue problem", USSR Computational Mathematics and Mathematical Physics 3: 637–657 . Also published in: Zhurnal Vychislitel'noi Matematiki i Matematicheskoi Fiziki 1 (4), 1961: 555–570
20. ^ See Golub & van Loan 1996, §7.3; Meyer 2000, §7.3
21. ^ Graham, D.; Midgley, N. (2000), "Graphical representation of particle shape using triangular diagrams: an Excel spreadsheet method", Earth Surface Processes and Landforms 25 (13): 1473–1477, doi:10.1002/1096-9837(200012)25:13<1473::AID-ESP158>3.0.CO;2-C
22. ^ Sneed, E. D.; Folk, R. L. (1958), "Pebbles in the lower Colorado River, Texas, a study of particle morphogenesis", Journal of Geology 66 (2): 114–150, doi:10.1086/626490
23. ^ Knox-Robinson, C; Gardoll, Stephen J (1998), "GIS-stereoplot: an interactive stereonet plotting module for ArcView 3.0 geographic information system", Computers & Geosciences 24 (3): 243, doi:10.1016/S0098-3004(97)00122-2
24. ^ Stereo32 software
26. ^ Xirouhakis, A.; Votsis, G.; Delopoulus, A. (2004), Estimation of 3D motion and structure of human faces (PDF), Online paper in PDF format, National Technical University of Athens
27. ^ Diekmann O, Heesterbeek JAP, Metz JAJ (1990), "On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations", Journal of Mathematical Biology 28 (4): 365–382, doi:10.1007/BF00178324, PMID 2117040
28. ^ Odo Diekmann and J. A. P. Heesterbeek (2000), Mathematical epidemiology of infectious diseases, Wiley series in mathematical and computational biology, West Sussex, England: John Wiley & Sons
• Korn, Granino A.; Korn, Theresa M. (2000), "Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review", New York: McGraw-Hill (1152 p., Dover Publications, 2 Revised edition), Bibcode:1968mhse.book.....K, ISBN 0-486-41147-8 .
• Lipschutz, Seymour (1991), Schaum's outline of theory and problems of linear algebra, Schaum's outline series (2nd ed.), New York, NY: McGraw-Hill Companies, ISBN 0-07-038007-4 .
• Friedberg, Stephen H.; Insel, Arnold J.; Spence, Lawrence E. (1989), Linear algebra (2nd ed.), Englewood Cliffs, NJ 07632: Prentice Hall, ISBN 0-13-537102-3 .
• Aldrich, John (2006), "Eigenvalue, eigenfunction, eigenvector, and related terms", in Jeff Miller (Editor), Earliest Known Uses of Some of the Words of Mathematics, retrieved 2006-08-22
• Strang, Gilbert (1993), Introduction to linear algebra, Wellesley-Cambridge Press, Wellesley, MA, ISBN 0-9614088-5-5 .
• Strang, Gilbert (2006), Linear algebra and its applications, Thomson, Brooks/Cole, Belmont, CA, ISBN 0-03-010567-6 .
• Cohen-Tannoudji, Claude (1977), "Chapter II. The mathematical tools of quantum mechanics", Quantum mechanics, John Wiley & Sons, ISBN 0-471-16432-1 .
• Fraleigh, John B.; Beauregard, Raymond A. (1995), Linear algebra (3rd ed.), Addison-Wesley Publishing Company, ISBN 0-201-83999-7 (international edition) .
• Hawkins, T. (1975), "Cauchy and the spectral theory of matrices", Historia Mathematica 2: 1–29, doi:10.1016/0315-0860(75)90032-4 .
• Horn, Roger A.; Johnson, Charles F. (1985), Matrix analysis, Cambridge University Press, ISBN 0-521-30586-1 (hardback), ISBN 0-521-38632-2 (paperback) Check |isbn= value (help) .
• Kline, Morris (1972), Mathematical thought from ancient to modern times, Oxford University Press, ISBN 0-19-501496-0 .
• Brown, Maureen (October 2004), Illuminating Patterns of Perception: An Overview of Q Methodology .
• Golub, Gene F.; van der Vorst, Henk A. (2000), "Eigenvalue computation in the 20th century", Journal of Computational and Applied Mathematics 123: 35–65, doi:10.1016/S0377-0427(00)00413-1 .
• Akivis, Max A.; Vladislav V. Goldberg (1969), Tensor calculus, Russian, Science Publishers, Moscow .
• Gelfand, I. M. (1971), Lecture notes in linear algebra, Russian, Science Publishers, Moscow .
• Alexandrov, Pavel S. (1968), Lecture notes in analytical geometry, Russian, Science Publishers, Moscow .
• Carter, Tamara A.; Tapia, Richard A.; Papaconstantinou, Anne, Linear Algebra: An Introduction to Linear Algebra for Pre-Calculus Students, Rice University, Online Edition, retrieved 2008-02-19 .
• Roman, Steven (2008), Advanced linear algebra (3rd ed.), New York, NY: Springer Science + Business Media, LLC, ISBN 978-0-387-72828-5 .
• Shilov, Georgi E. (1977), Linear algebra (translated and edited by Richard A. Silverman ed.), New York: Dover Publications, ISBN 0-486-63518-X .
• Kuttler, Kenneth (2007), An introduction to linear algebra (PDF), Online e-book in PDF format, Brigham Young University .
• Demmel, James W. (1997), Applied numerical linear algebra, SIAM, ISBN 0-89871-389-7 .
• Beezer, Robert A. (2006), A first course in linear algebra, Free online book under GNU licence, University of Puget Sound .
• Lancaster, P. (1973), Matrix theory, Russian, Moscow, Russia: Science Publishers .
• Halmos, Paul R. (1987), Finite-dimensional vector spaces (8th ed.), New York, NY: Springer-Verlag, ISBN 0-387-90093-4 .
• Larson, Ron; Edwards, Bruce H. (2003), Elementary linear algebra (5th ed.), Houghton Mifflin Company, ISBN 0-618-33567-6 .
• Curtis, Charles W., Linear Algebra: An Introductory Approach, 347 p., Springer; 4th ed. 1984. Corr. 7th printing edition (August 19, 1999), ISBN 0-387-90992-3.
• Sharipov, Ruslan A. (1996), Course of Linear Algebra and Multidimensional Geometry: the textbook, arXiv:math/0405323, ISBN 5-7477-0099-5 .
• Gohberg, Israel; Lancaster, Peter; Rodman, Leiba (2005), Indefinite linear algebra and applications, Basel-Boston-Berlin: Birkhäuser Verlag, ISBN 3-7643-7349-0 .
External links[edit]
Online calculators
Demonstration applets |
3f1ec498817e43f3 | Hydrogen atom
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Hydrogen 1.svg
Name, symbol protium, 1H
Neutrons 0
Protons 1
Nuclide data
Natural abundance 99.985%
Half-life Stable
Isotope mass 1.007825 u
Spin ½+
Excess energy 7288.969±0.001 keV
Binding energy 0.000±0.0000 keV
Depiction of a hydrogen atom showing the diameter as about twice the Bohr model radius. (Image not to scale)
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen constitutes about 75% of the elemental (baryonic) mass of the universe.[1]
In everyday life on Earth, isolated hydrogen atoms (usually called "atomic hydrogen" or, more precisely, "monatomic hydrogen") are extremely rare. Instead, hydrogen tends to combine with other atoms in compounds, or with itself to form ordinary (diatomic) hydrogen gas, H2. "Atomic hydrogen" and "hydrogen atom" in ordinary English use have overlapping, yet distinct, meanings. For example, a water molecule contains two hydrogen atoms, but does not contain atomic hydrogen (which would refer to isolated hydrogen atoms).
Production and reactivity[edit]
The H–H bond is one of the toughest bonds in chemistry, with a bond dissociation enthalpy of 435.88 kJ/mol at 298 K (25 °C; 77 °F). As a consequence of this strong bond, H2 dissociates to only a minor extent until higher temperatures. At 3,000 K (2,730 °C; 4,940 °F), the degree of dissociation is just 7.85%:[2]
H2 ⇌ 2 H
The most abundant isotope, hydrogen-1, protium, or light hydrogen, contains no neutrons; other isotopes of hydrogen, such as deuterium or tritium, contain one or more neutrons. The formulas below are valid for all three isotopes of hydrogen, but slightly different values of the Rydberg constant (correction formula given below) must be used for each hydrogen isotope.
Quantum theoretical analysis[edit]
In 1913, Niels Bohr obtained the spectral frequencies of the hydrogen atom after making a number of simplifying assumptions. These assumptions, the cornerstones of the Bohr model, were not fully correct but did yield fairly correct energy answers (with a relative error in the ground state ionization energy of around α2/4 or around 10−5). Bohr's results for the frequencies and underlying energy values were duplicated by the solution to the Schrödinger equation in 1925–1926. The solution to the Schrödinger equation for hydrogen is analytical, giving a simple expressions for the hydrogen energy levels and thus the frequencies of the hydrogen spectral lines. The solution of the Schrödinger equation goes much further than the Bohr model, because it also yields the shape of the electron's wave function ("orbital") for the various possible quantum-mechanical states, thus explaining the anisotropic character of atomic bonds.
The Schrödinger equation also applies to more complicated atoms and molecules. When there is more than one electron or nucleus the solution is not analytical and either computer calculations are necessary or simplifying assumptions must be made.
The Schrödinger equation is not fully accurate. The next improvement was the Dirac equation (see below).
Solution of Schrödinger equation: Overview of results[edit]
The solution of the Schrödinger equation (wave equations) for the hydrogen atom uses the fact that the Coulomb potential produced by the nucleus is isotropic (it is radially symmetric in space and only depends on the distance to the nucleus). Although the resulting energy eigenfunctions (the orbitals) are not necessarily isotropic themselves, their dependence on the angular coordinates follows completely generally from this isotropy of the underlying potential: the eigenstates of the Hamiltonian (that is, the energy eigenstates) can be chosen as simultaneous eigenstates of the angular momentum operator. This corresponds to the fact that angular momentum is conserved in the orbital motion of the electron around the nucleus. Therefore, the energy eigenstates may be classified by two angular momentum quantum numbers, and m (both are integers). The angular momentum quantum number = 0, 1, 2, ... determines the magnitude of the angular momentum. The magnetic quantum number m = −, ..., + determines the projection of the angular momentum on the (arbitrarily chosen) z-axis.
Note that the maximum value of the angular momentum quantum number is limited by the principal quantum number: it can run only up to n − 1, i.e. = 0, 1, ..., n − 1.
Due to angular momentum conservation, states of the same but different m have the same energy (this holds for all problems with rotational symmetry). In addition, for the hydrogen atom, states of the same n but different are also degenerate (i.e. they have the same energy). However, this is a specific property of hydrogen and is no longer true for more complicated atoms which have a (effective) potential differing from the form 1/r (due to the presence of the inner electrons shielding the nucleus potential).
Taking into account the spin of the electron adds a last quantum number, the projection of the electron's spin angular momentum along the z-axis, which can take on two values. Therefore, any eigenstate of the electron in the hydrogen atom is described fully by four quantum numbers. According to the usual rules of quantum mechanics, the actual state of the electron may be any superposition of these states. This explains also why the choice of z-axis for the directional quantization of the angular momentum vector is immaterial: an orbital of given and m′ obtained for another preferred axis z′ can always be represented as a suitable superposition of the various states of different m (but same l) that have been obtained for z.
Alternatives to the Schrödinger theory[edit]
In the language of Heisenberg's matrix mechanics, the hydrogen atom was first solved by Wolfgang Pauli[3] using a rotational symmetry in four dimension [O(4)-symmetry] generated by the angular momentum and the Laplace–Runge–Lenz vector. By extending the symmetry group O(4) to the dynamical group O(4,2), the entire spectrum and all transitions were embedded in a single irreducible group representation.[4]
In 1979 the (non relativistic) hydrogen atom was solved for the first time within Feynman's path integral formulation of quantum mechanics.[5][6] This work greatly extended the range of applicability of Feynman's method.
Mathematical summary of eigenstates of hydrogen atom[edit]
In 1928, Paul Dirac found an equation that was fully compatible with Special Relativity, and (as a consequence) made the wave function a 4-component "Dirac spinor" including "up" and "down" spin components, with both positive and "negative" energy (or matter and antimatter). The solution to this equation gave the following results, more accurate than the Schrödinger solution.
Energy levels[edit]
The energy levels of hydrogen, including fine structure (excluding Lamb shift and hyperfine structure), are given by the Sommerfeld expression:
\begin{array}{rl} E_{j\,n} & = -m_\text{e}c^2\left[1-\left(1+\left[\dfrac{\alpha}{n-j-\frac{1}{2}+\sqrt{\left(j+\frac{1}{2}\right)^2-\alpha^2}}\right]^2\right)^{-1/2}\right] \\ & \approx -\dfrac{m_\text{e}c^2\alpha^2}{2n^2} \left[1 + \dfrac{\alpha^2}{n^2}\left(\dfrac{n}{j+\frac{1}{2}} - \dfrac{3}{4} \right) \right] , \end{array}
where α is the fine-structure constant and j is the "total angular momentum" quantum number, which is equal to | ± 1/2| depending on the direction of the electron spin. The factor in square brackets in the last expression is nearly one; the extra term arises from relativistic effects (for details, see #Features going beyond the Schrödinger solution).
The value
\frac{m_{\text{e}} c^2\alpha^2}{2} = \frac{0.51\,\text{MeV}}{2 \cdot 137^2} = 13.6 \,\text{eV}
is called the Rydberg constant and was first found from the Bohr model as given by
-13.6 \,\text{eV} = -\frac{m_{\text{e}} e^4}{8 h^2 \varepsilon_0^2},
where me is the electron mass, e is the elementary charge, h is the Planck constant, and ε0 is the vacuum permittivity.
This constant is often used in atomic physics in the form of the Rydberg unit of energy:
1 \,\text{Ry} \equiv h c R_\infty = 13.605\;692\;53(30) \,\text{eV}.[7]
The exact value of the Rydberg constant above assumes that the nucleus is infinitely massive with respect to the electron. For hydrogen-1, hydrogen-2 (deuterium), and hydrogen-3 (tritium) the constant must be slightly modified to use the reduced mass of the system, rather than simply the mass of the electron. However, since the nucleus is much heavier than the electron, the values are nearly the same. The Rydberg constant RM for a hydrogen atom (one electron), R is given by: R_M = \frac{R_\infty}{1+m_{\text{e}}/M}, where M is the mass of the atomic nucleus. For hydrogen-1, the quantity m_{\text{e}}/M, is about 1/1836 (i.e. the electron-to-proton mass ratio). For deuterium and tritium, the ratios are about 1/3670 and 1/5497 respectively. These figures, when added to 1 in the denominator, represent very small corrections in the value of R, and thus only small corrections to all energy levels in corresponding hydrogen isotopes.
The normalized position wavefunctions, given in spherical coordinates are:
\psi_{n\ell m}(r,\vartheta,\varphi) = \sqrt {{\left ( \frac{2}{n a_0} \right )}^3\frac{(n-\ell-1)!}{2n(n+\ell)!} } e^{- \rho / 2} \rho^{\ell} L_{n-\ell-1}^{2\ell+1}(\rho) Y_{\ell}^{m}(\vartheta, \varphi )
3D illustration of the eigenstate \psi_{4,3,1}. Electrons in this state are 45% likely to be found within the solid body shown.
\rho = {2r \over {na_0}} ,
a_0 is the Bohr radius,
L_{n-\ell-1}^{2\ell+1}(\rho) is a generalized Laguerre polynomial of degree n − 1, and
Y_{\ell}^{m}(\vartheta, \varphi ) \, is a spherical harmonic function of degree and order m. Note that the generalized Laguerre polynomials are defined differently by different authors. The usage here is consistent with the definitions used by Messiah,[8] and Mathematica.[9] In other places, the Laguerre polynomial includes a factor of (n+\ell)!,[10] or the generalized Laguerre polynomial appearing in the hydrogen wave function is L_{n+\ell}^{2\ell+1}(\rho) instead.[11]
The quantum numbers can take the following values:
Additionally, these wavefunctions are normalized (i.e., the integral of their modulus square equals 1) and orthogonal:
\int_0^{\infty} r^2 dr\int_0^{\pi} \sin \vartheta d\vartheta \int_0^{2 \pi} d\varphi\; \psi^*_{n\ell m}(r,\vartheta,\varphi)\psi_{n'\ell'm'}(r,\vartheta,\varphi)=\langle n,\ell, m | n', \ell', m' \rangle = \delta_{nn'} \delta_{\ell\ell'} \delta_{mm'},
where | n, \ell, m \rangle is the representation of the wavefunction \psi_{n\ell m} in Dirac notation, and \delta is the Kronecker delta function.[12]
Angular momentum[edit]
The eigenvalues for Angular momentum operator:
L^2\, | n, \ell, m\rangle = {\hbar}^2 \ell(\ell+1)\, | n, \ell, m \rang
L_z\, | n, \ell, m \rang = \hbar m \,| n, \ell, m \rang.
Visualizing the hydrogen electron orbitals[edit]
Probability densities through the xz-plane for the electron at different quantum numbers (, across top; n, down side; m = 0)
The image to the right shows the first few hydrogen atom orbitals (energy eigenfunctions). These are cross-sections of the probability density that are color-coded (black represents zero density and white represents the highest density). The angular momentum (orbital) quantum number is denoted in each column, using the usual spectroscopic letter code (s means = 0, p means = 1, d means = 2). The main (principal) quantum number n (= 1, 2, 3, ...) is marked to the right of each row. For all pictures the magnetic quantum number m has been set to 0, and the cross-sectional plane is the xz-plane (z is the vertical axis). The probability density in three-dimensional space is obtained by rotating the one shown here around the z-axis.
The "ground state", i.e. the state of lowest energy, in which the electron is usually found, is the first one, the 1s state (principal quantum level n = 1, = 0).
An image with more orbitals is also available (up to higher numbers n and ).
Black lines occur in each but the first orbital: these are the nodes of the wavefunction, i.e. where the probability density is zero. (More precisely, the nodes are spherical harmonics that appear as a result of solving Schrödinger's equation in polar coordinates.)
The quantum numbers determine the layout of these nodes.[13] There are:
• n-1 total nodes,
• l of which are angular nodes:
• m angular nodes go around the \phi axis (in the xy plane). (The figure above does not show these nodes since it plots cross-sections through the xz-plane.)
• l-m (the remaining angular nodes) occur on the \theta (vertical) axis.
• n - l - 1 (the remaining non-angular nodes) are radial nodes.
Features going beyond the Schrödinger solution[edit]
• Although the mean speed of the electron in hydrogen is only 1/137th of the speed of light, many modern experiments are sufficiently precise that a complete theoretical explanation requires a fully relativistic treatment of the problem. A relativistic treatment results in a momentum increase of about 1 part in 37,000 for the electron. Since the electron's wavelength is determined by its momentum, orbitals containing higher speed electrons show contraction due to smaller wavelengths.
Both of these features (and more) are incorporated in the relativistic Dirac equation, with predictions that come still closer to experiment. Again the Dirac equation may be solved analytically in the special case of a two-body system, such as the hydrogen atom. The resulting solution quantum states now must be classified by the total angular momentum number j (arising through the coupling between electron spin and orbital angular momentum). States of the same j and the same n are still degenerate. Thus, direct analytical solution of Dirac equation predicts 2S(1/2) and 2P(1/2) levels of Hydrogen to have exactly the same energy, which is in a contradiction with observations (Lamb-Retherford experiment).
Hydrogen ion[edit]
Hydrogen is not found without its electron in ordinary chemistry (room temperatures and pressures), as ionized hydrogen is highly chemically reactive. When ionized hydrogen is written as "H+" as in the solvation of classical acids such as hydrochloric acid, the hydronium ion, H3O+, is meant, not a literal ionized single hydrogen atom. In that case, the acid transfers the proton to H2O to form H3O+.
See also[edit]
3. ^ Pauli, W (1926). "Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik". Zeitschrift für Physik 36: 336–363. Bibcode:1926ZPhy...36..336P. doi:10.1007/BF01450175.
4. ^ Kleinert H. (1968). "Group Dynamics of the Hydrogen Atom". Lectures in Theoretical Physics, edited by W.E. Brittin and A.O. Barut, Gordon and Breach, N.Y. 1968: 427–482.
5. ^ Duru I.H., Kleinert H. (1979). "Solution of the path integral for the H-atom". Physics Letters B 84 (2): 185–188. Bibcode:1979PhLB...84..185D. doi:10.1016/0370-2693(79)90280-6.
6. ^ Duru I.H., Kleinert H. (1982). "Quantum Mechanics of H-Atom from Path Integrals". Fortschr. Phys 30 (2): 401–435. doi:10.1002/prop.19820300802.
7. ^ P.J. Mohr, B.N. Taylor, and D.B. Newell (2011), "The 2010 CODATA Recommended Values of the Fundamental Physical Constants" (Web Version 6.0). This database was developed by J. Baker, M. Douma, and S. Kotochigova. Available: http://physics.nist.gov/constants. National Institute of Standards and Technology, Gaithersburg, MD 20899. Link to R, Link to hcR
8. ^ Messiah, Albert (1999). Quantum Mechanics. New York: Dover. p. 1136. ISBN 0-486-40924-4.
9. ^ LaguerreL. Wolfram Mathematica page
10. ^ Griffiths, David (1995). Introduction to Quantum Mechanics. New Jersey: Pearson Education, Inc. p. 152. ISBN 0-13-111892-7.
11. ^ Condon and Shortley (1963). The Theory of Atomic Spectra. London: Cambridge. p. 441.
12. ^ Introduction to Quantum Mechanics, Griffiths 4.89
13. ^ Summary of atomic quantum numbers. Lecture notes. 28 July 2006
• Griffiths, David J. (1995). Introduction to Quantum Mechanics. Prentice Hall. ISBN 0-13-111892-7. Section 4.2 deals with the hydrogen atom specifically, but all of Chapter 4 is relevant.
• Kleinert, H. (2009). Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th edition, Worldscibooks.com, World Scientific, Singapore (also available online physik.fu-berlin.de)
External links[edit]
(none, lightest possible)
Hydrogen atom is an
isotope of hydrogen
Decay product of:
Decay chain
of hydrogen atom
Decays to: |
163c7ef348b05ce7 | Adventures in Ethics and Science
At day camp yesterday, the sprogs (and their fellow campers) had a visitor:
Elder Free-Ride offspring; She was an astrophysicist. You know what that is, right?
She talked to us about studying light that comes from space, and all the different kinds of light there are traveling across space. There’s infrared, and ultraviolet, and even X-rays. And, of course, there’s white light that we can see with our eyes.
While there are many different kinds of lights, there are only some colors of light that our eyes can detect. A bunch of those are actually mixed together in white light. You can separate them with a prism.
The astrophysicists was telling us about separating the light coming from space. Because you wouldn’t want to put a big prism into orbit, they made thin sheets with many, many, many little prisms in them.
Dr. Free-Ride: A diffraction grating?
Elder Free-Ride offspring: I think so.
Younger Free-Ride offspring: Then we did an activity. We each got a cardboard tube and a piece of the degrading thing.
Dr. Free-Ride: The diffraction grating?
Younger Free-Ride offspring: Yeah, thin sheet with lots of little prisms. We got a silver sticker with a hole in it. We put the diffraction grating under the hole in the sticker, then covered one end of the tube with it.
When I looked through, it looked like a prism with lots of light all over. When I held up my finger at the end, it looked like a flower with reflections all around it.
Then we covered the open end up the tube with a sticker. I looked through and saw nothing.
Dr. Free-Ride: Because no light was getting into the tube?
Younger Free-Ride offspring: Yeah. And then we poked a hole in the middle and looked through. We could see a little rainbow on the side of the tube. Then we poked one more hole and looked through. Then three more and looked through. She said the smaller the holes, the better the rainbows.
We went outside and looked at the blacktop and didn’t see many rainbows. But then we pointed it at other objects that were red or green or blue and saw really good rainbows.
Elder Free-Ride offspring; The prism splits the white light into all the different colors that make up white light. Colored light is the opposite of colored pigments. If you mix lots of different colors of pigment, what you get is darker and darker. But if you mix red, orange, yellow, green, blue, and purple light, you get white light.
This site describing how to make a simple spectroscope seems pretty similar to the activity the sprogs describe.
1. #1 Super Sally
June 26, 2009
Spectroscopy was the most exciting lab for me in my (1st round of) college physics.
Now the sprogs can discuss with me the COBE/DIRBE IR observations I was working on before they were born, and start to appreciate how the observations were made and some of the visualizations I helped with. How exciting!
Who was the astrophysicist who visited?
Thanks for sharing.
2. #2 Uncle Fishy
June 26, 2009
I see a t-shirt with Schrödinger equations on it in somebody’s future. I wonder if I still have mine…
3. #3 Hap
June 29, 2009
No – I think a shirt with Maxwell’s equations would be a better start. |
68a7157eae991fa0 | Dirac equation
From Wikipedia, the free encyclopedia
(Redirected from Dirac Equation)
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In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-½ massive particles, for which parity is a symmetry, such as electrons and quarks, and is consistent with both the principles of quantum mechanics and the theory of special relativity,[1] and was the first theory to account fully for special relativity in the context of quantum mechanics.
It accounted for the fine details of the hydrogen spectrum in a completely rigorous way. The equation also implied the existence of a new form of matter, antimatter, previously unsuspected and unobserved, and actually predated its experimental discovery. It also provided a theoretical justification for the introduction of several-component wave functions in Pauli's phenomenological theory of spin; the wave functions in the Dirac theory are vectors of four complex numbers (known as bispinors), two of which resemble the Pauli wavefunction in the non-relativistic limit, in contrast to the Schrödinger equation which described wave functions of only one complex value. Moreover, in the limit of zero mass, the Dirac equation reduces to the Weyl equation.
Although Dirac did not at first fully appreciate the importance of his results, the entailed explanation of spin as a consequence of the union of quantum mechanics and relativity—and the eventual discovery of the positron—represent one of the great triumphs of theoretical physics. This accomplishment has been described as fully on a par with the works of Newton, Maxwell, and Einstein before him.[2] In the context of quantum field theory, the Dirac equation is reinterpreted to describe quantum fields corresponding to spin-½ particles.
Mathematical formulation[edit]
The Dirac equation in the form originally proposed by Dirac is:[3]
Dirac equation (Original)
where ψ = ψ(x, t) is the wave function for the electron of rest mass m with spacetime coordinates x, t. The p1, p2, p3 are the components of the momentum, understood to be the momentum operator in the Schrödinger theory. Also, c is the speed of light, and ħ is the Planck constant divided by 2π. These fundamental physical constants reflect special relativity and quantum mechanics, respectively.
Dirac's purpose in casting this equation was to explain the behavior of the relativistically moving electron, and so to allow the atom to be treated in a manner consistent with relativity. His rather modest hope was that the corrections introduced this way might have bearing on the problem of atomic spectra. Up until that time, attempts to make the old quantum theory of the atom compatible with the theory of relativity, attempts based on discretizing the angular momentum stored in the electron's possibly non-circular orbit of the atomic nucleus, had failed – and the new quantum mechanics of Heisenberg, Pauli, Jordan, Schrödinger, and Dirac himself had not developed sufficiently to treat this problem. Although Dirac's original intentions were satisfied, his equation had far deeper implications for the structure of matter, and introduced new mathematical classes of objects that are now essential elements of fundamental physics.
The new elements in this equation are the 4 × 4 matrices αk and β, and the four-component wave function ψ. There are four components in ψ because evaluation of it at any given point in configuration space is a bispinor. It is interpreted as a superposition of a spin-up electron, a spin-down electron, a spin-up positron, and a spin-down positron (see below for further discussion).
The 4 × 4 matrices αk and β are all Hermitian and have squares equal to the identity matrix:
and they all mutually anticommute (if i and j are distinct):
\alpha_i\alpha_j + \alpha_j\alpha_i = 0
\alpha_i\beta + \beta\alpha_i = 0
The single symbolic equation thus unravels into four coupled linear first-order partial differential equations for the four quantities that make up the wave function. These matrices, and the form of the wave function, have a deep mathematical significance. The algebraic structure represented by the gamma matrices had been created some 50 years earlier by the English mathematician W. K. Clifford. In turn, Clifford's ideas had emerged from the mid-19th century work of the German mathematician Hermann Grassmann in his Lineale Ausdehnungslehre (Theory of Linear Extensions). The latter had been regarded as well-nigh incomprehensible by most of his contemporaries. The appearance of something so seemingly abstract, at such a late date, and in such a direct physical manner, is one of the most remarkable chapters in the history of physics.
Making the Schrödinger equation relativistic[edit]
The Dirac equation is superficially similar to the Schrödinger equation for a massive free particle:
-\frac{\hbar^2}{2m}\nabla^2\phi = i\hbar\frac{\partial}{\partial t}\phi.
The left side represents the square of the momentum operator divided by twice the mass, which is the non-relativistic kinetic energy. Because relativity treats space and time as a whole, a relativistic generalization of this equation requires that space and time derivatives must enter symmetrically, as they do in the Maxwell equations that govern the behavior of light — the equations must be differentially of the same order in space and time. In relativity, the momentum and the energy are the space and time parts of a spacetime vector, the four-momentum, and they are related by the relativistically invariant relation
\frac{E^2}{c^2} - p^2 = m^2c^2
which says that the length of this four-vector is proportional to the rest mass m. Substituting the operator equivalents of the energy and momentum from the Schrödinger theory, we get an equation describing the propagation of waves, constructed from relativistically invariant objects,
\left(\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)\phi = \frac{m^2c^2}{\hbar^2}\phi
with the wave function ϕ being a relativistic scalar: a complex number which has the same numerical value in all frames of reference. The space and time derivatives both enter to second order. This has a telling consequence for the interpretation of the equation. Because the equation is second order in the time derivative, then by the nature of solving differential equations, one must specify both the initial values of the wave function itself and of its first time derivative, in order to solve definite problems. Because both may be specified more or less arbitrarily, the wave function cannot maintain its former role of determining the probability density of finding the electron in a given state of motion. In the Schrödinger theory, the probability density is given by the positive definite expression
and this density is convected according to the probability current vector
J = -\frac{i\hbar}{2m}(\phi^*\nabla\phi - \phi\nabla\phi^*)
with the conservation of probability current and density following from the continuity equation:
\nabla\cdot J + \frac{\partial\rho}{\partial t} = 0.
The fact that the density is positive definite and convected according to this continuity equation, implies that we may integrate the density over a certain domain and set the total to 1, and this condition will be maintained by the conservation law. A proper relativistic theory with a probability density current must also share this feature. Now, if we wish to maintain the notion of a convected density, then we must generalize the Schrödinger expression of the density and current so that the space and time derivatives again enter symmetrically in relation to the scalar wave function. We are allowed to keep the Schrödinger expression for the current, but must replace the probability density by the symmetrically formed expression
\rho = \frac{i\hbar}{2m}(\psi^*\partial_t\psi - \psi\partial_t\psi^*).
which now becomes the 4th component of a spacetime vector, and the entire probability 4-current density has the relativistically covariant expression
J^\mu = \frac{i\hbar}{2m}(\psi^*\partial^\mu\psi - \psi\partial^\mu\psi^*)
The continuity equation is as before. Everything is compatible with relativity now, but we see immediately that the expression for the density is no longer positive definite – the initial values of both ψ and tψ may be freely chosen, and the density may thus become negative, something that is impossible for a legitimate probability density. Thus we cannot get a simple generalization of the Schrödinger equation under the naive assumption that the wave function is a relativistic scalar, and the equation it satisfies, second order in time.
Although it is not a successful relativistic generalization of the Schrödinger equation, this equation is resurrected in the context of quantum field theory, where it is known as the Klein–Gordon equation, and describes a spinless particle field (e.g. pi meson). Historically, Schrödinger himself arrived at this equation before the one that bears his name, but soon discarded it. In the context of quantum field theory, the indefinite density is understood to correspond to the charge density, which can be positive or negative, and not the probability density.
Dirac's coup[edit]
Dirac thus thought to try an equation that was first order in both space and time. One could, for example, formally take the relativistic expression for the energy
E = c\sqrt{p^2 + m^2c^2}\,,
replace p by its operator equivalent, expand the square root in an infinite series of derivative operators, set up an eigenvalue problem, then solve the equation formally by iterations. Most physicists had little faith in such a process, even if it were technically possible.
As the story goes, Dirac was staring into the fireplace at Cambridge, pondering this problem, when he hit upon the idea of taking the square root of the wave operator thus:
\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2} = \left(A \partial_x + B \partial_y + C \partial_z + \frac{i}{c}D \partial_t\right)\left(A \partial_x + B \partial_y + C \partial_z + \frac{i}{c}D \partial_t\right).
On multiplying out the right side we see that, in order to get all the cross-terms such as xy to vanish, we must assume
AB + BA = 0, \;\ldots
A^2 = B^2 = \ldots = 1.\,
Dirac, who had just then been intensely involved with working out the foundations of Heisenberg's matrix mechanics, immediately understood that these conditions could be met if A, B, C and D are matrices, with the implication that the wave function has multiple components. This immediately explained the appearance of two-component wave functions in Pauli's phenomenological theory of spin, something that up until then had been regarded as mysterious, even to Pauli himself. However, one needs at least 4 × 4 matrices to set up a system with the properties required — so the wave function had four components, not two, as in the Pauli theory, or one, as in the bare Schrödinger theory. The four-component wave function represents a new class of mathematical object in physical theories that makes its first appearance here.
Given the factorization in terms of these matrices, one can now write down immediately an equation
\left(A\partial_x + B\partial_y + C\partial_z + \frac{i}{c}D\partial_t\right)\psi = \kappa\psi
with κ to be determined. Applying again the matrix operator on both side yields
\left(\nabla^2 - \frac{1}{c^2}\partial_t^2\right)\psi = \kappa^2\psi.
On taking κ = mc/ħ we find that all the components of the wave function individually satisfy the relativistic energy–momentum relation. Thus the sought-for equation that is first-order in both space and time is
\left(A\partial_x + B\partial_y + C\partial_z + \frac{i}{c}D\partial_t - \frac{mc}{\hbar}\right)\psi = 0.
A = i\beta \alpha_1, B = i\beta \alpha_2, C = i\beta \alpha_3, D = \beta \, ,
we get the Dirac equation as written above.
Covariant form and relativistic invariance[edit]
To demonstrate the relativistic invariance of the equation, it is advantageous to cast it into a form in which the space and time derivatives appear on an equal footing. New matrices are introduced as follows:
\gamma^0 = \beta \,
\gamma^k = \gamma^0 \alpha^k. \,
and the equation takes the form
Dirac equation
i \hbar \gamma^\mu \partial_\mu \psi - m c \psi = 0
where there is an implied summation over the values of the twice-repeated index μ = 0, 1, 2, 3. In practice one often writes the gamma matrices in terms of 2 × 2 sub-matrices taken from the Pauli matrices and the 2 × 2 identity matrix. Explicitly the standard representation is
\gamma^0 = \left(\begin{array}{cccc} I_2 & 0 \\ 0 & -I_2 \end{array}\right),
The complete system is summarized using the Minkowski metric on spacetime in the form
\{\gamma^\mu,\gamma^\nu\} = 2 \eta^{\mu\nu} \,
where the bracket expression
\{a, b\} = ab + ba
denotes the anticommutator. These are the defining relations of a Clifford algebra over a pseudo-orthogonal 4-d space with metric signature (+ − − −). The specific Clifford algebra employed in the Dirac equation is known today as the Dirac algebra. Although not recognized as such by Dirac at the time the equation was formulated, in hindsight the introduction of this geometric algebra represents an enormous stride forward in the development of quantum theory.
The Dirac equation may now be interpreted as an eigenvalue equation, where the rest mass is proportional to an eigenvalue of the 4-momentum operator, the proportionality constant being the speed of light:
P_\mathrm{op}\psi = mc\psi. \,
Using {\partial\!\!\!\big /} (pronounced: "d-slash"[4]) in Feynman slash notation, which includes the gamma matrices as well as a summation over the spinor components in the derivative itself, the Dirac equation becomes:
i \hbar {\partial\!\!\!\big /} \psi - m c \psi = 0
In practice, physicists often use units of measure such that ħ = c = 1, known as natural units. The equation then takes the simple form
Dirac equation (natural units)
(i{\partial\!\!\!\big /} - m) \psi = 0\,
A fundamental theorem states that if two distinct sets of matrices are given that both satisfy the Clifford relations, then they are connected to each other by a similarity transformation:
\gamma^{\mu\prime} = S^{-1} \gamma^\mu S.
If in addition the matrices are all unitary, as are the Dirac set, then S itself is unitary;
\gamma^{\mu\prime} = U^\dagger \gamma^\mu U.
The transformation U is unique up to a multiplicative factor of absolute value 1. Let us now imagine a Lorentz transformation to have been performed on the space and time coordinates, and on the derivative operators, which form a covariant vector. For the operator γμμ to remain invariant, the gammas must transform among themselves as a contravariant vector with respect to their spacetime index. These new gammas will themselves satisfy the Clifford relations, because of the orthogonality of the Lorentz transformation. By the fundamental theorem, we may replace the new set by the old set subject to a unitary transformation. In the new frame, remembering that the rest mass is a relativistic scalar, the Dirac equation will then take the form
( iU^\dagger \gamma^\mu U\partial_\mu^\prime - m)\psi(x^\prime,t^\prime) = 0
U^\dagger(i\gamma^\mu\partial_\mu^\prime - m)U \psi(x^\prime,t^\prime) = 0.
If we now define the transformed spinor
\psi^\prime = U\psi
then we have the transformed Dirac equation in a way that demonstrates manifest relativistic invariance:
(i\gamma^\mu\partial_\mu^\prime - m)\psi^\prime(x^\prime,t^\prime) = 0.
Thus, once we settle on any unitary representation of the gammas, it is final provided we transform the spinor according the unitary transformation that corresponds to the given Lorentz transformation. The various representations of the Dirac matrices employed will bring into focus particular aspects of the physical content in the Dirac wave function (see below). The representation shown here is known as the standard representation – in it, the wave function's upper two components go over into Pauli's 2-spinor wave function in the limit of low energies and small velocities in comparison to light.
The considerations above reveal the origin of the gammas in geometry, hearkening back to Grassmann's original motivation – they represent a fixed basis of unit vectors in spacetime. Similarly, products of the gammas such as γμγν represent oriented surface elements, and so on. With this in mind, we can find the form of the unit volume element on spacetime in terms of the gammas as follows. By definition, it is
V = \frac{1}{4!}\epsilon_{\mu\nu\alpha\beta}\gamma^\mu\gamma^\nu\gamma^\alpha\gamma^\beta.
For this to be an invariant, the epsilon symbol must be a tensor, and so must contain a factor of g, where g is the determinant of the metric tensor. Since this is negative, that factor is imaginary. Thus
V = i \gamma^0\gamma^1\gamma^2\gamma^3.\
This matrix is given the special symbol γ5, owing to its importance when one is considering improper transformations of spacetime, that is, those that change the orientation of the basis vectors. In the standard representation it is
\gamma_5 = \begin{pmatrix} 0 & I_{2} \\ I_{2} & 0 \end{pmatrix}.
This matrix will also be found to anticommute with the other four Dirac matrices:
\gamma^5 \gamma^\mu + \gamma^\mu \gamma^5 = 0
It takes a leading role when questions of parity arise, because the volume element as a directed magnitude changes sign under a spacetime reflection. Taking the positive square root above thus amounts to choosing a handedness convention on spacetime .
Conservation of probability current[edit]
By defining the adjoint spinor
\bar{\psi} = \psi^\dagger\gamma^0
where ψ is the conjugate transpose of ψ, and noticing that
(\gamma^\mu)^\dagger\gamma^0 = \gamma^0\gamma^\mu \,,
we obtain, by taking the Hermitian conjugate of the Dirac equation and multiplying from the right by γ0, the adjoint equation:
\bar{\psi}(-i\gamma^\mu\partial_\mu - m) = 0 \,
where μ is understood to act to the left. Multiplying the Dirac equation by ψ from the left, and the adjoint equation by ψ from the right, and subtracting, produces the law of conservation of the Dirac current:
\partial_\mu \left( \bar{\psi}\gamma^\mu\psi \right) = 0.
Now we see the great advantage of the first-order equation over the one Schrödinger had tried – this is the conserved current density required by relativistic invariance, only now its 4th component is positive definite and thus suitable for the role of a probability density:
J^0 = \bar{\psi}\gamma^0\psi = \psi^\dagger\psi.
Because the probability density now appears as the fourth component of a relativistic vector, and not a simple scalar as in the Schrödinger equation, it will be subject to the usual effects of the Lorentz transformations such as time dilation. Thus for example atomic processes that are observed as rates, will necessarily be adjusted in a way consistent with relativity, while those involving the measurement of energy and momentum, which themselves form a relativistic vector, will undergo parallel adjustment which preserves the relativistic covariance of the observed values.
See Dirac spinor for details of solutions to the Dirac equation. The fact that the energies of the solutions do not have a lower bound is unexpected – see the hole theory section below for more details.
Comparison with the Pauli theory[edit]
See also: Pauli equation
The necessity of introducing half-integral spin goes back experimentally to the results of the Stern–Gerlach experiment. A beam of atoms is run through a strong inhomogeneous magnetic field, which then splits into N parts depending on the intrinsic angular momentum of the atoms. It was found that for silver atoms, the beam was split in two—the ground state therefore could not be integral, because even if the intrinsic angular momentum of the atoms were as small as possible, 1, the beam would be split into three parts, corresponding to atoms with Lz = −1, 0, +1. The conclusion is that silver atoms have net intrinsic angular momentum of 12. Pauli set up a theory which explained this splitting by introducing a two-component wave function and a corresponding correction term in the Hamiltonian, representing a semi-classical coupling of this wave function to an applied magnetic field, as so:
H = \frac{1}{2m}\left(\sigma\cdot\left(p - \frac{e}{c}A\right)\right)^2 + e\phi.
Here A and φ represent the components of the electromagnetic four-potential, and the three sigmas are the Pauli matrices. On squaring out the first term, a residual interaction with the magnetic field is found, along with the usual classical Hamiltonian of a charged particle interacting with an applied field:
H = \frac{1}{2m}\left(p - \frac{e}{c}A\right)^2 + e\phi - \frac{e\hbar}{2mc}\sigma\cdot B.
This Hamiltonian is now a 2 × 2 matrix, so the Schrödinger equation based on it must use a two-component wave function. Pauli had introduced the 2 × 2 sigma matrices as pure phenomenology— Dirac now had a theoretical argument that implied that spin was somehow the consequence of the marriage of quantum mechanics to relativity. On introducing the external electromagnetic 4-vector potential into the Dirac equation in a similar way, known as minimal coupling, it takes the form (in natural units)
(i\gamma^\mu(\partial_\mu + ieA_\mu) - m) \psi = 0\,
A second application of the Dirac operator will now reproduce the Pauli term exactly as before, because the spatial Dirac matrices multiplied by i, have the same squaring and commutation properties as the Pauli matrices. What is more, the value of the gyromagnetic ratio of the electron, standing in front of Pauli's new term, is explained from first principles. This was a major achievement of the Dirac equation and gave physicists great faith in its overall correctness. There is more however. The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner. First the equation is written in the form of coupled equations for 2-spinors with the units restored:
\begin{pmatrix} (mc^2 - E + e \phi) & c\sigma\cdot \left(p - \frac{e}{c}A\right) \\ -c\sigma\cdot \left(p - \frac{e}{c}A\right) & \left(mc^2 + E - e \phi\right) \end{pmatrix} \begin{pmatrix} \psi_+ \\ \psi_- \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix}.
(E - e\phi) \psi_+ - c\sigma\cdot \left(p - \frac{e}{c}A\right) \psi_- = mc^2 \psi_+
-(E - e\phi) \psi_- + c\sigma\cdot \left(p - \frac{e}{c}A\right) \psi_+ = mc^2 \psi_-
Assuming the field is weak and the motion of the electron non-relativistic, we have the total energy of the electron approximately equal to its rest energy, and the momentum going over to the classical value,
E - e\phi \approx mc^2
p \approx m v
and so the second equation may be written
\psi_- \approx \frac{1}{2mc} \sigma\cdot \left(p - \frac{e}{c}A\right) \psi_+
which is of order v/c - thus at typical energies and velocities, the bottom components of the Dirac spinor in the standard representation are much suppressed in comparison to the top components. Substituting this expression into the first equation gives after some rearrangement
(E - mc^2) \psi_+ = \frac{1}{2m} \left[\sigma\cdot \left(p - \frac{e}{c}A\right)\right]^2 \psi_+ + e\phi \psi_+
The operator on the left represents the particle energy reduced by its rest energy, which is just the classical energy, so we recover Pauli's theory if we identify his 2-spinor with the top components of the Dirac spinor in the non-relativistic approximation. A further approximation gives the Schrödinger equation as the limit of the Pauli theory. Thus the Schrödinger equation may be seen as the far non-relativistic approximation of the Dirac equation when one may neglect spin and work only at low energies and velocities. This also was a great triumph for the new equation, as it traced the mysterious i that appears in it, and the necessity of a complex wave function, back to the geometry of spacetime through the Dirac algebra. It also highlights why the Schrödinger equation, although superficially in the form of a diffusion equation, actually represents the propagation of waves.
It should be strongly emphasized that this separation of the Dirac spinor into large and small components depends explicitly on a low-energy approximation. The entire Dirac spinor represents an irreducible whole, and the components we have just neglected to arrive at the Pauli theory will bring in new phenomena in the relativistic regime – antimatter and the idea of creation and annihilation of particles.
Comparison with the Weyl theory[edit]
In the limit m → 0, the Dirac equation reduces to the Weyl equation, which describes relativistic massless spin-1/2 particles.[5]
Dirac Lagrangian[edit]
Both the Dirac equation and the Adjoint Dirac equation can be obtained from (varying) the action with a specific Lagrangian density that is given by:
\mathcal{L}=i\hbar c\overline{\psi}\gamma^{\mu}\partial_{\mu}\psi-mc^{2}\overline{\psi}\psi
If one varies this with respect to ψ one gets the Adjoint Dirac equation. Meanwhile if one varies this with respect to ψ one gets the Dirac equation.
Physical interpretation[edit]
The Dirac theory, while providing a wealth of information that is accurately confirmed by experiments, nevertheless introduces a new physical paradigm that appears at first difficult to interpret and even paradoxical. Some of these issues of interpretation must be regarded as open questions.[citation needed]
Identification of observables[edit]
The critical physical question in a quantum theory is—what are the physically observable quantities defined by the theory? According to general principles, such quantities are defined by Hermitian operators that act on the Hilbert space of possible states of a system. The eigenvalues of these operators are then the possible results of measuring the corresponding physical quantity. In the Schrödinger theory, the simplest such object is the overall Hamiltonian, which represents the total energy of the system. If we wish to maintain this interpretation on passing to the Dirac theory, we must take the Hamiltonian to be
H = \gamma^0 \left[mc^2 + c \gamma^k \left(p_k-\frac{q}{c}A_k\right) \right] + qA^0.
where, as always, there is an implied summation over the twice-repeated index k = 1, 2, 3. This looks promising, because we see by inspection the rest energy of the particle and, in case A = 0, the energy of a charge placed in an electric potential qA0. What about the term involving the vector potential? In classical electrodynamics, the energy of a charge moving in an applied potential is
H = c\sqrt{\left(p - \frac{q}{c}A\right)^2 + m^2c^2} + qA^0.
Thus the Dirac Hamiltonian is fundamentally distinguished from its classical counterpart, and we must take great care to correctly identify what is an observable in this theory. Much of the apparent paradoxical behaviour implied by the Dirac equation amounts to a misidentification of these observables.
Hole theory[edit]
The negative E solutions to the equation are problematic, for it was assumed that the particle has a positive energy. Mathematically speaking, however, there seems to be no reason for us to reject the negative-energy solutions. Since they exist, we cannot simply ignore them, for once we include the interaction between the electron and the electromagnetic field, any electron placed in a positive-energy eigenstate would decay into negative-energy eigenstates of successively lower energy by emitting excess energy in the form of photons. Real electrons obviously do not behave in this way.
To cope with this problem, Dirac introduced the hypothesis, known as hole theory, that the vacuum is the many-body quantum state in which all the negative-energy electron eigenstates are occupied. This description of the vacuum as a "sea" of electrons is called the Dirac sea. Since the Pauli exclusion principle forbids electrons from occupying the same state, any additional electron would be forced to occupy a positive-energy eigenstate, and positive-energy electrons would be forbidden from decaying into negative-energy eigenstates.
If an electron is forbidden from simultaneously occupying positive-energy and negative-energy eigenstates, then the feature known as Zitterbewegung, which arises from the interference of positive-energy and negative-energy states, would have to be considered to be an unphysical prediction of time-dependent Dirac theory. This conclusion may be inferred from the explanation of hole theory given in the preceding paragraph. Recent results have been published in Nature [R. Gerritsma, G. Kirchmair, F. Zaehringer, E. Solano, R. Blatt, and C. Roos, Nature 463, 68-71 (2010)] in which the Zitterbewegung feature was simulated in a trapped-ion experiment. This experiment impacts the hole interpretation if one infers that the physics-laboratory experiment is not merely a check on the mathematical correctness of a Dirac-equation solution but the measurement of a real effect whose detectability in electron physics is still beyond reach.
Dirac further reasoned that if the negative-energy eigenstates are incompletely filled, each unoccupied eigenstate – called a hole – would behave like a positively charged particle. The hole possesses a positive energy, since energy is required to create a particle–hole pair from the vacuum. As noted above, Dirac initially thought that the hole might be the proton, but Hermann Weyl pointed out that the hole should behave as if it had the same mass as an electron, whereas the proton is over 1800 times heavier. The hole was eventually identified as the positron, experimentally discovered by Carl Anderson in 1932.
It is not entirely satisfactory to describe the "vacuum" using an infinite sea of negative-energy electrons. The infinitely negative contributions from the sea of negative-energy electrons has to be canceled by an infinite positive "bare" energy and the contribution to the charge density and current coming from the sea of negative-energy electrons is exactly canceled by an infinite positive "jellium" background so that the net electric charge density of the vacuum is zero. In quantum field theory, a Bogoliubov transformation on the creation and annihilation operators (turning an occupied negative-energy electron state into an unoccupied positive energy positron state and an unoccupied negative-energy electron state into an occupied positive energy positron state) allows us to bypass the Dirac sea formalism even though, formally, it is equivalent to it.
In certain applications of condensed matter physics, however, the underlying concepts of "hole theory" are valid. The sea of conduction electrons in an electrical conductor, called a Fermi sea, contains electrons with energies up to the chemical potential of the system. An unfilled state in the Fermi sea behaves like a positively-charged electron, though it is referred to as a "hole" rather than a "positron". The negative charge of the Fermi sea is balanced by the positively-charged ionic lattice of the material.
In quantum field theory[edit]
See also: Fermionic field
In quantum field theories such as quantum electrodynamics, the Dirac field is subject to a process of second quantization, which resolves some of the paradoxical features of the equation.
Other formulations[edit]
The Dirac equation can be formulated in a number of other ways.
As a differential equation in one real component[edit]
Generically (if a certain linear function of electromagnetic field does not vanish identically), three out of four components of the spinor function in the Dirac equation can be algebraically eliminated, yielding an equivalent fourth-order partial differential equation for just one component. Furthermore, this remaining component can be made real by a gauge transform.[6]
Curved spacetime[edit]
This article has developed the Dirac equation in flat spacetime according to special relativity. It is possible to formulate the Dirac equation in curved spacetime.
The algebra of physical space[edit]
This article developed the Dirac equation using four vectors and Schrödinger operators. The Dirac equation in the algebra of physical space uses a Clifford algebra over the real numbers, a type of geometric algebra.
See also[edit]
The Dirac Equation appears on the floor of Westminster Abbey on the plaque commemorating Paul Dirac's life, which was inaugurated on November 13, 1995.[7]
1. ^ P.W. Atkins (1974). Quanta: A handbook of concepts. Oxford University Press. p. 52. ISBN 0-19-855493-1.
2. ^ T.Hey, P.Walters (2009). The New Quantum Universe. Cambridge University Press. p. 228. ISBN 978-0-521-56457-1.
3. ^ Dirac, P.A.M. (1958 (reprinted in 2011)). Principles of Quantum Mechanics (4th ed.). Clarendon. p. 255. ISBN 978-0-19-852011-5. Check date values in: |date= (help)
4. ^ see for example Brian Pendleton: Quantum Theory 2012/2013, section 4.3 The Dirac Equation
5. ^ Tommy Ohlsson (22 September 2011). Relativistic Quantum Physics: From Advanced Quantum Mechanics to Introductory Quantum Field Theory. Cambridge University Press. p. 86. ISBN 978-1-139-50432-4. Retrieved 17 March 2013.
6. ^ Akhmeteli, Andrey (2011). "One real function instead of the Dirac spinor function" (PDF). Journal of Mathematical Physics 52 (8): 082303. arXiv:1008.4828. Bibcode:2011JMP....52h2303A. doi:10.1063/1.3624336.
7. ^ Gisela Dirac-Wahrenburg. "Paul Dirac". Dirac.ch. Retrieved 2013-07-12.
Selected papers[edit]
• Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons.
• Shankar, R. (1994). Principles of Quantum Mechanics (2nd ed.). Plenum.
• Bjorken, J D & Drell, S. Relativistic Quantum mechanics.
• Thaller, B. (1992). The Dirac Equation. Texts and Monographs in Physics. Springer.
• Schiff, L.I. (1968). Quantum Mechanics (3rd ed.). McGraw-Hill.
• Griffiths, D.J. (2008). Introduction to Elementary Particles (2nd ed.). Wiley-VCH. ISBN 978-3-527-40601-2.
External links[edit] |
66bfa8ab59b68fea | This Quantum World/Implications and applications/A quantum bouncing ball
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A quantum bouncing ball[edit]
As a specific example, consider the following potential:
V(z)=mgz\quad\hbox{if}\quad z>0\quad\hbox{and}\quad V(z)=\infty\quad\hbox{if}\quad z<0.
g is the gravitational acceleration at the floor. For z<0, the Schrödinger equation as given in the previous section tells us that d^2\psi(z)/dz^2=\infty unless \psi(z)=0. The only sensible solution for negative z is therefore \psi(z)=0. The requirement that V(z)=\infty for z<0 ensures that our perfectly elastic, frictionless quantum bouncer won't be found below the floor.
Since a picture is worth more than a thousand words, we won't solve the time-independent Schrödinger equation for this particular potential but merely plot its first eight solutions:
Quantum bouncer.png
Where would a classical bouncing ball subject to the same potential reverse its direction of motion? Observe the correlation between position and momentum (wavenumber).
All of these states are stationary; the probability of finding the quantum bouncer in any particular interval of the z axis is independent of time. So how do we get it to move?
Recall that any linear combination of solutions of the Schrödinger equation is another solution. Consider this linear combination of two stationary states:
Assuming that the coefficients A,B and the wave functions \psi_1(x),\psi_2(x) are real, we calculate the mean position of a particle associated with \psi(t,x):
The first two integrals are the (time-independent) mean positions of a particle associated with \psi_1(x)\,e^{i\omega_1t} and \psi_2(x)\,e^{i\omega_2t}, respectively. The last term equals
and this tells us that the particle's mean position oscillates with frequency \Delta\omega= \omega_2-\omega_1 and amplitude 2AB\int\!dx\,\psi_1x\psi_2 about the sum of the first two terms.
Visit this site to watch the time-dependence of the probability distribution associated with a quantum bouncer that is initially associated with a Gaussian distribution. |
beedd50fcc558dbc | The Many Interpretations of Quantum Mechanics
What is the ultimate nature of reality?
Are quantum effects constantly carving us into innumerable copies, each copy inhabiting a different version of the universe? Or do all those other worlds pop out of existence as mere might-have-beens? Do our particles surf on quantum waves? Or are we ultimately made of the quantum waves alone? Or do the waves merely represent how much information we could possess about the state of the world? And if the waves are just a kind of information, information about what? Or is the information all that there is—and all that we are?
And although quantum mechanics is primarily the physics of the very small—of atoms, electrons, photons and other such particles—the world is made up of those particles. If their individual reality is radically different from what we imagine then surely so too is the reality of the pebbles, people and planets that they make up.
As recounted by our December article, The Many Worlds of Hugh Everett by journalist Peter Byrne, 50 years ago the iconoclastic physics student Hugh Everett introduced the idea that quantum physics is incessantly splitting the universe into alternate branches. Byrne’s article talks about Everett’s life (did you know his son is the lead singer of the rock band Eels?) as well as about his theory and the “Copenhagen Interpretation” he aimed to supplant. But many other interpretations of quantum mechanics exist, and today Copenhagenists have more subtle variants to choose from than the one that Everett once called “a philosophic monstrosity.” Here is an all-too-short run-down on some of them.
The basic scenario an interpretation must address is when a quantum system is prepared in a combination of states known as a superposition. For example, a particle can be at both location A and B, or in the infamous thought experiment, Schrödinger’s quantum cat can be alive and dead at the same time. The problem is that when we observe or measure a superposition, we get but one result: our detector reports either “A” or “B,” not both; the cat would appear either very alive or very dead.
Copenhagen Interpretation
This interpretation (or variants of it) has long been the party line for quantum physicists. The Schrödinger equation describes how a wave function evolves smoothly and continuously over time, up until the point when our big, clunky measuring apparatus intervenes. The wave function enables us to predict, say, there’s a 60% probability we’ll detect the particle at location A. After we detect it at A or B, we have to represent the particle with a new wave function that conforms with the measurement result.
What bothers some people about this interpretation is the random, abrupt change in the wave function, which violates the Schrödinger equation, the very heart of quantum mechanics. Everett argued that this approach was philosophically a mess: it used two contradictory conceptual schemes to describe reality, the quantum one of wave functions and the classical one of us and our apparatus.
Many Worlds Interpretation
Everett’s theory. Also known as the relative state formulation.
The superposition of the particle spreads to the apparatus, and to us looking at the apparatus, and ultimately to the entire universe. The components of the resulting superposition are like parallel universes: in one we see outcome A, in another we see outcome B. All the branches coexist simultaneously, but because they are completely non-interacting the “A” copy of us is completely unaware of the “B” copy and vice versa. Mathematically, this universal superposition is what the Schrödinger equation predicts if you describe the whole universe with a wave function.
What bothers people about this interpretation is its conclusion that we are perpetually dividing into multiple copies, which may have ghastly implications as well as being bizarre.
Bohmian Interpretation
Also known as the De Broglie–Bohm interpretation or the pilot wave interpretation.
This theory postulates that every particle not only has a wave function but also exists as an actual particle riding along at some precise but unknown location on the wave and being guided by it. How the wave guides the particle is described by a new equation that is introduced to accompany the standard Schrödinger equation. The randomness of quantum measurements comes about because we cannot know exactly where a particle started out. The theory was proposed by David Bohm in 1952 (a few years before Everett’s theory), extending a theory of Louis De Broglie’s from 1927.
Changing the Rules
Some theorists seek to find a mechanism that causes the “collapse” of the wave function from a superposition of possibilities to a single outcome. For example, Roger Penrose has proposed that gravitational effects may play this role. Other models, such as the Ghirardi-Rimini-Weber theory, introduce specific modifications to the Schrödinger equation. By differing from standard quantum theory, such models in principle might be falsifiable by experiment (or conversely, standard theory could be falsified in their favor).
Decoherence Theory
This is not an interpretation, but it is an important element of the modern understanding of quantum mechanics. It expands upon the kind of mathematical analysis that led Everett to his interpretation, because it analyzes the effect that stray quantum interactions with the surrounding environment have on a system in a superposition. The chief conclusion is that the almost unstoppable loss of information through these channels “decoheres” a quantum superposition, making it more like an ordinary classical state. It explains very well why we see the classical world that we do, and clarifies the requirements to keep quantum effects manifest in the lab.
Copenhagenists can point to decoherence as an explanation of what makes large classical systems different from small quantum systems (in general, large systems decohere much more readily and rapidly than tiny ones). Everettians can point to it as a more complete explanation of how the parallel branches form and become independent. But best of all, decoherence can be studied experimentally, and a very active area of quantum research is confirming it and exploring it in ever greater detail.
Consistent Histories
This scheme analyzes sequences of states of a system (which may include the whole universe), to find what questions can be consistently answered about the system, such as “was the particle at A or B at time T?” The measurement problem, however, is not resolved: the question of which histories actually happen remains a matter of probabilities just as with the standard Copenhagenist approach.
Is it Real?
In some respects the decision between a Copenhagenist and an Everettian viewpoint boils down to a basic question: Is the wave function real or is it just information? If it is “real”—in some sense the universe really consists of quantum waves propagating around—then one tends to be driven to an Everettian viewpoint; the “collapses” that wave functions must undergo to produce the one reality that we see are too problematic. But if the wave function is just information, for example, a representation of what an experimenter knows about a system, then that “collapse” is completely natural. Imagine the standard classical scenario of flipping a coin. Before you look at it, your knowledge of its state is “50% chance of heads, 50% chance of tails.” When you look, your knowledge instantaneously changes to, say, “100% heads, 0% tails.”
“Shut Up and Calculate!”
Some physicists talk of the “shut up and calculate interpretation”: ignore the philosophical puzzle of how the classical and the quantum coexist and use the Schrödinger equation (and all the subsequent mathematical developments of quantum theory) to compute quantities of practical interest. These include energy levels of atoms; predictions for particle collider experiments; the properties of semiconductors, superconductors and other materials; and so on. It is all that most physicists ever need.
Transactional Interpretation
This interpretation has waves traveling forward and backward in time, setting up standing waves, for example between an emitter of a particle and its subsequent detector. It was proposed by John G. Cramer (physicist and science fiction author) in 1986 and claimed by him to provide insight into puzzles such as wave function collapse and the Schrödinger’s cat experiment. These insights have led Cramer to pursue an experiment to try to demonstrate the sending of signals backward in time (which most quantum physicists will tell you is impossible if standard quantum mechanics is correct).
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5025740f8511ffb9 | Take the 2-minute tour ×
In classical mechanics, $F=ma$ tells us how to evolve a system at time $t=t_0$ to $t=t_0+dt$.
In quantum mechanics, the Schrodinger equation gives us a similar recipe.
These equations are, in a certain sense, completely deterministic. Is it possible that nature only appears to be deterministic because the only language we know how to express physics is math (particularly equations), which (not to offend statisticians) seems to be particularly apt at describing deterministic systems?
In other words, are there possible time-evolution laws that are both non-deterministic and falsifiable?
If not, is determinism not falsifiable?
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2 Answers
"are there possible time-evolution laws that are both non-deterministic and falsifiable?"
Yes, they are called stochastic (differential) equations. The classic example is the Langevin equation, which is Newton's law with a random force.
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Interesting. I still feel that this is deterministic in some sense. By running Monte Carlo simulations, etc. one could in principle map out the probability distribution of this particle as a function of time. In this sense, it almost as deterministic as QM – hwlin Feb 23 '13 at 3:59
Yes, the probability distribution evolves deterministically. Whatever your non-deterministic mechanics are you can describe it by a deterministically evolving probability distribution on some appropriate space of states. – Michael Brown Feb 23 '13 at 4:19
@hwlin - To elaborate Michael's point a little bit, if something is not deterministic once, we can always take a massive ensemble of similar systems and make some kind of a statistical inference which is true on average but not necessarily in every case. This is how the whole of statistical mechanics came about, and leading from that, stochastic processes and non-equilibrium stat mech as well. So essentially, take a large enough sample of your non-deterministic thing, and we have the tools to tell you what will happen. :) – Kitchi Feb 23 '13 at 9:57
To elaborate a bit more. The Langevin equation for a single system is $\tilde{F}=m\tilde{a}$, which is non deterministic because both $\tilde{F}$ and $\tilde{a}$ are random. If we take averages on both sides we recover the deterministic $F=ma$ with $F=\langle\tilde{F}\rangle$ and $a=\langle\tilde{a}\rangle$. Thus the ensemble behaves deterministically because we are averaging out the random fluctuations, but each individual system continues being non-deterministic. – juanrga Feb 24 '13 at 13:55
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$F=ma$ only applies to a special class of classical systems. It does not apply to non-deterministic classical systems for which more general equations of motion are needed:
Poincaré resonances and the extension of classical dynamics
Poincaré resonances and the limits of trajectory dynamics
The same about the Schrödinger equation except that any ordinary textbook on QM already explains you in what situations you cannot use the Schrödinger equation to describe the evolution of the system under study
The quantum version of the above extension of classical dynamics is covered in The Liouville Space Extension of Quantum Mechanics
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839acea1bfc82a51 | elementary particle
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elementary particle
any of several entities, such as electrons, neutrons, or protons, that are less complex than atoms and are regarded as the constituents of all matter
Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005
Elementary particle
A particle that is not a compound of other particles. At one time the elementary particles of matter were the atoms of the chemical elements, but the atoms are now known to be compounds of the electron, proton, and neutron. In turn, the proton and neutron, and likewise all the other hadrons (strongly interacting particles), are now known to be compounds of quarks. It is convenient, however, to continue to call hadrons elementary particles to distinguish them from their compounds (atomic nuclei, for instance); this usage is also justified by the fact that quarks are not strictly particles, because, as far as is known, they cannot be isolated. The term fundamental particle can be used to denote particles that are truly fundamental constituents of matter and are not compounds in any sense. See Electron, Hadron, Neutron, Proton, Quarks
The known fundamental particles (see table) fall into two categories: the gauge bosons, comprising the photon, gluon, and weak bosons; and the fermions, comprising the quarks and leptons. The graviton, the quantum of the gravitational field, has been omitted from table since it plays no role in high-energy particle physics: it is firmly predicted by theory, but the prospect of direct observation is exceedingly remote. Of the gauge bosons, the photon has been known since the beginning of quantum mechanics. The heavy gauge bosons W ± and Z 0 were observed in 1983; their properties had been deduced from the weak interactions, for which they are responsible. The lightest (and stable) lepton, the electron (e), is the first known fundamental particle. The next found was the muon (μ, originally called the mu meson). The fundamental fermions are grouped into three families. Gluons and quarks are never seen as free particles; this phenomenon is known as confinement. Particles that are composed of quarks and gluons are called hadrons; essentially, mesons are composed of a quark-antiquark pair q, and baryons are three quarks qqq, bound together by the exchange of gluons. See Baryon, Gluons, Graviton, Intermediate vector boson, Lepton, Meson, Photon
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Particles with the properties of the quarks of the quark model (charges ±23e or ±13e and masses less than 300 MeV) have never been observed. Direct evidence both for quarks and for their confinement is given by the phenomenon of hadronic jets. For example, in high-energy deep-inelastic electron-proton scattering, in which the electron loses a sizable fraction of its energy, the observed cross section shows that the charge of the proton is carried by pointlike (radius less than 10-1 femtometer) particles of small mass. However, no such particles are seen in the final state of this process, or indeed of any other high-energy collision. What is seen is a narrow shower of hadrons. The interpretation is that the electron scatters off one of the quarks in the proton and gives it a large energy and momentum, the quark responding as though it were a free particle of mass much less than 100 MeV, consistent with the masses of the u and d quarks (see table). Later, through the production of quark-antiquark pairs, the energy and momentum of the struck quark is divided among a number of hadrons, mostly pions, a process called hadronization or fragmentation of the quark, which is to be distinguished from the decay of a free particle. The resulting shower of hadrons, whose total momentum vector is roughly that of the original quark, is called a hadronic jet (like a jet of water which breaks up into a spray of droplets). Such jets are also seen in other high-energy reactions, such as e+e- annihilation into hadrons, and also in pp collisions; they are the closest available phenomenon to the actual observation of a quark as a free particle.
To each kind of particle there corresponds an antiparticle, or conjugate particle, which has the same mass and spin, belongs to the conjugate representation (multiplet) of internal symmetry, and has opposite values of charge, I3, strangeness, and so forth (quantum numbers which are conserved additively). The product of the space parities of a particle and its antiparticle is +1 if the particle is a boson, -1 if a fermion. For instance, the electron e and its antiparticle, the positron e-, have the same masses and spins, and opposite charges and lepton number, and an S-wave state of e and e- has parity -1. Particles for which the antiparticle is the same as the particle are called self-conjugate; examples are the photon γ and the neutral pion &pgr;0. The equality of masses implies the equality of lifetimes of particle and antiparticle. Thus the positron is stable; however, in the presence of ordinary matter it soon annihilates with an electron, and thus is not a component of ordinary matter. See Antimatter, Positron
The interactions of particles are responsible for their scattering and transformations (decays and reactions). Because of interactions, an isolated particle may decay into other particles. Two particles passing near each other may transform, perhaps into the same particles but with changed momenta (elastic scattering) or into other particles (inelastic scattering). The rates or cross sections of these transformations, and so also the interactions responsible for them, fall into three groups: strong (typical decay rates of 1021–1023 s-1), electromagnetic (1016–1019 s-1), and weak (<1015 s-1). Strong interactions occur only between hadrons. Electromagnetic interactions result from the coupling of charge to the electromagnetic field. Weak interactions are usually unobservable in competition with strong or electromagnetic interactions. They are observable only when they do something which those much stronger interactions cannot do (forbidden by the selection rules); for instance, by changing flavors they can make a particle decay which would otherwise be stable, and by making parity-violating transition amplitudes they can produce an otherwise absent asymmetry in the angular distribution of a reaction. See Selection rules (physics)
Most particles are unstable and decay into smaller-mass particles. The only particles which appear to be stable are the massless particles (graviton, photon), the neutrinos (possibly massless), the electron, the proton, and the ground states of stable nuclei, atoms, and molecules. It is speculated that some or all of the neutrinos may be massive and unstable and that the proton (and therefore all nuclei) may be unstable. The present view is that the only massive particles which are strictly stable are the electron and the lightest neutrino(s). The electron is the lightest charged particle; its decay would be into neutral particles and could not conserve charge. Likewise, the lightest neutrino is the lightest fermion; its decay would be into bosons and could not conserve angular momentum. See Neutrino
The unstable elementary particles must be studied within a short time of their creation, which occurs in the collision of a fast (high-energy) particle with another particle. Such fast particles exist in nature, namely the cosmic rays, but their flux is small; thus most elementary particle research is based on high-energy particle accelerators. See Nuclear reaction, Particle accelerator, Particle detector
Hadrons can be divided into the quasistable (or hadronically stable) and the unstable. The quasistable hadrons are simply those that are too light to decay into other hadrons by way of the strong interactions, such decays being restricted by the requirement that isobaric spin I and flavors be conserved.
The unstable hadrons are also called particle resonances. Their lifetimes, of the order of 10-23 s, are much too short to be observed directly. Instead they appear, through the uncertainty principle, as spreads in the masses of the particles—that is, in their widths—just as in the case of nuclear resonances. See Uncertainty principle
A characteristic of the hadrons is that they are grouped into i-spin multiplets (for example, n, p; &pgr;-, &pgr;0, &pgr;+); the masses of the particles in each multiplet differ by only a few megaelectronvolts (MeV). The i-spin multiplets of hadrons themselves form groups (called supermultiplets) which were recognized in 1961 as multiplets (representations) of the group SU3 (now referred to as SU3flavor to distinguish this physical symmetry from SU3color). For instance, the lightest mesons (&eegr;, K, &pgr;) and baryons (&Lgr;, N, &Xgr;, &Sgr;) are each a set of eight particles having i-spins I = (0, 12, 12, 1) and hypercharges Y = (0,1, - 1,0) respectively; this pattern is that of the octet, {8}, representation of the group SU3. Again, the lowest-mass JP = &frac;32+ baryons (Δ, &Sgr;*, &Xgr;*, &OHgr;), ten particles with I = (&frac;32, 1, 12, 0) and Y = (1, 0, -1, -2), form a decuplet, {10}, representation of SU3. The spread of the masses in these groups is about a hundred times greater than in the i-spin multiplets, a few hundred MeV compared to a few MeV. According to the quark model, this SU3 symmetry and the pattern of charges in the SU3 multiplets result simply from the existence of a third kind (flavor) of quark, the s (strange) quark, with charge the same as the d quark, namely 13, together with the flavor independence of the glue force; that is, all three quarks u, d, and s have the same interaction with the glue field. The resulting flavor SU3 symmetry is broken by the relatively large mass of the s, approximately 150 MeV. The three quarks make up the fundamental triplet, {3}, representation of SU3.
Hadrons are known which contain yet more massive quarks, the c and the b (see the table). The resulting symmetry is badly broken, and the supermultiplets hardly recognizable.
It appears that the “glue” field which binds quarks together to make hadrons is a Yang-Mills (that is, a non-abelian) gauge field of an SU3 symmetry group, SU3color. This is an exact symmetry of nature. The quanta of the field are called gluons, and its quantum theory is called quantum chromodynamics (QCD). The gluon field resembles the electromagnetic field, but has an internal symmetry index (octet index) which runs over eight values; that is, there are really eight fields, corresponding to the eight parameters needed to specify an SU3 transformation. Just as the electromagnetic field is coupled to (that is, photons are emitted and absorbed by) the density and current of a conserved quantity, charge, the gluon field is coupled to color. The coupling of the gluon to a particle is fixed by the color of the particle (that is, what member of what color multiplet) and just one universal coupling constant g, analogous to the electronic unit of charge e. (The analogy breaks down in quantum theory, as discussed below; the quantity g is no longer constant but it is still universal.)
Since the long-range forces observed between hadrons are no different than those between other particles, hadrons must be colorless, that is, color singlet combinations of quarks, their colored constituents. The two simplest combinations of quarks which can be colorless are 1q2 and q1q2q3; these are found in nature as the basic structure of mesons and baryons, respectively. The exchange of gluons between any of the quarks in these colorless combinations gives rise to an attractive force, which binds them together.
Gluons are not colorless, and therefore they are coupled to themselves. This situation is very different from electromagnetism, where the photon does not carry charge. The consequence of this self-coupling of massless particles is a severe infrared (small momentum transfer or large distance) divergence of perturbation theory. In particular, the interaction between two colored particles through the gluon field, which in lowest order is an inverse-square Coulomb force, proportional to g2/r2 (where r is the distance between the particles), becomes stronger than this inverse-square force at larger r. A way of describing this is to say that the coupling constant g is effectively larger at larger r; this defines the so-called running coupling constant g(r). According to the first-order radiative correction, g(r) becomes infinite at a certain distance, the so-called scale parameter rc.
A specific form for the gluonic force between two colored particles, at large r, namely that it falls to a nonzero constant value λ, of the order of ℏcrc-2 (where ℏ is Planck's constant divided by 2&pgr;, and c is the speed of light), is suggested by a model, the superconductor analogy. This force is confining.
The conjecture is that the vacuum is like a superconductor with respect to color, with the interchange, however, of electric and magnetic quantities. That is, the vacuum acts like a color magnetic superconductor which confines color flux into bundles which have a diameter of order rc and an energy per unit length equal to λ of order ℏcrc-2. The color flux bundles run between colored particles; they can also form closed loops. These flux bundles are often idealized as having vanishing diameter and are then called strings. This idealization is obviously good only if the flux bundles are long compared to rc, and if their local radius of curvature is always much larger than rc.
According to the so-called naive quark model, hadrons are bound states of nonrelativistic (slowly moving) quarks, analogous to nuclei as bound states of nucleons. The interactions between the quarks are taken qualitatively from QCD, namely a confining central potential and (exactly analogous to electrodynamic interactions) spin-spin (hyperfine) and spin-orbit potentials; quantitatively, these potentials are adjusted to make the energy levels of the model system fit the observed hadron masses. This model should be valid for hadrons composed of heavy quarks but not for hadrons containing light quarks (u, d, s), but in fact it succeeds in giving a good description of many properties of all hadrons. One reason is that many of these properties follow from so-called angular physics, that is, symmetry-based physical principles that transcend the specific model. A meson is a bound state of a quark and an antiquark, q1q2. A baryon is a bound state of three quarks, q1q2q3.
The known heavy quarks are the c (charm), b (bottom), and t (top) quarks, whose masses are larger than the natural energy scale of QCD, ≈1 GeV. But because the width of the t is also larger than 1 GeV, the t quark decays before the QCD force acts on it, and thus before any well-defined hadron forms. So in the present context “heavy quarks” mean only c and b. A hadron which contains a single heavy quark resembles an atom; the heavy quark sits nearly at rest at the center, and is a static source of the color field, just as the atomic nucleus is a static source of the electric field. Just as an atom is changed very little (except in mass) if its nucleus is replaced by another of the same charge (an isotope), a heavy-quark hadron is changed very little (except in mass) if its heavy quark is replaced by another of the same color. This is called heavy-quark symmetry. So, for example, the D, D*, B, and B* mesons are similar, except in mass. This plays an important role in the quantitative analysis of their weak decays.
If a hadron contains two heavy quarks, then in a not too highly excited state the heavy quarks move slowly, compared to the speed of light c, and so the effect of the exchange of gluons between the quarks can be approximated (up to radiative corrections) by a potential energy which depends only on the positions of the quarks (local static potential); further, the wave function of the system satisfies the ordinary nonrelativistic Schrödinger equation. Consequently, the properties of hadrons composed of heavy quarks are rather easily calculated.
Mesons with the composition c and b are called charmonium and bottomonium, respectively. These names are based on the model of positronium, ee-; the generic name for flavorless mesons, q, is quarkonium. Since both heavy quarkonium and positronium are systems of a fermion bound to its antifermion by a central force, they are qualitatively very similar.
The electroweak theory, starting from the observation that both the electromagnetic and weak interactions result from the exchange of vector (spin-1) bosons, has unified these interactions into a spontaneously broken gauge theory. Similarly, the observation that the strong (hadronic) interactions are also due to the exchange of vector bosons (gluons) suggests that all these vector bosons (the photon, the three weak bosons, and the eight gluons) are quanta of the components of the gauge field of a large symmetry group, SU5 or larger. Such theories are called grand unification theories (GUTs). The large symmetry group of the grand unification theory must be spontaneously broken, making all the gauge bosons massive except the gluon octet and the photon, leaving SU3 × U1 (color × electromagnetism) as the apparent gauge symmetry of the world. See Grand unification theories
In these theories, the leptons and quarks occur together in multiplets of the large symmetry group. These multiplets are called families (or generations). The known fundamental fermions do seem to fall into three families (see table). Each family consists of a weak i-spin doublet of leptons (neutrino [charge 0] and charged lepton [charge +e]), and a color triplet of weak i-spin doublets of quarks (up-type [charge 23e] and down-type [charge -13e]).
elementary particle
[‚el·ə′men·trē ′pärd·i·kəl]
(particle physics)
A particle which, in the present state of knowledge, cannot be described as compound, and is thus one of the fundamental constituents of all matter. Also known as fundamental particle; particle; subnuclear particle.
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Helium atom ground state.
Smallest recognized division of a chemical element
Mass range1.67×10−27 to 4.52×10−25 kg
Electric chargezero (neutral), or ion charge
Diameter range62 pm (He) to 520 pm (Cs) (data page)
ComponentsElectrons and a compact nucleus of protons and neutrons
An atom is the smallest constituent unit of ordinary matter that constitutes a chemical element. Every solid, liquid, gas, and plasma is composed of neutral or ionized atoms. Atoms are extremely small, typically around 100 picometers across. They are so small that accurately predicting their behavior using classical physics—as if they were billiard balls, for example—is not possible due to quantum effects. Current atomic models use quantum principles to better explain and predict this behavior.
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. More than 99.94% of an atom's mass is in the nucleus. The protons have a positive electric charge, the electrons have a negative electric charge, and the neutrons have no electric charge. If the number of protons and electrons are equal, then the atom is electrically neutral. If an atom has more or fewer electrons than protons, then it has an overall negative or positive charge, respectively — these atoms are called ions.
The electrons of an atom are attracted to the protons in an atomic nucleus by the electromagnetic force. The protons and neutrons in the nucleus are attracted to each other by the nuclear force. This force is usually stronger than the electromagnetic force that repels the positively charged protons from one another. Under certain circumstances, the repelling electromagnetic force becomes stronger than the nuclear force. In this case, the nucleus splits and leaves behind different elements. This is a form of nuclear decay.
The number of protons in the nucleus is the atomic number and it defines to which chemical element the atom belongs. For example, any atom that contains 29 protons is copper. The number of neutrons defines the isotope of the element. Atoms can attach to one or more other atoms by chemical bonds to form chemical compounds such as molecules or crystals. The ability of atoms to associate and dissociate is responsible for most of the physical changes observed in nature. Chemistry is the discipline that studies these changes.
History of atomic theory
In philosophy
The basic idea that matter is made up of tiny indivisible particles is very old, appearing in many ancient cultures such as Greece and India. The word atomos, meaning "uncuttable", was coined by the ancient Greek philosophers Leucippus and his pupil Democritus (5th century BC). These ancient ideas were not based on scientific reasoning.[1][2][3][4]
Dalton's law of multiple proportions
Atoms and molecules as depicted in John Dalton's A New System of Chemical Philosophy vol. 1 (1808)
In the early 1800s, John Dalton compiled experimental data gathered by himself and other scientists and noticed that chemical elements seemed to combine by weight in ratios of small whole numbers. Dalton called this pattern the "law of multiple proportions". For instance, there are two types of tin oxide: one is 88.1% tin and 11.9% oxygen, and the other is 78.7% tin and 21.3% oxygen. Adjusting these figures, for every 100 g of tin there is either 13.5 g or 27 g of oxygen respectively. 13.5 and 27 form a ratio of 1:2, a ratio of small whole numbers. Similarly, there are two iron oxides in which for every 112 g of iron there is either 32 g or 48 g of oxygen respectively, which gives a ratio of 2:3. As a final example, there are three oxides of nitrogen in which for every 140 g of nitrogen, there is 80 g, 160 g, and 320 g of oxygen respectively, which gives a ratio of 1:2:4. This recurring pattern in the data suggested that elements always combine in multiples of basic indivisible units, which Dalton concluded were atoms. In the case of the tin oxides, for every one tin atom, there are either one or two oxygen atoms (SnO and SnO2). In the case of the iron oxides, for every two iron atoms, there are either two or three oxygen atoms (Fe2O2 and Fe2O3).[a] In the case of the nitrogen oxides, their formulas are N2O, NO, and NO2 respectively.[5][6][7][8]
Kinetic theory of gases
In the late 18th century, a number of scientists found that they could better explain the behavior of gases by describing them as collections of sub-microscopic particles and modelling their behavior using statistics and probability. Unlike Dalton's atomic theory, the kinetic theory of gases describes not how gases react chemically with each other to form compounds, but how they behave physically: diffusion, viscosity, conductivity, pressure, etc.
Brownian motion
In 1827, botanist Robert Brown used a microscope to look at dust grains floating in water and discovered that they moved about erratically, a phenomenon that became known as "Brownian motion". This was thought to be caused by water molecules knocking the grains about. In 1905, Albert Einstein proved the reality of these molecules and their motions by producing the first statistical physics analysis of Brownian motion.[9][10][11] French physicist Jean Perrin used Einstein's work to experimentally determine the mass and dimensions of molecules, thereby providing physical evidence for the particle nature of matter.[12]
Discovery of the electron
The Geiger-Marsden experiment:
In 1897, J.J. Thomson discovered that cathode rays are not electromagnetic waves but made of particles that are 1,800 times lighter than hydrogen (the lightest atom). Therefore, they were not atoms, but a new particle, the first subatomic particle to be discovered. He called these new particles corpuscles but they were later renamed electrons. Thomson also showed that electrons were identical to particles given off by photoelectric and radioactive materials.[13] It was quickly recognized that electrons are the particles that carry electric currents in metal wires, and carry the negative electric charge within atoms. Thus Thomson overturned the belief that atoms are the indivisible, fundamental particles of matter.[14] The misnomer "atom" is still used, even though atoms are not literally "uncuttable".
Discovery of the nucleus
J. J. Thomson postulated that the negatively-charged electrons were distributed throughout the atom in a uniform sea of positive charge. This was known as the plum pudding model. In 1909, Hans Geiger and Ernest Marsden, working under the direction of Ernest Rutherford, bombarded metal foil with alpha particles to observe how they scattered. They expected all the charged particles to pass straight through with little deflection, because Thomson's model said that the charges in the atom are so diffuse that their electric fields in the foil could not affect the alpha particles much. Yet Geiger and Marsden spotted alpha particles being deflected by angles greater than 90°, which was supposed to be impossible according to Thomson's model. To explain this, Rutherford proposed that the positive charge of the atom is concentrated in a tiny nucleus at the center. Only such an intense concentration of charge could produce an electric field strong enough to deflect alpha particles that much.[15]
Discovery of isotopes
While experimenting with the products of radioactive decay, in 1913 radiochemist Frederick Soddy discovered that there appeared to be more than one type of atom at each position on the periodic table.[16] The term isotope was coined by Margaret Todd as a suitable name for different atoms that belong to the same element. J.J. Thomson created a technique for isotope separation through his work on ionized gases, which subsequently led to the discovery of stable isotopes.[17]
Bohr model
The Bohr model of the atom, with an electron making instantaneous "quantum leaps" from one orbit to another with gain or loss of energy. This model of electrons in orbits is obsolete.
In 1913 the physicist Niels Bohr proposed a model in which the electrons of an atom were assumed to orbit the nucleus but could only do so in a finite set of orbits, and could jump between these orbits only in discrete changes of energy corresponding to absorption or radiation of a photon.[18] This quantization was used to explain why the electrons' orbits are stable (given that normally, charges in acceleration, including circular motion, lose kinetic energy which is emitted as electromagnetic radiation, see synchrotron radiation) and why elements absorb and emit electromagnetic radiation in discrete spectra.[19]
Chemical bonds between atoms were explained by Gilbert Newton Lewis in 1916, as the interactions between their constituent electrons.[21] As the chemical properties of the elements were known to largely repeat themselves according to the periodic law,[22] in 1919 the American chemist Irving Langmuir suggested that this could be explained if the electrons in an atom were connected or clustered in some manner. Groups of electrons were thought to occupy a set of electron shells about the nucleus.[23]
The Bohr model of the atom was the first complete physical model of the atom. It described the overall structure of the atom, how atoms bond to each other, and predicted the spectral lines of hydrogen. Bohr's model was not perfect and was soon superseded by the more accurate Schroedinger model (see below), but it was sufficient to evaporate any remaining doubts that matter is composed of atoms. For chemists, the idea of the atom had been a useful heuristic tool, but physicists had doubts as to whether matter really is made up of atoms as nobody had yet developed a complete physical model of the atom.
The Schrödinger model
In 1925 Werner Heisenberg published the first consistent mathematical formulation of quantum mechanics (matrix mechanics).[20] One year earlier, Louis de Broglie had proposed the de Broglie hypothesis: that all particles behave like waves to some extent,[25] and in 1926 Erwin Schrödinger used this idea to develop the Schrödinger equation, a mathematical model of the atom (wave mechanics) that described the electrons as three-dimensional waveforms rather than point particles.[26]
A consequence of using waveforms to describe particles is that it is mathematically impossible to obtain precise values for both the position and momentum of a particle at a given point in time; this became known as the uncertainty principle, formulated by Werner Heisenberg in 1927.[20] In this concept, for a given accuracy in measuring a position one could only obtain a range of probable values for momentum, and vice versa.[27] This model was able to explain observations of atomic behavior that previous models could not, such as certain structural and spectral patterns of atoms larger than hydrogen. Thus, the planetary model of the atom was discarded in favor of one that described atomic orbital zones around the nucleus where a given electron is most likely to be observed.[28][29]
Discovery of the neutron
Fission, high-energy physics and condensed matter
In 1938, the German chemist Otto Hahn, a student of Rutherford, directed neutrons onto uranium atoms expecting to get transuranium elements. Instead, his chemical experiments showed barium as a product.[32][33] A year later, Lise Meitner and her nephew Otto Frisch verified that Hahn's result were the first experimental nuclear fission.[34][35] In 1944, Hahn received the Nobel prize in chemistry. Despite Hahn's efforts, the contributions of Meitner and Frisch were not recognized.[36]
In the 1950s, the development of improved particle accelerators and particle detectors allowed scientists to study the impacts of atoms moving at high energies.[37] Neutrons and protons were found to be hadrons, or composites of smaller particles called quarks. The standard model of particle physics was developed that so far has successfully explained the properties of the nucleus in terms of these sub-atomic particles and the forces that govern their interactions.[38]
Subatomic particles
Neutrons have no electrical charge and have a free mass of 1,839 times the mass of the electron, or 1.6749×10−27 kg.[40][41] Neutrons are the heaviest of the three constituent particles, but their mass can be reduced by the nuclear binding energy. Neutrons and protons (collectively known as nucleons) have comparable dimensions—on the order of 2.5×10−15 m—although the 'surface' of these particles is not sharply defined.[42] The neutron was discovered in 1932 by the English physicist James Chadwick.
A nucleus that has a different number of protons than neutrons can potentially drop to a lower energy state through a radioactive decay that causes the number of protons and neutrons to more closely match. As a result, atoms with matching numbers of protons and neutrons are more stable against decay, but with increasing atomic number, the mutual repulsion of the protons requires an increasing proportion of neutrons to maintain the stability of the nucleus.[48]
The number of protons and neutrons in the atomic nucleus can be modified, although this can require very high energies because of the strong force. Nuclear fusion occurs when multiple atomic particles join to form a heavier nucleus, such as through the energetic collision of two nuclei. For example, at the core of the Sun protons require energies of 3 to 10 keV to overcome their mutual repulsion—the coulomb barrier—and fuse together into a single nucleus.[49] Nuclear fission is the opposite process, causing a nucleus to split into two smaller nuclei—usually through radioactive decay. The nucleus can also be modified through bombardment by high energy subatomic particles or photons. If this modifies the number of protons in a nucleus, the atom changes to a different chemical element.[50][51]
If the mass of the nucleus following a fusion reaction is less than the sum of the masses of the separate particles, then the difference between these two values can be emitted as a type of usable energy (such as a gamma ray, or the kinetic energy of a beta particle), as described by Albert Einstein's mass-energy equivalence formula, , where is the mass loss and is the speed of light. This deficit is part of the binding energy of the new nucleus, and it is the non-recoverable loss of the energy that causes the fused particles to remain together in a state that requires this energy to separate.[52]
Electron cloud
Electrons, like other particles, have properties of both a particle and a wave. The electron cloud is a region inside the potential well where each electron forms a type of three-dimensional standing wave—a wave form that does not move relative to the nucleus. This behavior is defined by an atomic orbital, a mathematical function that characterises the probability that an electron appears to be at a particular location when its position is measured.[54] Only a discrete (or quantized) set of these orbitals exist around the nucleus, as other possible wave patterns rapidly decay into a more stable form.[55] Orbitals can have one or more ring or node structures, and differ from each other in size, shape and orientation.[56]
The amount of energy needed to remove or add an electron—the electron binding energy—is far less than the binding energy of nucleons. For example, it requires only 13.6 eV to strip a ground-state electron from a hydrogen atom,[57] compared to 2.23 million eV for splitting a deuterium nucleus.[58] Atoms are electrically neutral if they have an equal number of protons and electrons. Atoms that have either a deficit or a surplus of electrons are called ions. Electrons that are farthest from the nucleus may be transferred to other nearby atoms or shared between atoms. By this mechanism, atoms are able to bond into molecules and other types of chemical compounds like ionic and covalent network crystals.[59]
Nuclear properties
By definition, any two atoms with an identical number of protons in their nuclei belong to the same chemical element. Atoms with equal numbers of protons but a different number of neutrons are different isotopes of the same element. For example, all hydrogen atoms admit exactly one proton, but isotopes exist with no neutrons (hydrogen-1, by far the most common form,[60] also called protium), one neutron (deuterium), two neutrons (tritium) and more than two neutrons. The known elements form a set of atomic numbers, from the single-proton element hydrogen up to the 118-proton element oganesson.[61] All known isotopes of elements with atomic numbers greater than 82 are radioactive, although the radioactivity of element 83 (bismuth) is so slight as to be practically negligible.[62][63]
About 339 nuclides occur naturally on Earth,[64] of which 252 (about 74%) have not been observed to decay, and are referred to as "stable isotopes". Only 90 nuclides are stable theoretically, while another 162 (bringing the total to 252) have not been observed to decay, even though in theory it is energetically possible. These are also formally classified as "stable". An additional 34 radioactive nuclides have half-lives longer than 100 million years, and are long-lived enough to have been present since the birth of the solar system. This collection of 286 nuclides are known as primordial nuclides. Finally, an additional 53 short-lived nuclides are known to occur naturally, as daughter products of primordial nuclide decay (such as radium from uranium), or as products of natural energetic processes on Earth, such as cosmic ray bombardment (for example, carbon-14).[65][note 1]
The actual mass of an atom at rest is often expressed in daltons (Da), also called the unified atomic mass unit (u). This unit is defined as a twelfth of the mass of a free neutral atom of carbon-12, which is approximately 1.66×10−27 kg.[68] Hydrogen-1 (the lightest isotope of hydrogen which is also the nuclide with the lowest mass) has an atomic weight of 1.007825 Da.[69] The value of this number is called the atomic mass. A given atom has an atomic mass approximately equal (within 1%) to its mass number times the atomic mass unit (for example the mass of a nitrogen-14 is roughly 14 Da), but this number will not be exactly an integer except (by definition) in the case of carbon-12.[70] The heaviest stable atom is lead-208,[62] with a mass of 207.9766521 Da.[71]
Shape and size
Atoms lack a well-defined outer boundary, so their dimensions are usually described in terms of an atomic radius. This is a measure of the distance out to which the electron cloud extends from the nucleus.[72] This assumes the atom to exhibit a spherical shape, which is only obeyed for atoms in vacuum or free space. Atomic radii may be derived from the distances between two nuclei when the two atoms are joined in a chemical bond. The radius varies with the location of an atom on the atomic chart, the type of chemical bond, the number of neighboring atoms (coordination number) and a quantum mechanical property known as spin.[73] On the periodic table of the elements, atom size tends to increase when moving down columns, but decrease when moving across rows (left to right).[74] Consequently, the smallest atom is helium with a radius of 32 pm, while one of the largest is caesium at 225 pm.[75]
When subjected to external forces, like electrical fields, the shape of an atom may deviate from spherical symmetry. The deformation depends on the field magnitude and the orbital type of outer shell electrons, as shown by group-theoretical considerations. Aspherical deviations might be elicited for instance in crystals, where large crystal-electrical fields may occur at low-symmetry lattice sites.[76][77] Significant ellipsoidal deformations have been shown to occur for sulfur ions[78] and chalcogen ions[79] in pyrite-type compounds.
Atomic dimensions are thousands of times smaller than the wavelengths of light (400–700 nm) so they cannot be viewed using an optical microscope, although individual atoms can be observed using a scanning tunneling microscope. To visualize the minuteness of the atom, consider that a typical human hair is about 1 million carbon atoms in width.[80] A single drop of water contains about 2 sextillion (2×1021) atoms of oxygen, and twice the number of hydrogen atoms.[81] A single carat diamond with a mass of 2×10−4 kg contains about 10 sextillion (1022) atoms of carbon.[note 2] If an apple were magnified to the size of the Earth, then the atoms in the apple would be approximately the size of the original apple.[82]
Radioactive decay
The most common forms of radioactive decay are:[84][85]
Magnetic moment
The nucleus of an atom will have no spin when it has even numbers of both neutrons and protons, but for other cases of odd numbers, the nucleus may have a spin. Normally nuclei with spin are aligned in random directions because of thermal equilibrium, but for certain elements (such as xenon-129) it is possible to polarize a significant proportion of the nuclear spin states so that they are aligned in the same direction—a condition called hyperpolarization. This has important applications in magnetic resonance imaging.[89][90]
Energy levels
The potential energy of an electron in an atom is negative relative to when the distance from the nucleus goes to infinity; its dependence on the electron's position reaches the minimum inside the nucleus, roughly in inverse proportion to the distance. In the quantum-mechanical model, a bound electron can occupy only a set of states centered on the nucleus, and each state corresponds to a specific energy level; see time-independent Schrödinger equation for a theoretical explanation. An energy level can be measured by the amount of energy needed to unbind the electron from the atom, and is usually given in units of electronvolts (eV). The lowest energy state of a bound electron is called the ground state, i.e. stationary state, while an electron transition to a higher level results in an excited state.[91] The electron's energy increases along with n because the (average) distance to the nucleus increases. Dependence of the energy on is caused not by the electrostatic potential of the nucleus, but by interaction between electrons.
For an electron to transition between two different states, e.g. ground state to first excited state, it must absorb or emit a photon at an energy matching the difference in the potential energy of those levels, according to the Niels Bohr model, what can be precisely calculated by the Schrödinger equation. Electrons jump between orbitals in a particle-like fashion. For example, if a single photon strikes the electrons, only a single electron changes states in response to the photon; see Electron properties.
The energy of an emitted photon is proportional to its frequency, so these specific energy levels appear as distinct bands in the electromagnetic spectrum.[92] Each element has a characteristic spectrum that can depend on the nuclear charge, subshells filled by electrons, the electromagnetic interactions between the electrons and other factors.[93]
An example of absorption lines in a spectrum
Valence and bonding behavior
Valency is the combining power of an element. It is determined by the number of bonds it can form to other atoms or groups.[99] The outermost electron shell of an atom in its uncombined state is known as the valence shell, and the electrons in that shell are called valence electrons. The number of valence electrons determines the bonding behavior with other atoms. Atoms tend to chemically react with each other in a manner that fills (or empties) their outer valence shells.[100] For example, a transfer of a single electron between atoms is a useful approximation for bonds that form between atoms with one-electron more than a filled shell, and others that are one-electron short of a full shell, such as occurs in the compound sodium chloride and other chemical ionic salts. Many elements display multiple valences, or tendencies to share differing numbers of electrons in different compounds. Thus, chemical bonding between these elements takes many forms of electron-sharing that are more than simple electron transfers. Examples include the element carbon and the organic compounds.[101]
Graphic illustrating the formation of a Bose-Einstein condensate
Quantities of atoms are found in different states of matter that depend on the physical conditions, such as temperature and pressure. By varying the conditions, materials can transition between solids, liquids, gases and plasmas.[104] Within a state, a material can also exist in different allotropes. An example of this is solid carbon, which can exist as graphite or diamond.[105] Gaseous allotropes exist as well, such as dioxygen and ozone.
At temperatures close to absolute zero, atoms can form a Bose-Einstein condensate, at which point quantum mechanical effects, which are normally only observed at the atomic scale, become apparent on a macroscopic scale.[106][107] This super-cooled collection of atoms then behaves as a single super atom, which may allow fundamental checks of quantum mechanical behavior.[108]
While atoms are too small to be seen, devices such as the scanning tunneling microscope (STM) enable their visualization at the surfaces of solids. The microscope uses the quantum tunneling phenomenon, which allows particles to pass through a barrier that would be insurmountable in the classical perspective. Electrons tunnel through the vacuum between two biased electrodes, providing a tunneling current that is exponentially dependent on their separation. One electrode is a sharp tip ideally ending with a single atom. At each point of the scan of the surface the tip's height is adjusted so as to keep the tunneling current at a set value. How much the tip moves to and away from the surface is interpreted as the height profile. For low bias, the microscope images the averaged electron orbitals across closely packed energy levels—the local density of the electronic states near the Fermi level.[109][110] Because of the distances involved, both electrodes need to be extremely stable; only then periodicities can be observed that correspond to individual atoms. The method alone is not chemically specific, and cannot identify the atomic species present at the surface.
Atoms can be easily identified by their mass. If an atom is ionized by removing one of its electrons, its trajectory when it passes through a magnetic field will bend. The radius by which the trajectory of a moving ion is turned by the magnetic field is determined by the mass of the atom. The mass spectrometer uses this principle to measure the mass-to-charge ratio of ions. If a sample contains multiple isotopes, the mass spectrometer can determine the proportion of each isotope in the sample by measuring the intensity of the different beams of ions. Techniques to vaporize atoms include inductively coupled plasma atomic emission spectroscopy and inductively coupled plasma mass spectrometry, both of which use a plasma to vaporize samples for analysis.[111]
Electron emission techniques such as X-ray photoelectron spectroscopy (XPS) and Auger electron spectroscopy (AES), which measure the binding energies of the core electrons, are used to identify the atomic species present in a sample in a non-destructive way. With proper focusing both can be made area-specific. Another such method is electron energy loss spectroscopy (EELS), which measures the energy loss of an electron beam within a transmission electron microscope when it interacts with a portion of a sample.
Spectra of excited states can be used to analyze the atomic composition of distant stars. Specific light wavelengths contained in the observed light from stars can be separated out and related to the quantized transitions in free gas atoms. These colors can be replicated using a gas-discharge lamp containing the same element.[113] Helium was discovered in this way in the spectrum of the Sun 23 years before it was found on Earth.[114]
Origin and current state
Baryonic matter forms about 4% of the total energy density of the observable Universe, with an average density of about 0.25 particles/m3 (mostly protons and electrons).[115] Within a galaxy such as the Milky Way, particles have a much higher concentration, with the density of matter in the interstellar medium (ISM) ranging from 105 to 109 atoms/m3.[116] The Sun is believed to be inside the Local Bubble, so the density in the solar neighborhood is only about 103 atoms/m3.[117] Stars form from dense clouds in the ISM, and the evolutionary processes of stars result in the steady enrichment of the ISM with elements more massive than hydrogen and helium.
Up to 95% of the Milky Way's baryonic matter are concentrated inside stars, where conditions are unfavorable for atomic matter. The total baryonic mass is about 10% of the mass of the galaxy;[118] the remainder of the mass is an unknown dark matter.[119] High temperature inside stars makes most "atoms" fully ionized, that is, separates all electrons from the nuclei. In stellar remnants—with exception of their surface layers—an immense pressure make electron shells impossible.
Periodic table showing the origin of each element. Elements from carbon up to sulfur may be made in small stars by the alpha process. Elements beyond iron are made in large stars with slow neutron capture (s-process). Elements heavier than iron may be made in neutron star mergers or supernovae after the r-process.
Isotopes such as lithium-6, as well as some beryllium and boron are generated in space through cosmic ray spallation.[125] This occurs when a high-energy proton strikes an atomic nucleus, causing large numbers of nucleons to be ejected.
Elements heavier than iron were produced in supernovae and colliding neutron stars through the r-process, and in AGB stars through the s-process, both of which involve the capture of neutrons by atomic nuclei.[126] Elements such as lead formed largely through the radioactive decay of heavier elements.[127]
There are a few trace atoms on Earth that were not present at the beginning (i.e., not "primordial"), nor are results of radioactive decay. Carbon-14 is continuously generated by cosmic rays in the atmosphere.[131] Some atoms on Earth have been artificially generated either deliberately or as by-products of nuclear reactors or explosions.[132][133] Of the transuranic elements—those with atomic numbers greater than 92—only plutonium and neptunium occur naturally on Earth.[134][135] Transuranic elements have radioactive lifetimes shorter than the current age of the Earth[136] and thus identifiable quantities of these elements have long since decayed, with the exception of traces of plutonium-244 possibly deposited by cosmic dust.[128] Natural deposits of plutonium and neptunium are produced by neutron capture in uranium ore.[137]
The Earth contains approximately 1.33×1050 atoms.[138] Although small numbers of independent atoms of noble gases exist, such as argon, neon, and helium, 99% of the atmosphere is bound in the form of molecules, including carbon dioxide and diatomic oxygen and nitrogen. At the surface of the Earth, an overwhelming majority of atoms combine to form various compounds, including water, salt, silicates and oxides. Atoms can also combine to create materials that do not consist of discrete molecules, including crystals and liquid or solid metals.[139][140] This atomic matter forms networked arrangements that lack the particular type of small-scale interrupted order associated with molecular matter.[141]
Rare and theoretical forms
Superheavy elements
All nuclides with atomic numbers higher than 82 (lead) are known to be radioactive. No nuclide with an atomic number exceeding 92 (uranium) exists on Earth as a primordial nuclide, and heavier elements generally have shorter half-lives. Nevertheless, an "island of stability" encompassing relatively long-lived isotopes of superheavy elements[142] with atomic numbers 110 to 114 might exist.[143] Predictions for the half-life of the most stable nuclide on the island range from a few minutes to millions of years.[144] In any case, superheavy elements (with Z > 104) would not exist due to increasing Coulomb repulsion (which results in spontaneous fission with increasingly short half-lives) in the absence of any stabilizing effects.[145]
Exotic matter
Other exotic atoms have been created by replacing one of the protons, neutrons or electrons with other particles that have the same charge. For example, an electron can be replaced by a more massive muon, forming a muonic atom. These types of atoms can be used to test fundamental predictions of physics.[150][151][152]
See also
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1. ^ Iron(II) oxide's formula is written here as Fe2O2 rather than the more conventional FeO because this better illustrates the explanation.
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Further reading
External links |
48a9e4dba534e237 | In The universe in a Helium droplet, Grigory Volovik relates the stability of a fermi surface to topology of a Green function. There he gives the example of a Fermi gas and says that the Green function for Fermi gas has form
$$G=\frac{1}{iw-v_{f}(P-P_{f})}$$ which has a singularity at $iw=0$ and $P=P_f$. The Volovik proposes that a topologial invariant for a Fermi surface can be found by calculating the winding number about this singularity using the invariant formula
$$N_1=\frac{1}{2\pi i}\oint dl G^{-1}\partial_lG $$
where $l$ corresponds to contour winding the singularity.
My questions is that singularity of Green function corresponds to the pole of a Green function, so if this is the case then $N_1$ also will not be zero for insulators. I cannot understand this.
The Green function you gave above is not the Green function for an insulator, which should generically has no pole if there is no Fermi surface.
The Volovik argument is a bit circular, since you know from the beginning that $p=p_{F}$ is the Fermi momentum, defining the Fermi surface. When $p\approx p_{F}$ you can infer the form of the Green function as you wrote in your question, and then from complex analysis you can write $N_{1}$. Then you realise $N_{1}$ is topologically non trivial since a perturbation $G=G_{0}+\delta G$ in $N_{1}$ will let it invariant (say differently $\delta N_{1}=N_{1}\left(G_{0}+\delta G\right)-N_{1}\left(G_{0}\right)=\mathcal{O}\left(\delta G^{2}\right)$ has no term linear in $\delta G$). So you promote $N_{1}$ to characterise the Fermi surface. In the case you gave $G^{-1}\left(z\right)=z+v_{F}\left(p-p_{F}\right)$ most certainly $N_{1}=\pm 1$ am I correct ?
I'm now wondering about the generic form of the Green function for an insulator. I feel it's something like $G^{-1}\left(z\right)=1$, but I do not understand why right now. I'd say it should be something sufficiently trivial analytically to have no density of state. The simplest way is to have no pole at all on the real axis. Clearly $N_{1}=0$ in this case.
Edit: Well, I wanted to see this in more details in fact. So let us try a simplistic model when a gap separates electron and hole bands.
A simple model
Suppose the Hamiltonian
with $p$ the momentum (instead of $p/\sqrt{2m}$), $\Delta$ a gap and $\mu$ a parameter (when going to many-body, it becomes the chemical potential). One sees that there is two bands (I adopt the condensed-matter terminology), say for electron and hole, both having the same effective mass and separated by the gap $\Delta$, which I suppose always positive. When $\mu>\Delta$, the Fermi surface is of electronic nature, whereas for $\mu<-\Delta$ only holes are present at the Fermi surface.
The associated Green functions are
with $\sigma=\pm$ representing the electron/hole ambivalence. Usually in condensed-matter, we prefer to discuss energy with the chemical potential taken as a reference, so we introduce $z=\omega-\mu$ (I drop $\hbar$). Then the Fermi surface is defined as the locus of the $p$'s lying at the chemical potential $\omega=\mu$. By construction, these loci are the pôles of $G_{\sigma}\left(z=0\right)$ above. One finds easily
for the pôles. The $\pm$ distinctions comes from the quadratic dispersion, so for each right-moving particle we have a left-moving one. This doubling is obviously essential in order to preserve the Galilean invariance, but it has nothing to do with the gap and the difference between metallic and insulating behaviour, so I drop the $\pm$ distinction from now.
Now we see the essential: for $\sigma = +1$ and $\mu>0$, there is no pôle along the real axis when $\mu<\Delta$. The same is true for $\mu<0$ and the hole sector $\sigma = -1$ if $\left|\mu\right|>\Delta$. In conclusion there is no pôle along the real axis in the gap. There are pôles along the real axis in the usual situation when the chemical potential lies above the gap $\mu > \Delta > 0$ in the electron sector and $-\mu>\Delta>0$ in the hole sector.
This is a generic argument : the Green function associated to an insulator has no pôle along the real axis, since the chemical potential lies inside a gap of forbidden momentum by construction. The Green function might have imaginary pôle inside the gap (as above in fact) which requires the momentum to be ill-defined (i.e. the momentum becomes imaginary). This can only happen by imposing boundary conditions since the "wave-function" is then evanescent. Such states are called for this reason edge states, or surface states.
Now to come back to the $N_{1}$ construction, we should define a generalised $N_{\sigma}$ which select whether we calculate the $N_1$ in the question using $G_{+}$ or $G_{-}$, and we will have
$$\begin{cases} \left|N_{1}\right|=+1\;;\;N_{-1}=0 & \mu>\Delta\\ N_{1}=N_{-1}=0 & \left|\mu\right|<\Delta\\ N_{1}=0\;;\;\left|N_{-1}\right|=+1 & \left|\mu\right|>\Delta \end{cases}$$
The sign in the metallic sector is of no real importance, it depends how one turns around the pôle. The important point is that $N_{\sigma}=0$ in the insulator sector, as is trivial the hole invariant when the chemical potential lies in the electron sector, and vice-versa $N_{1}=0$ when $\left|\mu\right|>\Delta$.
On the pôles of the Green functions
It seems the following remark is welcome (see the comment below from Meng-Cheng).
Usually, the spectrum properties of the system are obtained from the pôles of the Green function, taken as a function of $z=\omega-\mu$. For instance the Green function above $G_{\sigma}\left(z\right)=\left(z-\sigma\left(p^{2}+\Delta\right)+\mu\right)^{-1}$ has only one pôle $\hat{\omega}_{\sigma}=\sigma\left(p^{2}+\Delta\right)$ along the $\omega$-axis which corresponds to the eigen-energies of the system (the dispersion relation), and appears in the so-called spectral properties of the Green function [Economu]. As also remarked by Meng-Cheng, the difference between insulator and metal is then given by the possibility to have arbitrary low energy excitations. In the example, when $\Delta>0$ the lowest energy is $\Delta$ (i.e. $\hat{\omega}_{\sigma=1}\left(p=0\right)=\Delta$, which goes to zero for ungapped system (so for a normal metal).
In contrary, all the machinery developed in the previous section are concerned with the pôles of the function $G_{\sigma}\left(z=0\right)$ with respect to $p$. Only these later pôles of $G_{\sigma}\left(z=0\right)$ with respect to $p$ are associated to the Fermi surface [Horava].
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• $\begingroup$ I think Green function will also have poles for an insulator because the poles of the green function gives you only knowledge about the spectrum of the system it doesn't tell you about the system is insulating or not?@ FraSchelle am i right? $\endgroup$ – 079 Apr 2 '15 at 2:03
• $\begingroup$ @079 The Green function clearly has no pole along the real axis in the insulating region, see edit about a simple model to see how this happens. In fact you're right the Green function gives you knowledge about the spectrum of the Schrödinger equation. More precisely the Green function has a pole along the real axis for each discrete energy, eventually it has branch cuts for bands. So how could a Green function has a pole when there is no associated state, as for a trivial insulator ? $\endgroup$ – FraSchelle Apr 2 '15 at 5:19
• $\begingroup$ The Green function, as a function of frequency, have poles corresponding to single-particle excitations. In your example, the pole is $z=\sigma(p^2+\Delta)-\mu$. The difference between metal and insulator is whether that the location of the pole can be arbitrarily small frequency (just gapless v.s. gapped). $\endgroup$ – Meng Cheng Apr 2 '15 at 5:53
• $\begingroup$ @MengCheng You're perfectly right, but for the Volovik construction, the integral is defined in momentum, and so the poles are the poles with respect to p, not z. The spectral properties correspond to the poles in z indeed, as you say. I developed the above (boring) machinery to be sure to say no stupidity in the case of Volovik $N_{1}$ invariant (it has been done by Horava in fact, see arxiv.org/abs/hep-th/0503006 ). But any way to prove the stability of the Fermi surface in a better way is indeed welcome :-) Thanks for your comment $\endgroup$ – FraSchelle Apr 2 '15 at 9:45
• $\begingroup$ @MengCheng Ok, I understand now your previous comment. Indeed, the comment I made to user:079 was perfectly unclear... sorry for that. So let me state it correctly in the answer. Please check the edit, and tell me if you agree. $\endgroup$ – FraSchelle Apr 2 '15 at 11:46
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7960e303f281b366 | Erwin Schrödinger. Discussions of quantum foundations often seem to involve this fellow's much abused cat.
Erwin Schrödinger. Discussions of quantum foundations often seem to involve his much abused cat.
The group of physicists seriously engaged in studies of the “foundations” or “interpretation” of quantum theory is a small sliver of the broader physics community (perhaps a few hundred scientists among tens of thousands). Yet in my experience most scientists doing research in other areas of physics enjoy discussing foundational questions over coffee or beer.
The central question concerns quantum measurement. As often expressed, the axioms of quantum mechanics (see Sec. 2.1 of my notes here) distinguish two different ways for a quantum state to change. When the system is not being measured its state vector rotates continuously, as described by the Schrödinger equation. But when the system is measured its state “collapses” discontinuously. The Measurement Problem (or at least one version of it) is the challenge to explain why the mathematical description of measurement is different from the description of other physical processes.
My own views on such questions are rather unsophisticated and perhaps a bit muddled:
1) I know no good reason to disbelieve that all physical processes, including measurements, can be described by the Schrödinger equation.
2) But to describe measurement this way, we must include the observer as part of the evolving quantum system.
3) This formalism does not provide us observers with deterministic predictions for the outcomes of the measurements we perform. Therefore, we are forced to use probability theory to describe these outcomes.
4) Once we accept this role for probability (admittedly a big step), then the Born rule (the probability is proportional to the modulus squared of the wave function) follows from simple and elegant symmetry arguments. (These are described for example by Zurek – see also my class notes here. As a technical aside, what is special about the L2 norm is its rotational invariance, implying that the probability measure picks out no preferred basis in the Hilbert space.)
5) The “classical” world arises due to decoherence, that is, pervasive entanglement of an observed quantum system with its unobserved environment. Decoherence picks out a preferred basis in the Hilbert space, and this choice of basis is determined by properties of the Hamiltonian, in particular its spatial locality.
I like having it both ways — to view the overall evolution of system and observer as unitary, but still use probability theory for the purpose of describing the observer’s experience. The “collapse” of the state vector is really an update taking into account the observer’s knowledge; if we wished to we could instead describe the joint evolution of system and observer coherently without any discontinuous collapse.
If we insist on sticking with the coherent description at all times, we are forced to include in our description all the possible outcomes of all measurements along the way, which may seem extravagant. In practice, for the purpose of describing one observer’s experience, we don’t normally need to do that. Instead the observer updates her description as she learns more.
A related controversy concerns whether the quantum state is “ontic” (a mathematical description of physical reality) or “epistemic” (a description of what a particular observer knows about reality). I don’t really understand this question very well. Why can’t there be both a fundamental ontic state for the system and observer combined, and at the same time an (arguably less fundamental) epistemic state for the system alone which is continually updated in the light of the observer’s knowledge?
The viewpoint encapsulated by (1) – (5) is a version of what is sometimes called the Everett interpretation of quantum theory. It puzzles me somewhat that physicists I respect very much, who unlike me are serious devotees of quantum foundations (among them Carl Caves, Chris Fuchs, Adrian Kent, Tony Leggett, David Mermin, Rob Spekkens, … ), seem to find this viewpoint foolish, though perhaps I should not put words in their mouths. I admit it’s less precise than one might desire, and that one can feel a bit dizzy when thinking about a description of a physical system that includes oneself. I feel pretty comfortable with the Everett interpretation, though I try not to be dogmatic about it.
Anyway, I was inspired to post on this subject today due to a recent paper by Schlosshauer, Kofler, and Zeilinger, reporting the results of a poll taken at a quantum foundations conference in 2011. The poll probes the views of the conference participants regarding the interpretation of quantum theory. We should keep in mind that the physicists among the respondents (there were also philosophers and mathematicians) may be a highly biased sample of the general physics community; those attending a conference like this one are, of course, particularly passionate about foundational questions. A broader poll of physicists might have found rather different results. It’s a small sample as well (33 participants).
Overall I find the poll results rather hard to interpret, in part because many of the questions are deliberately ambiguous. But I was intrigued by the list, at the beginning of Sec. 4, of the statements supported by a majority of those surveyed:
1. Quantum information is a breath of fresh air for quantum foundations (76%).
2. Superpositions of macroscopically distinct states are in principle possible (67%).
3. Randomness is a fundamental concept in nature (64%).
4. Einstein’s view of quantum theory is wrong (64%).
5. The message of the observed violations of Bell’s inequalities is that local realism is untenable (64%).
6. Personal philosophical prejudice plays a large role in the choice of interpretation (58%).
7. The observer plays a fundamental role in the application of the formalism but plays no distinguished physical role (55%).
8. Physical objects have their properties well defined prior to and independent of measurement in some cases (52%).
9. The message of the observed violations of Bell’s inequalities is that unperformed measurements have no results (52%).
I’m surprised to find I agree with every one of these statements! Perhaps I am less out of sync with the quantum foundations crowd than I had imagined. Or is that an illusion?
I’m not sure about the value of polls like this one. But they are kind of fun anyway. |
0c337581ff0f28d4 | Solving a two-electron quantum dot model in terms of polynomial solutions of a Biconfluent Heun equation
, , and .
Annals of Physics (August 2014)
The effects on the non-relativistic dynamics of a system compound by two electrons interacting by a Coulomb potential and with an external harmonic oscillator potential, confined to move in a two dimensional Euclidean space, are investigated. In particular, it is shown that it is possible to determine exactly and in a closed form a finite portion of the energy spectrum and the associated eigenfunctions for the Schrödinger equation describing the relative motion of the electrons, by putting it into the form of a biconfluent Heun equation. In the same framework, another set of solutions of this type can be straightforwardly obtained for the case when the two electrons are submitted also to an external constant magnetic field.
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77f33a96e2260e6d | Archive for the ‘History’ Category
Yesterday I introduced Paul Dirac, number 10 in “The Guardian’s” list of the 10 best physicists. I mentioned that his main contributions to physics were (i) predicting antimatter, which he did in 1928, and (ii) producing an equation (now called the Dirac equation) which describes the behaviour of a sub-atomic particle such as an electron travelling at close to the speed of light (a so-called relativistic theory). This equation was also published in 1928.
The Dirac Equation
In 1928 Dirac wrote a paper in which he published what we now call the Dirac Equation.
The equation now known as the Dirac Equation describes the behaviour of an electron when travelling close to the speed of light. The equation now known as the Dirac Equation describes the behaviour of an electron when travelling close to the speed of light.
This is a relativistic form of Schrödinger’s wave equation for an electron. The wave equation was published by Erwin Schrödinger two years earlier in 1926, and describes how the quantum state of a physical system changes with time.
The Schrödinger eqation
The time dependent Schrödinger equation which describes the motion of an electron The time dependent Schrödinger equation which describes the motion of an electron
The various terms in this equation need some explaining. Starting with the terms to the left of the equality, and going from left to right, we have i is the imaginary number, remember i = \sqrt{-1}. The next term \hbar is just Planck’s constant divided by two times pi, i.e. \hbar = h/2\pi. The next term \partial/\partial t \text{ } \psi(\vec{r},t) is the partial derivative with respect to time of the wave function \psi(\vec{r},t).
Now, moving to the right hand side of the equality, we have
m which is the mass of the particle, V is its potential energy, \nabla^{2} is the Laplacian. The Laplacian, \nabla^{2} \psi(\vec{r},t) is simply the divergence of the gradient of the wave function, \nabla \cdot \nabla \psi(\vec{r},t).
In plain language, what the Schrödinger equation means “total energy equals kinetic energy plus potential energy”, but the terms take unfamiliar forms for reasons explained below.
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Today (January 30th) marks the 50th anniversary of the last time The Beatles played live together, in the infamous “rooftop” concert in 1969. Although they would go on to make one more studio album, Abbey Road in the summer of 1969; due to contractual and legal wranglings the rooftop concert, which was meant to be the conclusion to the movie they were shooting, would not come out until 1970 in the movie Let it Be.
It is also true to say that some of the songs on Abbey Road were performed “live” in the studio with very little overdubbing (as opposed to separate instrument parts being recorded separately as was done on e.g. Sgt. Pepper). But, the rooftop concert was the last time the greatest band in history were seen playing together, and has gone down in infamy. It has been copied by many, including the Irish band U2 who did a similar thing to record the video for their single “Where the Streets Have no Name” in 1987 in Los Angeles.
The Beatles were trying to think of a way to finish the movie that they had been shooting throughout January of 1969. They had discussed doing a live performance in all kinds of places; including on a boat, in the Roundhouse in London, and even in an amphitheatre in Greece. Finally, a few days before January 30th 1969, the idea of playing on the roof of their central-London offices was discussed. Whilst Paul and Ringo were in favour of this idea, and John was neutral, George was against it.
The decision to go ahead with playing on the roof was not made until the actual day. They took their equipment up onto the roof of their London offices at 3, Saville Row, and just start playing. No announcement was made, only The Beatles and their inner circle knew about the impromptu concert.
The concert consisted of the following songs :
1. “Get Back” (take one)
2. “Get Back” (take two)
3. “Don’t Let Me Down” (take one)
4. “I’ve Got a Feeling” (take one)
5. “One After 909”
6. “Dig a Pony”
7. “I’ve Got a Feeling” (take two)
8. “Don’t Let Me Down” (take two)
9. “Get Back” (take three)
People in the streets below initially had no idea what the music (“noise”) coming from the top of the building was, but of course younger people knew the building was the Beatles’ offices. However, they would not have recognised any of the songs, as these were not to come out for many more months. After the third song “Don’t Let Me Down”, the Police were called and came to shut the concert down. The band managed nine songs (five different songs, with three takes of “Get Back”, two takes of “Don’t Let Me Down”, and two takes of “I’ve Got a Feeling”) before the Police stopped them. Ringo Starr later said that he wanted to be dragged away from his drums by the Police, but no such dramatic ending happened.
At the end of the set John said
I’d like to thank you on behalf of the group and ourselves, and I hope we’ve passed the audition.
You can read more about the rooftop concert here.
Here is a YouTube video of “Get Back” (which may get taken down at any moment)
and here is a video on the Daily Motion website of the whole rooftop concert (again, it may get taken down at any moment).
Enjoy watching the greatest band ever perform live for the very last time!
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50 years ago today, on 21 October 1966, a tragedy happened in a small mining village in Wales which horrified the world. At 9:15am, Pantglas school in a place called Aberfan was engulfed by a river of coal debris. 116 children (more than half of the school’s pupils) and 28 adults were killed. Dozens more were rescued from the horror, with people from Aberfan and surrounding villages digging with their hands in a desperate attempt to save some lives.
The tragedy was due to a tip of coal waste (“slag heap” as they were often called) which had been piled on the side of the mountain against which the village nestles, and was entirely preventable. For months the local council had been warning the National Coal Board (NCB) of the risk, but the NCB had taken no notice.
In a tribunal held after the tragedy, the NCB were found guilty of negligence and of corporate manslaughter. However, they never paid a penny of compensation to the families, nor did they pay to have the numerous slag heaps rendered safe. Local families had to raise the money to do this themselves. After years of campaigning, in 1997 the newly-formed Welsh Assembly government finally repaid the families the money that they had raised. Some 10 years later the Welsh Assembly government paid the families a much larger sum, to correct for the inflation in the intervening 40 years.
I have been to the cemetery and memorial park in Aberfan. It is a beautiful tribute and memory to the tragedy that happened that wet October day in 1966.
Here is a very moving poem simply called Aberfan by Vera Rich, an English-born poet.
I have seen their eyes, the terrible, empty eyes
Of women in a glimmerless dawn, and the hands
Of men who have wrestled through long years with the dark
Underpinning of the mountains, strong hands that fight
In dumb faith that what was once flesh born of their flesh
And is earth of the earth, should rest in the earth of God,
Not that of the devil’s making…
The Tip had crouched like a plague-god, with the town,
A victim in reversion, held beneath
A vast, invisible paw… Not a lion to toss
A proud, volcano-mane of destruction, crouched
Like a rat, it waited…
I have seen their eyes, and the empty hands of men,
And they walk like victims of a second Flood
In a world no longer home, where the void of sky
Between tall mountains looms as a cenotaph
For a generation of laughter…
I have seen them
Walking, near-ghosts, wraiths from a half-formed legend
Of this more-than-Hamelin, where, on an autumn Friday,
Between nine and ten of the clock, death raised his flute
And the children followed…
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Today I thought I would share this great anti-war song – “Fortunate Son” by Creedence Clearwater Revival. It was released in September 1969, and is specifically about the lucky men who were born into families which, somehow, meant that they were not called up for the draft to fight in the Vietnam war.
These were the senators’ sons, the millionaires’ sons, the fortunate sons. Sons like George W. Bush, who miraculously found himself in the National Guard, far away from any danger, rather than in Vietnam fighting. I wonder why? Oh, maybe because his father, George H. Bush, had the political clout and importance to make sure his precious son didn’t go and fight in the jungles of Vietnam, unlike the poor white and black men who were drafted there.
As the draft went on, it became more and more apparent how many fortunate sons were avoiding going to war, thanks to their family’s influence in bending the rules. And how many poor blacks and whites had no choice, they were forced to go and would be jailed should they refuse. The Vietnam war was wrong on so many levels, but the inequity of the draft was certainly one of its wrongs.
“Fortunate Son” was released in September 1969, and talks of the privileged few who, somehow, avoided the Vietnam war draft.
“Fortunate Son” is rated at 99 in Rolling Stone Magazine’s list of the 500 greatest songs of all time. It really is a great song, I am surprised that I haven’t blogged about it before.
Some folks are born, made to wave the flag
Ooo, they’re red, white and blue
And when the band plays “Hail to the Chief”
Ooo, they point the cannon at you, Lord
Some folks are born, silver spoon in hand
Lord, don’t they help themselves, y’all
But when the taxman comes to the door
Lord, the house looks like a rummage sale, yeah
It ain’t me, it ain’t me, I ain’t no millionaire’s son, no, no
Yeah, yeah
Some folks inherit star spangled eyes
Ooh, they send you down to war, Lord
And when you ask ’em, “How much should we give?”
Ooh, they only answer “More! More! More!”, y’all
It ain’t me, it ain’t me, I ain’t no military son, son
It ain’t me, it ain’t me, I ain’t no fortunate one, one
Here is a video of the song. Enjoy!
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Just over 7 years ago, in early 2009, I bought a CD of some of Robert Kennedy’s greatest speeches. Whilst his brother John F. Kennedy gave some memorable speeches, for me Bobby Kennedy possibly surpassed JFK with his eloquence. One of his most moving and wonderful speeches has been passing through my mind these last two weeks or so; with the senseless shootings of innocent black people by police in the United States, the killing of the five policemen by an assassin in Houston, the horrific terrorist attack in Nice on Bastille Day which has killed at least 84 people, many of them children, and the failed coup in Turkey with over 100 dead. And, just as I was putting this blog together yesterday, the shooting of 3 more police officers in Baton Rouge.
Robert Kennedy (RFK) served as Attonery General under his brother’s Prisidency, but in 1965 he entered the Senate as one of the senators for New York. On 16 March 1968, RFK announced that he would run for the presidency, and set about touring the USA to garner support for his campaign. On the evening of 4 April, he was due to give a speech in Indianapolis when he learnt en-route of the assassination of Martin Luther King. He broke the news to the gathered crowd, many of whom had not heard the news until Bobby Kennedy told them. He gave a very moving and powerful speech on that evening, and I may blog about that particular speech another time.
But, today I am going to share the speech that he gave the day after MLK’s assassination, on 5 April 1968. The speech is entitled “The mindless menace of violence“, and it was delivered at the Cleveland Club in Ohio.
Kennedy toured the country as part of his campaign to become President of the United States, concentrating to a large part on some of the poorest communities in the country, where he met with dissaffected whites, blacks and latinos who had been left behind by the ‘American Dream’.
“this mindless menace of violence in America which again stains our land and every one of our lives.”
It is quite a long speech, nearly 10 minutes long, but bear with it and I think you will be struck by its eloquence. Bobby Kennedy wrote the speech himself, putting it together in the hours after the horror of MLK’s assassination had sunk into his mind.
The speech opens with these lines….
This is a time of shame and sorrow. It is not a day for politics. I have saved this one opportunity to speak briefly to you about this mindless menace of violence in America which again stains our land and every one of our lives.
Why? What has violence ever accomplished? What has it ever created? No martyr’s cause has ever been stilled by his assassin’s bullet.
But, Bobby Kennedy was also deeply concerned with the economic disparities in the United States, and with the sickening racism which had profoundly disturbed him. He later goes on to say…
This is the breaking of a man’s spirit by denying him the chance to stand as a father and as a man among other men. And this too afflicts us all. I have not come here to propose a set of specific remedies nor is there a single set. For a broad and adequate outline we know what must be done.
Followed immediately by these words…
The entire text can be found here at the John F. Kennedy presidential library website.
There are several versions of this mesmerising speech on YouTube, but many seem to have had an annoying soundtrack of some music added. I feel the added music detracts from hearing Bobby Kennedy’s words, which are powerful enough and do not need any music to make them more dramatic. So, the version I have included here is just RFK’s incredible words.
What strikes me most when I hear or read these words of Bobby Kennedy is how little progress we have made. One could argue that we have digressed; there are more mass shootings now in the USA than in the 1960s when these words were spoken. There is more terrorism and conflict than ever. And, in the presumptive Republican Party presidential nominee Donald Trump, we have a man who is the very antithesis of the wonderful ideals for which Bobby Kennedy stood.
I would say “enjoy” this video, but I am not sure that one can enjoy this speech. It is moving, harrowing, thought-provoking, upsetting, but also uplifting. To think that RFK was himself assassinated within a few months of giving this speech, it only adds poignancy to his words and highlights even more the truth and sadness of the mindless menace of violence
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Fifty years ago yesterday (17 May 1966), one of the seminal moments in 20th Century popular culture took place in the Manchester Free Trade Hall. Bob Dylan, who had burst onto the folk scene a few years before, was playing to a packed crowd towards the end of his gruelling 1966 World tour. The first half of his set was vintage Dylan, just the man (poet) and his guitar. The crowd were enraptured.
But, it all turned sour in the second half, when Dylan was joined by his band, The Hawks, and proceeded to do an ‘electric’ set. The crowd became restless. Many left; others booed, stamped their feet or started chanting. When he came back on to do his encore, things came to a head.
“Judas!” a man shouted.
“I don’t believe you.” Dylan replied. Then he started getting ready for the encore song. A few seconds later Dylan added
“You’re a liar!”
Then, he turned to his band and said “Play it fucking loud”, and they ripped into an angry version of Like a Rolling Stone. This is the moment as captured on film, it forms the closing scene of Martin Scorsese’s fascinating documentary No Direction Home.
There is also a very interesting in-depth audio documentary about this whole seminal incident, Ghosts of Electricity, made by Andy Kershaw for BBC Radio 1 and broadcast in 1999. It is available here on Andy Kershaw’s website.
Andy Kershaw’s fascinating documentary about the Bob Dylan “Judas” incident, which was originally broadcast in 1999 on BBC Radio 1.
The whole concert was recorded and circulated as a bootleg for many years. For some reason, it became known as the Royal Albert Hall Concert, even though it had happened at the Manchester Free Trade Hall; possibly because the 1966 World tour ended at the Royal Albert Hall on the 26 and 27 May. Dylan sanctioned an official release of the concert in 1998.
The cover for Bob Dylan’s “Royal Albert Hall Concert” CD, which includes the “Judas” heckle. In fact, the concert was recorded at the Manchester Free Trade Hall on 17 May 1966.
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One of the physicists in our book Ten Physicists Who Transformed Our Understanding of Reality (follow this link for more information on the book) is, not surprisingly, Isaac Newton. In fact, he is number 1 in the list. One could argue that he practically invented the subject of physics. We decided to call him the ‘father of physics’, with Galileo (whose life preceded Newton’s) being given the title of ‘grandfather’.
Newton was, clearly, a man of genius. But he was also a nasty, vindictive bastard (not to mince my words!). He didn’t really have any close friends in his life; there were plenty of people who admired him and respected him, and of course he had colleagues. But, apart from a niece whom he seemed to dote on in later life, and two men with whom he probably had love affairs, he was not a man who sought company. He was probably autistic, but lived at a time before such conditions were diagnosed or talked about.
Isaac Newton (1643-1727), the ‘father of physics’. He relished in feuding with other scientists
One sort of interaction that he did seem to enjoy with other people though was feuds. In fact, he seemed to thrive on feuding with other scientists. He loved to argue with others, which is not uncommon amongst academics. He had strong opinions which he liked to defend; this is normal. But, Newton took these disputes to an extreme; if he fell out with someone he would do everything he could to destroy that person.
Although I am sure that he had many ‘minor’ arguments, he had three main feuds with fellow scientists. These three men were
• Robert Hooke – curator of experiments at the Royal Society
• Gottfried (von) Leibniz – the German mathematician
• John Flamsteed – the first Astronomer Royal
In each case, he did his level best to destroy the other man. Each of these feuds is discussed in more detail in our book, but in this blogpost I will give a brief summary of his feud with Leibniz.
The feud came about because Newton refused to believe that Leibniz had independently come up with the mathematical idea of calculus. It was a recurring theme throughout Newton’s life that he sincerely believed that he was special. He had deep religious views (some would say extreme religious views). As part of these views, he believed that he had been specially chosen by God to understand things that others would never be able to understand.
Thus, when he heard that Leibniz had developed a mathematics similar to his own ‘theory of fluxions’ (as Newton called it), he naturally assumed that the German had stolen it from him. There then ensued a 30-year dispute between the two men, with Newton very much the aggressor.
Gottfried (von) Leibniz (1646-1716), German mathematician and co-inventor of calculus
It escalated from a dispute to a feud, and culminated in the Royal Society commissioning an ‘official investigation’ to establish propriety for the invention of calculus. When the report came out in 1713 it came out in Newton’s favour. But, by this time Newton was not only President of the Royal Society, but he had secretly authored the entire report. It was anything but impartial. Leibniz died the following year, a broken man from Newton’s relentless attacks.
One should, of course, be able to to admire a person for their work but not admire them in the least for the person that they were. Newton, in my mind, falls very firmly into this category. His contribution to physics is unparalleled, but I don’t think he was the kind of person one would want to know or even come across if one could help it!
Ten Physicists Who Transformed Our Understanding of Reality is available now. Follow this link to order
What is your favourite story about Newton?
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As I mentioned in this blog here, a few months ago I contributed some articles to a book called 30-Second Einstein, which will be published by Ivy Press in the not too distant future. One of the articles I wrote for the book was on Indian mathematical physicist Satyendra Bose. It is after Bose that ‘bosons’ are named (as in ‘the Higgs boson’), and also terms like ‘Bose-Einstein statistics’ and ‘Bose-Einstein condensate’. So, who was Satyendra Bose, and why is his name attached to these things?
Satyendra Bose was an Indian mathematical physicist after whom the 'boson' and Bose-Einstein statistics are named
Satyendra Bose was an Indian mathematical physicist after whom the ‘boson’ and Bose-Einstein statistics are named
Satyendra Bose was born in Calcutta, India, in 1894. He studied applied mathematics at Presidency College, Calcutta, obtaining a BSc in 1913 and an MSc in 1915. On both occasions, he graduated top of his class. In 1919, he made the first English translation of Einstein’s general theory of relativity, and by 1921 he had moved to Dhaka (in present-day Bangladesh) to become Reader (one step below full professor) in in the department of Physics.
It was whilst in Dhaka, in 1924, that he came up with the theory of how to count indistinguishable particles, such as photons (light particles). He showed that such particles follow statistics which are different from particles which can be distinguished. All his attempts to get his paper published failed, so in an act of some desperation he sent it to Einstein. The great man recognised the importance of Bose’s work immediately, translated it into German and got it published in Zeitschrift für Physik, one of the premier physics journals of the day.
Because of Einstein’s part in getting the theory published, we now know of this way of counting indistinguishable particles as Bose-Einstein statistics. We also name particles which obey this kind of statistics bosons; examples are the photon, the W and Z-particles (which mediate the weak nuclear force), and the most famous boson, the Higgs boson (responsible for mediating the property of mass via the Higgs field).
With the imminent partition of India when it was gaining independence from Britain, Bose returned to his native Calcutta where he spent the rest of his career. He died in 1974 at the age of 80.
You can read more about Satyendra Bose, Bose-Einstein statistics and Bose-Einstein condensates in 30-second Einstein, out soon from Ivy Press.
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In part 3 of this blog series I explained how Max Planck found a mathematical formula to fit the observed Blackbody spectrum, but that when he presented it to the German Physics Society on the 19th of October 1900 he had no physical explanation for his formula. Remember, the formula he found was
E_{\lambda} \; d \lambda = \frac{ A }{ \lambda^{5} } \frac{ 1 }{ (e^{a/\lambda T} -1) } \; d\lambda
if we express it in terms of wavelength intervals. If we express it in terms of frequency intervals it is
E_{\nu} \; d \nu = A^{\prime} \nu^{3} \frac{ 1 }{ (e^{ a^{\prime} \nu / T } - 1) } \; d\nu
Planck would spend six weeks trying to find a physical explanation for this equation. He struggled with the problem, and in the process was forced to abandon many aspects of 19th Century physics in both the fields of thermodynamics and electromagnetism which he had long cherished. I will recount his derivation – it is not the only one and maybe in coming blog posts I can show how his formula can be derived from other arguments, but this is the method Planck himself used.
Radiation in a cavity
As we saw in the derivation of the Rayleigh-Jeans law (see part 3 here, and links in that to parts 1 and 2), blackbody radiation can be modelled as an idealised cavity which radiates through a small hole. Importantly, the system is given enough time for the radiation and the material from which the cavity is made to come into thermal equilibrium with each other. This means that the walls of the cavity are giving energy to the radiation at the same rate that the radiation is giving energy to the walls.
Using classical physics, as we did in the derivation of the Rayleigh-Jeans law, we saw that the energy density (the energy per unit volume) is
\frac{du}{d\nu} = \left( \frac{ 8 \pi kT }{ c^{3} } \right) \nu^{2}
After trying to derive his equation based on standard thermodynamic arguments, which failed, Planck developed a model which, he found, was able to produce his equation. How did he do this?
Harmonic Oscillators
First, he knew from classical electromagnetic theory that an oscillating electron radiates (as it is accelerating), and he reasoned that when the cavity was in thermal equilibrium with the radiation in the cavity, the electrons in the walls of the cavity would oscillate and it was they that produced the radiation.
After much trial and error, he decided upon a model where the electrons were attached to massless springs. He could model the radiation of the electrons by modelling them as a whole series of harmonic oscillators, but with different spring stiffnesses to produce the different frequencies observed in the spectrum.
As we have seen (I derived it here), in classical physics the energy of a harmonic oscillator depends on both its amplitude of oscillation squared (E \propto A^{2}); and it also depends on its frequency of oscillation squared (E \propto \nu^{2}). The act of heating the cavity to a particular temperature is what, in Planck’s model, set the electrons oscillating; but whether a particular frequency oscillator was set in motion or not would depend on the temperature.
If it were oscillating, it would emit radiation into the cavity and absorb it from the cavity. He knew from the shape of the blackbody curve (and, by now, his equation which fitted it), that the energy density E d\nu at any particular frequency started off at zero for high frequencies (UV), then rose to a peak, and then dropped off again at low frequencies (in the infrared).
So, Planck imagined that the number of oscillators with a particular resonant frequency would determine how much energy came out in that frequency interval. He imagined that there were more oscillators with a frequency which corresponded to the maximum in the blackbody curve, and fewer oscillators at higher and lower frequencies. He then had to figure out how the total energy being radiated by the blackbody would be shared amongst all these oscillators, with different numbers oscillating at different frequencies.
He found that he could not derive his formula using the physics that he had long accepted as correct. If he assumed that the energy of each oscillator went as the square of the amplitude, as it does in classical physics, his formula was not reproduced. Instead, he could derive his formula for the blackbody radiation spectrum only if the oscillators absorbed and emitted packets of energy which were proportional to their frequency of oscillation, not to the square of the frequency as classical physics argued. In addition, he found that the energy could only come in certain sized chunks, so for an oscillator at frequency \nu, \; E = nh\nu, where n is an integer, and h is now known as Planck’s constant.
What does this mean? Well, in classical physics, an oscillator can have any energy, which for a particular oscillator vibrating at a particular frequency can be altered by changing the amplitude. Suppose we have an oscillator vibrating with an amplitude of 1 (in abitrary units), then because the energy goes as the square of the amplitude its energy is E=1^{2} =1. If we increase the amplitude to 2, the energy will now be E=2^{2} = 4. But, if we wanted an energy of 2, we would need an amplitude of \sqrt{2} = 1.414, and if we wanted an energy of 3 we would need an amplitude of \sqrt{3} = 1.73.
In classical physics, there is nothing to stop us having an amplitude of 1.74, which would give us an energy of 3.0276 (not 3), or an amplitude of 1.72 whichg would give us an energy of 2.9584 (not 3). But, what Planck found is that this was not allowed for his oscillators, they did not seem to obey the classical laws of physics. The energy could only be integers of h\nu, so E=0h\nu, 1h\nu, 2h\nu, 3h\nu, 4h\nu etc.
Then, as I said above, he further assumed that the total energy at a particular frequency was given by the energy of each oscillator at that frequency multiplied by the number of oscillators at that frequency. The frequency of a particular oscillator was, he imagined, determined by its stiffness (Hooke’s constant). The energy of a particular oscillator at a particular frequency could be varied by the amplitude of its oscillations.
Let us assume, just to illustrate the idea, that the value of h is 2. If the total energy in the blackbody at a particular frequency of, say, 10 (in arbitrary units) were 800 (also in arbitrary units), this would mean that the energy of each chunk (E=h \nu) was E = 2 \times 10 = 20. So, the number of chunks at that frequency would then be 800/20 = 40. 40 oscillators, each with an energy of 20, would be oscillating to give us our total energy of 800 at that frequency.
Because of this quantised energy, we can write that E_{n} = nh \nu, where n=0,1,2,3, \cdots.
The number of oscillators at each frequency
The next thing Planck needed to do was derive an expression for the number of oscillators at each frequency. Again, after much trial and error he found that he had to borrow an idea first proposed by Austrian physicist Ludwig Boltzmann to describe the most likely distribution of energies of atoms or molecules in a gas in thermal equilibrium. Boltzmann found that the number of atoms or molecules with a particular energy E was given by
N_{E} \propto e^{-E/kT}
where E is the energy of that state, T is the temperature of the gas and k is now known as Boltzmann’s constant. The equation is known as the Boltzmann distribution, and Planck used it to give the number of oscillators at each frequency. So, for example, if N_{0} is the number of oscillators with zero energy (in the so-called ground-state), then the numbers in the 1st, 2nd, 3rd etc. levels (N_{1}, N_{2}, N_{3},\cdots) are given by
N_{1} = N_{0} e^{ -E_{1}/kT }, \; N_{2} = N_{0} e^{ -E_{2}/kT }, \; N_{3} = N_{0} e^{ -E_{3}/kT }, \cdots
But, as E_{n} = nh \nu, we can write
N_{1} = N_{0} e^{ -h \nu /kT }, \; N_{2} = N_{0} e^{ -2h \nu /kT }, \; N_{3} = N_{0} e^{ -3h \nu /kT }, \cdots
Planck modelled blackbody radiation as a series of harmonic oscillators with equally spaced energy levels
To make it easier to write, we are going to substitute x = e^{ -h \nu / kT }, so we have
N_{1} = N_{0}x, \; N_{2} = N_{0} x^{2}, \; N_{3} = N_{0} x^{3}, \cdots
The total number of oscillators N_{tot} is given by
N_{tot} = N_{0} + N_{1} + N_{2} + N_{3} + \cdots = N_{0} ( 1 + x + x^{2} + x^{3} + \cdots)
Remember, this is the number of oscillators at each frequency, so the energy at each frequency is given by the number at each frequency multiplied by the energy of each oscillator at that frequency. So
E_{1}=N_{1} h \nu , \; E_{2} = N_{2} 2h \nu , \; E_{3} = N_{3} 3h \nu, \cdots
which we can now write as
E_{1} = h \nu N_{0}x, \; E_{2} = 2h \nu N_{0}x^{2}, \; E_{3} = 3h \nu N_{0}x^{3}, \cdots
The total energy E_{tot} is given by
E_{tot} = E_{0} + E_{1} + E_{2} + E_{3} + \cdots = N_{0} h \nu (0 + x + 2x^{2} + 3x^{3} + \cdots)
The average energy \langle E \rangle is given by
\langle E \rangle = \frac{ E_{tot} }{ N_{tot} } = \frac{ N_{0} h \nu (0 + x + 2x^{2} + 3x^{3} + \cdots) }{ N_{0} ( 1 + x + x^{2} + x^{3} + \cdots ) }
The two series inside the brackets can be summed. The sum of the series in the numerator, which we will call S_{1} is given by
S_{1} = \frac{ x - (n+1)x^{n+1} + nx^{n+2} }{ (1-x)^{2} }
(for the proof of this, see for example here)
The series in the denominator, which we will call S_{2}, is just a geometric progression. The sum of such a series is simply
S_{2} = \frac{ 1 - x^{n} }{ (1-x) }
Both series are in x, but remember x = e^{-h \nu / kT}. Also, both series are from a frequency of \nu = 0 \text{ to } \infty, and e^{-h \nu /kT} < 1, which means the sums converge and can be simplified.
S_{1} \rightarrow \frac{x}{ (1-x)^{2} } \text{ and } S_{2} \rightarrow \frac{ 1 }{(1-x)}
which means that \langle E \rangle = (h \nu S_{1})/S_{2} is given by
\langle E \rangle = \frac{ h \nu x }{ (1-x)^{2} } \times \frac{ (1-x) }{1} = \frac{h \nu x}{ (1-x) }
and so we can write that the average energy is
\boxed{ \langle E \rangle = \frac{h \nu}{( 1/x - 1) } = \frac{h \nu}{ (e^{h \nu/kT} - 1) } }
The radiance per frequency interval
In our derivation of the Rayleigh-Jeans law (in this blog here), we showed that, using classical physics, the energy density du per frequency interval was given by
du = \frac{ 8 \pi }{ c^{3} } kT \nu^{2} \, d \nu
where kT was the energy of each mode of the electromagnetic radiation. We need to replace the kT in this equation with the average energy for the harmonic oscillators that we have just derived above. So, we re-write the energy density as
du = \frac{ 8 \pi }{ c^{3} } \frac{ h \nu }{ (e^{h\nu/kT} - 1) } \nu^{2} \; d\nu = \frac{ 8 \pi h \nu^{3} }{ c^{3} } \frac{ 1 }{ (e^{h\nu/kT} - 1) } \; d\nu
du is the energy density per frequency interval (usually measured in Joules per metre cubed per Hertz), and by replacing kT with the average energy that we derived above the radiation curve does not go as \nu^{2} as in the Rayleigh-Jeans law, but rather reaches a maximum and turns over, avoiding the ultraviolet catastrophe.
It is more common to express the Planck radiation law in terms of the radiance per unit frequency, or the radiance per unit wavelength, which are written B_{\nu} and B_{\lambda} respectively. Radiance is the power per unit solid angle per unit area. So, as a first step to go from energy density to radiance we will divide by 4 \pi, the total solid angle. This gives
We want the power per unit area, not the energy per unit volume. To do this we first note that power is energy per unit time, and second that to go from unit volume to unit area we need to multiply by length. But, for EM radiation, length is just ct. So, we need to divide by t and multiply by ct, giving us that the radiance per frequency interval is
\boxed{ B_{\nu} = \frac{ 2h \nu^{3} }{ c^{2} } \frac{ 1 }{ (e^{h\nu/kT} - 1) } \; d\nu }
which is the way the Planck radiation law per frequency interval is usually written.
Radiance per unit wavelength interval
If you would prefer the radiance per wavelength interval, we note that \nu = c/\lambda and so d\nu = -c/\lambda^{2} \; d\lambda. Ignoring the minus sign (which is just telling us that as the frequency increases the wavelength decreases), and substituting for \nu and d\nu in terms of \lambda and d\lambda, we can write
B_{\lambda} = \frac{ 2h }{ c^{2} } \frac{ c^{3} }{ \lambda^{3} } \frac{ 1 }{ ( e^{hc/\lambda kT} - 1 ) } \frac{ c }{ \lambda^{2} } \; d\lambda
Tidying up, this gives
\boxed{ B_{\lambda} = \frac{ 2hc^{2} }{ \lambda^{5} } \frac{ 1 }{ ( e^{hc/\lambda kT} - 1 ) } \; d\lambda }
which is the way the Planck radiation law per wavelength interval is usually written.
To summarise, in order to reproduce the formula which he had empirically derived and presented in October 1900, Planck found that he he could only do so if he assumed that the radiation was produced by oscillating electrons, which he modelled as oscillating on a massless spring (so-called “harmonic oscillators”). The total energy at any given frequency would be given by the energy of a single oscillator at that frequency multiplied by the number of oscillators oscillating at that frequency.
However, he had to assume that
1. The energy of each oscillator was not related to either the square of the amplitude of oscillation or the square of the frequency of oscillation (as it would be in classical physics), but rather to the square of the amplitude and the frequency, E \propto \nu.
2. The energy of each oscillator could only be a multiple of some fundamental “chunk” of radiation, h \nu, so E_{n} = nh\nu where n=0,1,2,3,4 etc.
3. The number of oscillators with each energy E_{n} was given by the Boltzmann distribution, so N_{n} = N_{0} e^{-nh\nu/kT} where N_{0} is the number of oscillators in the lowest energy state.
In a way, we can imagine that the oscillators at higher frequencies (to the high-frequency side of the peak of the blackbody) are “frozen out”. The quantum of energy for a particular oscillator, given by E_{n}=nh\nu, is just too large to exist at the higher frequencies. This avoids the ultraviolet catastrophe which had stumped physicists up until this point.
By combining these assumptions, Planck was able in November 1900 to reproduce the exact equation which he had derived empirically in October 1900. In doing so he provided, for the first time, a physical explanation for the observed blackbody curve.
• Part 1 of this blogseries is here.
• Part 2 is here.
• Part 3 is here.
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There has been quite a bit of mention in the media this last week or so that it is 100 years since Albert Einstein published his ground-breaking theory of gravity – the general theory of relativity. Yet, there seems to be some confusion as to when this theory was first published, in some places you will see 1915, in others 1916. So, I thought I would try and clear up this confusion by explaining why both dates appear.
Albert Einstein in Berlin circa 1915 when his General Theory of Relativity was first published
Albert Einstein in Berlin circa 1915/16 when his General Theory of Relativity was first published
From equivalence to the field equations
Everyone knew that Einstein was working on a new theory of gravity. As I blogged about here, he had his insight into the equivalence between acceleration and gravity in 1907, and ever since then he had been developing his ideas to create a new theory of gravity.
He had come up with his principle of equivalence when he was asked in the autumn of 1907 to write a review article of his special theory of relativity (his 1905 theory) for Jahrbuch der Radioaktivitätthe (the Yearbook of Electronics and Radioactivity). That paper appeared in 1908 as Relativitätsprinzip und die aus demselben gezogenen Folgerungen (On the Relativity Principle and the Conclusions Drawn from It) (Jahrbuch der Radioaktivität, 4, 411–462).
In 1908 he got his first academic appointment, and did not return to thinking about a generalisation of special relativity until 1911. In 1911 he published a paper Einfluss der Schwerkraft auf die Ausbreitung des Lichtes (On the Influence of Gravitation on the Propagation of Light) (Annalen der Physik (ser. 4), 35, 898–908), in which he calculated for the first time the deflection of light produced by massive bodies. But, he also realised that, to properly develop his ideas of a new theory of gravity, he would need to learn some mathematics which was new to him. In 1912, he moved to Zurich to work at the ETH, his alma mater. He asked his friend Marcel Grossmann to help him learn this new mathematics, saying “You’ve got to help me or I’ll go crazy.”
Grossmann gave Einstein a book on non-Euclidean geometry. Euclidean geometry, the geometry of flat surfaces, is the geometry we learn in school. The geometry of curved surfaces, so-called Riemann geometry, had first been developed in the 1820s by German mathematician Carl Friedrich Gauss. By the 1850s another German mathematician, Bernhard Riemann developed this geometry of curved surfaces even further, and this was the Riemann geometry textbook which Grossmann gave to Einstein in 1912. Mastering this new mathematics proved very difficult for Einstein, but he knew that he needed to master it to be able to develop the equations for general relativity.
These equations were not ready until late 1915. Everyone knew Einstein was working on them, and in fact he was offered and accepted a job in Berlin in 1914 as Berlin wanted him on their staff when the new theory was published. The equations of general relativity were first presented on the 25th of November 1915, to the Prussian Academy of Sciences. The lecture Feldgleichungen der Gravitation (The Field Equations of Gravitation) was the fourth and last lecture that Einstein gave to the Prussian Academy on his new theory (Preussische Akademie der Wissenschaften, Sitzungsberichte, 1915 (part 2), 844–847), the previous three lectures, given on the 4th, 11th and 18th of November, had been leading up to this. But, in fact, Einstein did not have the field equations ready until the last few days before the fourth lecture!
The peer-reviewed paper of the theory (which also contains the field equations) did not appear until 1916 in volume 49 of Annalen der PhysikGrundlage der allgemeinen Relativitätstheorie (The Foundation of the General Theory of Relativity) Annalen der Physik (ser. 4), 49, 769–822. The paper was submitted by Einstein on the 20th of March 1916.
The beginning of Einstein's first paper on general relativity, which was received by Annalen der Physik on the 20th of March 1916 and
The beginning of Einstein’s first peer-reviewed paper on general relativity, which was received by Annalen der Physik on the 20th of March 1916
In a future blog, I will discuss Einstein’s field equations, but hopefully I have cleared up the confusion as to why some people refer to 1915 as the year of publication of the General Theory of Relativity, and some people choose 1916. Both are correct, which allows us to celebrate the centenary twice!
You can read more about Einstein’s development of the general theory of relativity in our book 10 Physicists Who Transformed Our Understanding of Reality. Order your copy here
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44d22c5d8897d7b7 | Friday, November 29, 2013
NMP and Consciousness
Alexander Wissner-Gross, a physicist at Harvard University and the Massachusetts Institute of Technology, and Cameron Freer, a mathematician at the University of Hawaii at Manoa, have developed a theory that they say describes many intelligent or cognitive behaviors, such as upright walking and tool use (see this and this ). The basic idea of the theory is that intelligent system collects information about large number of histories and preserves it. Thermodynamically this means large entropy so that the evolution of intelligence would be rather paradoxically evolution of highly entropic systems. According to standard view about Shannon entropy transformation of entropy to information (or the reduction of entropy to zero) requires a process selecting one of instances of thermal ensemble with a large number of degenerate states and one can wonder what is this selection process. This sounds almost like a paradox unless one accepts the existence of this process. I have considered the core of this almost paradox in TGD framework already earlier.
According to the popular article (see this) the model does not require explicit specification of intelligent behavior and the intelligent behavior relies on "causal entropic forces" (here one can counter argue that the selection process is necessary if one wants information gain). The theory requires that the system is able to collect information and predict future histories very quickly.
The prediction of future histories is one of the basic characters of life in TGD Universe made possible by zero energy ontology (ZEO) predicting that the thermodynamical arrow of geometric time is opposite for the quantum jumps reducing the zero energy state at upper and lower boundaries of causal diamond (CD) respectively. This prediction means quite a dramatic deviation from standard thermodynamics but is consistent with the notion of syntropy introduced by Italian theoretical physicist Fantappie already for more than half a century ago as well as with the reversed time arrow of dissipation appearing often in living matter.
The hierarchy of Planck constants makes possible negentropic entanglement and genuine information represented as negentropic entanglement in which superposed state pairs have interpretation as incidences ai↔ bi of a rule A↔ B: apart from possible phase the entanglement coefficients have same value 1/n1/2, where n=heff/h define the value of effective Planck constant and dimension for the effective covering of imbedding space. This picture generalizes also to the case of multipartite entanglement but predicts similar entanglement for all divisions of the system to two parts. There are however still some questions which are not completely settled and leave some room for imagination.
1. Negentropic entanglement is possible in the discrete degrees of freedom assignable to the n-fold covering of imbedding space allowing to describe situation formally. For heff/h=n one can introduce SU(n) as dynamical symmetry group and require that n-particle states are singlets under SU(n). This gives rise to n-particle states constructed by contracting n-dimensional permutation symbol contracted with many particle states assignable to the m factors. Spin-statistics connection might produce problems - at least it is non-trivial - since one possible interpretation is that the states carry fractional quantum numbers- in particular fractional fermion number and charges.
2. Is negentropic entanglement possible only in the new covering degrees of freedom or is it possible in more familiar angular momentum, electroweak, and color degrees of freedom?
1. One can imagine that also states that are singlets with respect to rotation group SO(3) and its covering SU(2) (2-particle singlet states constructed from two spin 1 states and spin singlet constructed from two fermions) could carry negentropic entanglement. The latter states are especially interesting biologically.
2. In TGD framework all space-time surfaces can be seen at least 2-fold coverings of M4 locally since boundary conditions do not seem to allow 3-surfaces with spatial boundaries so that finiteness of the space-time sheet requires covering structure in M4. This forces to ask whether this double covering could provide a geometric correlate for fermionic spin 1/2 suggested by quantum classical correspondence taken to extreme. Fermions are indeed fundamental particles in TGD framework and it would be nice if also 2-sheeted coverings would define fundamental building bricks of space-time.
3. Color group SU(3) for which color triplets defines singlets can be also considered. I have been even wondering whether quark color could actually correspond to 3-fold or 6-fold (color isospin corresponds to SU(2)) covering so that quarks would be dark leptons, which correspond n=3 coverings of CP2 and to fractionization of hypercharge and electromagnetic charge. The motivation came from the inclusions of hyper-finite factors of type II1 labelled by integer n≥ 3. If this were the case then only second H-chirality would be realized and leptonic spinors would be enough. What this would mean from the point of view of separate B and L conservation remains an open and interesting question. This kind of picture would allow to consider extremely simple genesis of matter from right-handed neutrinos only (see .
There are two objections against this naive picture. The fractionazion associated with heff should be same for all quantum numbers so that different fractionizations for color isospin and color hyper charge does not seem to be possible. One can of course ask whether the different quantum numbers could be fractionized independently and what this could mean geometrically. Second, really lethal looking objection is that fractional quark charges involve also shift of em charge so that neutrino does not remain neutral it becomes counterpart of u quark.
Negentropy Maximization Principle (NMP) resolves also the above mentioned almost paradox related to entropy contra intelligence. I have proposed analogous principle but relying on generation of negentropic entanglement and replacing entropy with number theoretic negentropy obeying modification of Shannon formula involving p-adic norm in the logarithm log(|p|p) of probability. The formula makes sense for probabilities which are rational or in algebraic extension of rational numbers and requires that the system is in the intersection of real and p-adic worlds. The dark matter matter with integer value of Planck constant and heff=nh predicts rational entanglement probabilities: their values are simply pi=1/n since the entanglement coefficients define a diagonal matrix proportional to unit matrix. Negentropic entanglement makes sense also for n-particle systems.
Negentropic entanglement corresponds therefore always to n× n density matrix proportional to unit matrix: this means maximal entanglement and maximal number theoretic entanglement negentropy for two entangled systems with number n of entangled states. n corresponds to Planck constant heff= n×h so that a connection with hierarchy of Planck constants is also obtained. Power of p-adic prime defines the largest prime power divisor of n. Individually negentropically entangled systems would be very entropic since there would be n energy-degenerate states with the same Boltzmann weight. Negentropic entanglement changes the situation: thermodynamics of course does not apply anymore. Hence TGD produces same prediction as thermodynamical model but avoids the almost paradox.
Thursday, November 28, 2013
Psychedelics induced experiences and magnetic body
Some background about psychedelics
Could instantaneous communications in cosmic scales be possible in TGD Universe?
Seth Lloyd on quantum life
The notion of quantum biology is becoming accepted notion although Wikipedia contains still nothing about its most important application (photosynthesis). I can be proud that I have been a pioneer of quantum biology for about two decades. TGD remains still one of the very few theories leaving the realm of standard quantum theory and suggesting besides the new view about space-time a generalization of quantum theory involving in an essential manner quantum theory of consciousness based on the identification of quantum jump as moment of consciousness. The new view about quantum theory involves a refined view about quantum measurement based on Negentropy Maximization Principle (NMP) kenociteallb/nmpc identified as the basic variational principle and zero energy ontology (ZEO) replacing ordinary standard energy ontology. The new view providing new vision about the relationship between subjective time and geometric time, about the arrow of time, and about second law.
The hierarchy of Planck constants having as space-time correlate effective (or real -depending on interpretation) n-sheeted coverings of 8-D imbedding space (or space-time) with heff=nh defining the value of (effective) Planck constant. p-Adic physics as physics of cognition is essential part of theory and together with the hierarchy of Planck constants closely related to the notion of negentropic entanglement characterizing living matter. Negentropic entanglement is maximal involving two-particle case tge entanglement of n states characterized by n× n unit matrix with n identified in terms of heff. Also maximal m-particle entanglement with 1<m≤ n is possible and one can write explicit formulas for the entangled states relating closely to the notion of exotic atom introduced earlier. The hierarchy of Planck constants as associated with dark matter so that dark matter is what makes living matter living in TGD Universe.
The concepts of many-sheeted space-time and topological field quantization imply that the concept of field body (magnetic body) becomes a crucial ew element in the understanding of living matter. Non-locality in even astrophysical scales becomes an essential piece of the description of living matter. Remote mental interactions making possible communication between biological and magnetic bodies become standard phenomena in living matter. The reconnection of magnetic flux tubes and phase transitions changing the value of heff and thus changing the length of magnetic flux tubes become a basic piece of biochemistry. Various macroscopic quantum phases such as dark electronic Cooper pairs and of protons and even ions as well as Bose-Einstein condensate of various dark bosonic objects with large value of heff are also central. They are associated with magnetic flux bodies (magnetic flux tubes) .
TGD implies a new, still developing, view about metabolism. Magnetic body as a carrier of metabolic energy and negentropic entanglement allows to understand the deeper role of metabolism in a unified manner. The notion of high energy phosphate bond assigned to ATP is one of the poorly understood notions of biochemistry. As a matter fact, all basic biomolecules are carriers of metabolic energy liberated as they are broken down in catabolism. It is usually thought that the covalent bonds containing shared valence electron pair between atoms involved carries this energy and that covalent bond reduces to standard quantum theory. TGD challenges this belief: covalent bond could in TGD framework correspond to magnetic flux tube associated with the bond having considerably larger size than the distance between atoms: similar picture has already earlier emerged in the model of nuclei as strings with colored flux tubes connecting nucleons and having length scale much longer than nuclei kenociteallb/nuclstring: this model also explains kenociteallb/padmass5 the puzzling observation that protonic charge radius seems to be somewhat larger than predicted kenocitebpnu/shrinkproton.
The metabolic energy quantum would be associated with large heff valence electron pair being identifiable as cyclotron energy in endogenous magnetic field for which the pioneering experiments of Blackman suggests value Bend=.2 Gauss as the first guess.Of course, entire spectrum of values coming as power of two multiples of this field strength can be considered. This would require rather high value of heff/h=n of order 108. Reconnection of flux tubes would make possible to transfer these electron pairs between molecules: actually a piece of flux tube containing the electron pair would be transferred in the process. This view allows to unify the model of metabolism with the view of DNA-cell membrane system as topological quantum computer with DNA nucleotides and lipids (or molecules assigned with them) by flux tubes.
Seth Lloyd represents three examples about situations in which quantum biology seems to be a "must": photosynthesis, navigation of birds, and odour perception. Photosynthesis represents the strongest and most quantitative support for quantum biology. Navitation and odour perception suggest strongly quantum theory model but leave the details of the model open.
I have applied TGD to numerous situations during years and also discussed simple TGD inspired models for all these three phenomena . The following represents briefly the core of Lloyd's talk and comparison with TGD based views. I do not of course have access to the data basis and can represent only a general vision rather than detailed numerical models. I share Lloyd's belief that quantum models provide the only manner to understand the data although models as such are not final. The authors of course want to publish their work and therefore cannot introduce explicitly notions like high temperature super-conductivity, which I believe are crucial besides purely TGD based concepts. What is however good that the models start from data and just look how to explain the data in quantum approach. Data lead to assumptions, which are not easy to defend in the framework of standard quantum theory. For instance, the presence of long-lived entangled pairs of electrons and electron and hole with wave functions possessing rather long coherence length and somehow isolated from entanglement destroying interactions with the external world emerge from the data. In TGD large value of heff/h and associated negentropic entangelement justifies these assumptions.
The incredible effectiveness of the first step of the photosynthesis after photon absorption
kenocitebbio/qharvesting is one of the key points of Lloyd in this talk. The organisms living deep under the surface of ocean are able to gather their metabolic energy using only the visible photons of black body radiation, whose typical photon energy is much lower than that of metabolic energy. In human eyes there is even mechanism preventing the detection of less than five photons at time.
The first step of photosynthesis after the capture of photon by harvesting antenna proteins has been a long standing mystery and here only quantum mechanical approach seems to provide the needed understanding. The light harvesting antenna proteins can be visualized as small disk like objects and are associated with a membrane like structure - so called thylakoid membrane similar to cell membrane. The absorption creates what is known as exciton - electron-hole pair, which is most naturally singlet. Photon has spin so that the exciton must have unit angular momentum. After its creation the electron of the exciton reaches by a random walk like process the reaction centre. Fromthe reaction centre the process continues as a stepwise electron transfer process leading eventually to the chemical storage of the photon energy.
The capture of photon occurs with some probability and also the process continues from reaction centre only with probability of about 5 per cent. The process with which the electrons reaches the reaction centre is however amazingly effective: effectives is above 95 per cent. This is mysterious since fort the classical random walk for exciton between the chromophores the time is proportional to the square root of distance measured as number of neighboring chromophores along the path.
The quantum proposal is that exciton is spin singlet state - this minimizes the interactions with photons - and performs quantum (random) walk to the reaction centre. The model assumes only experimental data as input and all parameters are fixed. Temperature remains the variable parameter. One can consider two extreme situations. At low energy limit the random walks tends to be stuck since external perturbations (mostly thermal photons) inducing the random walk process are not effective enough and quantum walk becomes so slow so that the exciton decays before it reaches the reaction centre. At high energy limit the thermal perturbations destroy quantum coherence and classical random walk results so that the efficiency becomes essentially zero. There is a temperature range where the transfer efficiency is near unity and time for reaching the reaction centre relatively short. This range has as midpoint room temperature.
If I have understood correctly, the model accepts as experimental facts the rather long lifetime of the exciton - few nanoseconds. In quantum-computerish this assumption translates to the statement that exciton belongs to a decoherence-free subspace so that external perturbations are not able to destroy the exciton too fast. Second assumption is that the exciton is de-localized over a ring ling like structure of size scale of 7 Angstroms (actually there are two rings of this kind, inner and outer and the wave function is assumed to be rotationally symmetric for the inner ring). This de-localization increases the probability of transfer to neighboring chromophore so that it is proportional to the square N2 of the number N of chromophores rather than N. The technical term expressing this is concatenated quantum code.
Skeptic would probably claim that coherence and stability of coherence are the weak points of the model. In TGD framework the assumption that electron-hole pair is negentropically entangled would guarantee its long life time. The reason is that NMP favors negentropic entanglement. Negentropic entanglement corresponds to entanglement associated with n-sheeted effective covering of imbedding space and n has interpretation in terms of effective Planck constant heff=nh. The naive guess is that coherence scale for the wave function of exciton scales up by factor n or kenosqrtn. This entangement need not have anything to do with spin but could relate to
large hbar.
I have earlier considered a slightly different proposal. Instead of exciton the negentropically entangled system would be Cooper pair of dark electrons. Note that the negentropic entanglement need not relate to the spin but to the n-fold covering although it could be assigned with spin too in which case the state would be spin singlet. The motivation came from the fact that the transfer of electrons to the reaction centre takes as pairs (see this). The TGD inspired interpretation of electron pair would be as dark Cooper pair. Two electron pairs would come from the splitting of two water molecules to O2, 4 protons and two electron pairs, and they would end up to P680 part of photosystem II (680 refers to maximally absorbed wavelength in nanometers) and from here to P680* as two pairs. This mechanism would require that the Cooper pairs absorbs the photon as single particle. In the case of dark Cooper pairs this might be naturally true. If this requires exchange of photon between the members of the pair, the rate for this process is of the order α2 lower.
Avian navigation
Second topic discussed by Seth Lloyd is avian navigation (see this). The challenge is to understand how birds (and also fishes) are able to utilize Earth's magnetic field in order to find their way during migration. In some cases the magnetite in the beak of the bird guides the way along magnetic field lines by inducing magnetic force, and the process can be understood at least partially. Consciousness theorist could of course wonder why these animals find year after year their exact birth place.
Robins however represent an example not so easy to understand. There are three input facts:
1. Robins are able to detect the orientation of BE but not its direction. They can also detect the angle between orientation and vertical to the Earth's surface and from this to deduce also the direction of BE.
2. Blue or green light is necessary for the successful detection of the orientation.
3. Oscillating em field with frequency of order MHz makes the robins totally disoriented.
The only model that seems to be able to explain the findings is that long-lived entangled pairs of electrons are created by the photon provided their energy is high enough. For red light the energy is 2 eV and is not yet quite enough. This suggests that the electrons originate from a pair of molecules or atoms of single molecule. It is not known what the molecules in question could be. The electrons of the pair are spinning in the magnetic field and this is suggested to cause the decay of the pair and second member (why not both?) of the pair would contribute to a current giving eventually rise to nerve pulse pattern.
Entangled long-lived electron pair should be created. Long lifetime is the problem. The proposed mechanism brings in mind the TGD based variant for the light harvesting mechanism of photosynthesis. Universality suggests that long lived dark negentropically entangled Cooper pairs are generated in both cases so that light harvesting is in question in both cases. These pairs assignable to membrane structures in both cases in turn would generate a supra current giving eventually rise to a generation of nerve pulses in the case of navigation and to electron transfer process in the case of photosynthesis. If the same mechanism is involved in both cases, the extreme effectiveness of this light harvesting process could make it possible for the birds to navigate even in dark. Electron has cyclotron frequency of about 1.5 MHz in the Earth's magnetic field and this makes easy to understand why oscillation with this frequency (resonance) induces disorientation by forcing the spinning of the dark Cooper pairs.
Why the energy of photon creating the dark electron Cooper pair should correspond to visible light? Cyclotron energy scale for the ordinary value of Planck constant is extremely small and corresponds to frequency in MHz range. For visible photons the frequency by order of magnitude 108 higher. Does this correspond to the value of heff? Similar order of magnitude estimate follows from several premises. If the scaling of h by n corresponds roughly to the scaling of p-adic scale by n1/2, one would have roughly 1015-fold (effective) covering of imbedding space which looks rather science-fictive! For electrons this would imply size of order cell size if dark scale corresponds to the p-adic scale. If the electrons are originally in bound states with binding energy of order eV, the value of heff could be much lower.
I smell the quantum
Quantum detection of odours was the third topic in Lloyd's talk. For decades it was believed that odor perception is based on lock and key mechanism. Human has 387 odour receptors and this would be the number of smells too. It has however turned out that humans can discriminate between about 104 smells and Luca Turin and his wife have written a book giving a catalogue of all these smells. It is clear that lock key mechanism is correct but something else is needed in order to understand the spectrum of odors.
The key observation of Turin is that the smells seems to be not purely chemically determined but is different for molecules consisting of atoms differing only by the weight of nucleus and thus being chemically identical. Therefore the vibrational spectrum of the molecule, which is typically in infrared, seems to be important. The proposal of Turin is that the process of odour perception involves the tunnelling of the vibraing electron from odour molecule. This tunnelling can be assisted by absorption of phonon coming from the receptor with frequency which corresponds to fundamental vibrational frequency or its multiple. The model has been tested in several cases. The latest test described by Lloyd is the one in which hydrogen in some molecule is replaced with deuteron, which is twice as heavy so that the vibrational frequency is reduced by a factor 1/kenosqrt2. Fruit flies took the role of odour perceivers and it turned out that they easily discriminate between the molecules.
I have considered earlier a somewhat different quantum model for odour perception by starting from the pioneering experimental work of Callahan kenocitebbio/Callahan, which led him to conclude that in the case of insects odour perception is "seeing" at infrared wavelengths. Infrared wavelengths correspond to vibrational energies for molecules so that this brings in the dependence on the square root of the inverse of the mass of the odorant and predicts that chemically identical molecules containing only different isotopes of atoms smell differently.Frequencies are same as in the model of Turin. Instead of phonons IR photons would play the key role serving as passwords exciting particular cyclotron state at particular magnetic tube. Similar mechanism could be at work in the case of ordinary vision.
Saturday, November 23, 2013
A new upper bound to electron's dipole moment as an additional blow against standard SUSY
A further blow against standard SUSY came for a couple of weeks ago. ACME collaboration has deduced a new upper bound on the electric dipole moment of electron, which is by order of magnitude smaller than the previous one. Jester and Lubos have more detailed commentaries.
The measurement of the dipole moment relies on a simple idea: electric dipole moment gives rise to additional precession if one has parallel magnetic and electric fields. The additional electric field is now that associated with the molecule containing electrons plus strong molecular electric field in the direction of spin quantization axes. One puts the molecules containing the electrons into magnetic field and measure the precession of spins by detecting the photons produced in the process. The deviation of the precession frequency from its value in magnetic field only should allow to deduce the upper bound for the dipole moment.
Semiclassically the non-vanishing dipole moment means asymmetric charge distribution with respect to the spin quantization axis. The electric dipole coupling term for Dirac spinors comes to effective action from radiative corrections and has the same form as magnetic dipole coupling involving sigma matrices except that one has an additional γ5 matrix bringing in CP breaking. The standard model prediction is of order de≈ 10-40 e× me,: this is by a factor 10-5 smaller than Planck length!
The new upper bound is de ≈ .87 × 10-32 e×me and still much larger than standard model prediction. Standard SUSY predicts typically non-vanishing dipole moment for electron. The estimate for the electron dipole moment coming from SUSYs and is by dimensional considerations of form de= c ℏ e× me/16π2M2, where c is of order unity and M is the mass scale for the new physics. The Feynman diagram in question involves the decay of electron to virtual neutrino and virtual chargino and the coupling of the latter to photon before absoption.
This upper bound provides a strong restriction on "garden variety" SUSY models (involving no fine tuning to make dipole moment smaller) and the scale at which SUSY could show itself becomes of order 10 TeV at least so that hopes for detecting SUSY at LHC should be rather meager. One can of course do fine tuning. "Naturality" idea does not favor fine tunings but is not in fashion nowadays: the existing theoretical models do not simply allow such luxury. The huge differences between elementary particle mass scales and quite "too long" proton lifetime represent basic example about "non-naturality" in the GUT framework. For an outsider like me this strongly suggests that although Higgs exist, Higgs mechanism provides only a parametrization of particle masses - maybe the only possible theoretical description in quantum field theory framework treating particles as point like - and must be eventually replaced with a genuine theory. For instance, Lubos does not see this fine tuning is not seen as reason for worrying too much. Personally I however feel worried since my old-fashioned view is that theoretical physicists must be able to make predictions rather than only run away the nasty data by repeated updating of the models so that they become more and more complicated.
Still about Equivalence Principle
Every time I have written about Equivalence Principle (briefly EP in the following ) in TGD framework I feel that it is finally fully and completely understood. But every time I am asked about EP and TGD, I feel uneasy and end up making just question "Could it be otherwise?". This experience repeated itself when Hamed made this question in previous posting.
Recall that EP in its Newtonian form states that gravitational and inertial masses are identical. Freely falling lift is the famous thought experiment leading to the vision of general relativity that gravitation is not a real force and in suitable local coordinate system can be eliminated locally. The more abstract statement is that particles falling freely in gravitational field move along geodesic lines. At the level of fields this leads Einstein's equations stating that energy momentum tensor is proportional Einstein tensor plus a possible cosmological term proportional to metric. Einstein's equations allow only the identification of gravitational and inertial energy momentum densities but do not allow to integrate these densities to four-momenta. Basically the problem is that translations are not symmetries anymore so that Noether theorem does not help. Hence it is very difficult to find a satisfactory definition of inertial and gravitational four-momenta. This difficulty was the basic motivation of TGD.
In TGD abstract gravitation for four-manifolds is replaced with sub-manifold gravity in M4× CP2 having also the symmetries of empty Minkowski space and one overcomes the mentioned problem. It is however far from clear whether one really obtains EP - even at long length scale limit! There are many questions in queue waiting for answer. What Equivalence Principle means in TGD? Just motion along geodesics in absence of non-gravitational forces or equivalence of gravitational and inertial masses? How to identify gravitational and inertial masses in TGD framework? Is it necessary to have both of them? Is gravitational mass something emerging only at the long leng scale limit of the theory? Does one obtain Einstein's equations or something more general at this limit - or perhaps quite generally?
What about quantum classical correspondence: are inertial and gravitational masses well-defined and non-tautologically identical at both quantum and classical level? Are quantal momenta (super conformal representations) and classical momenta (Noether charges for Kähler action) identical or does this apply at least to mass squared operators?
Quantum level
One can start from the fact that TGD is a generalization of string models and has generalization of super-conformal symmetries as symmetries. Quantal four-momentum is associated with quantum states - quantum superpositions of 3-surfaces in TGD framework. For the representations of super-conformal algebras (this includes both Virasoro and Kac-Moody type algebras) four-momentum appears automatically once one has Minkowski space and now one indeed has M4× CP2. One also obtains stringy mass formula. This happens also in TGD and p-adic thermodynamics and leads to excellent predictions for elementary particle masses and mass scales with minor input - one of them is five tensor factors in the representations of the super Virasoro algebra five tensor factors in the representations of the super Virasoro algebra. The details are more fuzzy since five tensor product factors for the Super Virasor algebra is the only constraint and it has turned possible to imagine many manners to satisfy the constraint. Here mathematician's helping hand would be extremely wellcome.
The basic question is obvious. Is there any need to identify both inertial and gravitational masses at the superconformal level? If so, can one achieve this?
1. There are two super-conformal algebras involved. The supersymplectic algebra associated with the imbedding space (boundary of CD) could correspond to inertial four-momentum since it acts at the level of imbedding space and the super Kac-Moody algebra associated with light-like 3-surfaces to gravitational four momentum since its action is at space-time level.
2. I have considered the possibility that the called coset representation for these algebras could lead to the identification of gravitational and inertial masses. Supersymplectic algebra can be said to contain the Kac-Moody algebra as sub-algebra since the isometries of light-cone boundary and CP2 can be imbedded as sub-algebra to the super-symplectic algebra. Could inertial and gravitational masses correspond to the four-momenta assignable to these two algebras? Coset representation would by definition identify inertial and gravitational super- conformal generators. In the case of scaling generator this would mean the identification of mass squared operators and in the case of they super counterparts identification of the four-momenta since the differences of super-conformal generators would annihilate physical states. The question whether one really obtains five tensor factors is far from trivial and here it is easy to fall to the sin of self deception.
A really cute feature of this approach is that p-adic thermodynamics for the vibrational part of either gravitational or inertial scaling generator does not mean breaking of super-conformal invariance since super conformal generators in the coset representation indeed annihilate the states although this is not the for the super-symplectic and super Kac-Moody representations separately. Note that quantum superposition of states with different values of mass squared and even energies makes sense in zero energy ontology.
3. Second option would combine these two algebras to a larger algebra with common four-momentum identified as gravitational four-momentum. The fact that this four-momentum does not follow from a quantal version of Noether theorem suggests the interpretation as gravitational momentum. In this case the simplest manner to understand the five tensor factors of conformal algebra would be by assigning them to color group SU(3), electroweak group SU(2)L× U(1)(2 factors), symplectic groups of CP2 and light-cone boundary δ M4+.
While writing a response to the question of Hamed the following question popped up. Could it be that classical four-momentum assignable to Kähler action using Noether theorem defines inertial four-momentum equal to the gravitational four-momentum identified as four-momentum assignable to super-conformal representation for the latter option? Gravitational four-momentum would certainly correspond naturally to super-conformal algebra just as in string models. The identification of classical and quantal four-momenta might make sense since translations form an Abelian algebra,or more generally Cartan sub-algebra of product of Poincare and color groups. Even weaker identification would be identification of inertial and conformal mass squared and color Casimir operators. EP would reduce to quantum classical correspondence just and General Coordinate Invariance (GCI) would force classical theory as an exact part of quantum theory! This would be elegant and minimize the number of conjectures but could be of course be wrong.
One can argue that p-adic thermodynamics for the vibrational part of the total scaling generator (essentially mass squared or conformal weight defining it) breaks conformal invariance badly. This objection might actually kill this option.
Classical four-momentum and classical realization of EP
In the classical case the situation is actually more complex than in quantum situations due to the extreme non-linearity of Kähler action making also impossible naive canonical quantization even at the level of principle so that quantal counterparts of classical Noether charges do not exist. One can however argue that quantum classical correspondence applies in the case of Cartan algebra. Four-momenta and color quantum numbers indeed define one possible Cartan sub-algebra of isometries.
By Noether theorem Kähler action gives rise to inertial four-momentum as classical conserved charges assignable to translations of M4 (rather than space space-time surface). Classical four-momentum is always assignable to 3-surface and its components are in one-one correspondence with Minkowski coordinates. It can be regarded as M4 vector and thus also imbedding space vector.
Quantum classical correspondence requires that the Noetherian four-momentum equals to the conformal four-momentum. This irrespectively of whether EP reduces to quantum classical correspondence or not.
Einstein's equations have been however successful. This forces to ask whether classical field equations for preferred extremals could imply that the inertial four-momentum density defined by Kähler action is expressible as a superposition of terms corresponding to Einstein tensor and cosmological term or its generalization. If EP reduces to quantum classical correspondence, one could say that not only quantum physics but also quantum classical correspondence is represented at the level of sub-manifold geometry.
In fact, exactly the same argument that led Einstein to his equations applies now. Einstein argued that energy momentum tensor has a vanishing covariant divergence: Einstein's equations are the generic manner to satisfy this condition. Exactly the same condition can be posed in sub-manifold gravity.
1. The condition that the energy momentum tensor associated with Kähler action has a vanishing covariant divergence is satisfied for known preferred extremals. Physically it states the vanishing of Lorentz 4-force associated with induced Kähler form defining a Maxwell field. The sum of electric and magnetic forces vanishes and the forces do not perform work. In static case the equations reduce to Beltrami equations stating that the rotor of magnetic field is parallel to the magnetic field. These equations are topologically highly interesting.
These conditions are satisfied if Einstein's equations with cosmological term hold true for the energy momentum tensor of Kähler action. The vanishing of trace of Kahler energy momentum tensor implies that curvature scalar is constant and expressible in terms of cosmological constant. If cosmological constant vanishes, curvature scalar vanishes (Reissner-Nordström metric is example of this situation and CP2 defines Euclidian metric with this property). Thus the preferred extremels would correspond to extremely restricted subset of abstract 4-geometries.
2. A more general possibility - which can be considered only in sub-manifold gravity - is that cosmological term is replaced by a combination of several projection operators to space-time surface instead of metric alone. Coefficients could even depend on position but satisfying consistency conditions guaranteeing that the energy momentum tensor is divergenceless. For instance, covariantly constant projection operators with constant coefficients can be considered.
If this picture is correct (do not forget the objection from p-adic thermodynamcs), what remains is the question whether quantum classical correspondence is true for four-momenta or at least mass squared and color Casimir operators. Technically the situation would be the same for both interpretations of EP. Basically the question is whether inertial-gravitational equivalence and quantal-classical equivalence are one and same thing or not.
Defending intuition
What is frustrating that the field equations of TGD are so incredibly non-linear that intuitive approach based on wild guesses and genuine thinking as opposed to blind application of calculational rules is the only working approach. I know quite well that for many colleagues intuitive thinking is the deadliest sin of all deadly sins. For them the ideal of theoretical physicist is a brainless monkey who has got the Rules. Certainly intuitive approach allows only to develop conjectures, which hopefully can be proven right or wrong and intuition can lead to wrong path unless it is accompanied by a critical attitude.
I am however an optimist. We know that it has been possible to develop perturbation theory for super symmetric version of Einstein action by using twistor Grassmann approach. Stringy variant of this approach with massless fermions as fundamental particles suggests itself in TGD. TGD Universe possesses huge symmetries and this should make also classical theory simple: I have indeed made proposal about how to construct general solutions of field equations in terms of what I call Hamilton-Jacobi structure. For these and many other reasons I continue to belief in the power of intuition.
Monday, November 18, 2013
Old web page address ceased to work again: situation should change in January!
The old homepage address has ceased to work again. As I have told, I learned too late that the web hotel owner is a criminal. It is quite possible that he receives "encouragement" from some finnish academic people who have done during these 35 years all they can to silence me. Thinking in a novel way in Finland is really dangerous activity! It turned out impossible to get any contact with this fellow to get the right to forward the visitors from the old address to the new one (which by the way differs from the old one only by replacement of ".com" with ".fi").
I am sorry for inconvenience. The situation should change in January.
Sunday, November 17, 2013
Constant torque as a manner to force phase transition increasing the value of Planck constant
The challenge is to identify physical mechanisms forcing the increase of effective Planck constant heff (whether to call it effective or not is to some extent matter of taste). The work with certain potential applications of TGD led to a discovery of a new mechanism possibly achieving this. The method would be simple: apply constant torque to a rotating system. I will leave it for the reader to rediscover how this can be achieved. It turns out that the considerations lead to considerable insights about how large heff phases are generated in living matter.
Could constant torque force the increase of heff?
Consider a rigid body allowed to rotated around some axes so that its state is characterized by a rotation angle φ.
Assumed that a constant torque τ is applied to the system.
1. The classical equations of motion are
I d2φ/dt2= τ .
This is true in an idealization as point particle characterized by its moment of inertia around the axis of rotation. Equations of motion are obtained from the variational principle
S= ∫ Ldt , L= I(dφ/dt)2/2- V(φ) , V(φ)= τφ .
Here φ denotes the rotational angle. The mathematical problem is that the potential function V(φ) is either many-valued or dis-continuous at φ= 2π.
2. Quantum mechanically the system corresponds to a Scrödinger equation
- hbar2/2I× ∂2Ψ/∂φ2 +τ φ Ψ = -i∂Ψ/∂ t .
In stationary situation one has
- hbar2/2I× ∂2Ψ/∂φ2 +τ φ Ψ = EΨ .
3. Wave function is expected to be continuous at φ=2π. The discontinuity of potential at φ= φ0 poses further strong conditions on the solutions: Ψ should vanish in a region containing the point φ0. Note that the value of φ0 can be chosen freely.
The intuitive picture is that the solutions correspond to strongly localized wave packets in accelerating motion. The wavepacket can for some time vanish in the region containing point φ0. What happens when this condition does not hold anymore?
• Dissipation is present in the system and therefore also state function reductions. Could state function reduction occur when the wave packet contains the point, where V(φ) is dis-continuous?
• Or are the solutions well-defined only in a space-time region with finite temporal extent T? In zero energy ontology (ZEO) this option is automatically realized since space-time sheets are restricted inside causal diamonds (CDs). Wave functions need to be well-defined only inside CD involved and would vanish at φ0. Therefore the mathematical problems related to the representation of accelerating wave packets in non-compact degrees of freedom could serve as a motivation for both CDs and ZEO.
There is however still a problem. The wave packet cannot be in accelerating motion even for single full turn. More turns are wanted. Should one give up the assumption that wave function is continuous at φ=φ0+ 2π and should one allow wave functions to be multivalued and satisfy the continuity condition Ψ(φ0)=Ψ(φ0+n2π), where n is some sufficiently large integer. This would mean the replacement of the configuration space (now circle) with its n-fold covering.
The introduction of the n-fold covering leads naturally to the hierarchy of Planck constants.
1. A natural question is whether constant torque τ could affect the system so that φ=0 ja φ=2π do not represent physically equivalent configurations anymore. Could it however happen that φ=0 ja φ= n2π for some value of n are still equivalent? One would have the analogy of many-sheeted Riemann surface.
2. In TGD framework 3-surfaces can indeed be analogous to n-sheeted Riemann surfaces. In other words, a rotation of 2π does not produce the original surface but one needs n2π rotation to achieve this. In fact, heff/h=n corresponds to this situation geometrically! Space-time itself becomes n-sheeted covering of itself: this property must be distinguished from many-sheetedness. Could constant torque provide a manner to force a situation making space-time n-sheeted and thus to create phases with large value of heff?
3. Schrödinger amplitude representing accelerated wave packet as a wavefunction in the n-fold covering would be n-valued in the ordinary Minkowski coordinates and would satisfy the boundary condition
Ψ(φ)= Ψ(φ+ n2π) .
Since V(φ) is not rotationally invariant this condition is too strong for stationary solutions.
4. This condition would mean Fourier analysis using the exponentials exp(imφ/n) with time dependent coefficients cm(t) whose time evolution is dicrated by Schröndinger equation. For ordinary Planck constant this would mean fractional values of angular momentum
Lz= m/n hbar .
If one has heff=nhbar, the spectrum of Lz is not affected. It would seem that constant torque forces the generation of a phase with large value of heff! From the estimate for how many turns the system rotates one can estimate the value of heff.
What about stationary solutions?
Giving up stationary seems the only option on basis of classical intuition. One can however ask whether also stationary solutions could make sense mathematically and could make possible completely new quantum phenomena.
1. In the stationary situation the boundary condition must be weakened to
Ψ(φ0)= Ψ(φ0+ n2π) .
Here the choice of φ0 characterizes the solution. This condition quantizes the energy. Normally only the value n=1 is possible.
2. The many-valuedness/discontinuity of V(φ) does not produce problems if the condition
Ψ(φ0,t)=Ψ(φ0+ n2π,t) =0 , & 0<t<T .
is satisfied. Schrödinger equation would be continuous at φ=φ0+n2π. The values of φ0 would correspond to a continuous state basis.
3. One would have two boundary conditions expected to fix the solution completely for given values of n and φ0. The solutions corresponding to different values of φ0 are not related by a rotation since V(φ) is not invariant under rotations. One obtains infinite number of continous solution families labelled by n and they correspond to different phases if heff is different from them.
The connection with WKB approximation and Airy functions
Stationary Schrödinger equation with constant force appears in WKB approximation and follows from a linearization of the potential function at non-stationary point. A good example is Schröndinger equation for a particle in the gravitational field of Earth. The solutions of this equation are Airy functions which appear also in the electrodynamical model for rainbow.
1. The standard form for the Schrödnger equation in stationary case is obtained using the following change of variables
u+e= kφ , k3=2τ I/hbar2 , e=2IE/hbar2k2 .
One obtains Airy equation
d2Ψ/du2- uΨ =0 .
The eigenvalue of energy does not appear explicitly in the equation. Boundary conditions transform to
Ψ(u0+ n2π k )= Ψ(u0) =0 .
2. In non-stationary case the change of variables is
u= kφ , k3=2τ I/hbar2 , v=(hbar2k2/2I)× t
One obtains
d2Ψ/du2- uΨ =i∂v Ψ .
Boundary conditions are
Ψ(u+ kn2π,v )= Ψ(u,v) , 0 ≤ v≤ hbar2k2/2I× T .
An interesting question is what heff=n× h means? Should one replace h with heff=nh as the condition that the spectrum of angular momentum remains unchanged requires. One would have k ∝ n-2/3 ja e∝ n4/3. One would obtain boundary conditions non-linear with respect to n.
Connection with living matter
The constant torque - or more generally non-oscillatory generalized force in some compact degrees of freedom - requires of a continual energy feed to the system. Continual energy feed serves as a basic condition for self-organization and for the evolution of states studied in non-equilibrium thermodynamics. Biology represents a fundamental example of this kind of situation. The energy feeded to the system represents metabolic energy and ADP-ATP process loads this energy to ATP molecules. Also now constant torque is involved: the ATP synthase molecule contains the analog of generator having a rotating shaft. Since metabolism and the generation of large heff phases are very closely related in TGD Universe, the natural proposal is that the rotating shaft forces the generation of large heff phases.
For details and background see the chapter Macroscopic quantum coherence and quantum metabolism as different sides of the same coin: part II" of "Biosystems as Conscious Holograms".
Addition: The old homepage address has ceased to work again. As I have told, I learned too late that the web hotel owner is a criminal. It is quite possible that he receives "encouragement" from some finnish academic people who have done during these 35 years all they can to silence me. It turned out impossible to get any contact with this fellow to get the right to forward the visitors from the old address to the new one (which by the way differs from the old one only by replacement of ".com" with ".fi"). The situation should change in January. I am sorry for inconvenience. Thinking in a novel way in Finland is really dangerous activity! |
d1743a1606f2d87e | Complex numbers: an introduction
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Complex numbers have fascinated me since high school. Usually, it’s where we are taught about natural numbers, integers, rational, irrational, and real numbers but never about complex numbers. This post is for those who might be interested in an easy introduction into the realm, or rather, plane of complex numbers. And they’re not without practical significance either: no electronic device such as the one you’re using to read this post could have been built without physicists, electrical engineers, and computer scientists knowing anything about the gift of complex numbers from sixteenth century mathematicians.
Blown away
‘There are such things as negative numbers’, explained my father to me when I must have been about six or seven years old since I was a second-year pupil in primary school. He explained the notion of negative possession when owing a certain number of marbles to someone which was greater than the number of marbles you physically carry with you. As this was one of those I-still-remember-where-I-was-when moments, like it was yesterday, I remember sitting on the floor besides the coffee table in the living room of our terraced house in the town of Emmeloord, which had been reclaimed just forty-three years earlier from the IJsselmeer, a lake formerly part of the North Sea.
I clearly remember feeling exactly the same when he had told me earlier our planet wasn’t flat and when my Mum told me in the car yet a few months earlier, that we were living on the sea floor. The cap of my mind was blown away, yet again. It took a while before I managed to fold my slow and wet brain lobes around the notion that negative numbers existed, even though you couldn’t see them in the real world like you could ‘see’ regular numbers such as in lengths or the number of marbles1.
I hastened to tell my primary school teacher excitedly about negative numbers. She just nodded and then told me to proceed with doing my homework on boring regular arithmetic. She had a point as I wasn’t very good at it.
Fast forward to when I must have been about fifteen or sixteen when I read about complex numbers in a popular textbook about quantum mechanics. The fact that they were called ‘complex’ may have triggered my curiosity as I assumed that term pertained to it being very difficult, but mostly because, apparently, so-called imaginary numbers are a thing! I had that exact same feeling again. The cap of my mind had melted. The whole notion seemed to radiate some kind of magical power. What sorcery was this? Could this be a doorway to extra dimensions?
The next day, I told my mathematics teacher, Mr Es – Es is not his actual name but it was his two-letter code in our high school timetable. I’ve always found it appropriate Es is also the symbol for the element Einsteinium in the periodic system. As his first name happened to be the same, my friends and I used to joke that we were on our way to the lessons of Albert Einstein.
Mr Es did what every good teacher does when a student tells you something they get enthusiastic about: he encouraged it – in his case by lending me his old textbook from when he was a first-year mathematics student in Amsterdam. It was an introductory text about complex numbers at the level of undergraduate mathematics.
The very textbook. (Click to enlarge.)
I’m ashamed to say I kept it. It was one of those instances where, after the nth time of moving house, I realised, oh my god, I still have this!? It’s also true that I treasured it. It carries a special meaning to me. It signifies how, at least once in my lifetime, I felt acknowledged in what stirred me deeply at the time. A thing I couldn’t really share with friends or anyone close in general, I suddenly shared with someone very clever whose name was denoted by the symbol for Einsteinium.
Thanks to the miracle of internet, we got back in touch, about twenty-five years later. I confessed I had always kept it and apologised. He had indeed wondered where it had been as he once wanted to show it to someone else. But I could keep it as he was cleaning out the attic anyway. And he was glad it had done something for me as he learnt about my current engagements in a bit of maths and physics.
I felt guilty. I still do. Someone else could have enjoyed it just as much as I have. And now I have prevented that from happening through his book. So, whoever you are, my sincerest apologies.
I hope, one day, I will be able to ignite sparks of joy for the beautiful mathematics of complex analysis to many others. I also hope you might experience at least a fraction of the amazement I felt and that the newly gained insight on the concept of ‘numbers’ might turn out to be beyond what you were able to imagine so far. So, let this be a beginning.
Number sets
A game of hopscotch drawn on the pavement with numbers on the tiles
We all know and love (or hate, depending) the natural numbers: the whole numbers we count things with. 1, 2, 3, etc. Some mathematicians will want to include the number 0 while others don’t. In any case, this mathematical set of numbers is called the natural numbers and is denoted by the symbol $\mathbb{N}$.
Then my father told me about the negative numbers, such as -1, -2, -3, etc. If you include the natural numbers and add to that these negative numbers, and add the number 0 to it (if you hadn’t already), then the result is an entirely new set of numbers called the integers, denoted by the symbol $\mathbb{Z}$.
To denote that the set $\mathbb{N}$ is part of the larger set $\mathbb{Z}$, people use this symbol for subset, $\subset$. They will write $\mathbb{N}\subset\mathbb{Z}$, the natural numbers are a subset of the integers.
Of course, there’s the ratio’s. The fractions. Between 1 and 2, there’s 1.5. So, in fraction-notation, that’s $\frac{3}{2}$. They’re obviously not whole numbers. They’re rational numbers because they can be represented by a ratio of integers. This number set is symbolised by $\mathbb{Q}$. We now have $$\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}.$$
It is interesting to note that, therefore, by this expression of subsets of subsets, even numbers such as 9 are rational numbers. On the surface, it’s not a fraction. Below the surface, however, it can be expressed as a ratio of integers: $9=\frac{9}{1}=\frac{18}{2}=\frac{36}{4}$, for example (and infinitely more).
But wait, there’s more. Fractions such as 1.5 and 3.2 are finite. What if the decimals don’t end? What if you can’t write a particular kind of numbers as ratios, such as with the number $\pi$ or $\sqrt{2}$? These numbers are called the irrational numbers. They are all the numbers which aren’t rational. There’s no symbol for that2.
Instead, there’s a symbol for all the natural numbers, the integers, the rational numbers, and the irrational numbers altogether3. They’re called the real numbers and this set is denoted by $\mathbb{R}$. This is the set we’re all used to working with. We now have $$\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}.$$
The set of real numbers $\mathbb{R}$ contains all the numbers. Or does it?
A diagram of all the number sets in the shape of ellipses. The ellipse of R containing the ellipse of Q containing the ellipse of Z containing the ellipse of N.
The secret of del Ferro, del Fiore, Tartaglia, and Cardano
Well, you guessed it. Here they come, the complex numbers. Let’s do just a tiny bit of maths. Remember what the quadratic of a number was? And what a square root was? What is the square root of 64, in other words, $\sqrt{64}$? Yes, that’s 8. Because 8 times 8, or 8 squared, or $8^2$ equals 64.
Okay, suppose $x^2 = 64$, what is $x$ then? Well, you do exactly the same thing, you un-square $x$ by taking its square root. And you have to do the same with the number after the equal sign. So, $\sqrt{x^2} = \sqrt{64}$, in other words, $x = 8$.
Maybe you remember this comes in handy when calculating the lengths of the edges of your piece of land. Suppose, the surface area of your square piece of land is 64 square kilometre (or square miles). What is the length of an edge of that land? That’s 8 kilometre (or miles).
All these calculations take place in the realm of $\mathbb{R}^+$, the positive part of all real numbers. Note that no surface area of a piece of land can be negative. In other words, a surface area of -64 square metres is nonsensical. Also, the square root of -64 has no solution. It’s not -8, because -8 times -8, or $(-8)^2$ is simply 64 again, because a negative number times a negative numbers equals a positive number as we proved in an earlier post.
Sometime in the sixteenth century, somewhere in Italy, Scipione del Ferro, professor of the University of Bologna, solved a slightly different kind of equation. It was a so-called cubic equation. Where we basically found the solution to a quadratic equation such as $x^2 = 64$ from the top of our heads, he found solutions for a cubic equation such as $x^3 + x^2 + 6x + 3 = 0.$ Del Ferro was known for not wanting to publish any of his proofs and solutions. He kept a secret notebook and that was it.
On his death bed, however, he told his pupil Antonio Maria del Fiore the secret to solving it. Del Fiore went on to challenge Niccolò Fontana Tartaglia, a mathematician residing in Venice at the time. Tartaglia had actually solved it himself before and trusted the formula to Gerolamo Cardano, the then Milan-based polymath and genius. Tartaglia messaged the solution in the form of a poem (no less!) but didn’t entrust the proof to him.
Of course, Cardano was able to reconstruct the proof anyway. As he learnt that del Ferro had also found the solution, he then proceeded to publish it all in his Ars Magna from 1545, much to the chagrin of Tartaglia.
So, what was the secret so many large minds had been secretive about? A new type of number.
Let’s take a simpler example. Suppose, we have the following simplistic quadratic equation: $x^2 – 4 = 0$. To solve it, we ‘move’ the 4 to the other side of the equal sign, by adding 4 to both sides: $x^2 – 4 + 4 = 0 + 4$, which simply becomes $x^2 = 4$. If you apply the square root to both sides, you get $\sqrt{x^2} = \sqrt{4}$. The solution to this equation is thus $x=2$ or $x=-2$ (because $-2\times -2 = 4$ too).
Good. Basically, the mathematicians of the sixteenth century opined that they should be able to solve a variation of this equation as well: $x^2 + 4 = 0$. Let’s bring the 4 again to the other side of the equal sign by subtracting 4 on both sides: $x^2 + 4 – 4 = 0 – 4$, which becomes $x^2 = -4$. Now, again, the question is, what is $x$?
Let’s try and apply the square root to both sides again: $\sqrt{x^2} = \sqrt{-4}$. Halt. Stop. What is the square root of -4? What is the square root of a negative number?
We have the same situation where we are to apply the square root of a negative surface area. The answer isn’t -2, because $-2\times -2 = 4$, not -4. What then?
Before del Ferro, Tartaglia, and Cardano, people would have said that there simply is no solution. Thanks to them, however, we can solve it. The answer lies in the following definition: $$i^2=-1.$$
This seemingly simple act enables us to solve $x^2=-4$. We can then write $x = 2i$ or $x = -2i$.
Let’s take our first solution, $x = 2i$. If we square this, we get $x^2 = (2i)^2$, which we can also write as $x^2 = 2^2i^2$. Now, since $i^2 = -1$, we can substitute that to get $x^2 = 2^2(-1)$, which is, of course, $x^2 = -4$. Ecco!
The same goes for the other solution, $x = -2i$. If we square this, we get $x^2 = (-2i)^2$, which we can write as $x^2 = (-2)^2i^2 = 4i^2 = 4(-1) = -4$. Ecco!
So, you may ask, what devilish entity is this $i^2=-1$? The letter $i$ stands for ‘imaginary’ and so, $i$ is a so-called imaginary number.
Now, because $i^2=-1$, you can also write4 that $i = \sqrt{-1}$. And that’s the crazy part: how can you calculate the square root of a negative number? How can you calculate the square root of a negative surface area? The answer is, you can’t. Not in the realm of the real numbers $\mathbb{R}$, that is. However, we’re not in Kansas anymore, Dorothy. We’re in a new land called the complex numbers. Bye $\mathbb{R}$, and welcome to $\mathbb{C}$.
Here are some examples of complex numbers: $2i$, $\frac{2}{3}i$, $i\sqrt{2}$, $i \pi$, $-0.25i$. What’s more, you can add these imaginary numbers to a real number such as 3, like so: $3 + 2i$ or $3 + \frac{2}{3}i$ etc. These sums are their own answer. They are complex numbers.
A complex number $z$ is of the form $z = a + bi$, where $a$ and $b$ are real numbers and $i^2 = -1$. The first real number, $a$, is called the real part of $z$. The last real number, $b$, is called the imaginary part of $z$. The set of all complex numbers is denoted by $\mathbb{C}$.
And so, we now have
Note that every real number is a complex number but not every complex number is a real number. That is what one thing being a subset of another thing means. For instance, the real number 9 is a complex number where $b=0$. In other words, the real number 9 can be written as the complex number $9 + 0i$, which is simply 9, which thus happens to be a real number too.
But $z = 3 + 2i$ is not a real number, because it has an imaginary part which is not equal to zero. So, $z$ is now exclusively a complex number.
A diagram of all the number sets in the shape of ellipses. The ellipse of C containing the ellipse of R containing the ellipse of Q containing the ellipse of Z containing the ellipse of N.
Complex plane
Graphically, all the real numbers of $\mathbb{R}$ can be thought of as a point on the number line.
A diagram depicting the real number line. Every point on this line represents a real number, such 0, 1, 2, 3 and the square root of 2, pi, and e.
So, where do complex numbers reside?
Owing to people such as Wallis, Wessel, Argand, Buée, Mourey, Warren, Français, Bellavitis, Gauss, and Euler[1], the idea to extend the real number line with an imaginary number line perpendicular to the real number line came to fruition. What you get is the so-called complex (geometric) plane, sometimes called the $z$-plane, Gauss plane or Argand plane.
So, a complex number such as $z = 3 + 2i$, ‘contains’ the real number $3$ along the real axis, and the imaginary part, along the imaginary axis, sits at $2i$. A complex number is therefore always represented by a point in a two-dimensional space. Note that all the numbers from all the subset of complex numbers, i.e. $\mathbb{R}$ all the way down to $\mathbb{N}$, can also be represented by a point in this same two-dimensional complex space – it’s just that they all reside on the real axis.
As you can -heh- imagine, doing calculations with complex numbers has become an exercise of geometry now! In fact, one of the most beautiful equations in mathematics (at least to my taste) pertains to trigonometry in the complex plane; it’s called Euler’s Formula.
A diagram representing the complex plane. Perpendicular to the real number line is now a so-called imaginary axis with numbers such as i, 2i, 3i, pi-i, i square root of 2, etc. A complex number is now a point in on that surface.
Not so imaginary
It’s unfortunate that this number $i$ and any real number multiplication of it are called imaginary numbers. It was the renowned French philosopher and mathematician René Descartes who coined the term imaginary numbers because he considered them to be illusory. In fact, even Cardano had described them as ‘some recondite third kind of thing’[2].
It’s unfortunate because ‘imaginary’ leads to semantic ambiguity. I get it: you would never see something like $\sqrt{-1}$ in the real world. But neither would you see $\sqrt{2}$ out in the wild, for that matter. And yet, it’s the exact length of the hypotenuse of a particular right triangle, which a skilled DIY person could make while you’re waiting. To me, ‘real’ numbers such as $\pi = 3.1415926535897 \dots$ without ever ending are as real as ‘imaginary’ numbers are (and vice versa). Circles are a real thing and $\pi$ can be used to do calculations on them. Well, with imaginary numbers you can do calculations on them just as well.
Complex numbers are used in a variety of sciences. In Einstein’s relativity, which makes GPS navigation possible, you could make use of so-called imaginary time. This sounds like a concept straight from a science-fiction novel, however, imaginary time is a well-defined concept. In fact, in a previous post, we used this to derive the central set of equations in relativity, called the Lorentz transformations. See how the word ‘imaginary’ might invoke unwanted ambiguity?
To make quantum mechanics work – the most successful theory to date – complex numbers are all over the place. Without them, the computer, mobile phone, tablet, TV, VCR, even your modern fridge – they wouldn’t have worked as no engineer would have been able to produce integrated circuits. The wave function is a complex function living in a complex separable Hilbert space, taking on complex probability amplitudes, evolving according to the Schrödinger equation, which itself is a complex equation.
In mathematics, one of the better-known areas of research where complex numbers play a central role is the study of complex dynamical systems. The featured image above is a detail of the famous Mandelbrot set. It’s a special collection of complex numbers, the projection of which you see plotted colourfully in the complex plane. The study of (complex) fractals also informs all kinds of patterns in nature and growth, even weather forecasts, and climate science – they’re all informed by complex-dynamical areas of mathematical interest. Also, we’ve used them in a previous post, calculating whether a lab centrifuge with $n$ available spots can be balanced out by a $k$ number of test tubes.
A fun application of complex numbers is computer games. To calculate rotations in three-dimensional space, computer scientists make use of quaternions, which are an extension of the complex plane. A quaternion is an expression of the form $a + bi + cj + dk$, where $a,b,c,d$ are any old real numbers, and $i^2=j^2=k^2=-1$. However, this is perhaps an interesting subject for another bit of maths and physics.
[1] Cooke, R. (2005) The history of mathematics : a brief course. 2nd edn. New York, N.Y.: Wiley.
[2] Open University (2014) Essential mathematics 1. Milton Keynes: Open University.
Featured image: Mandelbrot set – Step 6 of a zoom sequence by Wolfgang Beyer under CC BY-NC-SA 2.0; adapted to fit layout.
Hopscotch Game by ncassullo.
Niccolò Fontana Tartaglia. Rijksmuseum, Dutch National Museum. Public domain.
Girolamo Cardano. Wellcome Images under CC BY 4.0.
1. Inexplicably, I had never considered the fact that temperature could get below 0 ℃, which it still did, back then in The Netherlands. We used to enjoy an outdoor activity called ice skating, on frozen lakes, ponds, rivers, and ditches.[]
2. Often, mathematicians circumvent the lack of a symbol by writing something like $\mathbb{R} \backslash \mathbb{Q}.$[]
3. Yes, indeed, my dear fellow mathematician, you thought correctly, I am skipping transcendental numbers here (and algebraic numbers, for that matter). As all transcendental numbers are irrational numbers but not all irrational numbers are transcendental, I decided it over-complicated things in what was supposed to be an introductory text on complex enough numbers anyway.[]
4. Although, I actually prefer to use $i^2=-1$ over $i=\sqrt{-1}$ even though the latter has been mentioned in many school books. However, I believe it might lead to confusion. Since we have the rule that $\sqrt{a}\sqrt{b}=\sqrt{ab}$ where $a$ and $b$ are positive real numbers, you might try to apply this rule to negative real numbers, such as when $a=b=-1$. You would then get the incorrect statement $\sqrt{-1}\sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{1} = 1$, which is wrong as it should be equal to -1. That’s why I try to avoid using $i = \sqrt{-1}$ where I can.[] |
ec9f7786eee1304c | Science and democracy reviews.
Science and Democracy reviews
Commentaries, book 2.
Copyright © 2015: Richard Lung.
Table of Contents.
John Davidson: The Gospel of Jesus.
Barbara Thiering: Christ in Qumran
Renaissance man.
William Lovett: Chartism.
Jill Liddington: Rebel Girls.
The franchise plummets to 11+: Leslie Brewer: Vote for Richard.
HG Wells: pre-internet idea of a World Brain.
The sixth extinction.
The political system fails the eco-system.
The amateur lawyer and open source software.
“How the banks robbed the world.” (The 2000-02 Dotcom bubble.)
David Craig and Matthew Elliott: Fleeced! (The 2008-9 Credit Crunch.)
The menace of the nuclear tyranny
A default government pushes more nuclear power pollution. (25 july 2006)
Big Business leads New (for Nuclear) Labour to assault the future (26 july 2007).
The Nuclear Vested Interest and a Nuclear Winter. (2 feb. 2008).
Response to Tory party commitment to more nuclear… (13 february 2010.)
Journalist partisans for nuclear power. (Dec. 2010).
After-note (2015): The determined dishonesty of atomic energy.
Some women scientists who should have won nobel prizes.
Murray Gell-Mann: The Quark and the Jaguar.
Paul Erdös, The man who loved only numbers, by Paul Hoffmann.
Julian Barbour: The End Of Time.
In classical physics
In quantum mechanics
Brian Greene: The Elegant Universe.
Brian Greene: The Hidden Reality.
Lee Smolin: Three roads to quantum gravity.
Lee Smolin: the trouble with physics.
A plea to automate and test Binomial STV.
Guide to three book series by the author.
Also in the Commentaries series.
The Democracy Science series.
Collected verse in five books.
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Science and Democracy reviews.
Table of contents.
This is the second book of a short series of Commentaries. The first, Literary Liberties, also has a considerable democratic content. They are both, mainly, collections of reviews. But the way they came about is quite distinct. They were of different intent and different in nature.
The core of the literary reviews were written in preparation for chatting with the local book club. That was when book clubs were new. Mass entertainment was producing movies and tv series about them. Discussing novels was a novelty.
As is to be expected, the novelty wore off, and I stopped reviewing even works that were well worth the trouble. In any case, I only reviewed books that I could appreciate. If my reviews helped any author, in some small way, that was all right. I didn’t see any point in doing any-one down.
To supplement these reviews of modern writers, I made the effort to draw on my fading memories of some favorite writers in my youth. These tended to be democratic in out-look, and, if there was one respect, in which I was negative, it was towards negativity, so successful in blocking genuine reform, if not the many shams and impostures.
At about the turn of the century, indeed the turn of the millennium, when I was getting to know a little of contemporary literature, I determined to get some idea of where physics was going.
I didn’t just review popular expositions, for relaxation and enjoyment, I studied them, to get as clear an idea as possible as to their thinking.
This was all the more necessary, given that the physical theories are built on a framework of cutting-edge mathematics, which I could not hope to understand.
This is why I decided to move, to near the end of the book, the most difficult reviews, that perhaps attempted too much detail of the works of Barbour, and, to a lesser extent, of Greene, and even Smolin. My middle-aged commentaries, in places, need more concentration than my old head always was willing to give.
The central problem of theoretical physics remains to construct a unified theory that seamlessly includes gravity with the other three known forces of nature, electro-magnetism, the strong and weak nuclear forces.
Michael Faraday and James Clerk Maxwell had united electricity and magnetism, in the nineteenth century. Abdus Salam and Steven Weinberg combined electromagnetism and the weak force, in their electro-weak theory, verified at CERN, with the discovery of speculated “heavy photons,” whose role would replace that of the photon in electromagnetism or light. Later, the new large hadron collider, further discovering the Higgs particle, did much to substantiate the so-called standard model, which further includes the strong force.
The standard model developed a quantum chromodynamics, analgous to quantum electrodynamics. “QED, the strange theory of light and matter” is the title of a superb popular book by Richard Feynman to explain his theory, which combines the classical theory of special relativity with quantum mechanics.
This leaves trying to reconcile quantum mechanics with general relativity. The core of this book is a study of some few attempts by physicists to convey this work to the general public.
Over-shadowing that endeavor was the emerging suspicion that about 96% of the universe is composed of a previously unsuspected dark matter and dark energy, which responds to gravity but not electromagnetism. Hence the characterisation, dark. The motions of the galaxies could not be explained without it, short of changing the known laws of motion, given by Newton and Einstein.
Einstein theory of general relativity slowly became comprehensively substantiated, especially after Roger Penrose made its mathematics more accommodating for physicists.
This amateur found its technicalities too difficult to follow, beyond popular accounts, to say nothing, of subsequent pre-occupations like string theory.
General relativity was couched in continuous terms that could be extended indefinitely, such as to the infinitely dense points, known as singularities, that mass was supposed to collapse into, in a black hole, a collapsed star under pressure of its own gravity, once the nuclear reactions, by which stars eventually build-up, (what we know as) the chemical table of elements, have given out.
This conception of the singularity as a dimensioness point is incompatible with quantum theory, which deals in discrete quantities, known as quanta (or as I would say, quantums) which are always a multiple of a basic unit, subject to no further division.
In 1900, Max Planck discovered that radiation energy could only be explained in discrete terms of a basic quantum of energy. By the end of the century, physicists were attempting a comparable endeavor of explaining space and time in terms of discrete units, associated with theories of quantum gravity.
I don’t understand these things, of course, but some idea, however limited and imperfect, was better than blank ignorance and indifference to all attempts to follow the advances of natural science.
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John Davidson: The Gospel of Jesus.
In Search of His Original Teachings.
Table of contents.
The Gnostic Jesus.
God is to be found within.
The Word.
The “Word made flesh,” a living Son of God.
Mystic baptism.
Spiritual practise.
Way of life and mode of conduct.
The Gnostic Jesus.
I once read a collection of essays on mysticism by great physicists. One can understand why the founders of quantum mechanics would be mystified! John Davidson, also a physicist by profession, has gone further, with a great research into a sort of faded halo of almost lost texts surrounding the teachings of Jesus, together with an authoritative study of the nature of mysticism.
Other revolutions in Jesus scholarship seek to reveal a Jesus hidden from history: a survivor of crucifixion; a shroud imprinter; an Eastern sojourner; as well as an iconoclast, of ritual Judaism, for equality before God, etc.
John Davidson seeks another hidden Jesus, the gnostic teacher. The gnostics claimed that Jesus taught a secret lore (hinted-at in canonic scripture). They were suppressed as heretics, and little known of their writings, till the sensational find at Nag Hammadi, in 1945.
Davidson draws freely on these and other ancient non-canonical texts, explaining their mystical inspiration.
The first part of The Gospel of Jesus reads like any historian concerned to show how the canonic, indeed all, writers were subject to human error creeping into the manuscript copying, and to human limitations of understanding what they were writing.
Nor were the gospels life stories of Jesus. They had other concerns at heart. So, they cannot be taken for granted as historical documents. It makes sense to follow the evidence, critically, across prescribed lines.
A good introduction by Ian Wilson, Jesus, the Evidence, does just this.
The extant Christian gospels, from before the end of the second century, but not in the Bible, may be obtained from Andrew Bernhard (earlygospels.net). Not being a mystic or having any experience in that line, I didn’t see their significance, till reading Davidson.
His volume of over one thousand pages breathes new life into many suppressed, neglected, forgotten, damaged or fragmented manuscripts.
This background of mystic knowledge or gnosis is used to throw light on the less obvious of Jesuses purported sayings, especially in the spiritualised gospel of St John.
Davidson claims that Jesuses teachings are consistent with what other mystics have taught. There is a greater reality than that of every-day life, just as tidal froth is not the whole existence of an ocean, tho it might seem so to beach dwellers.
He gives examples of the “oceanic feeling,” mystic experiences of vastly expanded consciousness and well-being, reminiscent of William James, on The Varieties of Religious Experience.
John Davidson, in earlier books, Subtle Energy and The Web of Life, combined traditional Indian meditative experience with Western fringe science of the bodys energy fields. Davidson was a Cambridge physicist. But he was not just talking about human auras as electro-magnetic fields, akin to the Earths aurora. Rather, the implication is there are higher or subtler, less gross manifestations of existence, than the material one we are so absorbed in.
One of the introducers to these books admitted he didn’t relate well to the oriental terms. (Neither did I.) I think he meant the “chakras” and the like. As Carl Jung said, we in the West are like children compared to Eastern understanding of mind.
These earlier books put me off, at first. But the style of The Gospel of Jesus is accessible. There is no mystical or scientific jargon. Instead, Davidson introduces mysticism to us, thru the spiritual teacher the West is most familiar with.
The start of chapter 27 sums up:
In our exploration of his teachings, we have seen that Jesus taught some simple, fundamental mystic truths. God is to be found within, he said; the path to him is that of the Word; the Word is to be contacted through the “Word made flesh,” a living Son of God, by means of mystic baptism and spiritual practice. And, while practising these spiritual exercises, a certain way of life and mode of conduct is required. This, in essence, is the mystic teaching of Jesus and of all the other great mystic Saviours.
Now that Davidson has substantiated this thesis with such a wealth of corroboration, really a much shorter book would not come amiss, to spell out the above quotation. I am not qualified to do that. However, a few words about the above terms and conditions, of the mystic path, may help. My comments are just one uninitiated person hazarding at meanings. They are not meant to be taken as authoritative, or even necessarily right. Everyone can try their own definitions.
God is to be found within.
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God is the unified strength of love beyond imagination or sense. Hence “within” us, in a manner of speaking, because our logic and perception can only put together a view of the world in parts, rather than a god-like omniscience of seeing the whole picture. God is beyond all the categories of space and time, life and death, mind and matter, or whatever.
The Word.
The Word is familiar from the beginning of St John gospel. Davidson describes it as the creative power of God, for which he provides many other metaphors from the ancient mystical literature.
Throughout history, God’s creative Power has been called by a multitude of names and expressions. Amongst the Christian and allied literature alone, it has been called the Word of Life, the Word of God, the Creative Word, the Logos, the Image of God, the Wisdom of God, the Voice of God, the Cry, the Call, the Holy Name, the Holy Spirit, the Holy Ghost, the Power, the Nous, the Primal Thought, Idea or Mind of God, His Command, His Law, His Will and His Ordinances.
In the metaphorical language so beloved of the Middle East, it has also been described as the Living Water, the Bread of Life, Manna from Heaven, the Breath of Life, the Medicine of Life, the Herb of Life, the Tree of Life, the True Vine, the Root, the Seed, the Pearl, the Way, the Truth, the Letter and many other figures of speech.
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God sends his dearly beloved Son, a soul, who is his perfect representative, into the world, in human form, to save or redeem souls trapped in the material cycle of existence. The mystic view is that the body is a prison, we are all too willingly jailed-in by our passions. These are for short-lived pleasures, that usually have a down-side, leave us dissatisfied, are subject to diminishing returns, which may lead to hopeless misery, unless our lives can somehow be turned around.
Why do we need a Savior if life is so short, anyway? For Davidson, the answer is that the death of the body is not the end of our problems. We are immortal souls, and the passions, that consumed our minds, continue after we lose the body to temporarily satisfy them. Inevitably, those worldly passions draw us back to further corporeal existence. The baser the passions, the baser the existence.
Davidson concurs with the doctrine of reincarnation, to the extent that ones sins may transmigrate ones soul even into the body of a lower animal. He does point out that some animals are perfectly loving and true, whereas many humans are “bestial.” Presumably, their souls would swop bodily forms, in the karmic scheme of things. But how reincarnation might work leaves much to be understood.
The largely successful but illegitimate banishment of reincarnation from official Christianity is discussed in The Original Jesus by Gruber and Kersten.
(They describe the tremendous extent of the Buddhist mission and compare similar teachings to those of Christ, tales of whom are astonishingly anticipated, re the Hindu Krishna, as well as by other religious legends.)
Davidson says that every mystic had a master. To escape from the prison of the body, while we are still alive, is “an outside job.” It needs the help of a Savior to show us the escape route. And ones lifetime is the only chance to effect that escape (normally taking innumerable lifetimes). After death, a souls unreformed mind is simply drawn back to its spiritual level of corporeal existence.
This resembles those prisoners, who have become institutionalised. When they are set free, they simply stay where they are, or gravitate back to their old haunts. Or, if that is not permitted, they get themselves re-committed.
With regard to needing a master to spiritually reform ourselves, Davidson says:
If we advise others to do something which we do not do ourselves, then it is unlikely to have much effect. As the saying is, example is better than precept. Masters are always perfect examples of everything they teach. Hence, if a Master is to teach the necessity of a Master, it is necessary for him to have a Master, too. Later followers characteristically like to portray their Master as if he had no Master, for they do not like to think that their Saviour was ever in need of help himself. But in order for a Master to convince others that in order to find the kingdom of God it is necessary to have a Master, he himself must have a Master. Otherwise, his own life would contradict his teaching and few discriminating people would believe him.
Mystic baptism.
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In Gospel Truth, Russell Shorto discusses this process, with John the Baptist in the role of Jesuses inducer. “The Big Dipper,” as he calls him, introduced baptism, as a cathartic experience for the purging or cleansing of sins. Unlike the costly animal sacrifices at the Jerusalem Temple, John baptised in the holy river Jordan to redeem the pious poor. This alleviating their grinding poverty, no doubt, made the exploiting authorities his enemy. Jesus would be under suspicion, by association with the Baptist, and also destined for execution.
In his attitude to baptism, as well as the gospel healings, and indeed to the crucifixion, Davidson reveals himself to be a true son of the ancient gnostics. Like them, he is only interested in the existence of a spiritual Christ after the crucifixion. The orthodox tended to think in terms of a physically resurrected Jesus. This may be because Jesus was secretly revived but superstition triumphed over the nature of his re-appearance.
Hence, Christian graveyards, where the bodies of the dead are all laid out, to rise again on the day of judgment.
For Davidson, this question, of Christs prolonged stay on earth, would be a minor matter compared to Jesus, as the holder of the keys to eternal life.
The author substitutes physical happenings for spiritualist interpretations of them. Submerging or sprinkling people in water -- what good does that do? He reasonably asks. Religious ritual is regarded as a forgotten remnant of spiritual practise, which alone makes possible profoundly blessed other-worldly experience.
One interpretation of baptism is as symbolic of re-birth, not merely in physical water, but in the Living Water of heaven, achieved by initiation into the spiritual mysteries.
Spiritual practise.
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Davidson views miracles as a Masters ability, as Gods agent on earth, to re-create things. His sense of miracles is that they are both grandiose and largely futile:
As fascinating as it may be to witness physical miracles, the simple fact is that miracles in themselves do not confer spirituality. Spirituality comes through spiritual practise, through purification of the mind and the cleansing of its myriad impure tendencies, freeing it from the force of many ingrained habits. How can simply witnessing a miracle do that? Nor do miracles confer true faith in and reliance on God. Faith in God develops naturally as the ego is worn down.
It is understandable, then, that Davidson has no time for whether the still mysterious Turin shroud is genuine.
Davidson believes the Savior is more concerned with spiritual healing than healing the body, which soon dies, anyway. This gnostic transcendentalism perhaps loses touch with Jesuses humanity. As both Ian Wilson and Russell Shorto say, some of Jesuses miraculous cures are recognisable from modern cases of faith healing, hypnotism and exorcism. Some patients may have called to be released from psychosomatic and multiple-personality disorders, that oppressed them.
However, the gnostic John Davidson sees most significance in the mystics, including Jesus, using the miracles --
as metaphors for spiritual truths. We are all spiritually blind, deaf and dumb. We are crippled and have forgotten how to walk straight in this world. We are carrying a heavy burden of weaknesses and sins from which we need to be healed. Our will power is paralysed and withered by our attraction to the world of the senses. In fact, we have become spiritually dead and full of darkness -- we need to be raised from the dead, to come out of the tomb of the body, not after four days but after many ages. Spiritually, we "stinketh" with the accumulated sins of many lifetimes!
To accomplish this, we need a spiritual physician to help us overcome the feverish activities of the mind, to learn how to walk upon the stormy waters of this world, to cast out the devils and demons of human weakness from within ourselves and to overcome the Devil himself. With the help of a Son of God, we must bathe in the pool of Living Water and come up healed after many years of infirmity without anyone having previously helped us to take that dip. We need to eat the true Bread of Life and to drink the wine of divine love at the marriage of the soul with God.
True mystics are not looking for a following but for disciples dedicated to finding the mystic reality. In terms of the parable of the sower, they are looking only for the seed that bears fruit.
It is not enough, Davidson says of those who think they just need to gen-up on all the spiritual practises:
They would have no one to oversee the repayment of their karmic debt, no one to meet them on the inside, no one to guide them externally or internally if they got into difficulties, no one to shower blessings and inspiration upon them in so many ways. They would be trying to climb to the top of an unknown mountain on their own, without real knowledge of the way or how to climb, and they would be likely to get lost or worse. To go adventuring into the realms of one’s own being without the guidance of one who knows the way is simply foolhardy.
But what form of spiritual practice or mystic prayer do the Masters teach? Mystics say that the headquarters of the mind and soul in the human body is in the forehead, immediately behind and above the two eyes. This focus of attention has hence been called the eye centre, the centre of consciousness or the thinking centre…But there is nothing physical about its “location”…It is a mental or subtle centre.
From this point, the attention drops down into the body, spreading out and scattering into the world through the sense organs and the organs of activity. And the more a person’s attention strays away from this centre of consciousness, the less is their awareness and consciousness of what is happening to them. Consequently, the more a person is scattered into the world, the less do they realise it. This is a dangerous situation.
Davidson continues that even when the body is exhausted, the mind goes on, in waking or in sleep, subconsciously or in dreams, obsessing one with the world.
To train the mind from running wild, a meditation, such as mentally repeating certain words, is practised with the attention fixed at the eye centre.
Davidson compares this “labour” of the mind to a child unwilling to go to school but eventually unwilling to leave higher education.
Nevertheless, it only takes a “degree of concentration and stillness, even of the body” for consciousness to withdraw from the body, towards the eye-centre. There follows a description of the beginnings of how “by degrees, the soul and mind leave the body and enter the astral realms.” It is like death, except the meditator is in control and still connected to the body.
Meditation is in fact a metaphorical “death,” becoming dead to the desires of ones senses. Luke is quoted:
and take up his cross daily, and follow me.
Crucifixion was a slow torture to death. Likewise, in daily meditation one denies oneself and “dies” to the temptations of the senses.
Davidson says Jesus describes this meditation on the eye centre, as “if thine eye be single, thy whole body shall be full of light”; but an unscrupulous mind, polluted with worldly ambitions, finds it “full of darkness.”
As the “single eye” leads to the “astral worlds,” mystics, such as Jesus, liken it to a “strait” and “narrow” “gate.” That is why “it is easier for a camel to go through the eye of a needle than for a rich man to enter into the kingdom of God.” That is to say a man encumbered by the possessions and desires of this world, which he cannot take with him to another world.
The gate is also called a door. Meditation is the knocking on that door, that comes from seeking God. As usual, Davidson quotes from canonic and non-canonic mystic texts, to point-up the moral. Seek and you shall find, because the seeking means that God meant you to find. Indeed, the Master or Son of God, may be waiting on the astral side of that door, and himself knock to see if there is an aspirants soul ready to enter, to be taken thru the mystic realms, back to God.
Hence, the parables that enjoin the servants to be ready for the unexpected return of their Master.
Way of life and mode of conduct.
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To reach God means becoming one with the one God, the ultimate reality. Hence, the need to love God and ones fellow creatures. Whereas, the ego, partial for things of this world, is only a “counterfeit self,” that does not truly represent God, but is like a phoney politician, who puts personal and partisan ambitions before the general interest.
This applies to all of us, of limited sympathies. The imprisoned love, that is pride or tribalism, acts out desires that belittle, deprive or infringe on others. The deadly sins are like a plague of addictions, harmful to all of us, obsessed or victimised by them.
Spiritual practise attempts to do away with a self-centred attitude that leads to doing unjustly by fellow creatures. Meanwhile, one must act as ethically as one can, despite impulses to behave without consideration for others.
A typical example of Davidsons gnostic outlook interprets “righteousness” as “spirituality.” Trying to make this world better, he seems to think may be good karma but is no substitute for the quest for eternal bliss.
Tho, his love of animals is evident. A long chapter puts the case that the mystic path requires one to be a vegetarian. All the true masters would say so, he claims, citing many, and arguing they included Jesus. For instance, the fishes are not mentioned in earliest references to the miracle of the loaves.
Since I wrote this review a new paper-back edition has been published by Clear Books: www.clearpress.co.uk
And the author, John Davidson set up a website:
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Barbara Thiering: Christ in Qumran and Revelation.
Table of contents.
Mother and child and the four evangelists, [
**]from the Book of Kells.
Links to sections:
Jesus The Man.
Jesus Of The Apocalypse.
Jesus The Man.
Joshua or Jesus, the best known of the Christs or Annointed Ones, continues to inspire the devotion of much of the human race. Quiet scholars are no exception. In recent years, alone, traditional Christianity has been challenged by several “Copernican” revolutions in understanding or conception of its founder.
Ideas that were suppressed as heresies have returned like lost sheep. Whether they ever belonged to the Good Shepherd is another matter. It has not been settled properly whether the Turin shroud was his. The old rumor that Jesus survived the crucifixion has been revived, even if he wasn’t. Eastern traditions of Jesuses travels and teaching have been researched by a Muslim professor, Fida Hassnain, believing “Jesus belongs to the world,” and saddened by Christians desirous of suppressing such evidence.
New attention has been drawn to hidden meanings in the New Testament, most strikingly in the Book of Revelation, that could draw back the veil over Christs mission in its first century. Such interpretations, using the Dead Sea Scrolls as a touch-stone for a New Testament sub-text, seem much too ambitious. But a scholarly consensus on some new insights, from this approach, may yet be achieved.
John Davidson, in The Gospel of Jesus, makes him a latter-day gnostic. Barbara Thiering might be likened to a modern pharisee. That is to say, she has a rigorous sense of ritual conservatism combined with popular sentiments. The latter show in her view of Jesus as a hero, who breaks down Judaic exclusiveness, by admitting married men, gentiles, women and the crippled, on equal terms, into the religion of the one god.
It is in these terms of Jesus the universalist, that Thiering finds symbolic meaning in the miracles. Her primary source of inspiration is not Nag Hammadi gnosticism but Dead Sea Scrolls ritualism. They show two steps of initiation into the community, usually assumed to be the Essenes and supposed residing at the Qumran site, near the caves where most scrolls were found in 1947.
This consensus has been vigorously challenged by Norman Golb, in Who wrote the Dead Sea Scrolls? What we are looking at, here, he says, is a miscellany more likely to have come from the evacuated Jerusalem library.
Barbara Thiering attacks the consensus from the opposite point of view. Qumran is not by-passed for Jerusalem, rather, Jerusalem is by-passed for Qumran. Gospel incidents there and elsewhere are “de-coded” as happening at Qumran.
For example, the raising of Lazarus is held to mean that an excommunication was lifted. Up to the middle ages, the church treated a man, decreed spiritually dead to their community, as physically dead. That is, he was put in a burial cave, complete with grave-clothes. And where better than near Qumran, which is full of secure caves?
As a hint for this location, Thiering examines the parable of the rich man and Lazarus (Luke 16: 19-31).
Two years after an initial baptism, wine, the drink of the community (from Qumran or where-ever) was taken only by celibates, entering a full monastic life. Thiering sees the “miracle,” of turning water into wine, as Jesus allowing all to take communion, because all are equal in the sight of God.
This may not be so far-fetched, having considered how John the Baptist had by-passed the Temple priests. Jesus also did this, Thiering suggests, from a symbolic reading of his “miracle” of the loaves, as giving ordinary men the priestly tribe of Levi prerogative of distributing the communion bread.
“Walking on water” is deemed a jocular reference to the garment-laden priest using a pier to reach and bless a boats “catch,” by the fishers of men, the Gentiles who were thus ritually “saved.” Thiering says the “miracle” was that Jesus took over the exclusive role of the Levites, making the Jewish priesthood unnecessary.
Jesus was allegedly of the line of King David. The Jewish leading roles were the prophets, priests and kings. Thiering claims Jesus stepped out of his proper role to wear the holiest vestments of the high priest, privileged to enter the Holy of Holies. His garments “became dazzling white, such as no fuller on earth could whiten them.” (Mark 9:2) The fuller whitened the high priest robes with frankinsense.
In this passage of the gospels, there is also that authentic-sounding put-down remark against Jesus: The scriptures say prophets never come from Galilee. In other words, they didn’t believe him.
The Christians were those who accepted Jesus as “the high priest of our confession.” They were typically not of the holy land and not of the highest status in the monotheistic religion of the Jews, that Christs supposed priestly usurpation asserted for them.
Jesus Of The Apocalypse
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Following from work such as Jesus The Man, Thiering, on Jesus Of The Apocalypse, proposes “The life of Jesus after the crucifixion.”
Her basic method is the same. She works from ancient Jewish beliefs that history repeated itself. It was thought that if one could measure the cycles of time, one would be able to predict when previous situations re-occured. Such as, when was the right time for the Jews to successfully rebel from their current oppressors, in keeping with past insurrections.
From this, came an obsession with keeping time, which got translated into rigid ritual observances, typified in the Dead Sea Scrolls. There were similarities and differences between these and Christian writers.
Also, Thiering claims the Christian writers gave a new twist to this historicism. Instead of the past being a code for the future, the necessarily secret doings of the Christians were secretly codified as a sub-text to their writings. A precise religious calendar supposedly offered a precise context for interpretation.
Thiering, at her most plausible, is a shrewd commentary on the Clementine books, thought to be mere romances, popular at the time. She argues that they were Christian propaganda containing real history, with inconvenient details glossed over, of how a distinguished Roman family was converted to their religion.
But this insight depended on her native wit, not on breaking a formal code.
She links this story to the authorship of Revelation. Everything is linked in Thiering scholarship, which led a web reviewer to compare her to a novelist of a Tolkein-like world.
Nevertheless, one can’t help feeling her guesses are sometimes good. There is the famous number of the beast, in Revelation:
Here is wisdom. He who has understanding let him count the number of the beast, for it is a number of a man.
An abbreviated version of Thiering here pertains to the gnostic eastern monastic system. from which the zealots arose. Their head would be the man in question. They regarded the Christians as heretics disloyal to the nationalist cause. As in a modern school system, letters (in this case Hebrew) were used as numbers for grades. In this connection, Thiering explains how the 666 emerged, in counting the stages to the long years of oppressive study, with militaristic designs.
The Christians contemptuously rejected this as "the 666" -- the number of the beast.
[PS. In 2015, a tv program (which may have been Secrets of the Bible) showed how the numbers made up the name, Nero.]
The church father, Irenaeus associated the four gospels, of the evangelists, with the four living creatures of Ezekiel and Revelation. The association persisted in imagery and architecture. They drew Ezekiel chariot of God, leaving the Jerusalem temple, to comfort the exiles that they need no longer “sit down by the waters of Babylon and weep.”
A new comfort was needed, the four living creatures were to be the four gospels, like the four divisions of The Old Testament, drawing God to the Christian exiles.
According to Thiering, the canonical gospels were not the four books that happened to be selected late on, but were an early plan to emulate the Old with a New Testament.
The four horsemen of the apocalypse were the priestly teachers of the gospels to the Diaspora. Their banners represented the color of the season they taught for: white for summer, red for autumn, black for winter, green for spring.
The latter priest was named Death because of his power of excommunication.
I won’t repeat any more details.
What are we to make of Thierings enormously creative, sometimes shrewd, if often credulous, out-pouring of hypotheses about that elusive character “the real” Jesus?
One of her colleagues perhaps sums up professional opinion about her: She’s a nice lady but she’s wrong.
It’s easy to see why some of her claims are dismissed out of hand. Not only is Jesuses crucifixion re-located to Qumran but he is supposed to have ritually re-enacted the event in later years.
Jesus, a chronic ritualist?
Jesus, having survived the crucifixion (and marrying), is not a thesis peculiar to Thiering. It is part of fringe scholarship. And if Jesus Lived In India frequenting the old spice road from China to Rome, this might fit with Thiering claim, that the living Jesus, not a “vision,” was asked, by Peter, “Quo Vadis?”
“Where are you going?” is a question that would be asked of a man, unless we are to assume “the vision” was a vulnerable “reincarnation” of Jesus the man. But then why call him a vision?
The savior appears to have made a habit of these visionary appearances. Along with other writers, Thiering has a surviving Jesus meeting with Paul, tho this gets re-located into a rigid time-table of ritual observances, she believes prevailed.
That example is part of Thierings relentless removal of fairy tales, for the babes in Christ. Its replacement by a sort of sub-text conspiracy that reads like clock-work, once you’ve turned the key, is surely another sort of fairy tale. History is not well regulated.
Also the main characters in the canonical plot are allowed to take on the identities of minor characters. It coheres into a story of sorts, but there is little out-side evidence to keep the run-away imagination in check. So little is known about first century Christianity.
The scholarship of the lady is not in doubt. Even if every guess she makes is wrong, one can still get a new insight into ancient Jewish history. One has to admire her dedication, if not rely on her judgment.
Inevitably, Thiering replied to the rebuff she received: He’s a nice man but he hasn’t looked at the evidence.
But how to sift her results and to assess her approach, in different parts of the New Testament?
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Renaissance man.
Table of contents.
Dmitri Merezhkovsky: The romance of Leonardo da Vinci. [
**]Michael White: Leonardo, the first scientist.[
Leonardo and “the romantic agony.”
Lost knowledge, scientific and moral progress, and universal education.
It may be bad practise to review together works of biographical fiction and non-fiction. But to be objective about a life is perhaps as much a fiction as to treat him subjectively in a novel. The biography of “the first scientist” can be compared to the dissection of corpses to understand the living body.
The study of anatomy was a da Vinci forte. White concentrates on this experimental work, in laying claim to his achievement as a scientist. For all his artistry and operational skill, unsurpassed till modern times, Leonardo didn’t discover the circulation of the blood. He did make considerable advances in refuting wrong theories of light and sight.
Every specialist is allowed his enthusiasm. To claim Leonardo as the first scientist is understandable. Scientists claim Galileo as the first recognisably modern scientist. All White is doing is to push the genesis of modern science back somewhat to investigator, Leonardo.
HG Wells, and more recently, Umberto Eco, in The Name of the Rose, go back further still, to Roger Bacon as the prophet of a distant future of technological marvels issuing from the scientific method.
If we are going to have exaggerated claims, they don’t come any more heroic than that the history of Western science is a foot-note to the work of Archimedes. This remark was quoted in a BBC Horizon program on the re-discovery of a mathematical manuscript by him, actually a palimpsest. It will take long to decipher but already it has been discovered that Archimedes was much closer to modern methods of the calculus than previously realised.
Had his work been available in the renaissance, it would have been a real boost to mathematical science. Suggestions were made that man would be on Mars by now, and such like. Similar forecasts were made on finding, in an ancient Greek ship-wreck, a model planetarium, using differential gears, not re-discovered till the seventeenth century.
[P.S. I wrote this before the program, The 2000 year old computer, on research that attributed that planetarium to an origin in Syracuse, home of the Archimedes work-shop. “Science is a foot-note to Archimedes” no longer sounds such an extravagance,]
Archimedes is reputed to have used burning lenses as a weapon in defense of Syracuse. A book, by Robert Temple, called The Crystal Sun, knows of well over two hundred lenses from antiquity languishing in museums, unrecognised for what they were. More than likely, the telescope was a lost invention. The author characterised this, in Sherlock Holmes fashion, as the case of the disappearing telescope. The problem is that ancient records get translated according to what the ancients were only supposed to know.
One such classical translator was only located in retirement in a nursing home. When he was put in the picture, he gladly re-translated a puzzling passage, without having to worry about what its author could not have known.
For all that, we cannot be sure that this lost science and technology would have meant greater progress for mankind. As HG Wells, a prophet of science, said: moral progress has not kept up with scientific progress. An even greater imbalance between the two might have plunged civilization back into another dark age. Indeed the abuse of technological power is raping the planet, which is heading for ecological collapse.
The Archimedes of his time, most Leonardo researches were also lost to mankind -- a considerable set-back to the revival of science. Eventually, about half of his volumes would be traced.
But grievous tho this waste was, the truth is that vast opportunities are being lost all the time, because most of mankind goes without education, basic facilities or teaching to any worth-while standard of competance. Much greater progress depends on much greater justice, in allowing all people to contribute their native talents.
In april 2002, the World Resources Unit reported that forty per cent of the worlds remaining intact forests could disappear in 10 to 20 years, at the current rate of destruction, due to mining, illegal logging and urban sprawl.
At the same time, Oxfam claim the European Union and other industrialised countries swindle poor countries out of $100 billion per year, with unfair trade laws. Among their supporters was former British Labour government minister, Mo Mowlam.
In march 2002, Mexico, world leaders, including US president George W Bush, signed, at the UN conference on finance for development, to alleviate poverty and make education available to all. International aid groups were unhappy at the lack of deeds, as well as words.
The internet makes universalal education a realistic goal. Whereas a sustainable ecology must become the priority to sustain civilization.
Western politicians have recognised the need to tackle world poverty as a defense against violent disaffection. At time of writing, President Bush wants a vast new missile defense shield. But rockets may be as costly and as obsolete, as battle-ships became, when naval powers were still building ever grander models.
Leonardo and “the romantic agony.”
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Leonardo, like Archimedes, was much concerned with developing “secret weapons.” They featured in his letters of introduction to ducal employers. He appears to have had scruples against revealing his plans for submarine warfare. Nevertheless, Dmitri Merezhkovsky novelised a tension between the gentle Leonardo and the monstrous armaments designer. One of his apprentices finds this Jekyll and Hyde clash too much to bear.
The vegetarian Leonardo buys caged birds from the market to release them. (This might be a self-defeating exercise.) He also wishes to release mankind into the air with flying machines. The carpenter of flying machine plans is also their “test pilot,” characterised as another victim of apprenticeship to Leonardo. One of his models was recently built for display. It is too heavy to fly. Michael White merely comments that he came no-where near to flight.
White doesn’t discuss one respect in which Leonardo is rather too much like modern scientists, dependent on state or corporate funding for their research, namely as an ingenious servant of those in political power. Tho, he wanted to establish his independence to follow his own investigations.
Leonardo appeared to have that unattractive modern scientific out-look, that hides expedience behind detachment. Excuses can be found for him from his personal life. Being illegitimate, he did not have full parental recognition or rights. Any loyalty he might have had to Florence must have been quashed by the prosecution for sodomy.
He was acquitted but left the city. An Italian art critic suggested he had to leave in a hurry. It is just as likely that he shook the dust off his feet. Merezhkovsky dismisses the charge, refusing to hear any wrong of his hero. Michael White reckons Leonardo was homosexual. His evidence is by association and inference. Maybe he is right. But other characterisations are possible and just as likely to be false.
Of attractive youths among his companions, Leonardo himself was reputed to be of out-standing beauty. Merezhkovsky characterises these relations as fatherly. Perhaps he was giving parental affection, that his illegitimacy had denied him. Reaction to an inferior birthright explains his aristocratic pretensions and the dandys care for his appearance.
It is worth remembering that male may love female and pine for her company without sexual desire for her. Heterosexual love might be the natural concomitant of such an attachment but is still distinct from it. One has to be careful about jumping to conclusions, however obvious they may seem.
Merezhkovsky strikes just the right ironic note in suggesting that a man of Leonardos universal interests must surely have included carnal knowledge in his strivings to encompass all experience.
Michael Angelo faced the charge against Leonardo, from someone he refused sketches to. Michael Angelo regarded the prior Bichiellini as the only saint he had met. He was without the bigotry and ambition of Savonarola. On Michael Angelo, who lived for his art, the prior commented that no man could have created such a work without purity of heart.
Michael Angelos companions and apprentices were found commissions and prospered. Whereas Leonardo himself died a virtual exile. No public effort was made to preserve and publish his voluminous notes. He was widely regarded as a heretic, his work neglected or plundered.
It would be amazing if Merezhkovskys avowed “romance” of Leonardo were an accurate indication of his personal life. To his credit, he creates a credible human being out of an incredible prodigy. It is hard to over-state Leonardos profusion of talents. This review doesn’t attempt to give an impression of their variety. But Merezhkovskys super-man is rendered weak and vulnerable by the very isolating effect of his genius from the rest of mankind.
“Renaissance man,” that later ages have so much admired in Leonardo, simply took on so much that he brought relatively little to completion. Merezhkovsky makes this gap between ability and fulfilment, the cause of Leonardos self-reproach. Here is “the romantic agony” on a heroic scale. The author, who is a poet, depicts Leonardo like a force of nature unbent by Italys mountain storms and somewhat as futile in human affairs.
The Merezhkovsky romance teems with exotic manifestations not only from Leonardo.
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William Lovett: Chartism.
Table of contents.
In 1876, one hundred years after the American Declaration of Independence, the author, of a declaration of independence of his class, published his life story.
The working class organiser, William Lovett wrote the famous “six points” of the Peoples Charter, “for the equal representation of the people”:
Universal Suffrage; Equal Representation; Annual Parliaments; No Property Qualification; Vote by Ballot, and Payment of Members.
(Lovett owns these ideas were not original to himself.)
It used to be the custom to say that all these points but annual parliaments were achieved, long after Chartism disappeared.
The US House of Representatives has biennial parliaments. That early radicalism has not been realised generally, tho.
Later, some realised that equal representation required voting to be counted proportionally. In mostly corrupt electoral reforms, the parties took for themselves monopolies of the proportional count. The result has been that all people are equal but parties are more equal than others.
Parties have favored list systems that treat votes as their own personal property to allocate as they please to the candidates on their lists. This lack of democratic principle has brought about any number of arbitrary electoral fixes, pretending to be “PR” or “some form of proportional representation.”
In other words, voting systems that use party lists are akin to the old property qualification laws, in all their irrational holds over others.
William Lovett criticised the anomalies of Household Suffrage for
the thousand legal quibbles of house, tenement, land, rating, and taxing which have rendered the Reform Bill a nullity; and which have wasted a countless amount of time and money in the vain attempt to unravel their legal and technical mysteries. And that they might be assured that the adoption of a Household Suffrage would not settle the great question of representative right; for the excluded classes would keep up and prolong the agitation, and be more and more clamorous as the injustice towards them would be more apparent.
Much the same can be said for so-called proportional representation that only extends a Partisan Suffrage of the proportional count to the exclusion of every other possible prefered personal characteristic that candidates possess, by age, sex, race, creed, work, class, language, personality type or whatever.
Like household suffrage, voting systems of the world have become a chaos of legal quibbles and technical mysteries. This is especially true of list votes, which are so much fodder for the parties to share out the seats between themselves. Party lists usurp the guiding principle of the voters right to elect candidates (that is supplied by the transferable voting system, in a proportional count).
William Lovett included Female Suffrage in his draft of a Bill. He later regreted that other Chartists talked him out of it, as too unrealistic an aim.
Of the Working Mens Association, which he founded in 1836, he says:
And as our object is universal, so (consistent with justice) ought to be our means to compass it; and we know not of any means more efficient, than to enlist the sympathies and quicken the intellects of our wives and children to a knowledge of their rights and duties; for, as in the absence of knowledge, they are the most formidable obstacles to a man’s patriotic exertions, so when imbued with it will they prove his greatest auxiliaries. Read, therefore, talk, and politically and morally instruct your wives and children; let them, as far as possible, share in your pleasures, as they must in your cares;
The modern American movement of Kids Voting shows that educating children in political issues and making voting a family affair increases turn-out.
In 1837, Lovett prepared “what we believe to be a loyal and outspoken address” to the newly enthroned Queen Victoria. She was warned of the false counsel of Whig and Tory. With their exclusive interests, they would divide her from her people.
This was like an anticipation of Disraeli for Tory Democracy and Radicalism but with the working class taking the initiative to ally with the chief aristocrat. Victoria, like Wellington, however, was no believer in universal suffrage.
Six years after the 1832 Reform Bill, Lovett election address said:
But it has been urged, as a plea to keep up exclusive legislation, that the people are too ignorant to be trusted with the elective franchise. Are Englishmen less enlightened than Americans? – and has the exercise of their political liberty proved them not to have deserved it? – Nay, in our country, are the unrepresented as a body more ignorant than the present possessors of the franchise? – Can they possibly return more enemies to liberty, more self-interested legislators than are returned by the present constituency to Parliament? The ignorance of which they complain is the offspring of exclusive legislation, for the exclusive few from time immemorial have ever been intent in blocking up every avenue to knowledge. POLITICAL RIGHTS necessarily stimulate men to enquiry – give self-respect – lead them to know their duties as citizens – and, under a wise government, would be made the best corrective of vicious and intemperate habits.
This passage is still relevant. Public apathy is the logical outcome of politics being made an exclusive profession by politicians seeking a career out of it. Most governments have denied the voters an effective choice of representatives and individual policies. Instead, voters are patronised by the take-it-or-leave-it manifestos of the parties. No surprise, if so many people decide to leave it.
In 1840, Lovett founded the National Association for science and technics education, artistic recreations, libraries, cultural society with the aim:
to rescue our brethren from the thralldom of their own vices, and from servilely imitating the corruptions and vices of those above them.
Thorstein Veblen showed the profound truth of this observation, in The Theory of the Leisure Class.
Lovett addresses tend to be burdened with the grace notes of heroic rhetoric. But they have perception and clarity, and, if repetitious, are at least forceful. In other words, they are Tom Paine style, earnest with a desperate hope.
The lack of much sense of humor may be excused by the condition of the eighteenth and nineteenth century English working class. Lovett himself was lucky to find work at last among furniture-makers. For a while, this aristocracy of labor resented his presence in their closed shop.
English furniture was accurately joinered but lacked style. French furniture was superbly artistic but you could practically throw the drawers in. So, Lovett tells us, with a rare departure from seriousness.
Lovett, like Paine, abhored physical force to gain ones ends. The catastrophes of violent revolution have proved them right. Lovett was a “moral force” Chartist simply because force is amoral or without principle:
We are of the opinion that whatever is gained in England by force, by force must be sustained; but whatever springs from knowledge and justice will sustain itself.
In 1844, as secretary to the Democratic Friends of All Nations, he claimed:
Let but the same daring mind and resources which have so often warred with tyranny, and so often been worsted in the conflict, be once morally applied and directed, and citadels, armies, and dungeons will soon lose their power for evil.
This was to prove true of the downfall of East European Communist one-party states. (Tho, it seems the evils, of ethnic strife, also have been liberated. And corruption thrives on being privatised.)
Absolutism dreads “one word of truth.” And pioneer English reformers battled against the tax on knowledge, thru a stamp-dutied press; against social class education; and against secret diplomacy war conspiracies.
The reformers had their romantic hot-heads for revolutionary secrecy. Lovett recalled of the 1831 National Union of the Working Classes and Others:
we had no trifling number of such characters; and night after night was frequently devoted to prevent them, if possible, from running their own unreflecting heads into danger, and others along with them.
This mentality is well exemplified in A Radical Song, which reflects a blood-thirsty demoralisation after the Napoleonic wars. Its “freedom” is of the free-booter, the bully and the yob. Speaking of the Devil, one line (one can well believe in the light of history) reads:
And should he prepare us in hell a warm berth,
We’ll forestall him by making a hell upon earth.
Lovett believed in the moral force of being bold and honest in a just cause, as would enlist public sympathy, rather than be secretive and excite suspicion and persecution.
In 1845, Lovetts National Association address reasoned against anti-democratic conduct, as a means to a professed democratic end, by the physical-force Chartists.
In his 1838 Irish address, he complained
that the principles we advocate have been retarded, injured, or betrayed by leadership, more than by the open hostility of opponents.
Lovett 1836 Belgian address was the first international working mens address. Many followed, both to Europe and North America. One such speech to the French made five points, which deserve as much historic recognition as “the six points” of the Peoples Charter.
The five points are a prototype of the United Nations Charter:
1) a protest against all war as against morality, religion and human happiness;
2) a Conference of Nations, with representatives chosen by the peoples to settle national disputes by arbitration;
3) war expenses to go to education and the improvement of the people;
4) “to set an example to other nations of that justice, forbearance, morality and religion they preach to their own people.”
5) to set bounds of justice to territorial acquisition.
Another fertile idea, from Lovett, was a General Association of Progress to unite reformers in their diverse aims, rather than leave them divided and weak.
From 1849, Lovett turned most of his attention to education. For example, he didn’t think spelling should be taught as an irksome and disagreeable task but as a game and amusement. He knew that to learn work that is useful it had best be enjoyable for its own sake.
RH Tawney introduced The Life and Struggles of William Lovett with a diligent summary.
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Jill Liddington: Rebel Girls.
Their fight for the vote.
Table of contents
Yorkshire-woman, Charlotte Bronte: pioneer romancer of freedom for women.
Yorkshire rebel girls.[
This is a history of largely new evidence of Yorkshire women campaigning for womanhood suffrage. Those from other parts of the country, such as London or Lancashire, are featured mainly when they appear to speak in Yorkshire. And the deputations and demonstrations before the Westminster parliament are told much from the point of view of the Northern contingents.
The book cover has a clog and shawl girl of sixteen, possibly fotoed just as she is shouting “Votes” while escorted away by police. Her name and some family history is uncovered. There is the self-educated Lavena Saltonstall in polemical local news-paper letters. Florence Lockwood had a lonely aspiration to become an artist. She eventually finds companionship and emancipation from restricted social and political values. Many others are trailed by the detective historian. To her regret, some are no more than glimpsed.
The author has already written a book dealing with the regional contribution from Lancashire. Yorkshire was more disparate than the well-organised work-forces of its neighboring county. Liddington has had to wait a good many years before remaining pieces of evidence have come together well enough for this new book (2006). The importance of this work is as a historical corrective to any notion that women got the vote mainly thru the agitation of some middle class southerners led by the militant Pankhursts.
In over-centralised Britain, this is welcome historical news.
However, the militant influence is felt in reading this book of the north women campaigners. That shouldn’t disguise the fact that in the years up to 1914, the rebels are looking increasingly desperate. Their local organisations spring up but soon wither. A faction breaks away from the Pankhurst autocracy. A few organisers are struggling to keep going. And their adherents are tipping to sensational acts to attract attention. As the violent protest escalates, Liddington points out that the movement was lucky not to harm someone. In general, I agree with her assessment of the limitations of militancy.
Peaceful protest can stir publicity that draws peoples attention to an injustice. Women lobbyists tried to speak up but were pulled down and carried, even kicking and screaming, out of Westminster.
The peaceful leader Mrs Fawcett generously said that they had done more than their traditional organisation in a dozen years to promote the Cause.
We should be clear who the real militants were, in the first place. The government tried to repress the women, just exercising the right to protest. Disgracefully rough handling of women protesters was a bad example in civil behavior. The forced feeding was more extremist than ever the women worked themselves up to, before the Great War intervened with a real example in violence.
Lazy-minded government couldn’t be bothered to work out the next step to where its actions were taking it. They tried to break the spirit of women as independent intelligences, before it dawned thay were going to have to give them equal rights. The government got to nearly killing suffragette prisoners, before it dawned that government militancy would have to give way to civilised treatment.
Leonora Cohen smashed a glass case in the Tower of London and then, with her husbands advice, got herself acquitted by a jury, because the prosecution over-estimated the cost of the damage. Her avowed motive was the treachery of the Asquith government, pretending sympathy to female suffrage but leaving it out of the Reform Bill.
The government might have got the message from the jury about the mood of the country.
Up till then, Cohen could still claim some sort of moral high ground against the government. She had not victimised businesses, at least in this instance of glass breaking. But then she incited arson against empty property. (An empty property, they broke into, turned out not to be empty.) In taking the law into their own hands (empty property must be burnt), these arsonist suffragettes were being as arrogant and inconsiderate as the government.
One notices sometimes that governments fail to rise – if they ever do – to the moral level of reformers, before some of the idealists have dragged themselves down to the level of the reactionaries. It is as if they have to exchange roles before they can come to terms with each other, like those role-playing therapy groups to get off the emotional hang-ups from being trapped in a dysfunctional family.
Of course, there were plenty of women campaigners, who gave the “extreme wing” no more than their due. In the meantime, the earlier constitutional organisation led by Mrs Fawcett quietly built up impressively, to use their own image, from an acorn into a mighty oak with many branches, and scores of thousands of affiliate Friends. When they marched to London, they didn’t have to try to force their way into the seat of power, the powers agreed to see their deputations.
In Rebel Girls, the less well-connected move center-stage. Mary Gawthorpe, a self-educated working girl from the textile towns and villages of the West Riding, became one of the wittiest speakers on the campaign circuit.
An example of Churchills wit was anticipated by her. That is when Nancy Astor said if she were married to Winston she would put poison in his drink.
Churchill answered: Nancy, if I were married to you, I would take it!
This Churchillian riposte was even featured on the cover of a recent collection of political wit.
In the Edwardian era, a male heckler, of Mary Gawthorpe, made Nancys threat.
Mary anticipated Churchill: No need for that, friend. If I were married to you, I would take it.
The heckler, who had been doing his best to put her off, was made a laughing stock and left soon after.
The courage and good will of this little woman (she was less than five foot tall) didn’t spare her the mob violence that was often meted out to the suffragettes. She was kicked in the stomach and had to have an operation. The book doesn’t say whether this was why she had no children.
Mary worked with the youngest Pankhurst daughter, Adela, who was sent north to do the organising there, out of the southern spot-light. She, too, is somewhat side-lined from the standard 1931 history by Sylvia Pankhurst, The Suffragette Movement.
The emigrant Mary Gawthorpe helped promote the American edition but found Sylvia had only given her a foot-note. (One wonders if that was only an after-thought for the help she was getting.)
The personal lives of the rebel girls are the most sympathetic aspect of the volume. I mean something a bit more human than “social history.” “The masses” are what most of us are, let’s be frank, and their story is largely our story. Their aspirations and endeavors are heart-warming. And a book, such as this, is the best we can do to reach out to them.
The two main causes of John Stuart Mill.
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Before the quibbles, I would like to make an important point about “the vote” which women fought for. So far, in Britain there are half a dozen undemocratic voting methods where one democratic method would do.
John Stuart Mill entered Parliament to promote two causes: votes for women and proportional representation alias personal representation, that is where the vote is personly transferable by the voters (the single transferable vote) and not merely by party bosses presenting party lists we have to vote for as party blocs, like the vote for British Euro-elections, or for additional members to the Scots, Welsh and London parliaments or assemblies.
Tony Blairs Labour party got large numbers of women MPs by forcing them on local constituencies. (Ebbw Vale, being one of Labours safest seats, meant that Labour voters could rebel and elect a former Labour man as an Independent against a mandatory woman. But that option is not generally available without a daring rebel to vote for, and without splitting votes and letting in the least wanted candidate.)
David Cameron, “Blairs heir” has so far (in 2006) got to asking Tory local constituencies to have two women out of four final nominees. The monopolistic single member constituency means that the voters are then presented with an accomplished fact. Tory MP Ann Widdicombe opposed women candidates having their paths smoothed for them, making them into second-class MPs.
At the Power Inquiry conference in 2006, single members was the one feature Cameron was adamant against changing. This public relations man, one might truly call “Safe seats Dave” or “Rotten Boro Cameron,” will take any camera call to make the world a better place but will not start by transforming his own party from another mean-spirited little oligarchy.
The transferable vote in multi-member constituencies allows voters to order a personal choice among several candidates, whatever their party or gender or ethnic origin or any other personal quality and character. This gives a genuinely democratic proportional representation.
We have votes for women. We don’t have PR, by STV, the democratic voting method.
Minor points.
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Finally, some quibbles, which are not meant to detract from the value of Rebel Girls, but will pad this review.
Actually, this would not be a minor point as far as the honor of Winston Churchill and his family was concerned. The author comes up with the legend that Churchill as Home Secretary ordered the soldiers to fire on the Tonypandy miners. When Richard Burton was given a part as Churchill, he came up with this folk lore. It was dismissed by the entertainment magazine doing the write-up. I also vaguely remember some book introduction, going off-topic to refute this allegation.
I’m not going to go into this defamation further. Liddington should have done that herself. Every blunder in Churchills long life has been raked over, and if one so serious were true, it would be widely known and recognised. It is common sense that the charge is wrong. When you consider all the enemies that Churchill made, from every party in his own country, not to mention those in other countries, and the debunkers after his death, it is inconceivable that they would not have made this supposed shooting order stick, if they could.
Anyway, anyone is free to investigate the evidence for themselves. I don’t want to glorify the man but neither is this the place to go into his short-comings.
The author says the Hull womens suffrage organiser, Dr Mary Murdoch “was very probably a lesbian.” Maybe so, but the surmise, is based on no actual evidence. And it is presumptive so to label her, since it reflects on every woman, who may be only married to her calling, yet lives happily with another woman.
In its apparent belief that the essence of happiness with another is sexual congress, it may say more about the author than about Dr Murdoch.
The author, out of political correctness, seems to be throwing a bone to the lesbian lobby.
Political correctness seems to be a symptom of the party patronage of lobbies, rather than the proportional representation of the public interest, in the House of Commons or Communities, as well as a proportional representation of special interests in a second chamber.
Getting now to the trivia, Adela Pankhurst spelling “humor” and “honor” is changed to “humour” and “honour”. This copy-book correcting may be misplaced. The shorter spelling was coming into scholarly English use in Victorian times. Then Teddy Roosevelt included it in his American spelling reforms. This effected a back-lash that made it a point of honor for the British to use the longer spelling.
One of the beauties of the internet, which spans different countries with recognised English spelling variations, is that one can pick and choose ones spelling, without the book publishers going in terror of committing a spelling heresy.
Yet Liddington uses the American solecisms or redundancies “report back” and “co-conspiracy” as well as the English varsity solecism “come up from” etc.
17 september 2006.
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[*The franchise plummets to 11+. *
Leslie Brewer: _Vote for Richard. _]
(Serialised in The News Chronicle; first published in 1948 by Art and Educational Publishers Ltd. London and Glasgow.)
Table of contents.
Joshua Reynolds: Angels.
“And did women get their own way in the end? Did you and your friends succeed in making people treat you properly?”
‘Yes; after a long struggle.’
What a wonderful idea! If women were able to do things like that, then boys and girls could, quite easily, if only Richard could make them realise it. They had only to be complete nuisances for a month or so and, like women, who were once treated ‘as children.’ they would get votes and become Members of Parliament, and very soon put things to rights. Where women had succeeded, clearly boys and girls could. There must be hundreds of boys and girls up and down the country just longing for a chance to do something like this. They had been fools not to think of it before.
There’s many a true word spoken in jest. But an idea has to be laughed at, first, before it can be taken seriously. Logic leads the way: If not suffragettes, why not… “suffraginos” shall we say? A millenarial wish, perhaps. But before the end of the second millenium, some young English people did ask for a childrens parliament. And there is a youth parliament.
[PS. What’s more, the vote for sixteen year-olds has become a reality in Scotland, since the 2014 Independence referendum.]
The childrens story, by Leslie Brewer, Vote for Richard is a light-hearted fantasy on the first child to stand for parliament. In the end, Richard succeeds, surprisingly, in getting the franchise extended to everyone over ten. But the author aims more at readers of ten or below.
A British child failing the “eleven plus” exam could mark him or her for life. So, Brewer giving the vote for all of eleven years or more, is no more than just. Considering that a childs whole future may be determined by their performance in exams, it could be said that children are carrying their adult selfs on their backs. As parents are responsible for children, children are responsible for their adult selves.
Since children have such a big personal responsibility in competitive examinations (“the rat race”) it cannot be said they are not responsible enough to vote.
It’s also noticable that children are among the most aware of the ecological dangers to the future of the planet, above all, their future.
Still, todays commercialised kiddies might find Brewer childish. In the post-war austerity, a child demands extra cheese rations for pet mice.
It hardly compares with toddlers answering pagers. One did this in our local library, the other day. Mobile phones are banned. But he was so small he couldn’t be seen passing below the librarians counter.
As a matter of fact, the big business of childrens adverts on television etc, that has grown up since the second world war, is a fresh reason for todays children, affluent or deprived, finding a public voice of their own, independent of commercial brain-washing. The argument, once used against womens rights, that children need protecting from the hurly-burly of politics sounds weak, in the midst of the relentless parade of dazzling toys to fill their Christmas stockings and empty their pockets.
In some ways, most adults have as little control, as children, over the economy. Adults have no forum to represent the interests of their working lives. The British House of Lords is to house “the peoples peers,” meaning the appointers peers. An “independent Appointments” commission is a contradiction in terms, another shining example of feudal British hypocrisy. It is a common-place example of politicians giving democracy a bad name, by pretending an oligarchy is a democracy.
“Sophisticated” moderns are clueless about economic democracy. And Brewers childish jeu d’esprit may not be so silly, either.
The newspapers were full (Richard read them carefully) of talk about things which vitally concerned boys and girls. Should the Cane be Abolished in Schools? Should School Holidays be Shortened? Are Examinations Fair? Yet the strange thing was this: All sorts of people gave their opinions on these matters, but boys and girls, who were going to be caned, or examined, or have their holidays shortened, were never consulted…
About a quarter of the whole population of Britain was ignored. That, both Richard and Sally agreed, must be quickly remedied.
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RG Collingwood Autobiography bitterly recalls his mania for excelling in exams, to prove himself. Afterwards, he realised the vanity of such work and condemned his prize achievements as worthless. He believed children were “criminally overtaught.”
In China, from where the British imported the system of examination in the classics, the government itself was obliged to condemn this educational forced feeding, after a singularly tragic effect of parental pressure to achieve top marks.
In march 2000, Premier Zhu Rongji told teachers to stop piling homework on children. Stop cramming them with intellectual facts. Consider their all-round education. Strengthen their moral education and help them develop practical abilities and a spirit of innovation.
In Vote for Richard, a new party is formed, with its own salute of thumbing ones nose. He and Sally have just approached a paper to spread the news of this Childrens Party:
“I hope the Editor wasn’t joking,” said Sally. “He may have thought we were dangerous lunatics and was only humouring us to get us out of his office.”
In this dated childrens story, there is some timeless advice for reformers of whatever age or country:
"I know," said the Editor, "that some of you want to begin a reign of terror, here and now, to make people give you the vote. That's not -- if I may say so -- the British way to set about it. Try peaceful means first. Ask nicely. If the petition fails and Parliament does nothing to meet your demands -- well, that will be the time to consider other, tougher ways and means. But put the petition first, please...Meanwhile, you have to show that you are worthy of being given the vote...I know it is not going to be easy but..."
There follows “the week of virtue.” But the Chartist-type petition is rebuffed. The children go “off the good as gold standard.” They also have the backing of a manufacturer of water pistols.
The children show that they, too, can go on strike. In the post-war period, there is still considerable juvenile employment, in offices and hotels, as well as news boys.
But we are still twenty years away from the 1968 international revolt of students. (The lowering of the franchise age, to eighteen, would soon re-define students as young adults.) In 1948, school sit-ins would have been akin to a treasonous breach of quasi-military discipline.
Anyway, the Prime Minister has second thoughts. Brewer makes a delightful under-statement of it:
The news that boys and girls had secured the vote was received with mixed feelings in the country. Most people agreed, however, that the Prime Minister probably knew what he was doing, though it was, as The Times newspaper said, all a bit sudden.
(This competance view of prime ministers was in the days of Attlee and Churchill.)
When women got the vote, only one woman was elected an MP. The story again follows precedent. There’s only enough money to afford a campaign in one constituency (with “seven boarding schools and three orphanages”).
But Richard, the parliamentary candidate finds himself up against dirty tricks.
Like Robert Redford film, The Candidate, he gets in at last. Unlike the Redford character, packaged for popularity, he won’t have to ask: Now what do I do?
Like F Anstey, writing Vice Versa, Leslie Brewer Vote for Richard could make a great movie. A completely modern script would be needed, but Brewer, like Anstey, has shown the potential of his plot.
In the real world, extending the franchise to sixteen year olds is gradually moving onto parties agendas. Moreover, parties most popular with the mid-teens have the incentive to act, when they get the chance.
In ancient Rome, you were a man at fourteen.
H G Wells argued that people, just left school, remembered more of their education to exercise a vote. Compared as good citizens, the young lived in hope, unlike those broken in spirit by the age of forty.
The franchise may become part of teaching the young responsibilities, with their rights against assault or abuse. The need for the young to settle their disputes by informal child courts of law, rather than violence and expoitation of each other, suggests a highest child court of the land, being a childrens parliament.
The Kids Voting movement, in the USA, has shown that political participation takes education, like anything else important enough to have to be done. Parental help of their children, to learn about policies, also increases adult turn-out at the polls.
School-boys invented proportional representation?
And finally (written in 2015), let’s not forget that school-boys have a good claim to inventing the future of democracy.
1n 1821, Thomas Wright Hill, son of Rowland Hill, observed, at how children elected a committee, at his fathers school. The favorite pupils became the candidates that children formed queues behind. the most popular candidates, with the longest lines of support, would lose some support to next prefered candidates, till their queue was no longer than needed to take a seat.
The first winning candidates surplus voters would transfer their allegiance to their next favorites, till they, too, had just the right length of queue to take another seat.
The least popular candidates, who never reached this winning length of queue, would become a hopeless cause, and also lose their supporters to next prefered candidates.
If there were five seats, then all the pupils eventually would form into five equal queues, electing the five favorite candidates.
In the formal count, these equal queues are called quotas, being the elective proportion of the whole vote, necessary to elect a candidate, to ensure the voters equal representation.
This is the original and genuine form of “Proportional Representation (PR): As voters transfer their loyalties, from most prefered candidates with surplus votes and least prefered candidates, in deficit of a quota, this system is called the Single Transferable Vote (STV). It is the future of democracy, if democracy, and humanity, has a future.
This review appeared in 2001.
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HG Wells: pre-internet idea of a World Brain.
Table of contents
A dismissive critic of World Brain.
Postscript (2015)
The brain organisation of the modern world. (1937.)
A dismissive critic of World Brain.
From the web, I was reading an academic survey of Wells proposal of a digest or abstract of mankinds increasing knowledge. He wrote several works about this including World Brain (1938), The Idea of a World Encyclopedia (1936), Science and the World Mind (1942).
I’d never seen any of these works so I was relying on the article to learn something about them. The author was concerned to warn against what he believed to be the dangers of “social repression” in Wells conception.
Wells regarded himself, in the subtitle to his Experiment in Autobiography, a very ordinary brain, which approaches Winnie the Pooh (a bear of very little brain) in modesty
Never the less, the world brain has been justified by events. The sciences had to resort to journals which are abstracts of the increasingly unmanagable output of their professions.
As far back as his utopian science fiction, Men Like Gods, he envisaged publication available to all. Until the world wide web, this was just a dream. Yet, it seems unlikely that the internet will be enough to help education win the race against catastrophe. (One of Wells most famous pronouncements is that “Civilisation is a race between education and catastrophe.”)
Let’s give credit where it’s due. Amongst many other things, Wells foresaw that information would have to be assimilated on a world scale to promote the most efficient growth of knowledge and harmoniously foster human talent and progress. He drew attention to the need and pioneered its supply with his own encyclopedias to educate the world.
Rayward cuttingly regarded these as medieval, for some unknown but derogatory reason. Their obvious influence is the eighteenth century encyclopedists.
In accord with the Enlightenment spirit of progress, Wells promoted “the democracy of science” with a charter of scientific fellowship. This was shortly after the 1940 Sankey Declaration of Human Rights, which he also inspired. Not to admit the consistency of Wells commitment to free speech, free debate and publication is simply to misrepresent him.
The critic (W Boyd Rayward) excuses himself for citing Wells works out of context, with a “pastiche” of quotations. He notes Wells reservations about the relevance of universities to human problems, and repays the compliment.
The fact is, by selective quotations, you can prove practically anything about anybody. You could argue convincingly that Beethoven was a symphonic non-entity with reference to the once popular Battle Symphony, mentioning that he also wrote nine other symphonies.
Scholars have shown to their satisfaction that Jesus was a political insurrectionist. Or equally, academics have shown him to be a barefoot philosopher with an academic indifference to the world. A latter-day pharisee, Barbara Thiering, sees him as a pharisee, and the gnostic John Davidson sees him as a gnostic. You could say he has been shown to be all things to all men who have held up mirrors to him. People have done this about Jesus and he never wrote anything, or nothing he wrote has survived, so far as we know. Imagine how easy it is to condemn a man by his own words, if he wrote over a hundred books during more than fifty years of turmoil.
There is nothing wrong in devils advocacy. Wells was no saint, to be sure. But it would be more honest to admit the prosecution role. Mr Rayward quoting reminds of a compedium of worst verse from the great poets. These shadowings of the great poets are merely amusing flops because everyone knows where the balance of the truth is.
If the shadow of HG Wells looms large, that is in keeping with his times. Tho he was not as consistent and influential as John Stuart Mill in the nineteenth century, it is little known that Wells better non-fiction writings offer the first half of the twentieth century about the nearest thing to a (low-profile) democrat of distinction, when democracy seemed an unfashionable failure.
(My e-book, Scientific Method of Elections, gives bibliography and commentary mainly on his writings for proportional representation. It also includes the above-mentioned charters.)
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The real significance of an unbalanced portrayal is what it says about the portrayer rather than his subject – a failure of the portrayer to rise to his subject. He concludes that the idea of a “world brain” “may be interpreted as becoming an expression of totalitarian values and authoritarian control.” So it may.
But this is a fact: HG Wells is one of the liberators of the human mind. Wells worked as much as anyone, as essentially stated in the preamble to the 1940 Declaration of Human Rights, to re-assert the rights of the individual against every extension of political and economic control.
When all his faults and short-comings have been admitted, it must be said that Wells did try to improve democracy, in practical terms. Established wisdom seems mainly to serve established things. That is why one has to go back to Wells.
Wells admired Plato for The Republic and sometimes called his utopia “The New Republic.” The critic refered to A Modern Utopia (1905) for Wells four categories of human beings (kinetic, poietic, dull, base ). This is one of the dullest works Wells ever wrote, and Wells was rarely “dull.” Occasionally he was “base.” (Arent we all?)
The Early HG Wells was “poietic” (or “mythopoietic” – myth-making – as critic Bernard Bergonzli said in his study of his late Victorian science fiction). His later works were “kinetic” – he certainly “moved” a new declaration of rights, that influenced the UN Charter.
Wells early SF could also be called the work of a maker or poet in the familiar sense. This is testified by T S Eliot, who called “unforgettable” the sunrise on First Men In The Moon.
Wells follows Plato in categorising human types. He admired Plato for making him realise that society could be changed after ones own heart. To the end of his life, he found a place in his thinking for this Platonic way of dividing people up into crude classes, which is arguably the wrong way. And also most un-Wellsian in its philosophy. Wells was a nominalist and not a Platonic realist. He didn’t believe in the reality of concepts, but regarded them only as more or less useful labels or names. Normally, we would expect Wells to say that a classification of human beings into four types did no justice to human diversity.
Then again, as the limitations of old age descend, I suspect that party government and the Establishment are rather well characterised as the dull and the base.
Our academic critic may not have been aware that this categorical limitation in Wells thinking was not characteristic. But it is odd that another work he chooses to cite is a work of fiction, The Shape Of Things To Come. Again, this is not one of Wells many works for which I have a high regard. The early pages are just one of many examples, in his fiction and non-fiction alike, in which Wells displays his gift for social history.
That future “history,” was bound to be over-taken by events. It is peculiarly unsatisfying to read, maybe because the style is journalistic, giving an impression of being an epic of misreported events. It was more successful when cut down and re-cast imaginatively, as the film story, Things To Come.
The Shape Of Things To Come is just one of many fictions, in which a new elite takes over the running of the world. Who can deny that the world is still run by elites?
2005; revised and posted May 2006.
Postscript (October 2015):
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Ten years after countering a disparaging review of World Brain, I managed to read it for myself. Wells conveys that the schools and universities are falling down on the job of properly informing public debate. He admits that his own attempts were inadequate and that he was being provocative.
In 1937, the provocation succeeded only in receiving a mass denial from the teachers. This is the pack defense of the professions, as described by Margaret Heffernan, in Wilful Blindness.
However, I can personally vouch for the truth of Wells criticism of history taught in primary schools, as “1066 and all that.” I can still remember Anglo-French relations summed up as: first they won and then we won.
And that was twenty years after his objection to narrow nationalist egotism impressed on young minds, when still at a barbaric phase of development.
Wells outlined a curriculum of essential content for people like my parents, who never had a secondary education. They were warned, with a sort of horror, that my five year old self still remembers in old age, from my mothers former teacher, against sending me to the local state school. Like a Soviet election, this was the sole choice on offer from the local authority.
Anxious to spare me their own skimped and brutalised experiences, my parents sent me to a private religious primary school, they could ill afford. At first-year level, the experienced old teacher allowed me to write in my own immature large hand, saving me from being mistaught into illiteracy. Apart from that boon, I think the average level of teaching was poor, tho I can forgive the carefree laxness, while it lasted, before the stupid obsession with an examination system that usurped the education system.
If primary education did not leave me nearly as well-informed as it might, this was more than made-up-for by the moral education from the religious head, who acted like a judge, imposing absolute equality before the law, on our childish misdeeds.
My secondary education in history was a very different matter. It started off in Wellsian fashion with the ancient Middle Eastern civilisations. The next I remember was an un-Wellsian European feudalism of castles and peasants, a recognisable forerunner of the English class system.
By ordinary level exam time, we were moving into the recognisably modern world of “revolution, reaction, and reform,” including a very frank impression of Englands all-out legalised oppression of Ireland. (Gasp!)
For what would now be called sixth form college, our history teachers pleaded with the education board to teach modern British and European history, right up to 1939. My year was the first to benefit from this, in about 1966; definitely not 1066! Tho, authority decided to leave the girls in the dark ages.
We did not need to go into the Second World War, as we were already saturated with films and documentaries of it. I knew less about the political history of my post-war childhood years, as I found out at college, from reading British Political Parties, by Robert McKenzie.
From first to last, the quality of teaching was very variable. The salvation of bad teachers, poor or inadequate teachers, was that the child had to take who they were given. This may have been all right for them but it was, in my youthful experience, the biggest drawback to a good education.
And it was not just a question of sub-standard teachers. A teacher may be good for some children but unsuitable for others. That is the extra important reason why children should be allowed to choose teachers of essential subjects. Teachers have their own temperaments, tempos and mind-sets. Pupils need to be able to select the ones that they can best get along with. There is no substitute for the childs subjective choice of teacher. No one else can know for them what is the best way for them to learn.
I would qualify this observation of an over-taught childhood and youth. Where a limited choice was offered, teachers got to canvass for their particular subjects and noticably developed the skills associated with politicians. Whereas the child was still innocent of promotion politics, in which public interest disguises self-interest.
The great medieval English scholar, CS Lewis was notoriously good at fighting his corner against competition from new courses in modern English literature. Of course, Lewis had the popular touch. I once read, or stumbled thru, his erudite volume in the Oxford history of English literature series, not for any particular interest in late medieval poetry (beyond the fact that I myself was a poet) but just because I knew his writing was good company, during one long winter, with ones feet before the fire.
Like JB Priestley, I don’t really approve of professional qualifications in, what I would consider, leisure occupations, like modern literature, the movies or media studies. That’s not what made Dickens or Wells great, and I don’t think we’ve had their like since.
Some writers, like Kingsley Amis, have suspected that the academising of literature has coincided with its falling off. Leisure should be a release from, not a substitute for reality.
Equally, I don’t think science and technology courses should be crammed like a sort of mental coal-mining, or not all of them. We should be doing our best to attract as many people as possible into the practice of a thoughtful and inventive frame of mind, instead of degrees in pretentious fluff.
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I didn’t get to college on the strength of my advanced level grades. Part of the reason, I gained just one acceptance, may have been a generous desire to assist a disadvantaged individual. But I was unable to help myself. All I wanted to do was learn the technique of science to solve social problems, as it had succeeded so well in understanding nature.
I expect they would deny it out-right, that Pitirim Sorokin was making exactly the kind of identification of social research with social reform, that I was looking for, but was not what they wanted, so we were not pointed in that direction. When a student asked about Sorokin, the lecturer put us off with its difficulty, as only suitable for post-graduate study. Anyway, due to my deficiencies, as well as theirs, I didn’t find out about Sorokin, till pension age, which was rather late, to make amends.
By the course third year, I had found out about HG Wells. The course principal, a man of remarkably equable temperament, came in one day, to remark that he’d looked and not found any sociology in HG Wells writings, and asked me for a reference.
I suggested Tono-Bungay.
He never came back to me about this. In the literary criticisms of Wells, it was acknowledged to be a foremost sociological novel. If I said so, tho, I would have expected to cause a demarcation dispute. Any opinion that came from literature was effectively banned from the sovereign territory of academic sociology.
This closed shop was alleviated by the odd kindness, even at course end. A junior lecturer told me, concerning his paper in the final: And if you mention HG Wells more than six times, I swear I’ll fail you!
This set just the right tone. He knew, and I knew, I was struggling, so he threw me a bit of a life-line, as long as I didn’t take it too far.
Duly, having counted my sixth reference to HG Wells, in his exam paper, I prudently refrained from any more. And wondered what I was going to say next.
I suppose it was remarkable for a student to come up with an independent view, which was not only at odds with the teachers, but also happened to be right. Afterwards, it was passed on to me that someone had done a post-graduate thesis on “The sociology of HG Wells,” at London university. This was, in fact, his old university.
More remarkable still was what I had more to find out about Wells as a social reform researcher, over the years, right into old age, and while reading only the other day, his book on a progressive, properly organised world education.
The letters of HG Wells, edited by David C Smith, I only obtained in later years, showed that at the turn of the twentieth century, he lobbied hard for a chair in sociology. He was still in early middle age, mainly with a reputation for science fiction novels and short stories. It was as if his genius for fantasy wanted to secure some hold on reality.
In late middle age, he saw the position of politician, as a means of exerting some influence in the world of practical affairs. In the early 1920s, he was twice a Labour Party candidate for MP from the London university constituency, a two member system, using the single transferable vote.
Proportional representation by the single transferable vote in large constituencies, of say a dozen members, was the system that Wells so passionately urged, as the only really effective means of representation.
Of course, a two member system is scarcely proportional and was not enough for Wells to be elected by the conservative university mentality.
There is no doubt that had Wells become an MP, he would have resumed the attempts of John Stuart Mill to get a bill for STV (“Mr Hare’s system”) passed in the Commons.
This was as well as Wells preoccupation with replacing war by international law. In this respect, he thought the Conservatives too imperialist, compared to Labour and the Liberals, tho he was clear that the latter were as big a cheats as the former, in banishing proportional representation from practical politics.
Wells had the knowledge of effective representation in politics. He then went on to publicise the need for the effective representation of knowledge itself. In these and other things, he was making a great contribution to human progress and prosperity, admittedly in the tradition of thinkers like Mill and Diderot.
At seventy years, the World Brain or World Encyclopedia may be regarded as Wells third attempt to move from ideas to action. It was the difference from being an educator to an educationalist.
The review, that dismissed Wells as either, may perhaps be countered here.
In 1916, Wells novel, Mr Britling Sees It Through, was a top ten publishing success in the United States, as conveying the distress of the Great War. And it does do that, tho it is a very average sample of his novel writing.
That catching the national mood was far out-stripped by The Outline of History, in 1920. This was nothing less than a historic attempt to put international relations on a new footing as one story of all mankind.
In the United States, this was the number one bestseller for two years running, and still remained in the top ten, in the following year.
The writing of that encyclopedic history, in the space of a year or so, was the cause of some astonishment to those that knew. It made him so ill from over-work that he had to be sent off, immediately afterwards, on a long holiday to recuperate.
Some lady in America claimed to have written the history herself and the lawyers made a killing out of this delusion. But Wells showed no resentment against this injustice, that cost him so dear in work, health and wealth.
Wells later wrote two more specialised encyclopedias, of more limited appeal, on biology, and on human ecology, as he called it. Whatever Wells wrote is usually lightened by that intelligent mind.
Now a few words against the dismissing of a World Brain, as scarcely worth a mention as a precursor of the Internet.
The fact is that the phrase “world brain” unavoidably associates itself with the Internet, which is analgous to the neural network of the brain.
The following lengthy quotation refutes the claim that the World Brain scheme was totalitarian in inspiration. (Alluding to the conservatism of universities doesn’t make one a totalitarian.) And the rest of the quote gives as good an intimation, as any, of the Internet. So, I give Wells the last word.
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Quotation from the chapter:
The brain organisation of the modern world. (1937.)
I can imagine quite a number of obvious preposterous mischievous experiments, a terrible sort of world university consolidation, an improvised knowledge dictatorship. Heaven save us from that! We want nothing that will in any sense override the autonomy of institutions or the independence of individual intellectual workers. We want nothing that will invade the precious time and attempt to control the resources of the gifted individual specialist. He is too much distracted by elementary teaching and college administration already. We do not want to magnify and stereotype universities. Most of them with their gowns and degrees, their slavish imitation of the past, are too stereotyped already.
… I imagine it as a permanent institution – untrammelled by precedent, a new institution – something added to the world network of universities, linking and coordinating them with one another and with the general intelligence of the world.
… This Encyclopedic organisation need not be concentrated now in one place; it might have the form of a network. It would centralise mentally but perhaps not physically. Quite possibly it might to a large extent be duplicated.
If a thing is really to live it should grow rather than be made. It should never be something cut and dried. It should be the survivor of a series of trials and fresh beginnings – and it should always be amenable to further amendment.
… And while on the one hand we have this world-wide receptivity to work upon, on the other hand we have among the men of science in particular a very full realisation of the need for a more effective correlation of their work. It is not only that they cannot communicate their results to the world; they find great difficulty in communicating their results to one another.
… And for me at any rate this is no Utopian dream. It is a forecast, however inaccurate and insufficient, of an absolutely essential part of that world community to which I believe we are driving now. I do not believe there is any emergence for mankind from this age of disorder, distress and fear in which we are living, except by way of such a deliberate vast reorganisation of our intellectual life and our educational methods…
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The sixth extinction.
Table of contents
Section links
Humanity as a catastrophe to other species if not itself.
Natural causes of extinction and their human promotion.
Lack of democracy results in injustice, ignorance and incompetance.
Humanity as a catastrophe to other species if not itself.
Man is the cause of the sixth mass extermination of species in the history of this planet. It goes on daily. There is no program to identify and describe all species. We have no idea how many life forms there are. Many creatures are disappearing without ever having been discovered.
The vast evolutionary experiment all around us is a unique and irreplacable lesson in lifes potential, which we are ignorantly trampling over. Nature is our teacher in the foods and medicines of plants but is also ignored at our peril. Mankind might become its own victim, too.
Such is the message of a publication in 1995, by Richard Leakey and Roger Lewin: The Sixth Extinction. Biodiversity and its survival.
Richard Leakey speaks as a practical conservationist (of the hard-pressed elephants in Kenya) as well as a modern theorist radically qualifying classical Darwinism.
Bible-inspired Catastrophism has come back into prominence. Darwinian evolution is by gradual change, thru inherited small advantages in adapting to the environment. But environmental catastrophes cause indiscriminate mass extinctions. Survival then depends on wider distribution of groups of species (or clades) which fare better no matter how many species they contain. Smaller creatures are less vulnerable than large.
As disasters reduce evolution from a question of good genes to good luck, “the balance of nature,” radically disturbed, gives way to the unpredictable fluctuations in population dynamics that chaos theory shows can lead to a complete collapse of the eco-system.
Island ecology has been compared, to rain-forest fragmentation, to find the relation between the size of an area and the number of species. It has been empiricly found that the number of species doubles for every ten-fold increase in area. Leaving isolated conservation areas, in a sea of agriculture, may not be enough to save many species.
In february 2004, the Royal Society for the Protection of Birds said $25 billion more a year was needed to establish a working system of protected areas for wild life. The record of the developed countries is “appalling” and they were just dragging their feet.
Also, there is a humpty-dumpty effect that prevents eco-systems being put together again, once disbanded. Jim Drake found it is not enough to co-habit a community of species again, in whatever order was tried. To reach a persistent state, an eco-system had to pass thru a whole range of stages.
Humanity is the greatest catastrophic agent since an asteroid wiped out half of earth species, sixty-five million years ago. This time the explosion is the human population explosion.
By not recognising other life forms, we are saying they are not important enough. It is a mistake of human pride before an ecological fall. Or, for that matter, a moral failure from a religious fall. Biologists, and the community of scientists in general, are having to become like Biblical prophets warning of the catastrophic crash awaiting the human population explosion. The eco-system may not be able to adapt in time to rapid global warming, disrupting its stability, on which survival depends, with chaotic and unpredictable results.
Not to mention the stock-piles for biological warfare, there are natural air-borne viruses, for which there is no known cure, virulent to humans as well as other animals. These natural threats exist in the wild, like pandemic accidents waiting to happen. Ebola belongs to the hanta group of viruses, causing hemorrhagic fevers that kill at rates of 80% in a few days. [This review was written a decade before a desperate ebola out-break in West Africa.]
This is what happened in 1545 in Mexico, leaving 12 to 15 million dead, after a great drought. (Ebola may have killed more than half the chimpanzees and gorillas in much of central Africa.) Virus carriers such as mice spread the disease as they concentrate at water holes. When the rains return, their population explodes and other species contract the disease from breathing the dust, where their droppings are found.
Thus, the unstable swings between drought and flood caused by global warming and deforestation may expose human and other life to the full force of such disease “time bombs.”
Scientists at the London School of Hygiene and Tropical Medicine (in the British Medical Journal, february 2004) said many animal experiments may do little to treat human disease. Much research is poorly conducted and evaluated and in need of systematic review before new experiments. This was a boost for animal rights activists. (A counter-attack, the same day, came from the Royal Society, but that isn’t speaking in a specialist capacity.)
In october 2003, the animal welfare group, Compassion in World Farming sought to have modern chicken breeding and rearing out-lawed by the High Court. Giving free range was the more humane and healthy practise of animal husbandry.
Not having the respect, to learn from lifes diversity, is arrogance, akin to ignoring human rights. Neglecting the quality of life and education for all species diminishes all humanity, by failing to promote all lifes potential, and forcing or habituating a parasitic existence on our fellow creatures.
Natural causes of extinction and their human promotion.
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The sixth extinction is a new kind of extinction in that it is caused mainly by the invasive effect of a single species (man) on others, under-mining the life-support system between life forms, that is the eco-system. Man is not the only threat to survival of life on earth.
Asteroids have been mentioned. They have certainly hit earth before with more or less devastating effect. Current technology could not prevent the more unfortunate scenarios of a major asteroid strike. The problems of interception were, if anything, under-estimated by recent disaster movies on the subject. Asteroids are hard to spot because small. Some are dark and simply may not be seen. Some orbits, a sling-shot from the sun maybe, make it hard to see them coming with enough notice. No doubt, granted technical progress, the situation would become more hopeful, when the next big strike comes.
Time is of the essence, with regard to another repeated natural catastrophe. The statistics of volcanic eruptions shows that Earth is over-due for the big one. That is an earth explosion, probably from somewhere in the Pacific Ocean volcanic rim, sending shock waves and tidal waves around the globe, and blotting out the suns rays with an ashen layer of clouds for years on end. Crops would fail and animate life starve, including an estimated loss of one billion of the earths current six billion people.
A volcanic eruption is believed to have almost exterminated mankind. Genetics show that the human race is all descended from no more than a few thousand survivors, from pre-historic times. The worst known explosion caused 100km crater, 74,000 years ago at Toba in northern Sumatra, enveloping the planet in a “volcanic winter.” The regularity of the geyser “Old Faithful” in Yellowstone National Park, has been compared to the ticking of a volcanic time bomb, geology shows is over-due, for one of its regular mega-blasts, that would put an end to much of human and other life.
95% of active volcanoes are by the sea or on island chains. Seasonal shifts in sea levels can stress the earths crust enough to raise the incidence of eruptions. Bill McGuire team of European scientists showed, in a study of the huge sea level changes of the last ice age, that more explosive blasts occurred when sea levels were changing most rapidly, either up or down. He warned that this was likely to be the effect of global warming, raising the sea level in the coming century. (Life. The Guardian 6 may 2004.)
Warmer oceans could also act to release the West Antarctic ice shelf raising sea levels ten times the predicted increase. The sheet only rests on submerged islands, and some are volcanic with clear water over them, whose eruptions could help dislodge the shelf. Stephen Schneider says it has happened before and could happen again but no-body is quite sure when. Often under-water, volcanoes or earthquakes or the masses, they may dislodge into the sea, set up tsunamis.
If and when this happens, Geohazards professor Bill McGuire (on 12 october 2000) said “the human race will face the greatest natural catastrophe in its history.” That is, presumably, unless an other geohazard gets there first.
The tsunami was popularised (if that’s the right word) in the SF movie The Day After Tomorrow.
Sunday Times reviewer Cosmo Landesman said:
Unfortunately, this film also wants to be a post-9/11 tribute to survival and the human spirit, when it should be an unabashed tribute to human stupidity.
The biggest danger may be from climatic chain reactions. In particular, natural causes of extinction could be promoted by humans. Obviously, if the Earth is filled to capacity with human populations, there is not going to be much room for millions of people to move away from disaster areas, thru flood, fire, drought, disease, crop failure etc. Even in a period of natural stability, human conflicts over territory are serious and threatening enough.
Another instance, of the need for territorial safety margins, is the natural history of sudden climatic changes, which puzzled palaeontologists. Evidence has correlated these changes with shifts in the direction of ocean currents. The warm waters of the Gulf Stream have switched many times from their crossing the North Atlantic Ocean round Northern Europe.
Over the recent past, a twenty per cent decrease in current speed has been estimated. Global warming is causing the Arctic ice sheet and Greenland glaciers to melt and swelling the great Arctic-bound Siberian rivers to dump huger quantities of fresh water into the stream. This slows down and makes sink earlier the heavier salty water warmed in the south. This would cut short the Gulf Stream conveyor belt motion that continually re-supplies the shores of north-west Europe with warm water.
This energy warmth is worth a million power stations output to the British Isles. Its loss would give the region a climate like Canada at the same latitude. A country like Ireland, which retains its reduced post-famine population from the nineteenth century, should be able to sustain its population under a greatly reduced growing season. That is provided the Irish population does not greatly increase. The French governments Napoleonic delusions are pursuing a subsidised population expansion policy, likely to prove unfortunate.
When the Gulf Stream might stop is not known. In 2004, on the BBC Horizon, scientists best guess was in maybe fifty years, possibly as soon as twenty years. They don’t know whether the change would be gradual or as without warning as a switch.
A sudden change would be beyond the capacity of vegetation to adapt. It would be a crash in life-support systems. Britains sixty millions or more people already cannot feed themselves. The loss of the Gulf Stream would surround the island with icebergs. Like The Titantic, sinking without half enough lifeboats, Britain would not have half enough arable land to support its population.
These climatic changes are not just local problems. Soil-depth readings have shown that direction changes, in the Gulf Stream and its warm moist air, have also coincided with the desertifying of the globes equatorial rain-forests, the oxygen-producing lungs of the planet. The Hadley Centre forecasts that global warming will kill off tropical forests to such an extent that instead of soaking up carbon dioxide they will add more to the atmosphere than all the power stations and cars of the past 30 years.
Climatologists believe that the switch to a runaway global warming that happened 55 million years ago may be repeated under present conditions, that threaten mass releases of methane, from under the permafrost of warming Siberia and from crystal structures on continental shelves destabilised by warming oceans. This caused a mass extinction comparable to the end of the dinosaurs, 10 million years previously. Ocean modeler, Stefan Rahmstorf says, more recently, North Africa turned from a swamp into a desert in a few years. (Fred Pearce, “Nature plants doomsday devices,” The Guardian, 26 november 1998.)
Lack of democracy results in injustice, ignorance and incompetance.
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As well as causing global warming, 200 years of fossil fuel energy has also supported a much larger human population than normal. “If you plot the logarithm of the body sizes of mammals against the log of the population density, you get an inverse relationship…that bigger animals occur at lower densities than smaller ones.” (Tim Radford, The Guardian, 22 july 2004.)
Instead of well over 6 billion humans, there would be one or two million, compared to roughly the same numbers of one or two varieties of chimpanzees, gorillas and orang-utan. There are perhaps more than 400,000 great apes but human population increases by that amount every two days. The great apes and many other species are being squeezed out of existence: "humans and their livestock now consume 40% of the planets primary production, and the planets other seven million species must scramble for the rest." That includes about 4000 types of mammal.
Fatal problems may be caused by one climatic change. But many such changes are possible, inter-acting in ways too complex to understand. One moral is that too many people don’t have enough room on the planet to save themselves, whenever nature comes up with unpleasant surprises. Tim Radford and Paul Brown say:
After years of argument, not least from the Bush White House, it is hard to find a politician on the planet who does not agree with these basic scientific facts and the danger that they pose. The problem remains getting the international political will together to do something about it – both to prevent the situation getting rapidly worse and coping with the problems we have already created.
Science or knowledge has more to offer than just ecology in promoting the political will to save the eco-system. The so-called political will may not be representative of the public will after informed debate has taken place. That is to say a dictatorship is more liable to make mistakes than a parliamentary democracy. Science and democracy, properly understood, are one learning process, by which genuine progress can be made.
The eco-system is already being recklessly ravaged by commerce and threatened by escalating wars. The US department of Energy figures show that the United States and Australia emit most carbon but many other countries are not far behind and catching up. The American “Union of Concerned Scientists” (www.ucsusa.org) has protested against government obstruction of environmental health research.
Tho the US and Australian governments defied the Kyoto protocol, the Russian government signed the treaty making up the number of industrialised nations needed for the treaty to become international law.
The Environment Agency (30 july 2003) said higher fines related to company turn-over and more prosecutions are needed to stop firms polluting. Some of Britains biggest and best known firms are repeat offenders: one-fifth of fined firms in 2002.
For decades farmers have been allowed to get away with spraying pesticides up to peoples hedges, because voluntary talks, to farmers not to do it, just don’t work. People are not informed and feed-back is lacking. Georgina Downs and family suffered twenty years of induced illness. She prepared a case costing her thousands of pounds and taking three years. The Sunday Telegraph (8 august 2004) reported that Britains minister for rural affairs was “satisfied that the protection afforded to the public was perfectly adequate…citing for his support the views of a man who had not seen the evidence which prompted the inquiry in the first place.” Also in august, a report came out on a big increase in brain-related illnesses such as Alzheimer’s disease.
These David and Goliath campaigns, by such as Miss Downs, are wholly admirable but no substitute for changing the constitutional rules of the game to an effective political and economic democracy. Indeed, such people would have a realistic chance of becoming representatives (with the voter-centered electoral system of freely transferable voting) as well as having genuine representatives of policy and economy, respectively in the political and economic houses of parliament.
The Bush presidency refused to sign the Kyoto treaty for globally reducing greenhouse gas emissions causing global warming. So, the Inuit people of the Arctic are making a pioneering effort to hold the US government legally responsible for violating their human rights. Another faraway people, marginalised by the march of “progress,” say “America’s refusal to sign the Kyoto protocol will affect the entire security and freedom of future generations of Tuvaluans.” In certain coral islands, little more than 20 inches above the level of the South Pacific, ever more often, the sea wells up thru the porous rocks to submerge their land. (Mark Lynas, [_The Observer _]5 october 2003.)
Worldwatch Institute of US researchers say one quarter of the world population have entered the consumer class and enjoy life that used to belong to the rich. This will soon include more people in China than the USA. With it come the draw-backs of the Wests poor quality of life, the impoverished and polluted environment, unsustainable devouring of natural resources, stressful demand on time, diet and transport problems. Just one in three Americans say they are very happy, the same as in 1957, when the US was half as wealthy.
The king of Bhutan seeks not to promote gross national product but gross national happiness!
Tokyo United Nations University, in 2004, warned of a warmer and wetter world with more storms, rising sea levels, deforestation, increasing population. In 50 years, this should see major flooding affect twice as many, or 2 billion people. The UN World Water development report, in march 2003, predicts 7 billion could face water shortage, on average a fall of one third over 20 years. Every day, 6000 children under five die from diseases linked to dirty water.
The Democracy Center letter from Latin America explained how a transnational firm privatising water resulted in charge hikes that the poor could not pay, causing a revolt that was put down with casualties. The ousted transnational then moved a multi-million lawsuit against the third world country. Papers leaked to the BBC (25 february 2003) suggest the European Union is pressuring some of the worlds poorest countries to let multi-nationals take over their basic assets, privatising water and electricity companies. The World Development Movement says this is their real intention, despite EU claims they do not want to privatise state-owned firms.
16 october 2003, World Food Day, the UN special investigator, on the right to food, reported that the number going hungry increased, from 815 million in 2001, to 840 million in 2002. A child dies from the effects of hunger every 7 seconds. Every 4 seconds, someone goes blind for lack of vitamin A. This is an “outrage” in a world with enough food. In the same month, UNICEF reported that one billion children in the developing world, more than half its population are severely deprived. 647 million are in absolute poverty.
Perhaps, the greatest wasted resource is human intelligence. With global electronic communications, it would be possible also to educate every child thru media like the internet. They would have ideas we never thought of and skills we lack, that would be invaluable to all mankind.
In april 2004, BBC World tv asked 1500 viewers the most important problems: 52% said US power and large corporations; corruption; 50% wars and terrorism; 49% hunger; 44% climate change; 38% illiteracy. In march 2004, the campaign group Global Witness said laws should compel firms to disclose pay to governments. There is a global epidemic of financial scandals. Billions go unaccounted-for in some of the worlds poorest countries, especially in Africa. Collusion, of oil and mining companies with governments, for rich natural resources, acts like a curse to keep the locals in poverty.
As the poor are left in more poverty, the rich seem to attract riches. A whistle-blower alleged bribes paid by Britains biggest defence company to rich buyers to win big contracts. ([_The Sunday Times _]25 july 2004.) Since 2002, departments and enforcement agencies received more than 20 such allegations of corruption over-seas, albeit that may be the way things are done with the customers in question.
A G8 Summit of leaders from the most powerful national economies, to discuss Third World poverty, was conducted on “European Vision” – not a policy but a luxury liner. The previous year bill was £500 million. The cost for 2001 could have been far in excess of £100 million. Gordon Rayner (Daily Mail, 21 july 2001) reported:
This is the equivalent of the combined annual national debt repayments of Malawi, Mali, Mozambique and Burkina Faso. It is double the entire health budget of Tanzania, which has a massive Aids crisis – one of the subjects on the agenda…
Recent figures suggest that if all Third World debts were cancelled and repayment money was spent on healthcare, clean water and education instead, the lives of 19,000 children per day would be saved.
Aid workers have found out that such a beneficial switch of resources does not happen sometimes without making themselves highly unpopular and getting thrown out of work. Peter Griffiths, working for the World Bank, found that a free market model economy was being imposed on poor countries such as Sierra Leone that were not ready for it. Seven months previously, the World Bank had forced the country onto a floating exchange rate, which collapsed the value of its currency. Free market traders would not import rice that people could not pay for, even when the currency was ten times its current value. The withdrawal of rice subsidies would lead straight to famine.
Moreover: “Third World governments frequently fire consultants, saying that they are incompetant or that they cannot get on with the locals. The real reason is usually that they are about to expose corruption, or the misuse of aid money.” Aid organisers, politicians, civil servants, marketing board officials earn personal commissions from the buying and selling of grain in a famine.
It’s so much easier to blame a solitary whistle-blower. The country cannot afford to fall out with some wealthy global organisation. Griffiths ([_The Observer _]31 august 2003) remembered:
I knew – everybody in the aid industry knew – that only five years earlier Steve Lombard had prevented a famine in Tanzania. He had had to put all he had into this, tapping all his contacts around the world, because officials refused to act. The Tanzanians insisted that the United Nations Food and Agricultural Organisation fire him. FAO, the World Bank and the aid community did nothing to protect him. He was indignant, furious, betrayed. Over the next three years he drank himself to death.
The disasters that dictatorships bring upon people are well documented. A recent example, which confirms a depressing trend thru-out the world, was President Suharto planning to make Indonesia self-sufficient in rice again. He ignored scientific advice that using canals to drain Kalimantan peat swamp forest would be “an ecological and economic catastrophe.” (Fred Pearce, “Borneo’s chainsaw massacre,” The Guardian 18 february 1999.) Forest clearing was controlled by “Mafia-style organisations,” the chainsaw being “a license to print money.”
The results were uncontrolled fires spreading especially along the dried canal banks – there was a mass wearing of smog masks – and uncontrolled floods, that drowned plants and lost livelihoods, and endangered the main habitat for the orang-utan. Also at risk are sun bears, clouded leopards, 30 other mammal and 150 bird species as well as “plants and fish seen nowhere else.” With the collapse of its currency, Indonesia has been selling off its priceless natural assets. The release of the carbon from tropical peat swamps could add critically to global warming. And the peat has no minerals to grow rice.
When Suharto came…to ceremonially harvest the first rice crop, nothing had grown. So officials transplanted rice from elsewhere to fool him.
That is the classic consequence of autocratic rule. The intolerance of opposition and criticism is compounded by fear to admit to “the great teacher” that his plan has gone wrong. His underlings don’t want to be punished for incompetance, that he would never admit was his own.
This is an old story. Max Prangnell, Cal McCrystal and Hege Duckert reported (in The Sunday Times, 11 september 1988) reported that
a combination of human ignorance, greed, poverty and inertia has thrown the (Himalayas water) machine out of control. Growing populations with an increasing need for food and livelihood are stripping the forests from the habitable areas on the southern slopes, increasing the frequency and devastating power of the floods…
Some environmentalists claim that at least one plant or animal species becomes extinct every half hour… Tropical forests are the main dispensary of raw materials for medicines. One recent study, for instance, showed that 70% of the 3000 plants identified by the US National Cancer Institute as having anti-cancer properties come from rain forests.
Lost and forgotten Amazonian civilisation, wiped out by disease from Iberian conquerors, was recently found to have developed a renewable agriculture. The lack of this, in the past few thousand years, has been largely responsible for the relentless desertification of the planet from the early middle eastern to modern western civilisation. Found around formerly settled “jungle,” “Terra preta” or “dark earth” is, unlike the usual yellow earth, mixed with organic semi-burnt charcoal, retaining minerals during rains.
Unlike the ruinous nomadic slash and burn agriculture, terra preta is thought to have a bacterial basis that allows it to reproduce itself from leaf fall 20 years after being mined. BBC Horizon (20 december 2002) says this property is being researched to produce sustainable agriculture in the third world. They might have added the old and new worlds as possible beneficiaries.
It would be wise, as well as just, to educate the skill and ingenuity of the whole world: “educational democracy” if you like, as well as political and economic democracy. Power and wealth, controled by the few, promote ignorance of the needs and abilities of the many people, who would confer greater benefit to all.
autumn 2004
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The political system fails the eco-system.
Over thirty years of Green warnings and the hope for grass roots reforms.
Table of contents.
Links to sections:
The Club of Rome and The Limits To Growth. (From 1968)
Paul Harrison: The Third World Tomorrow (1980). And their “brain drain” today.
The illiterate English alfabet, illiteracy and lawlessness.
The Mad Officials.
The world is dying. What are you going to do about it? Sunday Times magazine (1989).
How to save the earth. Time magazine (2000).
In april 1968, a meeting was convened by Dr Aurelio Peccei, that was to be called the Club of Rome. They had no shared ideology. But all believed that current institutions and policies could not cope with “the present and future predicament of man.”
The world-wide stir created by the Club of Rome report, The Limits to Growth, reached even into Solzhenitsyn Letter to Soviet Leaders. There he prophesied that the huge arms build-up would all have to be scrapped and was an enormous waste of resources.
In July 2001, President George W Bush offered an agreement with President Putin to reduce somewhat their still over-whelming nuclear missile arsenals of about 10,000 war-heads each.
But the Bush administration caused concern with its treaty-breaching missile defense program and unwillingness to come to terms with the Kyoto agreement on limiting global warming. This American president was felt to be an oil profits leader, rather than having energy-efficient and pollution-minimising policies.
In 1972, the executive committee of the Club of Rome commented:
Short of a world effort, today’s already explosive gaps and inequalities will continue to grow larger. The outcome can only be disaster, whether due to the selfishness of individual countries that continue to act purely in their own interests, or to a power struggle between the developing and developed nations. The world system is simply not ample enough nor generous enough to accommodate much longer such egocentric and conflicting behavior by its inhabitants. The closer we come to the material limits to the planet, the more dificult this problem will be to tackle… The last thought we wish to offer is that man must explore himself -- his goals and values -- as much as the world he seeks to change. The dedication to both tasks must be unending. The crux of the matter is not only whether the human species will survive, but even more whether it can survive without falling into a state of worthless existence.
Long after publication, The Limits to Growth was given a working-over for its pessimism as to the amounts of non-renewable resources waiting to be found in the ground. Also belabored was the crudity of the socio-economic feed-back model, using the more limited computational resources of the period.
Such criticisms were anticipated. And the analysis served as a prototype for Green politics. A Blueprint for Survival also carried graphs of up-curves of pollution and down-curves of non-renewable resources. This manifesto coincided with the launch of an Ecology party, in the UK, which later followed the German example, by re-naming themselves the Green party.
The launch into conventional politics has been moderately successful. And there are already signs of green politics being neutralised by power politics, as happened to the socialist movement.
With justice, the general public seem to have little faith in the political system. Environmental organisations have replaced political parties for mass membership.
The parties, competing with each other to represent, are really monopolists, between themselves, of representation and therein is the disillusion with politics. They have robbed Parliament of its role as the nations decisive forum. This is widely perceived. Anthony Barnett, a deputy editor of Labours The New Statesman attacked the “…contempt in which the new governing elite holds MPs.” He cited how the PM “lectured” MPs that they were not in parliament to have ideas of their own but to follow party policy. This was in a Daily Mail article (19 february 2000), The Death Of The House. Under Mr Blair Parliament is an irrelevance and MPs are little more than a joke.
Nothing could be more revealing of how right-wing New Labour is as intolerantly doctrinaire as the old Labour Left. They amount to nothing more than a conspiracy of antagonism, even if it is partly a self-deceiving conspiracy, of which left and right may not themselves be fully aware. As with left and right wings within parties, the same applies between right and left wing parties, which merely leave the voters to choose either side of the same old authoritarian coin.
The disregard for parliament was trumpeted by Tony Blair announcing the British general election, not to the House, but to a childrens school. One of his ministers merely said it was “odd.” The Tory chairman spectacularly missed the point by criticising Blair for bringing children into politics! Those comments in themselves reveal how little esteemed are MPs.
Odd indeed! It is inconceivable that one of the great parliamentarians would have committed such a breach of courtesy, as if he were the only pebble on the beach.
Many campaigners turn to publicity-seeking action, for which they hope to secure popular approval and oblige the government to follow their lead. Democracy has been forced into some tortuous and dubious channels of expression.
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Inside The Third World, Paul Harrison pilgrimaged thru poverty. His 1980 sequel gave examples across the globe of how the poor are trying to pull themselves up by their own boot-straps. As a matter of fact, this old saying doesn’t apply, because the world poor are usually too poor to have any boots to strap.
Only the elites of third world countries can afford to be served by Western-style well-heeled professionals. The insistence on only the best standards of service obliges the world poor to go without, indefinitely.
Harrison discerns a change from this all-or-nothing approach. Hence, the rise of the bare-foot professional: the bare-foot business-man, the bare-foot doctor, family planner, township planner, ecologist, literacy teacher, and intermediate technologist inspired by Ghandi and Schumacher, etc.
Instead of pouring money into the bottomless pits of prestige projects, aid could be put into local self-help projects. A West African village is taught an Asian-style rice-growing project which happens to suit its particular ecology. Local artisans or black-smiths may be re-trained in relation to sometimes prefered factory products for agriculture or industry.
Traditional healers may be taught the basic lore of modern medicine, as summarised, for the bare-foot doctors of the Andes, in a 110-page Health Promoter’s Manual, using simple language with illustrations. Paul Harrison says:
The science of medicine itself has to be decolonised, de-mystified, de-professionalized. A new appropriate technology of simplified medicine has to be developed: low in cost, easily mastered by ordinary people, using local resources wherever possible and drawing on those traditional methods that are known to work. To quote Halfdan Mahler… “We must break the chain of dependence on unproved over-sophisticated and over-costly health technology,” and evolve an “essential health technology, a technology which people can understand and which the non-expert can apply.”
In 2001, the British medical profession deplored the continued influx of skilled immigrants to bolster the over-stretched National Health Service, because it deprived poor countries of their training.
The Daily Mail (21 july) says 15,400 British nurses are set to qualify in 2001. But this will be exceeded for the first time by the number of over-seas nurses recruited. A new high of 50,000 over-seas nurses will staff British hospitals, as a result of government recruitment and increased applications from abroad.
South Africa, Ghana and Jamaica have protested against the NHS “hoovering-up” their nurses.
An agreement has been signed with India, to siphon off their “surplus” of nurses. 6000 Indian nurses will earn far more money, some of which they may send home. Some may return to their home-land with greater expertise. This level will surpass the current highest number of applications from the Fillipines.
As with nurses, so with teachers, in 2001 Britain had the biggest shortage in 36 years, with 5000 vacancies expected. Moreover, head teachers said they were unhappy with perhaps 6000 accepted teachers, in England and Wales.
These short-falls are as nothing compared to the situation in India, South Africa, Namibia and Nigeria. Voluntary Service Overseas has accused Britain of “looting” teachers from developing countries. VSO chief Mike Goldring said:
Try telling the 40m Indian children with no access to education that British children are more deserving.
Harrison and Mahler egalitarian policy is also needed in the over-developed West.
The bulk of all our needs are basic needs, which may be met by basic solutions. The needy, themselves, are most of all in need of a no-frills service in every department of their lives.
Paul Harrison talks of educational experiments, often opposed, that cut out the “academic twaddle” for things people need to know. Growing a row of organic vegetables might be more healthy than too much homework.
Children of rich, as well as poor, countries might be taught the basics of first aid and hygiene, as part of “the national curriculum.”
The illiterate English alfabet, illiteracy and lawlessness.
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Schools could have a spell-as-you-speak rational English alfabet, of about the existing 26 letters, to abolish the functional illiteracy rate of over twenty per cent. Adam Smith said the professions are conspiracies against the public. In this respect, the literate may be the biggest closed shop of them all. Perhaps seventy per cent of the world population is illiterate. Equality starts here.
The lack of respect for democracy may be discerned in this basic issue. Literacy, equally available to all, partly depends on the liberty to spell rationally, and its fraternal tolerance by those who can spell conventionally. The testing for conventional spelling trivialises literacy teaching. It professionalises the preaching of mindless conformity, in the way we spell.
Perhaps, school testing in general has more to do with expensively promoting unquestioning mediocrity than anything else. Of course, this has been said by a “school” of radical educationalists. In 1969, Neil Postman and Charles Weingartner made their case with humor and humility, in Teaching as a subversive activity.
Literacy is the foundation of all the specialist forms of knowledge that the professions govern. Exclusive preserves might share their knowledge, at least of the essentials.
European Union teachers can no longer use the stick or cane on their pupils to enforce the will of the system. But the system enforces itself on the teachers, who are tested as much as the pupils, to see if they make the grade.
It is as if the examiners of orthodoxy fear any creative lack of conformity so much, that they must purge it in the teachers, as well as their pupils.
In 2001, a University study from Ulster (a province with the highest academic standards) said up to fifteen per cent of children are "functionally illiterate." Other studies have put the rate at over 20% for adults. As Huxley wrote, in Brave New World, the system is not made to suit the people. The people are forced to suit the system. The real source of functional illiteracy is not so much in the teachers and the taught as in the insistence on our “functionally illiterate” English alfabet.
In the first place, it is our unreformed alfabet that cannot spell properly. That is where the blame really belongs and with the prejudices that refuse to admit it. To put the blame on “poor teaching” is a mentally lazy excuse to do nothing to intelligently reform English spelling. Its absurdities are a convenient habit for the complacently literate, who don’t care about the trouble it causes the “illiterate” and the educational and economic inefficiency it is bound to generate.
For failing to see the real cause of illiteracy, throwing money at the problem will not solve it either. In september 2000, the Scottish executive allocated £22.5m. to end adult illiteracy within ten years. But the United States threw a mountain of money at illiteracy to no noticable effect for “the Great Society.”
It reminds me of many an old movie, of my childhood, that had a fortune spent on the costumes, sets and casting but neglected to find a decent script.
Teachers report that seriously disruptive pupils are often covering-up for poor skills. Indeed, as many as 60% of prisoners, in England and Wales, are illiterate.
Moreover, the unruliness of children is unfair on teachers. The right to children of freedom from fear (one of Four Freedoms by Franklin Roosevelt) should be shared by teachers. Children have responsibilities, as well as rights, which they must learn the sooner the better. This implies practical education of young people in the law, with childrens courts. (The amateur lawyer is the subject of a sequel chapter.)
This is an example of the need for education to teach youngsters how works the world they are going out into.
The Mad Officials.
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In 2001, the teachers were so over-burdened with the latest testing regime that they simply declared it unworkable. But the teachers are not the only demoralised profession. The home secretarys promise to reduce paper-work was received with a slow hand clap by the police federation. It drives experienced officers out of the force. Doctors, too, have a crippling load of official documentation to complete. The medical profession delivered an out-spoken vote of no-confidence in government reforms, at the time of the 2001 general election. The British Medical Association balloted the 36,000 General Practitioners. Of the two-thirds that answered, 86% said they would be prepared to resign, "unless ministers cut bureaucracy and give them more time with patients." ( Daily Mail 2 june 2001.)
At present, in Britain, the professions command the heights of a status society, with high inequalities of income in their favor. On the re-defined National Socio-economic Classification, The Daily Mail (17 march 2001) captioned: “The New Pecking Order…Do you know your place?” The emphasis was on grading occupations according to contracts, conditions, prospects and security.
More important is to define constituencies of work for their coherent role in the functioning of society. Instead of a pointless pecking order, there should be feed-back to the elected representatives of those vocational constituencies to the second chamber of government.
BBC Ceefax (26 july 2001) reported Management Today saying British bosses are the highest paid in Europe, by more than £100,000. Chief executives earned over half a million pounds, an increase of 29% since 1999. Only US bosses earn more, with average salaries of £1m. But British manufacturing workers are the lowest paid, and the cheapest to dismiss, in the developed world. At £20,475, they are below the national average wage, and they also have put more time in than most of Europe. They are still incredibly rich compared to the rest of the world, four-fifths of whose people dont earn any money at all.
A carrot and stick economy weildss the “carrot” of plutocracy with the “stick” of bureaucracy.
Business has also long complained about being tied up in red tape. In 1993, the Single Market imposed a huge burden of 218 harmonization directives, which, in many ways, left the level playing field as far away as ever. So says a book written on the follies of the administrative laws of the European Union and their excessive and ritualistic, rather than realistic, implementation by civil servants and inspectors in Britain.
A “checklist mentality” reeled-off all the points they’d been told to look out for, at college or seminar, demanding thousands be spent, and forcing shops and businesses to close down.
Yet, in this pre-occupation, inspectors lack of experience might lead them to over-look real risks posed to sought-after objectives of hygiene, safety, conservation, institutional caring or whatever.
The Mad Officials (1994) by Christopher Booker and Richard North gets its title from an essay by G K Chesterton, so quoted:
Booker and North said:
wherever the monster (of bureaucracy) impinged on the real world, it invariably had the same effect. It threw out clouds of deadening jargon; it tied people up in absurd paperwork and form-filling; it made ridiculous demands; it asserted its power in a blind, wilful way; it crushed enterprise and independence; at worst, it turned far too many of those who fell under its sway into nothing more than uncomprehending and often fearful victims.
There is a way out from the carrot and stick of plutocracy and bureaucracy: democracy, in the economy as well as the polity. It should mean greater economic equality and fraternity, as well as greater freedom from officialdom, for all classes.
The Parliamentary laws and administrative laws could be checked by a second chamber, representative of all occupations. This could redress the excesses of official administrative chores delegated to the public and private sectors. The occupations themselves, in concert with each other, must know the needs of their own work best, subject to the first chamber, the Commons, representing the interests of communities as a whole.
The closed shop, of the unions, was out-lawed by the European Union. But the professions, also should be more open. Their basic knowledge and most essential skills should be broadly based in the population, either thru a more practical general education or by a part-time work-force of trained amateurs on a basic income.
In 1980, Paul Harrison said “Reform will not be a Sunday school tea-party.”
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As prime minister, Mrs Margaret Thatcher once grouped Green activists among “the enemy within.” It is thought that Prince Charles made her more aware of environmental issues. At any rate, she changed her mind about including Britains antarctic survey vessel in her current round of cuts. The ship sailed again to discover a hole in the ozone layer over the south pole.
The PM convened a conference on ozone layer depletion, which would make mankind more vulnerable to skin cancer, if we continued to fully enjoy our freedom merely to walk in the sun. To mark the occasion, The Sunday Times decided it was high time its readers all woke up to the folly of destroying our eco-system:
We are all polluters on this planet. We burn fossil fuels, we create waste, we ravage natural resources with little or no regard for the consequences. But time is running out. Our planet is becoming despoiled, rotten, overcrowded and barren. We could all be contributing to the causes; we will certainly all suffer from the effects.
The magazine focused on chemical spills into air or sea, killing people or marine animals, by the thousands or scores of thousands. Or the systematic pumping of factory wastes into rivers and seas, such as the North Sea and Mediterranean. Poisons are dumped on other peoples door-steps, or dump-ships used, even if illegal.
The magazine mapped deforestation and over-population, with a global sample of some of the more out-rageous and life-bereaving pollution disasters. Richard Mabey did an article on “the roots of civilization:” trees are the pillars of green society. After citing Europes tree-intolerance, he described the white North American settlers destructiveness as “pogroms of an arrogance and violence that rival those in modern Amazonia.”
“The burgher that ate a rain forest” summed up the fact that “It takes 55 square feet of rain forest to raise enough beef to make a single American hamburger.”
Still fighting a losing battle are the re-foresters. Some of their work was featured, especially Vietnams national effort and that of the World Wide Fund for Nature (WWF).
In 1969, The Sunday Times magazine disclosed the exposing of Brazilian natives to disease, under the caption of “Genocide.” Survival International was founded as a result. Its director Robin Hanbury-Tenison gave one of the most closely written articles in the magazines 1989 green issue, about the continued persecution and betrayal of the natives, “whose understanding of the medical and nutritional resources of the rain forest is unrivalled.” Their land continues to be ruined, as shown in the familiar pictures of deserts of tree stumps.
The author put responsibility on 300 or so banks, trying to re-coup Brazil debts. Also, land reform is resisted by five per cent of the people holding 80% of the land.
Another sample of “greed, corruption and political ambition” featured tusk poaching and the threatened extinction of the African elephant.
The Sunday Times “The world is dying” rounded off with a survey of the Green campaign from such as Friends of the Earth, Greenpeace and the WWF.
(Not to forget an amusing, but serious, after-thought article on “poop-scoop” laws for dog-owners.)
How to save the earth. Time magazine (2000).
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Time magazine “Earth Day special edition 2000” has a good sample of spot-light articles, as one would expect. Besides being recent, at the time I wrote this, such a publication has its professional on-line counter-part. So, I confine myself to a brief discussion, here.
Time magazine established an Environment section in august 1969. That is little more than a year after the founder members of the Club of Rome met. In 1970, they covered Barry Commoner. His book, The Closing Circle. Confronting the Environment Crisis came out in 1971. (A new magazine, The Ecologist, devoted several pages to panning the book as “one-dimensional ecology.”) Commoner reviewed how the 1963 limited nuclear test ban treaty came about:
This unexpected event was a tribute to the political effectiveness of the scientists’ campaign to inform the public about fallout.
Radio-activity could be carelessly spread, while an information black-out was effectively imposed, as in war-time. PM, Harold Macmillan suppressed the truth about Britains first major radio-active leak from a nuclear power station. He feared the public would turn against nuclear power.
If so, it wouldn’t be the first time the intuition of the “lay-man” was more reliable than the experts.
By millenium end, the rape of the planet goes on and an information war or propaganda goes on to excuse it. Looking back, it has to be admitted that the media have informed the public reasonably well. In the early seventies, I once remarked (by letter maybe to an editor of The Ecologist) that there seemed to be more environmental stories. I was told that I was right, because a group of journalists had got together to promote such news.
The media can mobilise opinion, as well as neutrally inform the public. Time magazine honored “Heroes for the Planet,” sponsored by The Ford Motor Company which advertised its environmental credentials, in the Time Earth Day 2000 edition.
American individualism may be responsible for the cult of heroes. As CG Jung said, great historical events are profoundly unimportant. The individual is not only the passive observer and sufferer of events but the maker of epochs.
The Sunday Times magazine, in 1989, was equally bent on reform. But it appealed directly to everyone: What are you going to do about it?
Readers were not given inspiring role models to emulate.
Paul Harrison gave examples from the third world of remarkable individuals. Granted, that the inspiration was of a more social emphasis. Community self-help organisations, with some expert and financial aid would start improvements, supported by further consultation and co-operation.
When you look at all three approaches to saving the planet, they perhaps all have one thing in common. They are all attempts to stimulate change largely from outside the system. In that respect, they all agree with the Club of Rome initiative.
The establishment has got us into this mess and has to be disestablished sufficiently to get us out of it again.
“Business as usual” depends on promoting wasteful “getting and spending.” This conflicts with advert-dependant editors exhorting and mobilising ordinary people to be conservative of resources.
The mass media are also a part of the establishment, who know the rich and powerful personally. And there is perhaps some ambiguity in their minds. Do they really want to change the system enough to make the public interest effective?
Since Randolph Hearst, the media have short-cut between the people and their official leaders of parties or industry. If lawless means, sometimes, were employed, they became possible as democracy was proving to be not nearly as representative as it should be. To amend that, requires, at least, a knowledge of democratic voting method and an extension of constitutional politics to economics, with occupational representation.
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The amateur lawyer and open source software.
Table of contents.
Section links:
Other professions needing amateurs.
“the law’s delay” and the firms delay.
Pit-falls awaiting the amateur lawyer.
Costs awaiting the small claimant.
Open source software.
Other professions needing amateurs.
This is not a discussion for the experts but for people who know as little as me, if possible. Paul Harrison, in The Third World Tomorrow, showed why the expensive expert must make way for amateurs sufficiently trained to meet the essential needs of all in society, not just a rich elite. If they don’t, relief for the very poor is postponed indefinitely.
Harrison also pointed out the lessons for our over-developed Western models of society. Besides his book, I reviewed the relevance of general training in essentials to Britains over-strained health service and education system. Also, these and other public services, as well as businesses, are beleaguered by bureaucracy.
This chapter gives two more examples (computer programming and legal redress) of the general need to make practical knowledge more freely available.
“the law’s delay” and the firms delay.
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The Law has been called “the oldest closed shop of them all.” (This may have been the caption to a Telegraph supplement, as far back as the late 1960s.) For instance, there were complaints about the minimal redress from the Law Society of complaints about members of its profession.
A century earlier, in Bleak House, Dickens frankly stated that the cause of “the law’s delay” (as Shakespear called it) is that it pays.
Henry Fielding was a majistrate as well as a novelist. In Tom Jones, he has a character foolishly desire to sue, till her husband remonstrates that it would put a job the way of their lawyer relative, but it wouldn’t do them any good.
Nowadays, Britain has a “Citizens Charter” to ensure standards of civil service. You would think scenarios, like the following, might be avoided.
Your small claims case eventually comes up for trial, after some few months. You hope it won’t take more than an hour, at lawyers charges. But the judge decides it will take longer. The judge adjourns the case, for a later date in the crowded court time-table. On the second hearing, the judge can’t decide to give a verbal verdict in court. A good while later, a written verdict is made. You fail to get the refund, you already took several months trying to obtain from the firm.
The judge gives the defending firm yet another chance to make amends, on their terms. The firm is so big and busy that they fail to collect the product for re-servicing. The defendant had said they would do the servicing in their own time. They took so long, they’d evidently over-looked you.
You go back to the court and claim the cost of servicing, to be done by some other firm. The judge, at least, has allowed for this in his verdict. It costs you another fee. You claim this cost, too, which wakes up your seller vehemently to refuse to pay it, while promising to collect the product for re-servicing.
Meanwhile, you are put under pressure to give-in to the seller, because their local court has not collected the payment from them, such as the verdict gave you a right to, if the seller didn’t service the product.
The claimants local court told you, the claimant, to come back if you didn’t receive the service from the defendant. But when you return to your local court, you are told, instead, to write to the defending firms local court to collect the default payment for defendants failure to service.
After several letters and months later, the defendants local court tell you, the claimant to go back to your local court to make the request. Only on the prompting of your local court does the defendants court pay the verdicts default payment, plus court fee for having to demand it.
At some point in this run-around, that the courts give the claimant, a staff member apologises, tho she is undoubtedly one of the few people in the whole dismal affair, you find no fault with. However, some man doing business with the court, at the same time, over-hears, and not minding his own business, puts in a good word for the court system. His dismissive manner, about any need for the court to apologise, suggests a cosy relationship with them. No doubt, some people do very well out of the system, if not those who it is meant for.
We are talking about eighteen months failed attempts to get a resolution to a sub-standard sale, either on ones own terms or even the defendants terms.
Pit-falls awaiting the amateur lawyer.
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Traditionally, the law has been regarded as the friend of the rich. At the age of no more than five, when the family car was wrecked by a van ignoring the right of way, with snow and ice on the ground, I said: Why don’t you get a lawyer?
My father replied: You never want to have anything to do with lawyers.
Eventually, the law was changed for the less well-off to take minor complaints to court, without incurring heavy costs. Or that was the theory of “small claims courts.”
[Since writing this essay, I have become so used to the majisterial pronouncements of Judge Judy that it is hard to believe anything wrong about them.]
In my second letter to the Wakeham report, an awareness, that court procedure was not always for the best, formed part of a case for constitutional reform.
Here, I wish to warn the ordinary citizen of possible pit-falls to being ones own civil lawyer, say, in seeking recompense for faulty goods or poor service, of the kind brought up on the consumer watch-dog television and radio call-in programs.
The Daily Mail ran two online computer issues on the subject of: Does anyone care about the poor customer? The press depends on lavish advertising by chain retailers, who may have been investigated, more than once, as a monopoly. So, this verdict, after a flood of consumer sob stories, was not idle.
Supposing one, whom the gods would destroy, is mad enough to risk taking a claim to court. There are several things to consider before one begins. Firstly, is ones claim just? Was the person or firm, one is making a claim against, given a proper chance to make redress or compensate one or make reasonable amends? Does ones grievance really demand the drastic step of going to court? In general, has one got a good conscience about ones business dealings? Has oneself been fair in ones dealings with the defendant, one is claiming against?
To test ones claim, one can seek an appointment with the Citizens Advice Bureau. If they are not impressed, a judge or arbiter is unlikely to be. You may also learn, there, what law to plead with. In the first instance, advice may be about getting the firm to act on your complaint without necessarily having to go to court. Other help may get you to marshall your arguments, making them forceful and to the point.
(Much to the advisers displeasure, the courts may try to use them as unpaid lawyers for the citizen claimant, facing the professionals in the small claims court.)
Is there a law that unambiguously says claimants, or plaintiffs, are within their rights, say, to return sub-standard goods or have a refund for bad service? If you are taking on some big multiple firm, they will have a department of lawyers, just to cope with people like you. You have to give them proper notice that if they don’t give customer satisfaction, then you may take legal action.
They may not settle. Their reply may state the relevant Act of Parliament that covers your case. This may be a consumer protection act. But you don’t take their word for it. Remember, your opponents lawyers are the professionals and know all the tricks. You are the amateur, whose first time mistakes can and will doom your case. Instead, you’ve contacted the Trading Standards Officer. After you’ve explained your complaint, he may tell you, even before you ask, the exact act upon which your affidavit or formal complaint must be based.
You may find that the most relevant act is an amended version, more strongly in your favor than the original act. So, in this instance, you would follow the trading standards officers advice of submitting your affidavit under a given consumer protection act as amended. This amendment might involve giving the customer a longer time to return faulty goods, whose state is not apparent at first.
Of course, your big corporate opponent knows all about this. As a matter of course, any complaints you make back to their store may be met with frustration and delay. The longer you can be kept with the product, you are unhappy with, the more your consumer rights are eroded. The worse the service, the less strong your legal position to claim redress.
The firms staff may be nice enough people but, as just one more customer, you may be fair game in the battle to maintain their turn-over, profits and jobs, to earn the living we all seek.
Perhaps one of the worst mistakes of the amateur lawyer is to assume that all you have to do is present all your arguments and the judge will see you have an over-whelming case. More likely, he will be over-whelmed by all the verbiage and miss the most important points, in his decision, when it is too late to correct him.
Judges have mountains of evidence to traverse in their jobs and cannot be expected to remember everything about your little grievance. You have to guide the judge on the best trail thru your case, so he doesn’t lose his way to your main points.
The lesser points can be appended to your main statement, in case they are needed to answer questions put by the judge or defending lawyer.
Costs awaiting the small claimant.
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Having searched ones conscience whether one is really in the right, just as crucial is whether one has the evidence to prove it. If one cannot make a convincing case, there is no point in wasting ones money prosecuting it. Don’t be beguiled by small claims court leaflets saying one doesn’t need the kind of cast-iron case required for a full court of law.
More to the point, the leaflets insist that, for any technological product which fails to meet standards, the plaintiff or claimant will need an expert report made independently of ones own influence. It musn't be from family, friend or employee. This is expensive. In the late 1990s, you would have to pay a computer expert £40 per hour for the report. A richly paid English judge may think you should have had him on hand in court -- still at £40 per hour. If you wanted a lawyer to handle your case, that was then £80 per hour.
You will pay the going rate, anyway, for hiring the judge. If you lose the case, that fee is forfeit. In answering your summons to court, the firms lawyer may ask the judge for compensation of at least that amount, at the claimants expense.
Well might you say: I thought this was supposed to be a small claims court!
That is not the end of the expenses, the small claimant may be up against. There is no rule that the big firm may not bring its own employed expert witness. There really does seem to be one law for the little man and another for big business. A few firms so dominate the market, it hardly seems worth going to another ones expert staff. None of them are likely to side with the public.
The claimant may consult a small independent business for expert advice but it is not in their interest to go against the giants of their industry. Even so, before the case comes to court, the defendant may try to dissuade the claimant from seeking independent advice, expecting the customer to trust their own qualified employee. And that, despite the fact that the law denies the dissatisfied customer any use of dependants or associates for expert advice.
It is like a battle of wills, in which the defendants try to take over your case and conduct it, on their own terms.
The firm also sends its own lawyer, who asks the judge that travel expenses of up to several hundred pounds also be awardable to his firm against the claimant. Presumably, the courts know that if claimants had to pay these travel expenses to go to the firms local court, most people far from London, or wherever, would waive their statutory rights. The courts would lose much business and justice would be seen not to be done but to be localised.
The defendant firm may even use your case as a chance to run in a trainee lawyer, scribbling down the proceedings, as if his livelihood depended on it. That is three against one in the court room and your opponent is a provider of employment to the system you are being adjudicated under. The firms lawyers are one of them, as far as the profession of judges is concerned. Moreover, the firms expert is another professional, whose opinion might well also be deemed of more weight than the amateur claimant.
In all probability, the judge admittedly knows nothing about the technicalities of a given case. He readily turns to the only expert on hand, that of the defending firm, which may have been the way they wanted things all along. It may be that the claimants mass of evidence simply does not weigh, in the judges mind, against expert testimony. However much the defendants expert may stand on his dignity, the claimants case becomes only as good as his opponents technical witness allows it to be. Like a politician, he has to decide whether or not it is prudent to buck his firms party line, at all.
What claimant would wish to so put himself at the mercy of his opponents?
We live in a culture of professionalism. The small claims court is an experimental intrusion of amateurs, which judges may not think much of. Law and technology, in their ways, are highly qualified occupations. He may feel that the man in the street gets no more than he deserves for intruding into the preserve of specialists. The amateur may be regarded automatically as “no better than he should be.”
We come to the crux of why amateur claims may not work or be allowed to work. It reminds of local interest groups or communities trying to fight the decisions of their local authority to welcome some outside “developer” and their mega-bucks, like the post-colonialism of some multi-national corporation draining third world countries of their resources. Supposedly independent arbitration only seems, to the locals, to rubber-stamp the official case.
Such arbitration is like being delivered into the hands of ones enemies.
The claimant only wanted to return his purchase and get his money back, or claim a refund from a service that failed to deliver its promises. But his little contest seems to imply much more. It becomes an indictment of the claimant. He is cross-examined by the defending lawyer and occasionally by the judge.
If he answered all the questions, he may still find to his surprise and chagrin that a judges written verdict slights his evidence, and even his character, over a matter that could have been simply cleared up, were there any opportunity to do so. Some more tact may be required of a judge, here.
This is an issue distinct from the question of lodging an appeal against the judges decision. The sums involved in a small claim don’t justify re-trial.
Open source software.
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While the whole point of the courts is that two contending parties agree to defer to a judge or arbiters decision, the judge may defer to the authority of the large firms expert. For instance, a computer firm lays down the law that it will only refund or replace sold items that have hardware faults but not if they only have software faults or "problems" -- they tend not to use words like "faults."
There is nothing in English consumer law to justify this distinction. A fault is a fault. Yet the judge can over-rule statute law and may go along with computer firm law. After all, they are the experts, aren’t they?
Indeed they are. And judge and novice computer buyer don’t know anything to contradict them. But it turns out that software “problems,” to use the euphemism, may not be so straight-forward. Apparently, one reason is, in a phrase, closed source software.
Anyone who has built a web site knows that to change the look of it, you have to view its source page. But you cannot do that with the almost universal Microsoft Windows Operating System. Its source is closed.
In general, computer programmers had best be able to see the source of the software, its actual logical structure, to debug it. At any rate, the actual writer of a program is in the best position to put it right. It turns out that computer specialists may not find correcting software programs a routine job. They are liable to charge you a couple of hours just to look at it, all at full rates, without promising results afterwards.
A joke about why hackers, under false pretences, obtained Microsoft source codes is that Windows has so many bugs in it, they were driven to desperate measures to put them right. At one point, the Microsoft corporation was reported -- by The Guardian -- as saying that Linux was (their) public enemy number one. Its code author jokingly talks about "world domination."
Linux is an open source operating system. Don’t ask me more. Obviously, I aren’t a programmer. But I haven’t the slightest doubt that the future is with open source. Tho, at the time of writing, even Linux seems not yet user-friendly enough to become the norm for unskilled home users.
If they were wise, Microsoft would make their Windows Operating System into open software, while they’re still ahead. Otherwise, we have the same old story of the supposed advantage of an imposed uniform standard over the creative freedom to modify.
[PS: Soon after this essay was written, Microsoft began to share their source code with trusted partners in government and business. But it’s still generally closed source, which caused problems for electronic self-publishers, when they tried to convert their book from the word processor program, Microsoft Word to e-book format. I used the open source html editor, Amaya, for my web-site and later, as a near approach to e-book coding requirements.
Electronic self-publishing is one of the great advances of amateurs into a virtually inaccessible profession.]
The conflict may be compared with the dead hand of convention that rests on English spelling, in all its aberrations, which leave so many illiterate.
Similarly, the availability of open source operating systems or other programs would allow people to learn better how they work. Rather than treating computer programs as magic, more people would become accustomed to see their logic. Program-literacy would be stimulated.
The moral, once again, is the need to spread important skills, like advocacy or programming, thru-out society. The spread of literacy made writers less of an exclusive profession. Technological advances are likely to repeat this creative enfranchisement in the visual and musical arts. The same needs to be done, in essentials, for all the professional skills that largely affect society.
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“How the banks robbed the world.”[
**]The 2000-02 Dotcom bubble.
Table of contents.
New York Stock Exchange, late 19th century.
Section link:
the Robber Banks,
Weird Nature and Horizon on Easter Island.
The 9 january 2003 was an evening for moralists on BBC2. Weird Nature showed how ingeniously life adapts to its environment. Man, too, is amongst the most unique adaptors. Popular biology used to burst with pride at the thought of mans opposable thumb and its brain-stimulating dexterity. Human forelimbs were freed, for this purpose, by uprightness -- physical uprightness, that is. (Subsequent programs, that night, were to cast doubt on the moral uprightness.) This is exceptional, except perhaps from a species of mangrove swamp monkeys that have learned to walk in the water, if not walk on water. Some think that deep wading is how man learnt the trick of his "funny walk."
The next BBC2 show was a new serving of Easter Island, from the Horizon science series. The series shows how scientists go about their work. I don’t describe that here, just mentioning a few findings of this program. The main observation was how the islanders mass produced stone statues, and were able to transport them around the island, incidentally committing ecological mass suicide, by cutting down all the trees and losing the soil.
A moral, there, for this island earth and the inadaptible weird nature of man.
Easter Island was the worlds largest bird colony, which disappeared upon the trial of mans arrival, as did the surrounding abundance of edible sea life.
Ethologists, like Tinbergen, showed, in trials on birds, in preference to their own eggs, the greater stimulus of over-size egg mock-ups to brood on. The natural behavior of some birds seems to have gone on from egg-rolling to stone-rolling.
Human idol worship may be less big-brained than bird-brained. Indeed, it is evident that the islanders did have a giantist fetish, carving bigger and bigger statues.
Maybe, the more man believes he is a religious spirit, removed from the influence of the environment, the more he is obeying the stimulus of some primitive instinct, that may not be adaptive in the circumstances.
When man finds himself on a longer environmental tether than other animals, he often uses the extra rope, he has been given, to hang himself with.
Over-confidence, in the godly protection of the statues, is suggested by the fact that these “living ancestors” in stone were turned upon, many being toppled, when the population was reduced to starvation.
The later cult of a “bird-man” suggests a more humble worship of mans integral part of nature. This cult was a ritual competition which replaced war-fares destruction with an orderly food distribution.
By the way, religions need not be always heedless of the good of the world. May-be the belief in re-incarnation, not only held by Indians, is more environmentally friendly than some scientists or positivists naive religion of there being “nothing” after death of the body and its senses. This “nothing” is akin to a sort of nirvana or independence from wordly attachments, that some spiritual teaching holds only to be achieved by much moral trial and error, even thru many life-times.
The islanders, isolated for a millenium, were as shocked by three ships, as the Earth would be, by visiting space-ships. By the time the Dutch arrived on Easter day in 1722, the natives had saved themselves, only to be almost exterminated by Western disease and slavery and more disease.
The scientists on Horizon pointed out the parable of Easter Island for the modern world, as relentlessly destroying its irreplacable natural resources, disrupting and threatening a collapse of the global eco-system.
Cue BBC2 program three, that evenings viewing, with the curiously forth-right title of “How the banks robbed the world.”
“How the banks robbed the world.”
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The following account is indebted to the BBC2 program, titled above. (Their web-site is: www.bbc.co.uk/business) People, including myself, are so ignorant of “high finance” that I hope the program-makers won’t mind this abridged re-telling of their research.
My Democracy Science web-site, and the e-book series that comes from it, is largely about the democratic alternative to the force and fraud in political economy, that the following story of the banks scandal exemplifies.
Clinton became President with the populist policy of limiting executives pay to one million dollars a year. So, the practise grew of giving company bosses stock options, at a set price no matter what the market price. They could get huge profits by pushing up the stock price of their firms.
This looked like a capitalist incentive for those running the firm to make it thrive. But it didn’t work out that way. The BBC2 program traced events in a series of five “scams.”
Scam one: secret loans.
The system of share options tempted executives to cook the books to make money. Profits were inflated by including predicted profits. That didn’t bring in cash. But the banks were eager to help in side-stepping the accounting rules and tax rules. The apparent attitude was: “Give me a rule and I’ll work around it.”
Citibank funneled $125m cash into a secret off-shore bank. “Delta Energy” then pretended to buy gas from Enron. Enron claimed this cash as income in its accounts. Enron shareholders were deceived. By another fake deal, Citibank got its money back plus interest. This sham transaction was the first of many such secret loans.
The share price rose; the executives won again.
According to Robert Roach, Chief Investigator, Senate Sub-committee on Investigations: Enron could not have engaged in the deceptions, it did, without the full knowledge and full assistance of the financial institutions.
They provided the means and the funding and were as much as anyone to blame.
Scam two: kick-backs.
Because WorldCom was buying up so many companies, the banks had a lot of lucrative deals to fight over. Some were so greedy, they dreamt-up a second scam: kick-backs.
A Citibank subsidiary, Salomon launched a new share flotation. Most were disappointed of the expected profits from this so-called Initial Public Offering. But for an executive making $2m profit, it was almost like giving him cash.
Salomon was rewarded by the handling of a record-breaking $37bn take-over, making WorldCom the worlds second telecom company. They boasted of perhaps taking over British Telecom, the defeated bidder.
For starters, a Securities lawyer is suing Salomon over such IPOs as amounting to bribery.
Scam three: the “Pied Piper” financial adviser.
Financial analysts are supposed to give unbiased advice to the public, and not be compromised by interests, the public are unaware of, as was the case. Banks could employ an analyst, whose pay was linked explicitly to the value of the banking deals he brought in. The BBC2 program gave a snippet of how the “Pied Piper” financial adviser made out anyone would be dumb beyond consideration not to buy into WorldCom.
It was stated that wherever he went, banking clients would follow with multi-million dollar deals. With his support, the WorldCom share price quadrupled over three years.
Another financial advisor, who refused to write rosy reports about Enron, was gotten sacked from two Wall Street firms. A favorable analyst replaced him.
Scam four: shell companies.
Citibank shoveled nearly $4bn into Enron in secret loans dressed up as deals. This was still not enough to cover-up Enron debts. So, a complex of secret companies was set-up as an accounting manouvre to hide debts and create very controversial earnings.
It took a professor of economics to sort out the tangle between some 4,300 such “boxes.” Not one appeared to match a real business purpose.
Merrill Lynch had the lucrative job of raising finance for one of the key shell companies, designed to hide Enron debts. Half Wall Street was invited and told the purpose of the company was to buy up Enron businesses not making enough money and under-mining the share price.
They were promised such fabulous rates of return that many of the bankers invested millions personally. The investors presence was being bought. Because of the scam, Enron claimed profits of $1bn, when there were in fact none. The chief of Enron had $180m in share options.
The WorldCom chief had $325m in share options with his company boasting profits of $2½bn. Yet “incredibly” he still needed cash. Citigroup gave him a nearly $500m dollars personal loan to help buy thousands of acres of American forest.
This should have been disclosed to World Com shareholders, with respect to the law on Securities Fraud.
Within six months, Citigroup was chosen to under-write a $4bn WorldCom bond issue. Citigroup made $15m on the deal. A year later, it happened again.
Scam five: ignoring costs.
In march 2000, the Stock Market collapsed. The “Pied Piper” changed his analysis to one of revenues, instead of profits, ignoring costs altogether.
Costs went out of control as prices fell. WorldCom executives began to record day to day costs as spending on assets, to account for them as long-term spending, and so boost short-term profits. This mis-accounted $9bn.
Executives were secretly selling stock and the Pied Piper was still urging the buying of shares even as they were collapsing.
Scam six: the three-shell game.
The BBC2 program doesn’t name this as a sixth scam but I am still closely following their account. The three-shell game is a fair-ground attraction where the public has to guess under which of three shells a nut has been put. The game promoter tries to deceive by sleight of hand and win the bet.
When the public were shown an investment version of the three-shell game, they were deceived even from knowing they were being drawn into a game of deceit. Citibank set up a new company called Yosemite to persuade out-side investors to lend cash to Enron. This time, Enron used the cash to pay off debts to Citibank. So, the public and not Citibank would lose money when Enron collapsed.
The BBC2 program showed a man playing the three-shell game, with three shells, labeled Yosemite, Enron and Citibank, hiding the publics bank-roll.
Scam seven: letting the banks off lightly.
I’ve called the lenient treatment of the offending banks a seventh scam.
The shell companies accountants apparently became too clever for their own good. The economics professor pointed out that they failed to make one box own 3% of another box, as it should have. Hence, a legal demand was made that two boxes be combined, resulting in a $1.2bn reduction in stock-holder equity.
This was “the first step in the avalanche” that led to investigations, stock price collapse and bankruptcy. Enron borrowed billions but couldn’t save itself.
Other company accounting scandals were exposed. It seemed everybody had been at it. The biggest falls came from where executives were given the biggest options.
They remind of the birds prefering to roost on “eggs” the size of an American foot-ball, or the Easter Islanders, destroying their environment, to build bigger and bigger statues.
The belief that “greed is good” worked its ruin on the Stock Market.
A largely corporate-staffed administration was obliged to promise a corporate clean-up. President George W Bush is regarded, by a blue-collar representative like Michael Moore, as the front man for corporate America. The President said on 9 july 2002: The business pages of American newspapers should not read like a scandal sheet.
Hauled before Congress, WorldCom executives took the fifth amendment. Some WorldCom and Enron executives have been arrested for trial. A case is under-way with regard to the biggest fish. But the boss of Citibank wont face charges. Their disbarred analyst, “the Pied Piper” has been discharged from their employ with what amounts to a scores-of-millions dollar sweetener.
The New York Attorney General is credited with some of the toughest banking reforms since the nineteen-thirties. Wall Street is barred from bribing company bosses with share issues. New rules are meant to ensure the independence of analysts.
The chief investigator believed the abuses, of any number of rules, will go on, while financiers lack a moral compass.
The banks have been fined: Merrill Lynch $100m, CSFB $150m. The rest have also settled. The biggest fine of $300m was to Citibank. Americas biggest financial services company can afford it. In the current year, they will make $16bn profit.
The BBC2 program says Citigroup did the most to help WorldCom and Enron deceive the world. Its boss had most to gain with an options package of almost $1bn.
2002 was the year the depths of Wall Street corruption was finally exposed. Only because of the big financial houses could WorldCom and Enron destroy $240 bn of investors money.
Sarah Teslik, of the Council of Institutional Investors, concluded: The people who can least afford to lose money have lost collectively billions of dollars -- because of fraud, because of greed -- that has been transfered out of their pay checks and out of their pensions to the pockets both of the corrupt executives and the Wall Street investment bankers who enabled them.
To top.
David Craig and Matthew Elliott: Fleeced!
How we’ve been betrayed by the politicians, bureaucrats and bankers and how much they’ve cost us.[
**]£50,000 taken from every person in Britain.
Table of contents.
A is for Austerity after the Avalanche of bad debt.
B is for Bureaucracy and Bad government.
The untouchable elites.
C is for Crash diet after Britains Credit Crunch.
P is for Power to the People.
[_November 2003, Deputy leader, Liberal Democrats, Vince Cable: _]
The growth of the British economy is sustained by consumer spending pinned against record levels of personal debt, which is secured, if at all, against house prices which the Bank of England describes as well above equilibrium level. What action will the Chancellor take on the problem of consumer debt?
Chancellor Gordon Brown: We have been right about the prospects for growth in the British economy and the right honourable gentleman has been wrong.
A is for Austerity after the Avalanche of bad debt.
The headline estimate of £50,000 debt imposed per man, woman and child is perhaps a severe under-estimate of the bank robbery of Britain. The authors, estimate, cited below, is about three trillion debt. But the Mail on Sunday supplement put it at nearer five trillion, with an obscene graphic in mountains of squandered bank notes. (And that’s only what’s come to light.)
You may have despaired, as I have, of ever keeping track of the serial mismanagement of Britain. There is a positive advantage in numbing the public with one disaster after another, so that the last is over-shadowed by the latest. When much of the populations computerised confidential banking details went missing, the new premier Gordon Brown assured us it would soon be forgotten. And so it would have been, with all the rest of the blunders, if he hadn’t been so dismissive of this particular one.
This 2009 book, Fleeced!, follows several titles, by these writers, sometimes with other co-authors, on the epic of inefficient spending by New Labour, since 1997. It isn’t all that partisan. One of the laughs is a wry rant on trying to lead a Tory horse to drink.
One of the authors advised the Tories to have a policy of ring-fencing vital services, while cutting out the layers of managerial fat. David Craig emfasised they must never say something like: we will ring-fence health or education, implying that the bureaucratic waste would be spared. But sure enough, that was just what they soon came out with.
In fact, I’ve just seen written in the standard David-Cameron commandeered side of the local Tory leaflet for the 2010 general election: “We will protect spending on the NHS and improve it for everyone.”
With the authors, thru-out, you have to laugh that you might not cry.
Reviews of these books, like: Squandered; Plundering the Public Sector; Rip-off!, have called them “terrifying,” “horrifying,” “shocking,” etc. We should be grateful that there are journalist accountants with a dogged determination to hold government accountable.
A really terrifying, horrifying, shocking fact is, tho, that Labour and Tory parties are only beginning to do their worst to this country, in their commitment to more nuclear power stations. Sixty years of fission energy provides evidence enough that scientists and technologists, and their political masters, can be as ignorant and impractical as anyone towards human survival and prosperity. If The Taxpayers Alliance, which Elliott heads, cannot see that, their worthy work is all but in vain.
[This review was written about a year before the enforced Japanese evacuations from the contaminated land around the disabled Fukushima nuclear plants.]
Craig and Elliott say: In New Labours decade of tax and spend, over a trillion more was spent than under Tory government levels.
We should remind ourselves a little of what went before. As I heard someone say at the time of the Tories: This government doesn’t want you to have anything for nothing.
Worse still, the poll tax – taxing you just because you exist, as one Labour MP put it.
One would think from the authors that all Labour spending was bad. In fairness, I would have to say this was not entirely true. A highly visible example is spending on public libraries. If our local central library was anything to go by, the Tories starved it of funds over many years. The stock was getting ever more dated, gaps in the shelves, hardly any new books.
I did find some of Labour public library spending wasteful and inefficient. Replacing every library, in a standard county way, was not to the advantage of every library. Our central library was austere and utilitarian pre-war, it is true. Personly, I prefered that to the luxury refurbishing. And the big oak bookcases were replaced by much smaller, from shoulder-high to crouch-yourself shelves. This further reduced book capacity, as did the room taken-up for computers.
These are relatively small matters but typical of government that all such decisions go on above the heads of the local people who use the services.
Nevertheless, I find it hard to believe that the Tories would have brought the libraries into the electronic age. Not nearly as readily as Labour did. Tory privatisation dogma was turning people away from the public library service by their neglect, perhaps as a pretext to roll it up altogether as out-dated. I doubt the Tories would have allowed any free access to the web. It has since been reduced and curtailed to minimal levels.
Labour purpose-built modern clinics, like decentralised mini-hospitals, to replace the stuffy little waiting room for the GPs office, typicly in some housing terrace.
A private-public conflict is a line of divide and rule by which the Labour-Tory duopoly persist in cornering power for themselves. As this book shows, Labour has built-up an army of largely dubious dependents in the public sector. The 2010 Labour election slogan of resisting Tory cuts seems to be an appeal to this bought support.
New Labour politicians were caught, by Greg Palast, in “lobbygate” stings, trying to sell themselves. So, it would not be surprising if mercenary politicians treated voters as mercenaries.
Moreover, the meddlesome control-and-fine inspectorate reads like a government turning Prussian rather than democratic. And this perhaps in Tory-controled council areas, so that it, in effect, becomes Labour-Tory duopoly policy.
We hear a lot of ambiguous cant against “neo-liberalism” but Labour became “neo-Prussian.”
It is fair to say that, unlike Fleeced!, the Press, unavailingly gave vent to chronic rage, against private firms uncontroled increases in excessive executive salaries. public sector emulators.
The authors forget how sustained were Fat-Cat attacks by the Press, when they criticise the Westminster journalists lobby for being managed by the government, feeding them scoops, so they would over-look public spending failures. After all, private sector greed has been emulated, if not matched, by the excessive public sector salaries.
The anxiety of Tony Blair, becoming PM, as well as Peter Mandelson, to encourage people (including himself) to become rich, encouraged corporate plunder. Inequality, not wealth creation, has been his legacy, in the private and the public sector. Of course, the authors are right to pick on the bankers as becoming pre-eminent instruments of inequality and injustice from the private sector.
The first time the financial market lost more than half its value was the great Crash of 1929 to 1930. But in the last 35 years, there have been three more: 1973-4 with the oil price hike; 2000-02 the dotcom bubble; 2008-9 the credit crunch.
After 2000-02 Dotcom bubble burst, money fled shares into housing, which pushed up prices and encouraged borrowing.
House prices boomed in USA and UK, Spain and Ireland.
Government encouraged home ownership and relaxed credit rules. Inflation in house prices, left out of the governments calculation, contributed to artificially low interest rates. Prices fell from increased industrial production in Asia. Low interest rates encouraged financial institutions into riskier investments, all essentially the same but generating huge fees. Doubts about their real value led to crisis.
Another illusive growth was in Britains public sector to over 6 million. About 5 million have final salary pensions, double that of the private sector, which is closing them down to new employees. Originally part of Tory privatising ideology, from the eighties, much of pensions mis-selling was encouraging to switch out of final salary schemes into risky investments or those likely to perform more poorly.
Companies selling out to pension-management companies creates a conflict of interest in the managers desire to maximise their own profits rather than the pensioners. Almost all pension savings tax benefits eaten-up by charges.
A citizen would need £50,000 a year for their whole working life to earn an equivalent pension to an MP. Raising their pay to £100,000 per annum (p.a.), since they’ve been found-out on expenses, would likewise increase their pensions.
The most generous benefits system in the world encouraged uncontroled immigration from all over the world. At the other extreme, of the world immigration scale to Britain, are countries that shoot, on sight, border crossers.
Meanwhile, Britains manufacturing, to pay for it all, declined, about 15% under New Labour, unlike China, India, Germany and Holland.
In the bust, approaching half a trillion pounds were wiped-out in UK shares. Another half a trillion pounds or more are to be found for public sector pensions. Then there’s the estimated tax-payers loss of at least £200 billion from the banks bail-out.
All in all a boom and bust loss of over 3 trillion, it will take decades to pay.
[PS: This turned-out to be an under-estimate.]
The worlds financial instruments are valued at many times that of national economies, especially since George Soros made a billion pounds, in one day, by helping Britain crash out of the European exchange rate mechanism in 1992.
New Labour gave honors to at least 23 bankers, including 7 life peerages; 3 as government ministers. And asked another 37 to work on commissions, quangos and advisory bodies. At one point, 3 financiers, worth £500 million, gave nearly 40% of their party donations.
With too much money to be made out of the financial bubble, credit ratings made investments look rosy. Finance is getting more complex and less transparent, including insurance against risky loans, such as mortgages that couldn’t be paid; over-selling and over-borrowing; the spreading of potential defaults thru the financial system.
Banks, with worthless investments, start collapsing. Confidence flees causing a chain reaction. Regulations for more capital adequacy, after allowing too little, over-compensated, inadvertantly leading to the credit crunch.
“What Gordon didn’t tell you”: while using public money to stave off the banks collapse, the claim, this is to encourage bank lending, ignores that the state is also telling the banks to build up their capital against their enormous losses and pay into a fund against further bank runs. Hence the woefuly poor interest offered for savers, so their prudence is fleeced thru inflation. And good businesses are starved of the credit they need to keep going. Vince Cable kept hearing of this, thru his constituency MP work.
Shares leak value because insiders get the best from knowing when to buy and sell at crucial upturns and down-turns of the market prices. Correspondingly, the public may be the losers. Fees and commissions accumulate into hefty cuts out of the eventual returns. The public may not know the nature of the products, they are being sold or mis-sold on pretence they are secure.
The executives get massive bonuses no matter how badly their companies perform.
Pensions liabilities of many companies are larger than their market value. Risks are being placed on the employees. Government can only pay for inflation-proofed public sector pension liabilities by charging the taxpayers. About 34,000 public sector pensions millionaires may double or even triple from New Labour extras.
B is for Bureaucracy and Bad government.
To top
Over half a million civil and public servants, with signs of steady increase, despite government protestations.
They are often transfered to quangos, whose number of employees rose from a million in 1998 to one and a half million by 2006. Spending up from £49 billion to £130 billion. Yet, powers transfered to regions and to EU. Regional Development Agencies were duplications of English Partnerships over regeneration. Many over-lapping agencies, against disadvantage, under the Department for Communities and Local Government.
Constant disturbing reshuffling of segments of departments made them look smaller on paper, as with the Cabinet Office. Constant abolishing and refashioning of ministries dislocates effective government.
Number of NHS managers doubled since 1997, despite 30% decrease in beds. Brussels laws doubled from 40% to 80% in ten years. Two-thirds reduction in proportion of laws matched by two-thirds increase in MPs salaries and expenses. New public sector managerial class created, one for every two administrative staff, checking targets, ticking boxes, writing reports, attending luxurious conferences, spreading best practise, instead of practising it, and not wasting actual workers time with inititiatives, to make themselves look active and improve only their own financial well-being.
The attempt to cut staff, with by-product of a redundancy bonanza, defeated, while hiring more people. An army of consultants cost £2.8 billion a year.
Public Accounts Committee member called the efficiency savings “an enormous amount of smoke and mirrors in the whole of the public service.”
(Reviewer comment: unelected and unaccountable officialdom doesn’t work. It’s time for representative democracy of all vocations and occupations, so that they can keep a check on themselves and each other.)
The civil service answer was that savings had been made regardless of the costs of making them, often more expensive than the savings.
The PAC said such answers could have come out of the satire: “Yes, Minister.”
Pay for front line services kept under control. (Only be a matter of time before driven to disruptive but ineffective strikes.) But top ten civil servants earn over £200,000 pa. Top ten quangocrats earn from one and one-quarter million pounds to three-quarter million pounds. British Nuclear Fuels include two of them. Over 60 earn more than top civil servants and at least 169 earn more than PM.
BBC top salaries from over £800,000 to £400,000. BBC bidding-up for sports sometimes artificially increases costs. Their charter is not to compete with commercial broadcasting but supply other service.
Organisation for Economic Co-operation and Development. (OECD) showed Britains public spending on the scale of nations, from 1997 at 17, by 2007 to 11, if not higher after. Education: reading fell from place 7 to 17; maths 8 to 24; science 4 to 14, from 2000, before tens of billions thrown at it, by 2006.
But Britain is high on crime tables and proportion of unskilled work force.
Despite more than doubling health spending from £45 billion to £105 billion a year, survival rates for cancer and strokes should be about 17,000 less, going by comparable European countries. (“Wasting Lives.” Taxpayers Alliance.)
NHS website admits 34,000 die unecessarily and 25,000 a year needlessly disabled in hospitals each year. Money squandered on new managerial class with no health-care training, living in their own world of target-setting and form-filling. Managers doubled from 20,000 to 40,000 from 1997 to 2009, costing over £3 billion a year more. Plus £600 million a year on managment consultants to show the managers how to manage. There’s 5000 more managers than medical consultants. Directors of finance, marketing, strategy, communication have lavish pay and pensions. It’s Parkinsons law with a vengeance.
“Achievments” might be doing more harm than good, as ritual of meeting targets can be fatal, just to make the figures look good. No patient has to wait more than 4 hours in Accident and Emergency. But it’s claimed some patients held back and dying in ambulances rather than break the 4 hour dead-line.
Hospital buildings, under the governments inordinately expensive and failing Private Finance Initiatives, cost over £10 billion.
The national health service ethic is replaced by a “managerial cover-up culture” reminiscent of the Bhopal and Exxon Valdez disasters. Margaret Haywood, whistle-blower against denial of basic care, was struck-off by Nursing and Midwifery Council in 2009. All the nurses in a hospital with high mortality were afraid to take part in a tv under-cover documentary.
This contrasts the pay and pensions boosts to one suspended on lavish salary, a chief executive of a hospital, criticised by the Healthcare Commission, finding anywhere from 400 to 1200 needless hospital deaths with shocking and appalling care: “hundreds of patients died because the trust’s board was more interested in meeting government targets and attaining elite foundation status than in patient care.”
Most government ministers have no experience of management in general or of what their departments do in particular. Most of them have never learned that money has to be earned before it can be spent. Typicly, politicians are lawyers, lecturers, trade union representatives, or political advisers straight from university. They are versed in a protective layer of management-speak or gobbledygook.
European Central Bank study estimated if UK public services were as efficient as in USA, Australia, Luxembourg, Ireland, Japan, Switzerland, there would be the same level of service for about 15% less cost. A trillion pounds wasted and probably another trillion to be wasted, before it can be brought back under control, if ever.
Instead of a positive feedback from wise investment, a negative feedback of wasted money, rising social breakdown, millions of unskilled who’ve never worked, rising benefit costs, increased taxes, reduced competitiveness, falling wealth, greater borrowing, higher taxes, greater burdens on households and businesses. The real costs of waste are almost unimaginable.
Prestige projects for politicians spend public money of no value to public: political or profiteering gimmicks. Pretence of low cost results in over-spending that suppliers can get away with, because of administration attitude that it’s only public money, and inexperience in auditing. Most civil servants and politicians move on from financial disasters and would rather avoid the blame than face-up to the failure.
Regular ritual, of Public Accounts Committee calling evasive civil servants, is to vent histrionic outrage. Serial poor value projects are usually described as the “worst” this or that.
The millenium dome only came 9th on cost-overruns. The 2012 Olympics and the NHS IT system by far the worst wastes. The Olympics involved building many facilities already well provided for. At least ten of the Olympic managers won gold by earning more than the PM.
In Plundering the Public Sector, the authors “explained in possibly painful detail exactly what was wrong with the whole [NHS IT] project; why it would cost billions more than budgeted; why it would be at least ten years late; why it would never work; and why it wasn’t ever necessary in the first place.”
The project boss left – apparently to Australia, as far as possible away without leaving the planet.
Two of the four suppliers couldn’t be induced, by all the billions, to have anything further to do with this highway to hell. It still doesn’t work and many hospitals won’t touch it. The money is still being wasted. They’ve started, so they will go-on regardless of cost or value. No-one has courage to pull the plug, lest government lose face, tho the NHS would benefit from the transfer of spending.
The governments reflex response to any problem is to set up an independent committee or watch-dog. In 2008, ten of the largest, set up since 1997, cost almost £1 billion. With expanding budgets and staff.
The utility bills go up steeply, nevertheless. Ofgem and Ofwat fail to protect: foreign energy and water companies earn four to five times the profits in Britain compared to their properly regulated home markets.
Qualifications authority spent over £1 billion, while exam results so discredited, that university admissions departments no longer recognise them as valid.
Regulatory capture, with poachers, become game-keepers, over-see businesses such as oil, tobacco, nuclear power, pharmaceuticals.
This is now a problem in public sector, such as health care and financial services.
In 2001, the National Patient Safety Agency has 292 staff, £30 million p.a. budget. After spending well over £100 million, it still didn’t know, by 2006, how many patients harmed by medical error. The Public Accounts Committee (PAC) said: the NPSA is “dysfunctional” and “not value for money.” The NPSA is one of several budget-busting regulators, some re-organised.
The Health Protection Agency produced vast amounts of literature, including about hospital-acquired infections. Meanwhile, 30,000 deaths in Britain. Comparable rate, had these victims been in countries like Belgium, Denmark and Sweden, would have been less than 600.
In 1997, Labour manifesto promised to cut administration to strengthen front line. Instead, £450 million p.a. more for regulators.
Gordon Browns tripartite system of financial regulation didn't give clear line of command and responsibility to one organisation in a crisis. The Bank of England was given the wrong target of monitoring the Consumer Price Index instead of the Retail Price Index, thus living in a fools paradise that inflation was only about 2% p.a. while no regulators noticed the unsustainable house inflation bubble.
The Financial Services Authority had over-seen selling of financial products, like savings, pensions, investments, unit trusts, mortgages, Ponzi schemes, etc. In any case, there was a whole series of mis-selling scandals, met by “FSA apathy,” to quote the Press. Likewise, “Toothless FSA leaves us all at the mercy of the banks.” Giving the FSA the job of macro-economic policy, the market stability of financial institutions, was not suited to its dubious skills.
In the crisis, a Bank of England excuse was that it was given the job of over-seeing the stability of the system, not individual institutions, the job of the FSA. (The Treasury designed the over-all structure of regulation.)
174 of the FSA staff received 6-figure salaries and practicly everyone in the building got a record bonus during the crisis, when they actually had some real work to do. Government allowed it to take its budget from £300 million to £415 million p.a. They admitted they would pay more than necessary to recruit more staff.
As in the USA, former leading financiers influence regulatory advice.
After the 1929 crash, the Glass-Steagall Act separated high street saving and lending banking from high risk investment banking, to protect from financial gamblers being bailed out with public money when they lost, because they were too big to go down without taking the country with them.
The untouchable elites.
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[*Local, like national, government out-of-touch. *]
Joan Bakewell, given some authority, commented, on councils intending to cut pensioners free bus passes, that people had a right to expect some return on all the money they had paid in council taxes.
Councils doubled council tax, yet want more and say front-line services will have to be cut.
By 2007-8, 1021 people, in the 469 local authorities, were earning over £100,000 p.a. The ten highest receiving more than the PM. Not all councils complied with Freedom of Information over their pay.
Local government employees limited to 2% rise, while top officials average was 6% of a much bigger salary, amounting to £7328 p.a. This also increases their pensions correspondingly. Many councils pay a recruitment company owned by the Society of Local Authority Chief Executives and Senior Managers.
Middle managment earning over £50,000 p.a., went-up eleven times, under New Labour, to about 38,000 staff.
Since 1997, cost of managers wages and their accessories, such as offices, secretaries, assistants, expenses and pensions, probably multiplied tenfold (or more) from about £400 million to £4 billion p.a. Council tax amounts to £24 billion. Had managers only increased five-fold instead of eleven-fold, council tax could have been reduced 10%.
Councillors allowance increases regularly exceed by five to ten times, the rate of inflation. In 386 local authorities providing information, they pocket an average of £10,000 p.a. Being a member of a police authority can add £10,000 to £15,000 and of a fire authority £5000 to £10,000 p.a.
More than 3500 councillors have joined Local Government Pension Scheme, as if they were salaried employees enabled earn pension benefits, inflation-proofed with early retirement thrown in.
Management-speak has invaded local government adverts for frilly jobs, consultants. Foreign and domestic holidays come thru twinning towns and seminars.
Plundering Politicians.
The following unworthily small sample of MPs fiddles is mainly taken from “Fleeced!”
In july 2009, MPs awarded themselves a £25 a night subsistence allowance payable without receipts when staying away from their main home. This could net some thousands a year tax free over their mortgage, rent, food, utility bills and council tax allowances.
This was right after the may and june Daily Telegraph uncensored version of the redacted MPs expenses given by Parliament.
Gordon Brown was one who flipped his designated second home, shortly after entering Downing Street. David Cameron, who took out a £350 000 mortgage, for a large house in Oxfordshire, took close to maximum allowed.
Choice of second home was often influenced by which could give the £24,000 p.a. allowance. In seven years, Jacqui Smith MP perhaps cost £2 million in salaries and expenses.
Balls and Cooper switched to second home the residence from where their children went to school.
Claims, on mortgages already paid, raised questions of criminal offenses.
“Flipping” is when MPs change which is their main home, so they can claim for furnishing and contents. Gordon Brown, flipper, used his Westminster flat as his second home, despite having a Downing Street flat. After moving into Downing Street, he flipped to his house overlooking the Firth of Forth.
Alistair Darling changed his designated second home 4 times in 4 years.
Geoff Hoon, as former MEP, maybe had a head-start in the expenses stakes. He built up a property empire thru expenses; flipped 4 times in 4 years. Finally caught, in march 2010, in another lobbygate sting with other Labour ex-ministers Patricia Hewitt, Stephen Byers, and another Labour MP, Margaret Moran.
[PS. Patricia Hewitt was among those cleared of impropriety.]
Moran is standing down after “a furore over her expenses.” She is also the chairman of an all-party Parliamentary group on the information society “which – highly unusually – is registered as a company.” (Mail on Sunday, 28 March 2010: MPs and peers run private company selling “influence over government policy”). Its corporate members pay more than £120,000 p.a. and fund expenses for junkets abroad.
Hazel Blears claimed for three properties and nights spent in a series of hotels in just one year.
John Bercow (Speaker) flipped twice in a year, both times avoiding capital gains tax from two house sales.
Kitty Ussher (Lab) flipped for a month during sale of property.
In five years, 27 outer-London MPs made claims averaging £63,000. But another 22, living similar distances, made no second-home claims.
Married MPs the Keens were known in Westminster as “Mr and Mrs Expenses”.
Married Andrew Mackay and Julie Kirkbride, each designated a separate second home. This meant that, between them, they had no main home but two second homes. Kirkbrides sister employed as a secretary, tho living 125 miles away. Her brother allowed stay in constituency home, against Commons rules, as it was funded out of expenses.
Alan Duncan, gardening expenses claimant, had his new flower bed cut in shape of a £ sign. One of half a dozen Tory MPs submitted gardening and grounds claims “in error” and repaid.
Barbara Follett claimed £25,000 on security guards as expenses. Why didn’t she change Labour policy, if it is so ineffective?
Parliamentary Commissioner for Standards, John Lyon dismissed 93 out of 113 complaints and resolved one in a year.
His predecessor, Elizabeth Filkin led high profile investigations into Keith Vaz, John Major and William Hague. It seems, she was subject to a whispering campaign against her, and obliged to re-apply for her job.
The Sunday Times, january 2009, did a sting, of the House of Lords, showing some members willing to change legislation in exchange for lobbying fees.
Allowances amount to substantial incomes of £56,000 to £66,000 taken by some Lords. Some tabled no questions or almost none. One peer claimed over £40,000, tabled no questions, spoke 9 times and voted twice in a year. Others nearly as bad value.
Several peers have registered as their first homes their French homes to allow generous expenses for visiting London.
The appropriately named Lord Ryder claimed over £100,000 by claiming that a converted stables, in his parents country estate, was his main home.
The Sunlight Centre for Open Politics complained against Lord Rennard, former chief executive of LIb Dems on second home claims.
The Bank Robbers.
To top
Once again, labour – the impoverished working class in Britain’s old industries, a large percentage of them miners – was being asked to bear the cost of capital’s mistakes.
Historians would condemn the crisis of the summer of 1931 as “the bankers’ ramp.” The flight from sterling on 11 August was not precipitated by the budget deficit – the millions being paid out in unemployment benefit – but by the speculative activities of London’s bankers.
…The London bankers were caught out, facing short-term foreign liabilities estimated at over £400 million.. It was the Bank of England’s decision to allow them to draw on the gold reserve that had caused sterling to run down.
The upshot was a National Government and a National Emergency.
Black Diamonds, by Catherine Bailey, in 2007 (the year before the Credit Crunch).
In 2007, the bankers appeal, to government for a bail-out, involved a capital guarantee that could be far in excess of the national income.
The assets and thus liabilities of Iceland banks were about ten times their national earnings (GDP). In UK, about four and a half times.
In 2007, the Royal bank of Scotland had total liabilities £1.9 trillion; Barclays £1.2 trillion; Lloyds-TSB and HBOS each £1 trillion. In all, UK GDP itself was just £1.4 trillion.
But the largest US banks, Bank of America, Morgan Chase, Citigroup, only had liabilities (in pounds) of about 1 trillion compared to US GDP of 14 trillion.
On just a few deals, RBS lost more than UK annual defense budget.
In a few months from 2008-9, Barclays share price, which later recovered, lost more value than UK spending on police and criminal justice in a year.
Banks bailed-out, including RBS and Northern Rock, went on to give bosses and staff extravagant bonuses; also huge earnings and bonuses to bosses of Lloyds-TSB and Bradford & Bingley.
In april 2008, the FSA prevented institutional shareholders of the RBS voting against Sir Fred Goodwin, “Fred the shred,” misguidedly fearing a mass rebellion, against the board, would rock the boat.
FSA claimed it was already looking into HBOS lending, a former HBOS head of Group Regulatory Risk, Mr Moore, warned against. The FSA did not explain how the bank still collapsed.
The government plays to the gallery, blustering about regulation to reduce risk, while fighting the European Union regulation proposals, they fear would reduce profits, to get the nationalised banks off their hands.
Vince Cable: “It is clear that the conditions set by the government over the original capitalisation were a sham. No effective monitoring and controls were put in place to ensure that the money went where it was intended.” That is in lending to home-owners and business. The disappearance of public money was largely to be expected because it was also telling the banks to build up capital against the worsening economic situation caused by their not lending to business.
Anything from £200 billion to £500 billion goes to banks to insure against potential losses in return for them to increase lending by about £40 billion.
Britains financial sector is about 9.4% of GDP; the Swiss is over 12%. Perhaps better to have let a few banks go to the wall with their big bosses massive pay-offs. Emergency legislation could have handed over the banking job, say, to the supermarket chains, to lend money, while the banking system was purged.
The top five US banks paid out $38 billion dollars in bonuses in the crash year of 2007-8, up from $36 billion of previous year record profits.
This dislocation of results and rewards explained by one condescending Wall-Streeter: “Joe-six-pack is never going to get this, but if we don’t pay the bonuses we lose the talent.”
Bosses making millions bankrupting Britains banks. Taxpayers have been voluteered to pick up the bill. Trashing one or two banks has done many bosses no harm at all, as future managers will have noticed.
There used to be five accounting firms, till Arthur Andersen was caught shredding incriminating evidence in the Enron scam. The four earn millions selling consultancy services to the banks, they are supposed to be auditing, creating a conflict of interest against whistle-blowing.
Deloittes appeared to raise no concern over RBS, which was to become the worlds biggest bankruptcy, but have been re-appointed auditors.
The big four auditors control the world market, can set high prices, passed on to the share-holders. They dominate the committees for standards and (non-)liabilities of accountants.
Bankers, regulatora and auditors were playing an “elaborate game” with “detailed and complex rules absolving any of the players of responsibility for anything, yet they all became fantastically rich from plundering our money.”
The authors, Craig and Elliott, conclude:
never again should ordinary people’s liabilities to any failing financial institution exceed more than five per cent of GDP. As for us being exposed to losses that were potentially greater than the country’s GDP at just one badly run bank, this is so absurd that it hardly seems believable that our leaders allowed it to happen. Unfortunately for us, our politicians’ and regulators’ self-interest has become so entwined with the interests of a few bankers that our government has pursued and is continuing to pursue policies which seem to favour financiers over ordinary people. Nothing any of our politicians or regulators has said so far gives any confidence that this imbalance, where the interests of over 60 million people are so cynically subordinated to those of a small but influential elite, will ever change.
C is for Crash diet after Britain’s Credit Crunch.
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The public sector grew from 38% in 1997 to 48% by 2009. Studies show 10% increase liable to reduce GDP by 1.5% and thus undermine ability to pay off debt. This trend almost mirrors the period from 1964 to 1976, when Labour had to go to the IMF in return for public spending cuts.
The need is to ring-fence front-line services, not Tory plan to bring in armies of accountants and consultants, franticly lobbying to look for department cuts, thus adding to costs, rather than reducing them, and putting-off making obvious and necessary savings.
The Inertia of Large Numbers: Wander round a department, attend a few pointless meetings; assess the poor quality of decision-making; notice the level of inactivity.
The levels of managers cannot see that they are the problem and only can cut down essential services. e.g. police cutting down front-line officers, while awarding managers huge bonuses.
500,000 out of 800,000 new posts have little to do with providing direct services to the public. Not hiring them would have saved £20 billion as well as their pension liabilities.
We will be told that tough decisions have to be made, meaning that the elite expect to reduce the quality of life for the many, so that the few can continue their excesses.
On the contrary, here are half a dozen points to cut waste and inefficiency without damaging and possibly relieving beneficial services:
1) £7 billion. Immediate savings in cost of bureaucracy: stop all bonuses, early retirement or redundancy pay-offs and recruitment of all but front-line services, such as police officers, doctors, nurses other medical specialists, teachers, and a few categories of manual workers for maintenance work. A public sector pay freeze for all those earning say over £40,000 p.a. Savings of £1 billion p.a. are a start, out of £670 billion p.a.
Declare a national emergency, like the Heath government 3-day week and put all public sector managerial and administrative staff, not dealing directly with the public, on a 4-day week. This would include all council and all NHS executives and managers and most admin in the main government departments, like health, education, the Cabinet Office, business enterprise and regulatory reform (or whatever it’s called this week), environment and many others, and almost all staff working for regulators and quangos.
Admin processing things like driving licenses, pensions, benefits, passports etc should be exempted but their managers should be put on a 4-day week. Least disruptive probably to take every friday off.
Private sector companies have used reduced numbers of shifts and shorter working weeks to save money while protecting jobs.
Letting the “multitudes of policy advisors, executives, managers, communications professionals, diversity officers, community relations specialists, involvement officers, diet advisors, racial awareness specialists, equality experts and administrators take every Friday off” should save about £4.5 billion a year, including £360 million from 40,000 managers in the NHS alone.
After about six months adaptation, reduce at least half the managers to a 3-day week, eventually bringing-about savings of say £6.8 billion p.a., without any costly retirement payments or redundancy packages. And without having to fight unfair dismissal cases.
If they don’t like it, let them go voluntarily for productive work in the private sector, to make wealth instead of spend it. Their hanging-on would show that their working lives are still preferable to the stress and insecurity and inadequate pensions of the private sector.
Cutting jobs is old-fashioned thinking costing more than it would save, at least in the short term. Later, government could look at the need for all the jobs.
The authors suggest reducing Parliament to sitting a 3-day week since most laws come from Brussels. They would halve the number of MPs and merge constituencies.
(Reviewer comment: But this would tighten the control of the two-party system and hence their pay-masters. Bigger single-member constituencies are harder for any but one of two parties to win. This explains fewer constituencies being Tory party policy.)
2) £11 billion. Make managers manage instead of having so many of them telling others what to do in meetings and documents. Move the transformers and improvers to line-management. Just imagine all they’d save if they actually used all their supposed expertise.
End departments hunting around for ways to spend their budgets, and make their jobs depend on cost-effectiveness.
Move spending decisions close to where the public can see it is their money being spent and give the locals power of choice. Give a school its own budget, to alert parents to useless spending on a superfluous official rather than say another teacher or equipment.
Local policing and court budgets with elected police chiefs would pressure crime-fighting rather than bureaucratic bonuses for political correctness.
[PS. After 2010, the police commissioners, on huge salaries, were elected on turn-outs around 17% of the vote. A police representative in the House of Lords complained the Supplementary Vote was too restrictive of choice, compared to the Alternative Vote.
AV was used to elect the Labour leader, Jeremy Corbyn, in 2015, four years after the bulk of the Labour party defied their leader, Ed Miliband, in opposing it.]
3). £5 billion, a saving of 1%, by liberating front-line workers, who know best how to make improvements, yet made afraid to make them, by layers of management, all keen to keep their badly-run empires from prying eyes.
Value for Money unit under the Treasury should be obliged to investigate, within a month, all front-line suggestions and their recommendations rewarded at 5% of the savings up to £100,000. Management shouldn't be so rewarded because making continuous improvements is their job they are already paid for.
4) £5 billion from better buying. Of £170 billion, a 2% procurement improvement (4 to 5% is quite common in private sector initiatives). Despite the Office of Government Commerce providing jargonised versions of freely available advice, the private sector will over-charge inexperienced government buyers.
Since the American Civil War, and rejuventated in 1986, the False Claims Act allows citizens to sue against fraud or corruption in government contracts and programs. Whistle-blowers allowed between 15% and 25% of the money saved the government. A 1% saving means £1.7 billion p.a. saved. The main value in US is admitted to be deterence worth hundreds of billions.
5) Low-hanging fruit. £23 billion plus £4 billion p.a.
Kill “Connecting for Health,” saving up to £10 billion. But ensure computer compatibility standards and that any locally-bought computer system has at least 10 other willing buyers or users. Would also stimulate the almost destroyed British healthcare computing industry and earn exports.
Austerity Olympics could have saved £8 billion by cutting on needless duplication of facilities etc.
Pay for consultants or interim managers should be cut by £2 billion out of £2.8 billion. No department should be allowed more than 0.1% spending on them. Consultants should have to itemise their projects and the exact benefits to be gained from them.
Scrap ID cards, save £5 billion. Worst of both worlds the government goes ahead but most people don’t have to take part for the supposed benefits. Scrap Contact Point childrens database and computor system. It cost £200 million but at least we wouldn’t have to pay £44 million p.a. to run it.
Cancel all government advertising, especially for jobs and put them on a website.
Bring the army home from questionable and unwinnable wars to help-out with the social problems caused by the governments benefits-dependent generation.
Scrap Private Finance Initiatives. Public buyers out-smarted.
6) £8 billion savings in longer term.
Prevent an investors strike with no-one willing to buy government debt, and having to beg the IMF, who will make the government control its spending, anyway. And prevent losing credit status, which would make borrowing more costly and increase the nations problem. The following means necessary:
Emergency legislation makes public sector employees, on, say, over £50,000 p.a., get pensions based on average rather than final salaries with their retirement ages raised to 65.
[Reviewer comment: Early retirement makes way for youth employment.]
All lump-sum payments subject to full income tax. The Lifetime Earnings Limit for a special tax should have an equivalent in public sector.
Any public sector worker, getting £25,000 p.a. or more, should be disqualified from the basic state pension. They already get enough public money, saving about £400 million p.a. In all, public sector pensions cut by about £2 billion to £3 billion p.a.
Prosecute the bankers for financial wrongdoings. Maybe only would yield £50 million to £100 million but would send a message.
Bring benefits culture under control. French politicians constantly complain about how it is drawing mass immigration from all over the world. Costs more than government gets from income tax.
British passport should depend on working full time five years without benefits. Immigration on needs, not on who wishes to come here.
Cash benefits restricted to people holding British passports, who live here permanently.
New age of responsibility and self-sufficiency.
Keep fees for less needed courses, not essential skills.
No benefits or council house provision for under-21s. This should save £5 billion out of £140 billion in benefits.
Lots of unnecessary complicated legislation immensely costly and bureaucratic, such as 500 inclusion officers.
Darling made a small cut in VAT that was pointless and cost businesses an immense amount of unnecessary work.
Massive admin and enforcment machine for BBC TV license could be avoided by pay out of general tax.
Corporation tax evaded by larger companies going to other countries, so smaller businesses pay unfair share. Simplicity of using just one tax, VAT, hundreds of millions saved in admin and billions in lost tax could be collected.
[This reviewer doesn’t favor Value Added Tax, which may penalise essentials, as well as luxuries. VAT is essentially a toll tax, which Adam Smith argued against, in principle, as a halter on free trade.]
P is for Power to the People.
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Unions down from 13 million to 6 million in 30 years: 15% in private sector; 58% public sector. Public sector unions routed government attempt to raise pension age to same as private sector.
New Labour managed the Westminster lobby of journalists, favoring the friendly and pushing out the over-critical, notably Andrew Gilligan, former BBC journalist for correctly reporting deceit leading to Iraq war.
For a decade or so, Press failed to reveal how gross sums thrown at public services without much noticable improvement.
Media are moving from information to infotainment. Dumbing down, sound-bites. Lack of critical questioning of politicians.
The Freedom of Information Act passed by late 2000 didn’t come into effect till 2005, almost the longest legislative delay in living memory. In the intervening four years, there were constant rumors of departments cleaning up their archives and removing info in the name of efficiency.
Some success in exposing waste in British government, but FoI needs extending to discover how hundreds of bodies are using taxpayers money: city academies, regional development bodies, the nationalised banks, quangos and fake charities.
Above all, the unredeemed European Union needs transparency.
By 2008, a strengthened Corporate Manslaughter Act (Corporate Homicide in Scotland) failed in its former or later form to be used to prosecute the many needless deaths from negligence over infections under the Maidstone and Tunbridge Wells NHS Trust, 2005-6, and, from 2005-8, the Mid-Staffordshire NHS Trust.
The Human Rights Act of 1998 into force by 2000, makes it unlawful for bodies to contravene the European Convention on Human Rights. It gives rights but neglects responsibilities. Thus, a right to education neglects the responsibility to be educated to contribute to society. And a right to marry and have children neglects the responsibility to support them without depending on public (other peoples) money.
Craig and Elliott:
Like so many laws introduced by this government, it can seem as if the HRA has been mainly hijacked by the feckless, idle, greedy and criminal rather than serving the interests of the huge majority of the population.
A European Court ruling has “put the right to dignity of foreign nationals above the right to life of EU citizens.” The people who seem to have done best out of human rights are the lawyers, especially the Matrix group, where Tony Blairs wife Cherie Booth worked. Matrix lists at least eleven major areas of law affected by HR laws.
Thankfully, ordinary people also cottoning on. In 2008, a judge ruled that article 2, right to life, allowed sue the Ministry of Defense for faulty equipment causing unnecessary deaths in the army.
This could apply to gross failures to protect patients from infection or citizens from known violent offenders.
Article 41. The Right to Good Administration.
Wasteful and corrupt EU spending has caused the auditors to refuse to sign-off their accounts for fourteen years in succession. Any member country or group of citizens might prosecute the EU for failing good administration and in the required reasonable time.
Perhaps the squandering by the NHS IT system is another case in point.
Under British law, directors do not have a duty of care. Shareholders may bring class actions to sue directors for negligence. This includes many unit trusts and pension funds, yet none seem to have shown any appetite to sue the banks and their well-rewarded executives.
The Bank of England, the Treasury and “even the ever supine FSA” should be considering bringing civil lawsuits against many
for negligence, breach of fiduciary duty or violation of investment regulations by publishing potentially misleading information about the financial condition of the banks over which they presided. Even though some charges might be difficult to prove, faced by years of potentially ruinous litigation, many of our great financiers might be prepared to settle out of court. Given the vast wealth that these people have accumulated over the years, this action could rake in tens of millions for taxpayers and send a message to other people that the public, through their representatives, will not tolerate the kinds of behaviour that we have seen over the last few years.
Predictably but disappointingly, the politicians and the bureaucrats, who are paid so generously by us, seem unwilling to turn on their friends in banking.”
The “charmed circle” are almost never prosecuted while “the public are groaning under the thousands of new laws” introduced by New Labour.
Private prosecutions are costly and risky and would depend on wealthy philanthropists.
Equality and Human Rights Commission, 2007, merging three other quangos. 482 people, on £60 million p.a., have been used for trivial but very costly complaints.
The authors even encourage other trivial complaints to show how ridiculous it is.
(This reviewer does not approve of this sabotage. The complaints system of changing the burden of proof from presumption of innocence, to proof of non-discrimination, seems a bad precedent. We should not go along with it, even in jest. Besides it only courages a waste of public money.)
[_The post-democratic age. _]
Most laws made by 27 unelected European Commissioners in Brussels with 45,000 unelected officials and 100,000 part-time advisors. The EU Parliament can only make small amendments to about half of their laws.
Is this the death of democracy? the authors ask.
They suggest primaries, the Recall.
About 405 out of 646 MPs are in safe seats. In remaining 241, the swing has to be pretty large to worry most of these MPs. The 2001 general election saw almost no change.
The authors seem unaware that proportional representation does not have to mean party lists and more safe seats. The single transferable vote (STV) can make elections personly represent voters.
In British Columbia, the corresponding body to Britains Taxpayers Alliance was the first large group to come-out in favor of STV, during the deliberations of the Citizens Assembly on Electoral Reform.
7 april 2010.
[PS: It is ironic that I gave this hint about promoting STV to Matthew Elliott of the Taxpayers Alliance, because, in 2011, he headed the victorious, and I have to say, infamous, No-to-AV referendum campaign.
Nevertheless, the Alternative Vote was not a system to be landed with, as Australia has been, for its federal lower chamber. As the Australians say, all the Alternative Vote does is put the (over-all majority) post in First past The Post.]
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A default government pushes more nuclear power pollution.
Table of contents
“We don’t want to scare the country to death.”
Dwight D Eisenhower, in 1953, forestalling tactless truths about nuclear stockpiles of destruction.
Rear Admiral Daniel Gallery… asserted it was wrong for a civilized society like the United States to have as its broad purpose in war “simply destruction and annihilation of the enemy.” That kind of war was not as simple as the prophets of the “10 day” atomic blitz seemed to think…“Levelling large cities has a tendency to alienate the affections of the inhabitants and does not create an atmosphere of good will after the war.”
"...on the President's desk when he took office in January 1953...was the report of a special commission...Forecasting to the year 1975, the study predicted oil shortages and concluded: "Nuclear fuels, for various technical reasons, are unlikely ever to bear more than about one-fifth the load...It is time for aggressive research in the whole field of solar energy -- an effort in which the United States could make an immense contribution to the welfare of the world."
Quotations from The Nuclear Barons, first published in 1981, by Peter Pringle and James Spigelman.
Section links:
Misplaced ambitions.
The return of the radioactivists.
The dependent energy review (2006).
The renewed nuclear reign of terror.
The nuclear weapons connection.
A biased Horizon: when science is misused for propaganda.
Some energy alternatives.
Misplaced ambitions.
More nuclear fission power stations are the worst energy option of all: deadly dangerous, insecure, costly, inefficient and inequitable. That the Labour government should be promoting them in its 2006 energy review makes no sense at all. This debate should never have been re-opened after the 2003 energy review. The policy reversal may be put down to the anti-democratic centralism of over-bearing leaders serving monopolistic opportunism. A conclusion to be drawn from their fatuous decision is that politics pursues vested interests, such as the nuclear lobby.
Who will prevent future generations forever being left with corporate government legacies of poisonous wastes? The government is unscrupulous in pursuing more nuclear power. But the public, in a free society, could prevent them on this and other attempts to over-ride their wishes. Not only government incompetance is to blame but also public ineffectiveness.
A peoples response to this situation is that the public interest must be made foremost in politics. Two minimum but neglected conditions for this are firstly, for all official elections, the democratic voting system, the Single Transferable Vote, which seems to terrify most politicians more than another Chernobyl. Secondly, the principle of “equality of lobbying” implies universal vocational suffrage. All occupational and professional elections could include proportional representation by STV to the second chamber of government.
No-one could fail to be impressed by the energy released from an atomic bomb, not least the physicists and engineers and administrators who released it. Seeking to make amends,“atoms for peace” has been hand in glove with bomb production from the early years. Both became conventional wisdom. I remember in my impressionable youth believing that nuclear energy was the science fiction-like debue of an awesome power of the future. That is not entirely wrong but it has proved seriously premature.
Fission energy stations are the Professor Branestawm of big government and big business. They are one means of many for powering a dynamo or electric generator. But so-called “nuclear electricity” is just the stupidest possible means of turbine-turning. Using radioactivity, to boil water for steam turbines, loads the planet with highly diffusive and more or less permanent pollution.
Every other means of generating electricity, from harnessing the various renewable natural forces down to the wind-up radio or wind-up torch, are models of sanity and practicality, in comparison.
The argument that nuclear power reduces carbon emissions is specious, because its pollution is far deadlier than the carbon emissions of conventionly fueled power stations. And chemical pollutants might just as well be contained from fossil fuels, so that a new generation of coal and gas power stations could then be labeled environmentally friendly.
Coal-fired power could be modernised to run more efficiently at higher temperatures and pressures, cutting carbon emissions. This would be radicly cheaper than a new power station. This is not to forget that fossil fuels are inherently dirty and that organic chemicals really are much too valuable to burn, when renewable energies are available.
However, with new carbon-capture technology, carbon dioxide, from coal and gas, may be pumped into the ground, making such power stations minimal contributors to global warming. Again, renewables are preferable, because we cannot be sure carbon sequestration is secure. As to accidental carbon emissions from such up-graded stations, it would be no serious matter. Whereas if there are radio-active emissions, it may be a matter of life and death, for who is in the way of their dispersion, or whether the land and water they blight will be habitable, harvestable or drinkable for the forseeable future.
There is an other reason why we should not go for a new generation of nuclear power stations, which I’ve not heard mentioned in Britains current energy debate. So, I’ll mention it here to give it some prominence on this page. Current nuclear technology is, of course, fission energy. It takes ten or twelve years to commission a new station. They wouldn’t all be built at once.
In a few decades, they will be obsolete. The real new nuclear generation of power stations has just begun in France, site of the worlds first fusion energy station. This is the biggest joint scientific venture on the planet, second only to the inter-national space station. This has no radio-active by-products, with their unsolved problems of storage or contamination, weapons or security and costs. Tho, the fusion reaction would still need a radio-active trigger.
If humans endure to build space-ships, they would be powered by fusion reactors, probably using helium-3. They would have other independent sources of power, such as solar sails, that already power some satellites. If the fusion reactor breaks down and you have no other means of propulsion, you may be not only lost in space but dead in space. Helium-3 is not obtainable on earth but could be mined from the moon. The helium-3 process does not need a fission trigger reaction.
Any risk of radioactive contamination, in the confined area of a space-ship, would surely be eliminated from the design, because there is no-where else for a crew to go.
The moral is the same for space-ship Earth. We should not be building more fission reactors to spread their permanent poison over the planet. The poison, like less durable but still lingering chemical poisons, may or may not be harmful at low levels but that isn’t any reason not to prevent their build up.
Most people may feel they could live with fusion power. Nevertheless, fusion is not the ideal form of domestic energy production. It removes most of the danger but danger is not the only issue. It is extremely centralised power and as such vulnerable. It also makes a population dependent on it, vulnerable and over-charged. Small is Beautiful, as E F Schumacher writes.
Decentralised energy would be transmission-efficient and less costly for local consumption. Energy independence is in the interests of the people, the human race as a whole, rather than conglomerates making monopolistic profits from the centralised supplies of power stations, conventional or nuclear, that we have now.
Friends of the Earth may be right in prefering tidal lagoons as less damaging to the environment than one grandiose Severn barrage. Britain has by far the best tidal power potential in the world. Making use of many tidal lagoons around the British Isles would be more transmission-efficient and less vulnerable to accident and sewage build-up. It would also spare one of the most important bird migration spots in the world. The wonder-of-the-world can turn out to be a monumental folly. It’s called: putting too many eggs in one basket.
The return of the radioactivists.
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Just before the 2005 election, The Independent carried a leak of the Prime Ministers intention of a new energy review, to legitimise more nuclear power stations, rejected by the 2003 review. Labour kept it quiet so as not to lose anti-nuclear votes to the Liberal Democrats. But Labour could lose another million votes or so, in the next general election on this issue alone, if they elect the likes of Gordon Brown as the new leader. This would deservedly lose them the election. On 14 april 2006, Liberal Democrat David Howarth said: “Going through a long review process, only to come up with whatever answer Tony Blair wants to hear, is no use to anyone.”
Trade and Industry secretary Alistair Darling said he wants to make it easier to replace ageing power plants “to meet our energy needs.” Presumably, this means Confederation of British Industry (CBI) energy demands. Also, such as the white collar-union Amicus. Maybe a third of Labour MPs support more nuclear power. It is a jobs issue in often marginal constituencies. People will vote with their pay packets. But this is not necessarily in the long term interest of everybody.
In Parliaments energy review debate, a Labour MP said 17,000 of the 40,000 nuclear workers were in his constituency. He said they would take another nuclear power station. He appeared willing to take on more than one more. But it might be charitable not to presume on such gallantry. What cannot be cured must be endured. And, it seemed from this MP, might as well be endured with some braggadocio.
It’s one of the cases against single member constituencies. If there’s a nuclear power station in the constituency, on which a living depends, then they can sway who gets to be MP. With a multi-member constituency, candidates have to face opposing arbitrators, and can make rational choices the public can respect, rather than be in the pocket of one vested interest.
That’s only one example of the general vulnerability of single member constituencies to particular interests, regardless of the public interest. It may apply to constituencies with prisons, if prisoners are given votes in accordance with new European Rights directives.
We already knew the government want to “streamline” major planning inquiries. In january 2006, The Independent learnt that “senior nuclear industry figures also want to strip public inquiries of the power to investigate the safety of Britain’s new generation of nuclear reactors.”
The Prime Minister twice addressed the CBI on more nuclear power, suggesting who is pulling the strings here. Greenpeace activists disrupted the first meeting. And the CBIs Sir Digby Jones, just like a self-serving government minister, went on about the PM being the democraticly elected government.
Actually, he and his government are nothing of the sort. It is a usurping government with a usurping review. For one thing, Blair is not a president and would most probably not have been elected a third time. His governing party was elected only by default on 35% of the votes. They are not there with the consent of a majority of the British people. They are there because of a deficient electoral system, which is too beneficial to the two main parties for either of them to have the decency, honesty or integrity to democratise it. Tho, it is perfectly within the wit of man to have a democratic voting system, rather than various party-controlled shams they pass for voting systems.
Reported on 17 may 2006, Blair addressed again the CBI, to say that a new generation of nuclear power stations was “back on the agenda with a vengeance.” It is up to this generation not to have to take “Blairs vengeance” on future generations from permanent contamination by radio-activity. Blairs vengeance is just a melodramatic gloss on a failed experiments refusal to suffer private loss, as long as the public may be made to suffer.
Despite any follow-my-leader effect, an IBM poll, for The Sunday Telegraph, 21 may 2006, showed that 47% opposed new nuclear power stations in Britain. 40% were in support. 12% said they did not know. 1% refused to answer. 56% of men but only 26% of women supported new reactors. The report by Melissa Kite didn't say how many women were opposed. But 60% of women were said to be fearful of nuclear power stations. 49% of people said they were fearful of them and 47% said they were not.
British women have been culturally permitted to show their emotions more than men. The foot-baller Paul Gascoigne crying, over an unjust penalty at the World Cup, was voted Britains most memorable sporting moment. For a man to show distress, in public, was traditionly held to be an admission of weakness. Women may be less frightened of showing justifiable fear.
Kite said Westminster was taken by surprise because the PM had pre-empted the out-come of his own energy review. Greenpeace director Stephen Tindale also accused the PM of prejudging his review. The PMs bias leant weight to the widespread criticism that the review was a front all along.
Why did the PM take the offensive, giving offense to many? The Sunday Telegraph suggested several things like “sticking up two fingers to his own party and the CND.”
Well, he’d already done that, when he pawned the Labour party, without even its treasurer knowing, to scrape his phoney victory in the 2005 general election. This financial usurpation is just one symptom of general elections turned into a degenerate presidential election for a party leader, without electoral reform to an effective choice of parliamentary representatives.
The Telegraph also suggested he was asserting his authority by setting the agenda. This is what an authoritarian, who has announced “the end of the liberal consensus” in British politics, would do. As The Times Simon Jenkins said, Blair and Brown are “natural authoritarians.” The assumption of power is the enemy of questioning assumptions by which people learn. Knowledge depends on freedom as freedom depends on knowledge. Seeking to suppress or brush aside that freedom turns back or perverts human progress.
The PMs out-burst over-shadowed and over-rid a recent rejection of nuclear power by the governments own advisors, chaired by Sir Jonathon Porritt. The Independent summed their verdict on fission energy as: dangerous, expensive and unwanted. The growing waste disposal problem has not been solved. It is a target for deadly attack and use in nuclear weapons. They warned that the public could end up footing the huge bill, as they had for the previous economicly failed generation of nuclear power stations.
The public would, in any case, be effectively paying a huge policing bill of defending the indefensible. An expensive nuclear programme would divert limited resources and attention from a great variety of useful solutions with important contributions to energy production and saving.
Asked what he thought of Blair not waiting for his energy review evidence, minister Malcolm Wicks said: “Well, he is the Prime Minister.”
That’s a response that invites a fresh look at that office and its powers, which have burgeoned with the lack of democracy in general elections. The single members are monopoly nominations of a local party. Local party choice over-rides local individual choice. And national party choice over-rides local party choice. In the absence of presidential elections, general elections are substantially a choice between party leaders, because offering a mostly ineffective choice of local representatives.
Previous environmental advisor, to Margaret Beckett, said, on 10 july 2006, on radio, that Tony Blair favoring more nuclear power stations had been the worst kept secret in politics.
Jonathan Leake said:
although nuclear power provides about a fifth of Britain's electricity, this translates into only 7% of the nation's total energy needs.
About a third of the energy that we consume is in the form of oil and petrol for transport while the rest – mainly gas and coal – is used by industry and for heating buildings. Nuclear energy simply cannot replace fossil fuels for such purposes…
[The 2003 energy white paper] set out licenses restricting companies’ carbon emissions, grants for energy saving insulation and a range of measures that could all be used to reduce demand without affecting the economy or people’s lifestyles. Such measures could, it suggested, slash 25m from the 183m tons of annual carbon emissions in Britain.
The day before the Blairite energy review was due, The Trade and Industry Committee expressed concern that its out-come should not be rushed thru without consultation. They suggested energy short-falls may have been over-estimated and that prolonging the life of some existing nuclear power stations would be better than rushing into a new generation of nuclear plants. And they criticised the government for failing to carry out a “full assessment” of energy needs. MPs urged the government to ensure it has “broad support” for its policies and criticised it for failing to build a cross-party consensus.
But this is the problem with government under an exclusive political system. The partisan electoral system and whipping system, the lobby system empower exclusive vested interests with closed minds incapable of good judgment. Until people learn this lesson, they are always going to have this problem.
Politicly correct talk about “inclusion” is hypocrisy under party oligarchy. The PMs “Respect” agenda belies its intentions by setting up yet another wasteful bureaucracy. Most politicians do anything but empower the people with, for a start, the democratic voting system and a vocationly representative second chamber, which actually would respect everybody politicly and economicly. Place-holding politicians appear without self-respect, let alone for any-one else.
A University of East Anglia poll, reported, on 17 january 2006, said 63% accepted a mix of renewable and nuclear energy. This was interpreted as Britons "accepting nuclear power." But it is not clear from the other statistics that it means that at all. 62% said it doesn't matter what the public thinks as new stations will be built anyway.
Obviously, such a state of mind is not conducive to taking the trouble to assess for oneself what is best for the countrys future, as ones efforts will be wasted anyway.
As the 2006 Power Inquiry said: “The current system is killing politics in Britain.” And one might add that top-down decision-taking may kill more than politics.
In the same study, 78% thought renewable technologies and energy efficiency were better ways of tackling global warming. That suggests that Britain "accepting nuclear power" is a case of putting up with what big business and government are determined to shove on the country.
54% said they'd accept nuclear power stations if they helped to fight climate change. This is one of my favorite candidates for "There are lies, damned lies and statistics" (Disraeli.) The conditional question is based on a false premise. Critics say carbon dioxide emissions would only be reduced three to four per cent, at a giant cost, from new nuclear generators, that could be better spent. Hardly a recommendation for meeting the global warming emergency.
[PS. In 2015, EDF admitted, as soon as it was given the go-ahead, that it wouldn’t be on schedule.]
The question is like pointing a climate-change “gun” at the interviewees head and demanding nuclear power stations or else. The surprise is that nearly half the questioners refused to be intimidated by the bluff.
There is what lawyers call “a leading question.” In other words, nuclear power versus climate change is a mis-leading question.
The “leading question” really meaning a misleading question reminds me of the Labour party accepting “the principle” of its Plant report that there should be different voting systems for different political bodies. That is to say their “principle” is they don’t have any principle. The Labour party in government don’t know the difference between law and anarchy. To quote John Reid about his new ministry: They are “not fit for purpose.”
On 11 july 2006, the Green Party came up with their own poll of 500 Britons. They found almost 9 out of 10 reject the nuclear option. 98% back greater investment in renewable energy. And 99% said that more should be done to promote energy-saving measures in the home.
The Greens said: “This puts paid to any suggestion that nuclear power is accepted.”
**]The dependent energy review (2006).
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The 2006 energy review was no more independent than the so-called “Independent” Commission on Voting Systems, chaired by Roy Jenkins. On 11 july 2006, Alistair Darling heralded the energy review by saying nuclear power had always been part of the energy mix and “should remain so.” This statement is illogical. It doesn’t follow that because something has been, it should remain. And if it’s not meant to be a reason, then it’s mere assertion, which there is no reason to follow. Because the Minister, or his Prime Minister, says so, is not a reason for doing something. We do not have to acquiesce in an ignorant politics of unquestioned authority.
To this Pangloss government, its existing in its present state made it the best of all possible worlds, that must be protected from change.
The Tory opposition spokesman said the energy review was not so much carbon free as content free. The Liberal Democrat spokeman welcomed the positive aspects of the report on increasing the use of renewables from 4% to 20% by 2020 and energy conservation in households and appliances. But he accused the government of "surrendering" to the nuclear lobby instead of building a cross-party policy on energy. The Lib Dems are against the building of new nuclear stations. The current Tory thinking is that they should only be "a last resort."
Darling interpreted or misinterpreted this as meaning that the Tory leader didn’t want nuclear till later. It must be admitted that Lib Dem policy is more forth-right. They are renouncing any more of these hostages to fortune. Whereas the Tory reformers haven’t recognised there is no point in having even a few more hostages to fortune.
Darling took comfort from recently leaked Tory party e-mails that there was a rebellion against the Zac Goldsmith anti-nuclear stance. The Tories showed themselves to be as divided as Labour. This was Darlings excuse against a cross-party consensus on energy policy.
The Tory pro-nuclear rebels gave the Labour government a big let-off. David Cameron is going to have to get a grip on this issue, if it isn’t to undermine confidence in his leadership and his party. Blair and Brown have already revealed themselves as lost souls to nuclear power, not to mention other kinds of power.
[PS. This essay was written during the Cameron pretence they would be “the greenest government ever,” when he was palling with huskies, in the arctic, to scrape for votes from Green and Liberal Democrat supporters. Normal Tory turpitude has since been resumed.]
The government are going to make planning permission for windmills and nuclear power stations easier. Some local authorities already give automatic clearance for domestic wind-mills. Anyway, that shouldn’t be much of a problem. That is not much of a bargain if it is also not much of a problem for some consortium to put a nuclear power station by your house. So much for government even-handedness between big business and the public.
Yet, politicians dare not give permission to build a reactor in a built-up area. If it blows, it takes the city with it. Also, some research on the density of cancer-related illnesses, suggested their radio-active leaks might be deleterious to health. The government can only bully and bribe sparser communities into having to put up with nuclear fission hazards.
The Labour government has been cited as wanting six new nuclear power stations. They hope to follow (according to Finnish Greenpeace) the “stupid” decision of Finland, which has ample renewable energy sources but, after repeated pressure, needlessly paid for the multi-billion pound costs of turbine-turning radio-activity. This sends the wrong signal to Britain, which also has a wealth of renewable energy resources. The government no doubt counts on people having to make the best of its impositions.
A few new nuclear stations would make a negligible contribution to combating carbon dioxide emissions. So, that isn’t the real reason for it. It is also not so big a contribution that could not be spanned otherwise by Britains potential, tidal, wave, wind and geo-thermal, hydro-electric and solar power resources, plus energy conservation. So, that’s not the real reason for it.
Darling gave a possible clue: maintaining the status quo. The government is not really looking at a change in direction that would be in the public interest. These leaders are really agents of business as usual, regardless of the general interest. Also, vested interests reason for being is to aggrandise themselves, not put the general welfare first. So, one could expect a few nuclear stations to be expanded up to the original ambitious twenty threatened. Economies of duplication would be cited, a dishonest limit on ambition forgotten, after the foot is in the door.
Joan Ruddock MP asked how long would nuclear power take to make a contribution and how big would the contribution be. Darling just brushed the questions aside, by saying he didn’t think they were important. By refusal to answer her questions, he ungraciously conceded that new nuclear power stations have no significant contribution to make against global warming, indeed waste limited resources. And that the real reasons for them are the usual one that government puts the interests of the big business lobbies before the public interest.
To appreciate Parliament, one only had to listen to how MPs questions probed the weak points in the government position. Darling said the nuclear industry would pay its full share in the commissioning and decommissioning of plants. Some MPs wanted a more explicit definition of full share, such as 100% of costs. But the minister wouldn't be pinned down.
Another MP wanted to know if this nuclear self-financing included security.
It didn’t. The minister couldn’t wriggle out of that one, for fear of scaring away private investments in fission plants. He tried to make his admission as unobtrusive as possible, by merely saying he didn’t agree with the questioner. But this means that limited resources for public protection are diverted and concentrated on these white elephants. Private profits are massively subsidised at public expense to the detriment of public safety.
Another MP asked what about the supposed profit from nuclear, if the price of uranium rose. And another MP asked was it not inevitable that once nuclear power was in place, the country would have no alternative but to accept the going price, not being able to do without. The Liberal Democrats likened more nuclear power stations to another stealth tax.
[PS. This question proved prophetic. The Tory government was determined to secure a nuclear deal at any price. On 24-09-2015, an energy analyst interviewed on the BBC said wind energy was already cheaper. This was on the occasion that wind energy produced a quarter of the nations energy, for the first time producing more than coal at one fifth.]
Ned Temko, of The Observer reported, on 9 april 2006, that the government would cap companies liabilities and guarantee a minimum energy price before business risk takers would take any risks with nuclear power.
The environmentalist Tom Burke claimed: “since the Treasury will never agree to pay for the power stations, the electricity market will have to be rigged for 30 years to guarantee a return for nuclear investors.”
Jonathan Leake concluded:
Three decades of bigger energy bills for homes and businesses: will that be Blair’s real legacy?
[PS. This is what happened under “Blairs heir, David Cameron, a thirty-five year guarantee of ten per cent profits to the French nuclear industry.]
Michael Meacher MP wanted to know from the government why the new nuclear stations, given that we already don’t know what to do with all the radio-active waste, the huge insecurity and uneconomic expense to the public. Some MPs tried to shout him down. The Speaker had to call for order to let him be heard.
Darling appeared to try to forestall his question by saying “I know where you’re coming from.”
The Minister as witch doctor sounded as if he’d seen an approaching asteroid from outer space, and hoped to divert it by saying “I know where you’re coming from.” Darlings reaction personified the charade of private-interests government. It pretends it is doing something but really does little more than hope Earth misses the “asteroids” of vested interests and their disasters.
The threats of these disasters were illustrated in two stories, I happened to see on the same day, 23 july 2006. One was in The Sunday Post. It revealed that several truck loads of radio-active waste had been side-lined in a heavily populated area for eight years. This was excused on the grounds that the British government couldn’t come to some sort of agreement with the Egyptian government.
The other story was of a radio-active truck-load intercepted on the Bulgarian border, on its way from a British firm, with apparent export approval, to Iran. The contents were refered to the Bulgarian atomic agency. According to the Mail account, the lead containers were destined for the Iranian Ministry of Defence. They contained Americium-beryllium capable of use for manufacturing a “dirty bomb.” Such material is “mainly found in spent reactor-fuel elements and is not at all easy to get hold of.”
A similar incident happened in august 2005, this time concerning a ton of zirconium silicate.
These chance news items high-light why Meacher found ominous all the nuclear waste sloshing around the country and the world.
Michael Meacher might have made a good Labour party leader. He espoused radical remedies before they became fashionable enough for David Cameron to make a sensation by Toryising them. Meacher was reviled for what now earns Cameron credit of being a Nice Man, Pity about the Party. To date, only one little-known Labour MP, John McDonnell, has put up against the spend-thrift Gordon Browns coming leadership “coronation.”
**]The renewed nuclear reign of terror.
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Private profits at social costs have already been responsible for serious radio-active pollution of the planet, just as business products have chemicly contaminated the world. A september 2005 study, of the chemical industrys legacy, found lingering traces of everyday chemicals, in mothers and children, which can lead to birth and growth defects.
In June 2005, it was revealed part of a Thermal Oxide Reprocessing plant at Sellafield Cumbria could be closed for months due to a leak undiscovered for up to eight months. Safety regulators claimed the discharge could result in criminal charges. These were brought in May 2006.
In august 2005, the Nuclear Decommissioning Authority wanted to speed up the cleaning of twenty civil nuclear sites from 125 years to 25 years. It gave a cost for decommissioning waste and its storage of up to £56 billion. The figure in The Sunday Telegraph, 21 May 2006, is £70 bn, which the taxpayer is already having to pay for. This is as well as the many billions of subsidies that nursed nuclear power over fifty years.
Even Malcolm Wicks, the energy minister, in charge of the current review, said this is a “disgrace.” And, I stand corrected, if he was not the man who notoriously said he wouldn’t have any “prejudice” over nuclear power.
Radioactivity afflicts alike the prejudiced and unprejudiced.
Elliott Morley was reshuffled out of the environment ministry, as one can understand from the following sane remark:
To have new nuclear power is going to involve very large sums of money. If nuclear power was so great then you would have the private sector willing to invest in it. The reality is that economically the risks are great and the returns are low.
In april 2006, Florida Power called in the Federal Bureau of Investigation and offered $100,000 reward to find out who drilled a small hole in a cooling system pipe for one of its reactors, and whether or not it was an accident.
Just consider that accident or no, in the light of apologies for more nuclear power stations. James Lovelock, of the “Gaia” concept, at first alleged that these plants would not be subject to sabotage – because he said so, one supposes. Later he wrote an article for Readers Digest, in which the punch-line was that the concrete casing of a reactor core could not be penetrated by a crashing air-craft. Yet Florida Power goes into red alert because some-one happened to drill a little hole in a pipe. Invulnerable indeed!
5 July 2006, applying under the Freedom of Information Act, revealed that the Nuclear Safety Directorate issued warnings over unexplained cracks in reactor cores of UK power stations including Hinkley Point B. British Energy was also criticised but no immediate public risk was found.
So, there it is! Reactor core coats in future allegedly cannot be cracked. But for the present reactor cores themselves have, well, cracks.
The Independent, 14 january 2006, reported a nuclear physicist as saying: “The public have the right to know the danger. The government says the terrorism threat is real.” He predicted an attack on a nuclear power station could kill over two million. The report continued:
The worst-case scenario could see 2,500 kg of caesium-137, the most dangerous isotope, escape – 100 times more than that released in the 1986 Chernobyl disaster.
On 18 april 2006, official UN figures predicted 4000 extra cancer deaths from Chernobyl fall-out. Greenpeace claimed that recent studies estimated there will be 100,000 extra, many in the Ukraine, Belarus and Russia. The BBC docu-drama on the 20th anniversary re-called the terrifying disaster that Soviet scientists did not know whether they could avert. Unchecked, Chernobyl would have resulted in a massive thermo-nuclear reaction, with millions of casualties, and amongst other things, the permanent poisoning of the water supply from two of the great river systems.
27 april 2006, security specialists told the Committee on Radioactive Waste Management that ministers must act against terrorist attack. “Deep disposal” was recommended, but where, they would not say.
The golden rule is that if you don’t want it in your back yard, then you shouldn’t inflict it on any one else. So, there’s no point in producing more unwanted radioactivity. Our undemocratic government is moving inevitably to over-riding local communities on waste disposal, as it promotes the producing of more.
The Mail on Sunday subsequently carried an article by Jason Lewis that Britains shortage of scientists meant an influx of foreign experts had to be screened. 18,000 last year meant the Office for Civil Nuclear Security was already struggling with the work-load. The head of the Office warned that Blairs plan to build a new wave of nuclear plants posed a major risk of terrorism.
“It would make no sense to authorise someone to construct a site who then passed that knowledge to someone with malicious intent.”
Britains intelligence service, MI6 admits its recruitment drive has resulted in attempts at inflitration.
The report continued: “This month the Prime Minister struck a deal with France to create a new wave of atomic power stations in the UK.”
The Mail, also revealed that Chancellor Gordon Browns brother Andrew is on board of the French nuclear industry. No wonder then that he supports an extension of nuclear power in Britain. The Mail also revealed that the American nuclear firm, whose cover-up was told in the movie “Silkwood,” had bought on board Tory tv personality and newspaper columnist Michael Portillo.
The nuclear weapons connection.
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In 1945, the Hiroshima atomic bomb killed some 80,000. In the following months, some 60,000 died of radiation poisoning. At Nagasaki, the second nuclear bomb, killed 39,000 out-right with another 75,000 dying from radiation poisoning.
These were only minor fission explosions compared to the hydrogen fusion bomb tested a few years later.
The Mail carried an article recently of a big increase in recruitment at Britains nuclear weapons facility. All this is going on, as if it was administrative routine, when it signals major policy decisions taken without leave of the public.
In the same paper, on 25 june 2006, Suzanne Moore commented on Gordon Brown:
We are to have a replacement for Trident whether it works or not, whether the military wants it or not (many don’t), without a debate.
This is the biggest spending commitment Brown has ever made…
An increasing number of people, not just on the Left, feel that no one is representing their views in Parliament at all. This decision, which appears to have already been made, is not a deterrent to anything except a properly functioning democracy.
A few pages on, William Rees-Mogg defends Browns decision. He is well stocked with dreadful memories of the Cold War lasting some forty years after 1945. Indeed, The Sunday Times of 9 july 2006, carried a piece about the increasingly Soviet style repression of opposition opinion. It’s fair to say, he thinks, that in a world of increasing proliferation of nuclear weapons among unstable nations, it makes sense for Britain to maintain its own.
Rees-Mogg says that Britain is Americas junior partner with its deterrent – America supplies the missile system – whereas France has an independent nuclear deterrent.
Actually, the United States did give France secret help with developing the neutron bomb. We remember President Chirac, with reckless national pride in coming to office, setting off nuclear explosions in a Pacific atoll – some-one elses irradiated back yard. Crack the foundation of the island and you would have major radio-active ocean contamination.
A Commons committee suggested that Britain no longer needed Tridents 24-hour state of alert and called for a scaled-down deterrent. There no longer is an expansionist, much less a Stalinist, Soviet Union.
On the 10 july 2006, twenty bishops wrote in The Independent that Trident was “evil” and that “possession and use are profoundly anti-God acts.” Nuclear warfare would kill millions of innocents and rain sickness on the earth. The guilty would be the best prepared in their bunkers.
The bishops said the money would be better spent on helping developing countries. It would show courage that could be respected. This could spread good will, prosperity and progress. That is, if, as usual, public money isn’t thrown away on corrupt and undemocratic administrations.
I repeat the need for the moral power of example in the proper democratic standard of voting system, and in a two-chamber representation of the economy as well as the polity.
The nuclear submarine has been the capital ship for over forty years time. That’s an unusual length of time in a faster changing world. And the Trident replacement is being projected for decades ahead.
The battleship was the capital ship from the turn of the twentieth century up to the second world war, when events proved that, within a forty-year span, it had been superseded by the aircraft carrier. Meanwhile, magnificent battleships were still being built, tho these armoured dinosaurs would be sunk like floating tin baths. The Bismarck was a marvel of German naval engineering but its fate was sealed when its steering was jammed by a torpedo from a few obsolete carrier-planes called “stringbags.” All the contending naval powers lost costly battleships by post-Jutland battle rules of engagement. The world moves on.
Perhaps the moral is that if the world is serious about containing war, it must limit the means to fight it, by due process of international law. And outstanding national grievances must be attended to. As the bishops say, money is better spent on plow-shares than swords.
The G8 powers met in St Petersburg with “global energy security” top of their agenda. On 9 july 2006, Teletext reported a leaked action plan for mass expansion of nuclear power for G8 countries with a network of nuclear fuel plants along with reactor sales to developing countries.
Typically, some plan, that the politicians know many people don’t want, has to be leaked before we find out about it.
The title of this page refered to “a default government” meaning my own country with its spurious electoral system. But a world-wide default government is the periodic world council of the G8 premiers, when it routinely takes fateful decisions over every-ones heads.
For domination in action, it is as if the aggression of nuclear warfare is ritually displaced on the populace thru the pollution threats from unwanted new nuclear plants.
The 8 july Washington Post reports the Bush administration will pay Russia billions “to dump spent nuclear fuel there.” This agreement promises to be unpopular across the Russian political spectrum. American government protests, against declining democratic standards in Russia, look like a bad act, when they are evading domestic protests against nuclear power, by dumping its waste on a country undefended by Americas constitutional tradition.
British attempts to reproach the Russian presidency were met by Vladimir Putin retorting that at least he didn’t sell seats in the legislature. Peer nominations, who happen to be party donors, go on under the smug delusion that Britain is a democracy.
What is wrong with politics?
Ministers have become Fixers and the Prime Minister has become the Prime Fixer. Politics have become party wars between lobby alliances. Force and fraud, as against freedom and reason, have become institutionalised in an obsolete constitution. Public-spirited causes have found party politics so futile that they are mainly extra-parliamentary pressure groups. This disaffection is an index of the electoral inefficiency of representation, that allows party government to be hi-jacked by unpopular policies.
The whole political and economic system from the ground up needs opening to the general public. I’ve dealt with some neglected essentials, tho by no means all issues, such as campaign finance reform, that does not owe the parties a meal ticket. There’s not much here about freedom of information, which is under renewed threat after being belatedly introduced. There’s not much about parliamentary procedure and the balance of power between the branches of government, or a Bill of Rights. Others are more expert on these and other constitutional reforms.
A biased Horizon: when science is misused for propaganda.
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The early advocates of nuclear power promised it would bring electricity “too cheap to meter.” As late as 1977, perhaps up to the eve of the Three-Mile Island melt-down, The McGraw Hill Encyclopedia of Science and Technology, which consists of authoritative articles by hundreds of experts, gave a table of probabilities for fatalities. They ranged from one in several thousand for motor accidents to one in a quarter of a million for tornados or hurricanes. But the rate given for nuclear reactors (over 100 plants) was one in five billion.
This was given as an average chance per year.
So much for the geological eras that life needs to be protected against radio-active waste.
There already had been some deaths kept secret and maybe many more.
On 13 july 2006, a BBC 2 Horizon program appeared on evidence challenging the risk of radiation exposure. The Chernobyl wild-life researcher himself said he had doubted his own findings. This means that tho his findings may be valid, they are puzzling in the light of other findings, and that scientists do not yet know all the ins and outs of their discipline. This research was not an occasion for jumping to conclusions.
Unfortunately, the program makers ran with the study to minimise the Chernobyl accident and conclude that more nuclear power might be desirable after all, allegedly to combat global warming. This was very timely political backing by “Science” with a big-s for the just published government energy review (2006). Instead of the Government becoming scientific, Science became propagandist.
Good information about the exact risks of radiation exposure remains desirable. But Horizon just took one side of the conflicting evidence and jumped to conclusions that will not promote the public image of radiation experts. The book, The Nuclear Barons, shows they have been discredited before for unreliable assessments.
Mass abortions, carried out on women in the after-math of the Chernobyl explosion, may have been largely unnecessary. There are, however, many serious reports of evidence of mass deaths and illnesses, despite official suppression and cover-up. (See the Chernobyl 20th anniversary web page.) The truth is great and will prevail. We are still waiting.
Horizon didn’t mention that the Soviet Union already had a disaster with radio-active fall-out from some sort of explosion at a nuclear waste dump. No public protection measures were taken till radiation sickness appeared. Mass evacuations and foodstuff destructions followed. Two years later, a physicist driving thru (advisedly at speed with windows shut) recalled:
“It was like the moon for many hundreds of square kilometers, useless and unproductive for a very long time, maybe hundreds of years.”
A decade later, local doctors were still advising pregnant women to have abortions.
The Kyshtym explosion, at end 1957 or start 1958, was covered-up but the experience may have motivated policy at Chernobyl. A speaker on Horizon used the term “hysterical” to describe public opinion on radiation hazards. The term “radio-phobia” was also trotted out. So, it is perhaps fitting to recall what Peter Pringle and James Spigelman, in The Nuclear Barons, describe as “a hysterical reaction from Western nuclear advocates” in 1976, to news of Kyshtym.
In the future, mankind should not be putting itself into situations where it has to make marginal decisions on levels of radio-active fall-out. Human beings and their welfare should not be a secondary consideration to nuclear energy investments.
Having said that, the Horizon characterisation of the Chernobyl accident was just plain wrong. To quote from the wind-up statement: "Chernobyl was as about as bad as a power station accident gets -- a complete melt down of the reactor core --".
The truth is the consequences could have been incomparably worse. The core “melt down” may become an unstoppable temperature increase characterised as “The China Syndrome.” American critics warned of uncontainable radioactive pollution, fancifully pictured as sinking right thru the Earth to China.
The Horizon program says most of the Chernobyl accident deaths were to the clean-up workers, citing 47. Unofficial sources have put the death rates much higher among the thousands who were conscripted.
And Horizon doesn’t refer to the heroic men who gave their lives, in containing the reactor melt-down, to prevent millions of people from dying from a nuclear holocaust. The 2006 BBC docu-drama on Chernobyl offsets the BBC Horizon misrepresentation. As the 20th anniversary docu-drama said: It was like 1941 all over again.
All Horizon could do was dismiss an alleged 56 deaths over-all, as “less than the weekly death toll on Britain’s roads.”
Never again! should have been the program message, if they’d had their priorities right.
The Horizon presentation of the Chernobyl accident measured the decline of radiation from the source. But there was no mention that it was pure luck that the radiation was not blown onto Kiev. One of the directions, it blew, blinded a Polish farmer, as The Sunday Times reported.
The balance of the official Chernobyl death toll was made up from nine deaths of children from thyroid cancer. You would think from Horizon that was it. However, Bernice Davison in The Telegraph, 22 april 2006, reported in Minsk
the new and large children’s cancer hospital, which specialises in looking after “Chernobyl victims”. For it was Belarus that bore the brunt of the radioactive cloud that poured north after the Chernobyl explosion…
Leukaemia and thyroid cancer rates (especially in children) in countries across eastern and northern Europe increased,..
28 countries are donating billions of dollars and limitless expertise to building a further new overcoat for this troublesome building.
I had expected the reactor to be cordoned off and abandoned, but workers were being disgorged from buses outside, preparing to cross the road for their next shift. Hundreds of people – electricians, carpenters, doctors, hydrologists, miners, meteorologists, scientists, cooks and cleaners – work each day in the heart of the dead zone, still trying to contain and clean up the reactor…for 15 days at a stretch,…
The Chernobyl sarcophagus will remain radioactive for at least 100,000 years. And the world is having to slave to rebuild it after only 20 years. The most enduring of human monuments, the Egyptian pyramids were built 5000 or 6000 years ago and their civilizations are long forgotten.
Perhaps future legends will say that the earths radioactive hot-spots were the work of certain conceited but malicious apes, who called themselves “wise” but were just too clever for themselves.
There is a danger that a Chernobyl could happen to one of Indias many unsafe plants, described as disasters waiting to happen, in highly populated areas. The humanitarian relief problem could strain world efforts, as never before, in an age when mankind already can hardly cope with all the global emergencies.
This is virtually all India has to show for an enormity of misapplied effort and expense over nuclear power. That is fearful hazards and, of course, the bomb. This drove Pakistan to make its own nuclear bomb and test fire missile systems. And Pakistans nuclear secrets were illegally passed on. So, Indias bomb hardly enhanced national security. Indias folly is not so different from that of the West, except that its poverty was less able to bear it.
So, peaceful nuclear power has been the road to nuclear weapons proliferation. The Third World wasted its substance on its own Cold War.
A recent Swedish research found a higher than expected long-term effect of Chernobyl on cancer levels. (Unlike Finland, Sweden has had the sense to go for renewable energies instead of fission energy.) The purpose of research is not about what levels of radio-active leakage a nuclear plant can get away with, so that investments are not threatened.
The Horizon conclusions on nuclear power might be likened to making some program, not to worry about ozone layer depletion, and not muzzle the chemicals industry, because low levels of ultra-violet radiation could be beneficial rather than carcinogenic.
But it was scientists who discovered the hole in the ozone layer and action against it has become one of their causes.
Physicists, however, were responsible for discovering nuclear energy and some seem to feel they have to justify fission energy at least for peaceful use. Scientists, above all, as their progressive profession demands, should be able to admit mistakes. There should be no mistakes too big to admit.
Some energy alternatives.
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Taking subsidies and environmental costs – including the to-be colossal global warming costs – into account, relatively poorly invested renewable energies are far cheaper than fossil or nuclear fuels.
Costings conclusion of an e-mail letter by Aidan Constable to The Guardian, Life, 10 feb. 2005.
Costs of nuclear reactors were made to appear lower by an estimate based on supposed achievement by an 8th reactor in a series. Also construction costs more in wealthy countries. First of a kind design-cost increases, delays and cost over-runs are endemic to massive technical projects. Supposed performance levels are higher than those typically achieved. Also to be costed are risks from terrorism, nuclear weapons proliferation and accidents.
MIT estimated increasing nuclear power world electricity from 17 to 19 per cent by 2050 would mean nearly trebling capacity or 1000 to 1500 more plants. But known supplies of uranium only last another 85 years at 2002 levels of use.
Graham Sinden of Oxford universitys Environmental Change Institute has researched, for the Carbon Trust, a viable mix of alternative energies to meet continuous demand, Looking at past weather records, he estimated that the best mix was 65% wind, 25% domestic Combined heat and Power (dCHP) boilers, producing electricity as they heat water, and 10% solar cells.
Wind is most important because it blows most in winter and the evening when demand highest. The dCHP also produces more at peak times with combined demand for hot water and heating. Solar helps when the production of the other two are lowest.
The wind or solar generators need to be dispersed so that they produce electricity if wind is blowing or sun shining somewhere, if not always in the windiest or sunniest parts.
Sinden worked out the need for stand-by capacity would be reduced from 90% to just 11%. Sinden also points out that a combined wave and tide system works better in meeting demand than tide alone, which is predictable but variable.
Altogether, Sinden reckoned that more than half of Britains electricity could ultimately be derivable from intermittent renewables.
Oliver Tickell, The Guardian, Life, 12-05-2005.
Andrew Simms said:
A flexible, safe, secure and climate friendly energy supply can be delivered by renewables. A broad combination of wind, solar and geothermal power tapped into with a range of micro, small, medium and large scale technologies, applied flexibly, could more than meet all our needs.
Thomas Edison first built a power plant in 1882. He believed in a decentralised energy industry. In 1907, 59% of American electricity was from small scale generation: more secure supply less prone to black-outs, more energy efficient than a national grid. OFGEM says the National Grid loses power as heat that costs the UK nearly £1 bn a year.
The Network for Alternative Technology and Technology Assessment was that if ten million consumers installed 2kW of microgen solar PV or wind systems, they would supply as much power as a UK nuclear program.
The Ashden awards for sustainable energy (www.ashendenawards.org ) judged two winners. ALI energy, a Scottish program of biomass heating, geothermal heat pumps, wind and solar energy, plans to make Argyll the first part of Britain entirely on renewables. The island of Gigha built a wind-farm, producing 75% of their electricity, which was the first community-owned and grid-connected.
Another award went to the Edinburgh-based Swift roof-top turbines. This is a small quiet wind generator capable of producing much of average house-hold electricity, and also supplying back to the grid. 4000 were ordered for 2006. The company is currently tooling up for mass production.
Guardian Life 30-06-2005. Andrew Simms “Power to the people.” Co-author of Mirage and Oasis: energy choices in an age of global warming.
A Russian wind turbine is being developed by scientists from the Makeyev State Rocket Centre near Miass. The blades, made of light glass fibre, move around twice wind speed, which is slow enough for birds to see, and almost silent. It looks like an egg-beater, is much cheaper than conventional design and of wider application, such as fitting to top of a house. The US company Empire Magnetics supplies the turbine alternators. It is being commercially developed with funding from US Department of the Environment.
Guardian Dispatch 25-11-2004.
Infra-red solar cells.
Edward Sargent and colleagues from University of Toronto, in Nature Materials, report creation of tiny semi-conductor crystals that can soak up infra-red light, half the suns energy, producing much more electricity than conventional solar cells. New nanocrystals as plastic solar cells are efficient and cost-effective. They are cheap enough to produce, large scale, and small enough to remain in solution such as paint, or they could be contained in tarmac or textiles. One-thousandsth of the US is paved with roads, which could supply all US energy needs if it could convert the suns power into electricity.
The new technology should be available within 10 years.
Guardian Dispatch 13-01-2005.
Stanford university scientists global wind map.
In Journal of Geophysical Research-Atmospheres, Cristina Archer and Mark Jacobson analysed wind speeds from around 7500 surface stations and 500 wind balloon stations to work out speeds at 80m height of modern wind turbines. They found 13% of sites, with winds of at least 6.9m per second. Wind could generate power enough for world energy demands.
At around 72 terrawatts (72 × 1bn watts) of power, this is equivalent to more than 500 nuclear reactors or thousands of coal fired plants. North America has greatest potential. Some of strongest winds in North Europe are along the North Sea. South tip of South America and Tasmania also recorded sustained strong winds.
Guardian Life Dispatch 19 – 05 – 2005.
Greg Barker the Tory environment spokesman was quoted by Sunday Telegraph, 21 May 2006, as saying:
… decentralised energy (DE)… may offer the best way of using the market to stimulate the necessary research, development and innovation required to…harness…renewable energy technologies…also…delivering energy to consumers in a far more efficient method…
DE could offer a truly substantial reduction in UK CO2 emissions... also...enhanced energy security -- less susceptibility to power failure cascades, terrorist attack or energy dependence on other states.
DE implies local combined heat and power generators and household roof turbines, and perhaps solar panels, with surplus electricity sellable back to companies, as in Germany. Ultimately half of electricity would be locally generated rather than from the National Grid.
In the 2006 Energy review debate, Alistair Darling couldn’t contemplate the prospect of winding down the National Grid. Again the attitude was, it had always been there and served well.
Jonathan Leake says:
Britain wastes more than half the power it produces through generation and transmission losses in the National Grid. Inefficient homes and businesses lose another 13%. Better transmission systems and insulated homes could reverse the growth in demand.
[PS. In 2015, visiting the EU, Nicola Sturgeon featured the need for a European-wide energy grid to maximise the utilisation of wind power, much of which comes from the North Sea turbines.]
The above reports on energy alternatives are doubtlesss only a tiny sample. The few innovations mentioned here don’t all know about each other, so that there must be scope for greater integration and more effective use of renewable sources of energy.
The lack of resourcefulness and imagination of so-called leaders makes them look about qualified to work a tread-mill.
A moral of the nuclear debate is that you have to conclude that governments fight policies as they fight elections. The purpose, of our undemocratic voting methods, is not to represent the people but to win power. The purpose of policy debates is not to represent the public realities but the private interests that drive the parties. Politicians are not interested in the true representation of issues any more than they are in the true representation of the peoples judgment thereon.
The truly representative voting system (transferable voting) and two truly representative chambers, political and economic, are necessary, but not sufficient, conditions for bringing honest debate into political economy.
In writing this page, I haven’t taken my cue from the environmentalists, tho I’m interested to know just what they make of the latest efforts of the nuclear pushers and apologists. I felt an obligation to counter, as much as possible, further spoiling of the planet for future lives, who are defenseless against present recklessness.
“Twenty Years After Chernobyl – April 26, 1986.”
This gave many more reports of Chernobyl-related mass deaths and illnesses. Recommended as an alternative to BBC Horizon condescension and complacency. Also there are many in-depth links and coverage of related issues.
“The International Campaign for Justice in Bhopal.” Has a web site on lack of safety standards in the nuclear, as well as the chemical, industry. Bhopal, of course, was scene of the worlds worst chemical pollution accident. Indias obsolete plants are among the worlds worst, with many serious accidents and near disasters, covered-up by government secrecy.
Peter Pringle and James Spigelman (first published 1981): The Nuclear Barons. The inside story of how they created our nuclear nightmare.
Jonathan Leake, The Sunday Times, 27 november 2005: Now for Blair’s dodgy nuclear dossier.
like its predecessor that was used to justify the invasion of Iraq, it will not be an independent inquiry but one led by members of Blair’s own strategy unit…
July 2006; modified 27 july ’06.
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Big Business leads New (for Nuclear) Labour to assault the future.
Table of contents
26 July 2007.
A forum message, on the site of the Porritt report, said critics of nuclear power were 20 years out of date with the advent of thorium power. On following this lead, it turned out that this was a fifteen year project – according to its advocates, such as Prof. Egil Lillestol trying to persuade Norway. In other words, its advocates are perhaps 20 years ahead of themselves but it sounds better to make out its critics are 20 years behind themselves.
On that time scale, thorium power is no answer to what the consensus of scientists regard as the immediate problem of reducing global warming.
On the Treehugger site there is a discussion: Thorium solves global energy shortage? The thorium process is exempt from the melt-down problem and produces uranium too contaminated for the chain reaction of nuclear weapons. A discussion member told of how his father had tried to promote the thorium alternative but that the industry wasn’t interested, apparently because they wanted the nuclear weapons capability of uranium fission energy plants. The plutonium by-products were really end-products. Indeed, for human and animal life on the planet, they could be: The End.
There are several possible technical alternatives to producing thorium power. For instance, one design uses liquid lead that could pose a contamination problem but it may also have its advantages. From all the complexities, certain salient facts emerge. Thorium power will produce radioactive waste lasting 500 years. This would normally rule it out as a prudent energy source. But it could also incinerate plutonium waste, from conventional nuclear stations, lasting geological eras.
For this reason alone, thorium power stations probably would have to be built, provided they can reduce the stockpiles of the most long-lasting wastes. Normally, we wouldn’t contemplate the production of 500 year-lasting ecological time “bombs,” but that would be better than 100,000 year waste-storage problems. We do have the moral obligation to lift this curse on Earths descendants. Thorium power appears to be a lesser evil that can mitigate a great evil. However, environmentalists will have to confirm that thorium power does have this benefit. In particular, only the appropriate designs for this purpose must be guaranteed.
Also environmentalists will have to oppose the expansion of conventional nuclear power, using the excuse that future thorium stations can clear up afterwards. They will also have to counter a new propaganda to promote thorium power, from a lesser evil helping somewhat to clear up a great evil, to a nuclear wonder solution.
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8-9 July 2007.
Now at last the New in New Labour stands for something: Nuclear Labour or poisoning the planet.
Investors wouldn’t come forward if they were liable to the possible catastrophes from new nuclear power stations. The public, exposed to these unnecessary dangers, will have to pay for the calamities they may engender, for the benefit of artificially created private profits. And the industry gets special benefit of state security, being a vulnerable target, instead of everyone benefiting equally from the states duty to protect its citizens.
The Inter-governmental Panel on Climate Change recently played down a part for nuclear power against global warming. Moreover, any large scale program was prohibited by the unsolved problem of safe long term fissile waste storage and the increased danger of spreading nuclear weapons from more material produced by more nuclear power stations.
In july 2007, Al Gore, as unofficial world ambassador for climate change awareness, echoed the scientific consensus, in a BBC tv interview, that nuclear power has only a small part to play in combating global warming. If they can solve the problems, fine, he added.
Gordon Browns last budget announced a two per cent cut in income tax. When all the small print was taken into account, the reduction was denounced as a conjuring trick. Well, I’m no accountant but I guess that the fabled income tax reduction was window dressing to disguise the real reduction of two per cent in corporation tax, for his real bosses.
The prodigal chancellor, or “the credit card chancellor,” as Michael Howard called Gordon Brown, formed a government in mid-2007 that entrenches the nuclear power lobby. In Blair and Brown, we have had not leaders but lobbyists. Browns brother Andrew is a director in the French nuclear industry. Blair did a deal with it, without consulting the British people about a fissile future.
A socialist point of view from Julie Hyland claims:
Tory spokesman Greg Barker commented “The nuclear lobby appears to have an arm-lock on New Labour…” when then planning minister Yvette Cooper created a quango, for big projects like nuclear power stations, to brush off local opposition. Her father Tony Cooper was recently chairman of the Nuclear Industry Association and is now Director of the Nuclear Decommissioning Authority. Her husband is another minister, Ed Balls, “close confidant of Gordon Brown.” (Mail on Sunday, 20 May 2007.)
Mr Brown is a family friend of the former left-wing MP Martin O’Neill. Lord O’Neill has become chairman of the Nuclear Industry Association – it’s like nuclear “musical chairs” between the Brown clan. I don’t dislike the man. I abhor what he and his alleged “cronies” are planning. The Financial Mail (17 June 2007) reports: “it sends the clearest possible signal that Labour is now pro-nuclear.”
Briefly before his appointment as Browns chancellor, so that it looked as if he were flying a kite, Alistair Darling displayed his ignorance of the alternatives to nuclear energy. He dismissed windmills as an “eyesore.” That is a subjective opinion and typical of arbitrary rulers to be so guided. Radioactive contamination is a sight more of a sore than a figurative eyesore and that is an objective fact, he chooses to ignore.
When he was Tory leader, Michael Howard announced, before the 2005 general election, his intention to start a new generation of nuclear power stations. It was wrong to try to impose this “Faustian bargain” on the public. But at least he was honest enough to admit his intentions. Blair and Brown New Labour hid the same intentions, till after the election, which is about as honorable as radioactive rape of the planet.
One of Browns outside ministerial recruits is Sir Digby Jones, who was chief of the CBI when Tony Blair chose, post-election, to publicise his commitment to nuclear power. Even by deceit, New Labour only managed 35% of the votes against a more disliked Tory party openly with a pro-nuclear policy. And Digby Jones was canting, about Blairs democraticly elected government, against the Greenpeace demonstrators. To be sure, Mr Blair chose the most sympathetic audience in the CBI, private gain never having been strong on social conscience.
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Current Tory leader, David Cameron made a welcome change of direction towards decentralised alternative energies and conservations. But their manifesto looks like a compromise cobbled together with the radioactivists, rather than a coherent policy. Another former Tory leader, William Hague has sought to by-pass him by calling for cross-party support on nuclear power. So much for voter choice!
Tory policy priorities include security of supply, and a level playing field between different energies. But the fact is that nuclear itself is not secure, not now or ever. And the level playing field over-looks the fantastic amounts invested over fifty years on nuclear energy, which has still failed to provide security from its by-products.
It looks as tho the Tories remain divided and untrustworthy on this issue. Only the Liberal Democrats, which have a long record on green policies, definitely promise that they won't build more nuclear power stations. But they and the Green party are kept out of parliament by the wasted vote ruse of First Past The Post. With preference voting (and a proportional count) the public could also prefer candidates of any party who were either pro- or anti-nuclear.
[PS. After the 2010 general election we found what Lib Dem “No 2 nuclear power” meant: Nothing.]
If there was a financially balanced debate, and the voters had a fair choice, then the people could decide for themselves on all issues, including nuclear power. But this isn’t the case. We have a black-mail system of voting, and lobby corruption of government. A democracy needs democratic voting system (PR by STV) and Equality of Lobbying by occupational proportional representation in the second chamber (which need not be in London).
We even know that fission energy has no long term future, only long term liabilities, because it will eventually be replaced by nuclear fusion energy, which doesn’t produce harmful radioactivity.
Nuclear power as fission energy was fantasticly over-rated and over-subsidised. Its misguided aspirations will not readily be abandoned, its losses will not readily be cut, no matter how criminally irresponsible the consequences for future generations potentially into geological eras of time.
The long-lasting radio-active by-products of fission energy were known to the scientists but not the politicians by the time the first atomic bomb was dropped. We do not know whether this knowledge would have changed the military decision. We do know that it does not change the decision of modern politicians and business men to amass radio-active waste without solving the long-term storage problem.
Ever-lasting poison is being dumped into the futures back-yard, because the unborn are helpless to prevent it. Indeed, anyone is liable to be dumped on, if they have not a robust enough constitutional law to prevent it.
The Mail reported (27 May 2007) “A dash for nuclear power…by the government…a committee of experts will decide where tons of toxic nuclear waste could be buried.” Yet no long-term seal has been manufactured for anything remotely like the time spans involved before some wastes lose their harmful radioactivity. Contamination may do irrepairable harm, after hiding away and ignoring disintegrating receptacles.
The Labour government has assured the British people that they won’t have to take local nuclear waste storage, if they don’t want to. This assurance can be taken with as much confidence as “Mr 45 minutes” other assurances. Translated, the “assurance” means: you’ll have to be strong enough to resist. The weakest will go to the wall.
Someone, somewhere will have to take the rising tide of nuclear bilge pumped out by new stations if they are built. Already showing they are strong enough to resist, the Scots with their parliament have vetoed new nuclear power stations. But Scotland will surely come under pressure to allow dangerous waste deposits in their least populated part of the UK.
The Sunday Telegraph editorial of 18 february 2007, carried a caption “Insultingly consulted.” Ministers consultation exercises on nuclear power and road pricing, afterwards made clear they would disregard the result. “Voters understandably feel that this is worse than not having been asked at all.”
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Robber barons of big business are invasion-forcing more nuclear power stations like Norman castles of occupation on a hostile populace. This odious lobby looks like a co-ordinated attack by a government on its people.
Big Business Brown gathers round him a “business council for Britain” of plutocrats, rather as William the Conqueror gathered his baronial council of state that eventually became the political parliament. Perhaps in another thousand years, Browns business council will “evolve” an economic second chamber of government, for a notional equality of lobbying, from its Thameside radioactive swamp.
Meanwhile, the Financial Times reports that Browns accession to leadership
Who said Gordon Brown couldn’t do public relations? He has got Baronness Williams to be his nuclear proliferation advisor. She must have the easiest job in political history. All she has to say to Brown is one word: Stop!
That’s the only useful advice anyone can give to Gordo the great proliferator. Instead, he has made Shirl, the pleasant and acceptable face of his nuclear proliferation: a public relations coup.
Gordon Brown defended a new generation of Trident nuclear submarines by having us imagine that these would defend Britain from the likes of North Korea – North Korea?! If only Brown and co would missile to the other side of the globe and stop there. It must be admitted that we need defending from lunatic dictators but the universal possession of constitutional safeguards, such as democratic voting system and Equality of Lobbying, would better prevent the likes of Brown from polluting the future. Whereas, the universal possession of nuclear weapons will almost certainly lay radio-active waste to the Earth.
Oh yes, Browns constitutional reforms were careful to avoid any commitment to a more democratic voting system. And Equality of Lobbying remains science fiction. Brown reform proposals amounted to dumping the Blair ballast of constitutional wrongs. But Brown and opposition leader Cameron shied at the first opportunity to prevent a constitutional wrong by not lifting a finger against the Freedom of Information (Amendment) Act, by which MPs exempted themselves from the publics right to know of acts in their own name. Either leaders disapproval, with its power of promotion over selfish politicians, could have stopped easily this private members bill.
The parties and their leaders also avoid freedom of voting choice, from democratic electoral system, that could elect more representative MPs than the corrupt safe-seats do.
The alacrity of the move to replace Trident suggests the real motive is to stimulate the flagging nuclear industry, without public debate about defense and energy alternatives. Never mind that Britains Trident secrets were stolen in the USA. What is that compared with the needs of business?
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The Nuclear Vested Interest and a Nuclear Winter.
The need to build up the immune system of the constitution against parasite politics.
Table of contents
Section links:
“Our solar-powered future”
Parasite politics as alibi and bribe
To build-up the immune system of the constitution
British government renewable energy sins of omission
The government nuclear energy sins of commission
Nuclear Winter
“Our solar-powered future”
The day, a report showed that the government neglected even to prevent council house plumbing from taking another life, the government announced more nuclear plants, whose products threaten the survival of human and all other developed life on the planet, while the energy minister insisted on their safety.
The Blair-Brown act waited till after the 2005 British election to force more nuclear plants, knowing its vote-losing unpopularity. This is unacceptable and I hope the British public will not be crushed under Browns nuclear steam-roller, now or henceforth, whatever deals governments make with the nuclear industry, over their heads.
The most important factor in the future of safe energy supplies was given in New Scientist, 8 december 2007. Their lead article was called “Here comes the sun. Our solar-powered future.” Photovoltaic cells, to produce electricity from solar rays, are the current biggest investment in the world. The advance of research is such that, at its present progress, in few years, they will become commercially competitive with current energy sources.
Meanwhile, the British government is recklessly determined to impose the vested interest in more nuclear power stations, before sufficient opposition can gather to stop their disasterous ill-judgment for environment and economy alike.
There will be no excuses.
Not mentioned in that New Scientist article is the longer term prospect of a paint of miniature solar cells. Theoreticly, if painted on all the roads it could satisfy US energy needs. This might take more or less as long to develop as it takes to build a nuclear power station.
Originally solar power looked limited to silicon cells with their theoretical limit of efficiency of thirty per cent. Efficiency levels have been improved from a few per cent to over twenty per cent. But other kinds of cells have been invented with very high efficiency levels. The current problem is to trade-off efficiency with cheapness of mass production.
My first anti-nuclear page began by quoting an official US government report, over 50 years ago, saying nuclear power would never contribute more than 20 per cent energy production. The US should invest for an aggressive research into solar power that would be of tremendous benefit to mankind. Recently, American President George W Bush put by a miserable hundred and some million dollars for that purpose.
A previous 2007 New Scientist article reported on the dangerous disintegration of US radioactive storage, begun barely half a century ago but a threat for geological eras. The government has to waste billions of public money on indefinite radioactive storage up-grades. And this is just the richest country in the world, that can afford it.
Meanwhile, great industrial powers, like Germany and Japan, are intensively funding solar power research. New Scientist reported that in Germany domestic solar-power users get mandatory refunds from power firms for their surplus solar electricity supplied back to the grid. The editor urged that Britain adopt similar imaginative policies.
Parasite politics as alibi and bribe
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Britain is evidently run for the benefit of a nuclear industry caucus.
Energy spokesman, David Howarth was the Liberal Democrat MP, I believe who remarked on Tony Blairs last months as premier, as being marked by a determination to flag-wave for nuclear power at every opportunity, usually with the remark that nuclear power is “back with a vengeance.”
Blairs nose-thumbing exit seems to have puzzled more people than myself.
Was this a typical case of post-ministerial place-seeking a position on a board? Indeed, Mr Blair was criticised in the Press for taking a position with a firm benefiting financially from the Iraq invasion.
My later speculation as to Blairs nuclear propaganda was that he was diverting attention from the real driving force behind nuclear power – creating an alibi, as it were – for Gordon Brown and his government by nuclear caucus. Blair and Brown are old allies. They fell out when Blair out-stayed his welcome and Brown grew impatient to succeed him. So, to make up to Brown, Blair indulged the pet craze of a fellow control freak. Blair provocatively took the flak for an unpopular policy: nuclear power “back with a vengeance.”
Blairs reward would be continued contact and influence with the man still in power. This is the network politics of a global elite remote from democratic accountability. Mr Blair actually excused his board appointment as being merited by his connections.
[PS. Blairs radioactivity followed directly on Browns brother being given a job with EDF.]
Gordon Brown was supposed to be a more serious politician than Tony Blair. But the lack of seriousness of Brown and Cameron was evident on the day of the momentous decision for more nuclear plants. Both party leaders were conveniently taking time off to grandstand with two of the countries top sportsmen. Not only Cameron can say he is “Blairs heir.” This makes of publicity an alibi to distract from controversial decisions.
A group of scientists condemned the Brown government decision to build new nuclear power stations as undemocratic and possibly illegal. Also in january 2008, The Guardian reported a row about “financial sweeteners” being offered to induce offers from private firms.
The energy minister has put no cap on the number of new nuclear power stations that may be built. This was a predictable consequence of the desire to profit by economies of scale. And I did predict this on my first page against the Labour governments obvious determination to change the 2003 energy review that decided on a renewable energies future.
Basicly the Labour government wants Britains energy production to go the French way rather than the German way. Britain has not been given the choice. As far as government might be concerned, big is beautiful, and the people are a prey to corporate feeding frenzies. It just so happens that Gordon Browns brother Andrew is a director in the French nuclear power industry. My second page, on nuclear power, also noted Brown friends and colleagues heading atomic energy authorities and smoothing (steam-rollering) the planning difficulties.
The only radioactive dump in Britain is near a village which has been offered a further bribe. BBC news mentioned seventy five million pounds bribe for taking low-level waste and a billion bribe for high-level waste-takers. It is right to speak of bribes in the context of generations unborn having to take the consequences of immoral dumping on their future habitats. When present generations take money to ignore future generations predicaments, that is a bribe.
It is barbaric to dump radioactive time bombs on the future. The nuclear industry may leave warning signs. They will only last so long. They may not be understood. There may not be anything that can be done about them. The situation is comparable to the terrorist planting of bombs. Sometimes there are warnings. Sometimes they are in time. Sometimes they can be acted on…
Who needs enemies, when you’ve got a government that puts the nuclear industrys meal ticket first.
It is one thing to honor the offices of government, it is another to honor uncriticly the decisions of their frail incumbents. It appears that the English are still too prone to honor an authority, however ill come by. It is time England stood up for itself, as indeed Scotland resists more nuclear plants.
To build-up the immune system of the constitution
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I won’t labor the point of the sub-title of this page: The need to build up the immune system of the constitution against parasite politics.
It is my too-oft repeated theme, that the voters need effective choice of representatives and their policies, such as on energy. This would be provided by the single transferable vote for all official elections. STV is the democratic method and the scientific method of elections.
The motto is: Britain has half a dozen undemocratic voting methods where the transferable voting method would do.
Plainly, the series of campaign financing irregularities by politicians means that there has to be a limit on spending far beyond the means needed to merely present an honest argument that the public can make an intelligent decision on. It shouldn’t be a propaganda battle like the Common Market referendum that allowed the business interests of the Yes campaigns bad deal (for the country) to spend twice as much as the No campaign. You couldn’t get away from the smiling-family bill-boards.
The result was a comparable proportion of votes cast, for the two sides. The argument against disproportionate spending (above a reasonable publicity level for both sides) is decisive. If you don’t need over-spending to win your case, prove it by not indulging, and so strengthen the legitimacy of a win. The correlation of greater spending with US presidential victories undermines their legitimacy.
Public polls continue to dismiss the impudence of the two-party oligarchy in claiming (involuntary) state funding for political parties. The two main parties glorify their parasitism as needed for “democracy”. (They already claim twenty million one way or another. And it still isn’t enough to over-come their unpopularity.)
Besides STV as scientific elections, an other constitutional protection against parasite politics would be two-chamber representation of the scientific relation between theory and practise. Vocational representation in the second chamber (also by STV) would bring specialist experience to test the political laws of the Commons or communities.
A vested interest, like atomic energy, by offering key posts and buying connections, may hi-jack the government, or indeed the two-party oligarchy, to force and prolong its failure on the nation. This is at the expense of other interests such as renewables research funding. Other interests in the second chamber, if democraticly represented, would have a powerful platform to resist the nuclear steam-roller driven by the government.
Of course that would not suit the all-powerful executive. But people were long since sick of it, in both national and in local government with their winner-takes-all voting systems.
British government renewable energy sins of omission
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Germany, one of the great engineering nations, has decided to phase out nuclear power. France, which is not, is over-whelmingly reliant on it. Neither Germany nor France have been rich in energy. Even France made use of their tidal power resources by building a tidal barrier, some forty years ago. Britain has the best tidal power resources in the world and, in all that time, has built nothing.
Why bother when there were all those valuable petro-chemicals off-shore, which, to use as fuel, was like burning money? That was after the government allowed itself to be black-mailed with the Common Fisheries Policy to join the Common Market. The Establishment gave away and depleted Churchills “sea of fish” around an over-populated island that cannot feed itself.
In january 2006, the Carbon Trust identified, in Britain, 8 out of 20 possible sites in the world for tidal power stations. The Severn, Dee, Solway and Humber are ideal sites. Power could be provided 10 hours a day while the tide goes in and out. They estimated 3% of the nations electricity suppliable by 2020, and up to 20%.
With regard to other renewables, there was the odd hydro-electic station, after which, British government lay back exhausted of innovation. British government energy policy is guilty of sins of omission, neglect of renewables, and sins of commission, a centralised energy over-lordship and serfdom to radioactive containment.
British government, as in France, has been the most highly centralised in Europe. And it seems to be able only to conceive energy policy on national terms. It’s only recently that David Cameron talked of decentralising energy. Alistair Darling, spoke on the governments dependent energy review. (The Labour government wouldn’t allow an independent energy review in case it gave the answer they didn’t want again). Darling spoke of the national grid as if it was a national treasure. But it wastes some thirty per cent of energy in transmission, rather than meeting peoples needs locally and economicly.
The centralists mistake grandiosity for progress, like some impoverished Third World dictatorship neglecting their peoples needs in favor of prestige projects. When the centralists even think about tidal power, it is in terms of a huge Severn barrage. This could meet 5% of the national energy supply but it would be both high impact and vulnerable. Tho incomparably better than nuclear power, a more resilient option might be tidal lagoons for the estuaries right round the British Isles supplied locally.
Power and policy-making is concentrated in the nation state, as if it is medieval and reactionary and beneath them to consider the modern technological version of the mill for producing local electricity from running water. But remote and over-bearing nationalism and inter-nationalism, such as the European Union, on balance, looks more of a menace than an assistance to ordinary people’s needs.
And let’s not forget the prematurely abandoned funding into wave power. This technology was called Salter’s ducks. One can mention other possibilities like biomass and geothermal energy and so on. Green campaigners talk about an energy mix of renewables to cover all our needs.
The broad refusal to use Britains abundant gifts of nature makes nonsense of the energy minister John Hutton claiming that we have to use the resources we’ve got, as if nuclear was the only option. He was speaking to Jon Snow on Channel 4 when the nuclear decision was announced. A few days before that, on The Politics Show, he did come out in favor of wind power turbines stationed round the British coast to supply Britains homes. The Independent reported this as a reversal of policy from giving to nuclear power what wind power could do without the waste that cannot be disposed of.
On the same show, the Tory spokesman insinuated that nuclear power, like wind power, might be subsidised. It sounds like a private fund-seeking party feeling the pinch from a disaffected public. The BBC presenter showed a previous film of him saying that nuclear power was only a method of last resort. The spokesman appeared to shrug off the embarrassment of one evidently not to be taken seriously. He didn’t try to explain his turn-about nor contradict the presenters assessment that nuclear power was fine with the Tories, now.
Of course nuclear power has had fifty years of subsidies. The funding of wind-power and renewables in general is minute by comparison. The Liberal Democrat energy spokesman said Europe was far ahead of us on renewable energy research and that the British governments efforts were “pathetic.” And nuclear power is still a deadly failure, thru its undisposable by-products, that threaten the survival of mankind and vertebrate life in general on the planet.
The government nuclear energy sins of commission
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Wind power remains expensive, tho, as John Hutton said, it might become less so. Nuclear power is incomparably the most expensive energy source because of the unwanted by-products and social costs that economics, as distinct from ecology and evolution, selfishly ignores. Liberty depends on eternal vigilance. But radioactive waste will take away liberty with the eternal vigilance required for protection from it.
Defense is diverted from defending the citizens to defending the nuclear power stations and their offensive by-products. That is as well as being a hidden but massive military and police subsidy of nuclear powers unreal profits to their deluded investors.
And the mining of uranium for nuclear power is not carbon-neutral. While nuclear power absorbs huge resources, that could be better used than to contribute a minimal 4% against global warming.
The Scottish National Party coalition refused planning permission for more nuclear power stations in Scotland. Giving the impression of the government hiding behind the name Westminster, it was reported that “Westminster” (The UK parliament) called “Holyrood” (the Scottish parliament) “irresponsible.” The government don’t know the meaning of the word. The trouble is that the British executive dominates its legislature.
The executive has also ignored the third branch of government, the judiciary. When Gordon Brown became prime minister, without a contest, his first question time announced more nuclear power. This was despite a judge ruling that Labour government energy enquiry (called to over-turn a previous enquiry) had not properly consulted the public. The judge called for a further enquiry which the new PM was ignoring. Greenpeace sent him a solicitors letter.
SNP leader Alex Salmond said, in The Scotsman, Scotland had no need of nuclear electricity last year. (Before becoming first minister Salmond had proposed putting a million pound surcharge on the British government transporting nuclear weapons in Scotland. – They are not toys.)
In october 2007, Dounreay, the Caithness nuclear plant was found to have about a hundred radioactive hot-spots on the surrounding beach. A clean-up plan involves a multi-million pound dredging of the local sea-bed.
In july 2007, an inquiry was held into the removal of body organs, from 65 workers, apparently without their families consent. They died between 1962 and 1991, and were mostly at Sellafield nuclear re-processing plant in Cumbria.
Thanks to the Scottish first minister, it makes a change to hear the British government having to bleat about someone being “irresponsible,” because for once it is as powerless as ordinary citizens. It should do the powerful good to feel powerless. But it is doubtful whether they would ever get used to feeling the frustration that public-spirited people feel from being ignored over the well-being of all future generations.
In april 2006, 35 groups, including Greenpeace and the National Farmers Union urged the government to go green. They called for a 2015 dead-line for all new buildings to be carbon-neutral, to eliminate inefficient products from the market, cut energy demand and boost renewable energy.
Campaigners need to get together in such alliances to have a better chance of being heard. A Democratic Policy-making Alliance is needed, in particular, against the nuclear steam-roller driven by the Brown government, being waved thru by Camerons Tories. Now the Tories have nothing to lose vote-wise to a Labour party more nuke-crazed than they honestly were under Michael Howard, before the 2005 general election.
An Electoral Reform Society campaign asked whether we think there is something rotten with our democracy. During the week that Peter Hain was another politician to be investigated for campaign funding irregularities, the Brown government quietly dropped his party commitment to a referendum on proportional representation. BBC Newsweek mentioned this went almost unnoticed.
Nuclear Winter
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Gordon Browns pronouncement, that a new generation of Trident nuclear submarines is needed against the likes of N Korea, is nuclear investors eye-wash. It has more to do with boosting a flagging industry, while the armed services have taken unnecessary casualties and are being stretched beyond their limits.
Try telling China and Japan and other neighbors that it is acceptable to retaliate against N Korea. The rush of nations to go nuclear, via the nuclear power back door, makes local nuclear war more likely.
Also, a question has been raised about the long-term stability of nuclear weapons. It is of the natural order of things for systems to degrade and become more error-prone. (The second law of thermodynamics.) There is a storage and maintenance and security problem, which is often just too much trouble and expense. But it would be irresponsible for world leaders to just hope for the best.
The astronomer Carl Sagan and colleagues researched the probability that a local nuclear exchange was enough for Nuclear Winter over the whole globe, a menace to all vertebrate life on the planet. This would throw up a dark cloud covering the skies. For lack of sun-light, the crops could not grow.
Previous estimates had assumed that governments could retire to their bunkers, supported by the army, for about three months. And that when the fall-out washed out of the air and the water supply, it would be safe for them to come out and start their blunders all over again. An other factor left out was the massive increase in the use of synthetic materials. A global Bhopal would result from burning chemical substances released into the atmosphere.
Gordon Brown proposed a uranium bank to facilitate the dotting of more nuclear plants all around the world. These would be the most lethal kind of bank for war to invest in mankinds ending.
Escalations of a local conflict are also probable, as in the two world wars. Sagan pointed out that warring nations could not afford to leave others out of the destruction, lest they move in and take over what was left.
The world has to become a civil society, outlawing wars as gang muggings. The penalties for neglecting a robust inter-national law are too serious to ignore. Britain needs a national debate not Browns nuclear business as usual. Indeed, we need an inter-national conference, as the Campaign for Nuclear Disarmament has urged.
5 February 2008.
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Response to Tory party commitment to more nuclear power stations and their fifty-year subsidised command economy failure.
Table of contents
(The following is a reply to a Conservative party answer to a letter.)
I consulted, as advised, Conservative Party policy which said that nuclear power was conditional on it being economic or paying its way.
Gordon Brown pretended to do this but secretly subsidised it, caving in to nuclear industry demands. (As reported in The Guardian fairly recently.
By the way, I think misleading Parliament and the public should have been a resigning matter. What has happened to public standards of honesty?)
On Channel 4 tv, Peter Snow asked the EDF spokesman how they were going to finance nuclear power because the private sector would not. He replied they had their own resources.
That means the French state firms hand perpetually in the pockets of the French taxpayers. Now they’ve got Gordon Brown to draw-on the British peoples pockets. This is to say nothing of the British government being in the pockets of the nuclear industry (“nuclear croneyism” as your party spokesman once so rightly called it.)
Judging by an authoritative reply to me, rather than what the Conservative web-site says, the Tory party fully intends to keep on supporting the nuclear crony government in dragging Britain down with Frances failed command economy in energy, which unfairly draws on public money, as if it were a bottomless pit, and stifles initatives for sustainable alternatives.
That is the failed model that brought collapse in Eastern Europe.
To prepare us for this, there was an article in Financial Mail on Sunday, a few weeks ago, saying that British energy bills would go up to four or five thousand a year (allegedly) to pay for the climate-change-combating energy supplies from nuclear power and renewable energies.
This is misleading. A Total Energy Audit of nuclear power shows it neither economic nor carbon-neutral (as witnessed by the FRSC, PR Rowland, in letter to Guardian science supplement of the time).
American capitalists, Citigroup report, New Nuclear – The Economics Say No, warned against nuclear plants as “corporate killers.” The free market won’t invest in more nuclear power. Like Walt Patterson, they have learned from past lessons.
Only governments, spending other peoiples money, are foolish enough to invest in the nuclear industrys high risks and low returns. People, who have had to earn their money, are putting it into renewables, especially photovoltaic cells research, which as New Scientist said, is “our solar powered future.”
I would ask again the Conservative Party to discuss energy solutions with the veteran expert Walt Patterson, I’ve already linked to.
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If the Tory party goes the Labour nuclear cronies way, it is predictable that more nuclear power stations will incur huge costs and very likely health hazards, and more than likely more emergency alerts on the scale of one to ten. The highest emergencies would end this little island as a nation. Even (officially) the highest, level 7, as at Chernobyl, created an area of ten-thousand square kilometres declared too dangerous for human habitation, tho much remained occupied and farmed. (Clive Ponting: A Green History of the World).
Meanwhile, scientific research in renewables will progress despite government by nuclear vested interest, seeking to stifle competition from wind farms, and probably imported from countries more enlightened than our own. People will naturally want to insulate themselves from the crippling nuclear costs and move towards energy independence by local micro-generation. Demand will also bring down renewables costs.
A further desire for political independence from the heedless Labour-Tory duopoly should be a likely side-effect. It is difficult to imagine how Britain could have been worse served than by this policy-united duopoly.
The Conservative party has copied the rhetoric of Liberal Democrats decentralised energy policy while following Labours centralist energy policy.
Ive made some further comments, below, to this Conservative policy reply with its standard arguments.
The Tory statement: this is a political issue, not a scientific or technical one, does not admit of dispute and suggests that it does not stand up to the facts. A group of British scientists, the nuclear consultation group, are opposed to the steam-rollering of more nuclear power. They called it undemocratic and questioned its very legality (in a Guardian report, 4 january 2008).
They warn that questions about the risks from radiation, disposal of nuclear waste and vulnerability to a terrorist attack have not been addressed – even though the government was ordered last February to repeat a public consultation on energy supply, after its exercise was declared unlawful by a high court judge.
The comment that new plants would be safe from crashing aircraft is an assertion already repeated by James Lovelock. On the contrary, another scientist warned of the grimmest possibilities, for instance, in the release of Caesium 137 fall-out from Windscale/Sellafield, after a terrorist attack. (It was reported in the Guardian science supplement of the time.)
The reality is that a nuclear power plant goes into trauma if so much as a drill-hole in a pipe is discovered: a Florida plant went into red alert over this, without the slightest idea of who did the damage or why. (Reported on teletext).
I thank you (official Conservative policy) again for your viewpoint but my considered opinion is that it reads like a nuclear industry Press hand-out, because it is so lacking in supporting evidence or objective distance as to be no more than wishful thinking that these deadly serious problems will go away.
It’s worse than a comic book fantasy. Even Superman is vulnerable to kryptonite.
The nuclear industry dare not admit to being fallible, because the possible consequences are too terrible to contemplate.
You Conservatives talk about radioactive contamination as just being one of many risks, mentioning amonst others, water contamination. But had the Chernobyl melt-down not been contained (every kind of expensive specialist, from the world over, is still working on it, full tilt, a quarter century later, as reported in a holiday feature in The Telegraph) a continental river system would have been made undrinkable with radioactive pollution for 12,000 years (as spoken in the BBC drama documentary).
By the way, all special interests should be represented in a second chamber of government, so they can combine to check those among them, like the fifty-year subsidised failure of a nuclear industry, whose only recourse is to lobby parties against the general interest.
Yours sincerely,
Richard Lung.
13 february 2010.
Postscript (15, 18 feb. 2010):
See also, Paul Brown: Voodoo Economics and the Doomed Nuclear Renaissance. A research paper.
“,,,the shareholders keep taking the profits and the taxpayer foots the bill.”
My favorite quote is from the Liberal Democrat MP, John Leech:
Nuclear Power Plants May Well Cost The Earth.
Source: manchester-libdems.org.uk/news
Alas, before his party lost its conscience to coalition.
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Journalist partisans for nuclear power.
Table of contents
The betrayal of balanced debate.
Abandoning standards of honesty.
The betrayal of balanced debate.
I’ve never really shared the opinions of journalists.
But one has to put up with that.
No doubt it is folly to complain now.
This is just to put on the record my opinion of their folly and failure to protect the public interest. The conviction that they are simply not doing their job properly was brought home to me by their propaganda for more nuclear power stations.
They are the idol of the journalist Christopher Booker. And he has the run of the right wing press, The Telegraph, The Times, The Mail and I don’t know what else. I even heard him intoning reverently for nuclear power on a UKIP CD.
It just needed a wind turbine, in France, to catch fire for his conditioned reflex: nuclear power.
(You’d be really worried if a nuclear power station caught fire, and it has happened, even here, “Oh, island most blessed.”)
When a wind turbine blade fell off, his colleague Peter Hitchens broke out into a carbon-copy ritual denunciation of wind power. One of his choice metaphors was of hamsters on treadmills.
[PS. On 24 september 2015, the BBC reported those wind turbine hamsters produced 25% of Britains energy.]
Booker and Hitchens and the rest of the anti-Green ranters were in step with the nuclear industrys wish for the government to cut back on this, the main energy competition unfolding at present (tho not in the future, given the progress of research into photovoltaic cells).
I also saw a nuclear energy spokesman denial of this threat to rivals (in 2010 in The Guardian). But he didn’t deny, indeed made clear, that they would be handing the waste over for the government to look after (for the next few geological eras).
Hitchens column did a plug for “Chris Booker’s” new energy book. Two other journalists for The Mail, Tom Utley and Max Hastings came out for nuclear power, falling on the “Green fanatics” like so many unqualified dominoes.
Before the end of 2010, two more Mail dominoes were spotted (note the pun), one of them Richard Littlejohn, following their party line against turbines. These authorities want to forbid wind-driven turbines in favor of turbines driven by mass exterminatory nuclear fuels.
Our knowledge, thru the British reactionary Press, of the Greens today – stereotyped as “beards and sandals” (for instance in about the first entry of The Mail science blog so-called) – is a bit like our knowledge of the Gnostics in the classical world: we know of them only thru the attacks of the censorious established church.
The worst of it is that you don’t see the case made by experts (Walt Patterson and Jonathan Porritt come to mind) for alternative energies and conservation and the phasing out of nuclear power. In 2010, Greenpeace brought out such a plan but I haven’t noticed the mainstream media giving it any attention.
I think they are reduced to local supporters trying to engage small audiences.
In The Mail, blog moderators seem indistinguishable from censors, as to my criticisms against atomic fission.
Quite apart from anything else, such ignorance is annoying in its arrogance.
The Guardian has its intemperate nuker in George Monbiot.
At least since The Independent was rescued by new owners, there have been pro-nuclear editorials. And a particularly feeble assessment of a so-called consultation over a new nuclear power station.
One comment will give the tone: on top of everything, the public were concerned to hear that a wind turbine might fall on the existing nuclear power workers there.
The last straw, indeed!
The pro-nuclear Ben Goldacre assessed an EDF consultation as that if you scare people enough with unemployment, they will be pro-nuclear.
The worst example (that’s been admitted) seems to be The Sun, whose owners told a former editor to leave out the Liberal Democrats, who were the one significant and most vociferous force against more nuclear power.
That is until they joined the Tories in coalition in 2010.
A Sky News interviewer kept prodding the new energy minister, Chris Huhne, about nuclear power, including subsidies for it.
Huhne had to point out the obvious that nuclear power has been on the go for a long time and didn’t deserve subsidies. Wind turbines were an infant industry, and therefore given some help to get on their feet.
At the Lib Dems first party conference in power, for the first time, the lobbyists were there in force, setting out their stalls. Whatever happened to integrity (not to mention economic democracy)?
The Tories, Labour, UKIP, the BNP remain zealously pro-nuclear. [PS. Soon to be joined by the Lib Dems in coalition with the Tories]
As to the scientific community (I’m not talking about the odd zealot), who forgot their doctrinal neutrality to become the false prophets of a nuclear utopia, that turned into a Frankenstein monster, what happened to them?
Each nuclear power station produces thirty tons per year of extremely high grade nuclear waste, says Michio Kaku (Physics of the Impossible).
Between the Two Cultures, of the humanities mandarins and the scientific neutrals, to quote Eldridge Cleaver: Those who say, don’t know, and those, who know, aren’t saying.
Abandoning standards of honesty.
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The media in Britain are what they have become in America: too much centralised control by reactionaries who can fabricate with impunity.
One national broadcaster even won a court case to the effect that they didn’t have to tell the truth, because it was not enforced by law, only offered as a guideline.
This degradation set-in when “a real media man,” Ronald Reagan took over the White House. In 1987, he abolished the Fairness doctrine, that required broadcasting both sides of a debate, including controversial issues of public interest. He vetoed the attempt of Congress to maintain the status quo of fair play.
So, here was a man, elected on a platform that “big government makes little people,” who made sure that big business makes people small. Deregulation of local autonomy led to corporate centralisation of the media as just another business without public obligations. Information monopoly misleads and closes-in the publics horizons.
A notable result was the compliance of US broadcasters over the second Iraq war, ignoring any dissenting voices.
Judging by Noam Chomsky, veteran critic of the Vietnam war, the American media are as conformist as the British. Here, the simulated battle between Left and Right goes on like the big-enders versus the little-enders, Jonathan Swift imagined in Gullivers Travels. Their partisan propaganda merely serves the ends of a survival tribalism.
Life’s a scramble and it will be for hundreds of years yet (HG Wells).
We could start going in the right direction again with good and wise laws against lying, stealing and cheating. Ending cheating, for example, by replacing fraudulent with genuine election methods. Laws against lying (the Fairness doctrine in broadcasting) and laws against stealing (like the Glass-Steagall Act), respectively repealed by Presidents Reagan and Clinton, should be re-instated.
Consumer advocate Ralph Nader called President Barack Obama loan guarantee for more nuclear power stations “a monumental mistake.”
The complaint was made that fairness was an excuse to harass the Right. No doubt they would not welcome a more even playing field. In other words, nothing must impede the darlings of fortune.
Doesn’t everyone just snatch their opportunities, anyway?
No doubt, fairness is too often a mirage. Unfortunately, such a mind-set brings countries to their knees, as the 2008-9 credit crunch has demonstrated.
22 december 2010. Minor addition, 2 january 2011.
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The determined dishonesty of atomic energy.
Table of contents
After the 2011 Japanese tsunami destroyed the Fukushima nuclear reactors, I wrote no more essays against nuclear power. Events had spoken louder than words!
This failure did not chasten the nuclear lobby and its supporters. Their most extreme propagandists perversely and disgracefully proclaimed that now they knew nuclear power was safe. (For instance, at least a couple of journalists, including a blogging science teacher, in The Mail, and Monbiot in The Guardian.) This was only days after the disaster, when it was not possible to know the truth.
If only arrogance could keep nuclear power stations safe, humanity would have no worries about them.
Of course, that was the purpose of the arrogance, to stifle legitimate worries of the populace about nuclear power. The Guardian Comment is Free was full of it.
A similar unteachable attitude of the British government and its officials was exposed, when The Guardian obtained emails that the energy department was anxious to play down the Fukushima disaster, to prevent adverse public opinion challenging its immovable intentions to build more nuclear power stations in Britain. That is to say in England and Wales, because Scotland won’t have them.
Angela Merkel was going to go back on phasing out nuclear power in Germany, until the Fukushima crisis made her change her mind. She has an educated interest in science, lacking in the British cabinet and legislature. Perish the thought that a British elective dictatorship could ever be induced to be guided by the evidence of events! The Titanic is unsinkable!
For my own part, before the Japanese tsunami, I already had made my views known, presciently, as it turned out. Yet I would never be an expert and there were plenty of others, much better informed, who I could only trail along after, as a secondary or tertiary source.
Moreover, the evidence remained unclear for the extent of, and potential for greater harm from, Japans nuclear tragedy, in the wake of the tsunami misery. For instance, in 2015, Japanese television reported that the extent of radioactivity, escaping into the atmosphere, had been under-estimated. And that says nothing about leaks and flushings of radioactive contamination into land and sea.
However, I have picked up a few salient points, from both supporters and opponents of nuclear power, as well as general reading, which are perhaps worth recording here.
The first atomic pile or nuclear reactor was built to understand how a chain reaction worked, in order for the Manhattan project to know how to build an atomic bomb.
As far as nuclear energy was concerned, from first to last, civilian needs were subordinated to military objectives. Indeed, the former has typically covered for the latter. Nuclear power has been the spin-off and accessory to nuclear weapons.
This was certainly the case in Britain, where the mess from the fifties nuclear weapons scramble still has to be cleared-up, in Sellafield, if it can be. It is suspected to be the case in Iran, secretly and illegally helped by Pakistan. Enenews alleged that the US presidency secretly and illegally armed Japan with nuclear weapons under cover of its nuclear power program.
I read on Comment is Free that even a scientist, who designed the atomic pile, for military research into destructive potential, knew this was not the optimum nuclear reaction for peaceful civilian energy purposes.
India has large deposits of thorium and has researched this nuclear reaction option. It was alleged (on CiF) that this was stalled by the Clinton administration offering favorable terms with its own uranium fission reactors.
Whether thorium power, or other not too offensive nuclear options, are feasible remains unproven. I know by my own specialty of election science, the self-interested wilfully ignorant human determination to corrupt and degrade even the obvious. So, for all I know, there may be a niche for a fairly civilised form of nuclear power. Or there may not. But private fortunes should not be begging governments to hi-jack public funds for its research.
If a peace-friendly nuclear power could be developed, with minimal levels of toxic waste, at least, it would undermine the fraudulent excuse of governments claiming to want (uranium fission) nuclear power just for peaceful purposes.
In any event, nuclear power has been the most outrageous example of private profits at social costs. It is so dangerous as to be uninsurable. The public pays for the contingency of being lethally irradiated and made more or less terminally ill. The public pays, from here to eternity, for the nuclear waste disposal problem, which remains unsolved.
If “the world is full of half baked solutions,” as a letter writer to The Guardian said, talking about Internet banking, this “solution” isn’t even half baked.
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The hubris of scientists promoted their own prestige in a nuclear utopia of unlimited energy. Even the equable Arthur C Clarke was caught up in it. (Greetings, Carbon-Based Bipeds! Collected essays 1934-1998.) Such was the craze for “atomics” in the 1950s, made possible by suppressing the inconvenient truth that this unlimited energy brought with it the potential for unlimited sickness.
Governments were signing up their peoples to a devils bargain, without their consent.
Harold Macmillan suppressed news of the Windscale reactor catching fire. Only the foresight of a chimney filter, derided as “Cockcroft’s Folly,” prevented a disastrous escape of fall-out. As it was, contaminated milk was disposed-of and cattle slaughtered.
Energy Minister, Anthony Wedgwood-Benn parroted the phrase that nuclear power would be “too cheap to meter.”
When I mentioned this, my veteran left-wing friend, Dorothy Cowlin recalled that she had found it hard to forgive him for that.
Investigations, into the atomic bombings of Japan, revealing that they resulted in the whole range of cancers, were kept secret. (Stated in a 70th anniversary tv program, on the bombing of Hiroshima.)
Alice Stewart discovered that x-rays of unborn children induced fatal cancers. This unwelcome news did not make Stewart a household name, as it should have done.
When an overseas colleague enlisted her aid, in researching the health of nuclear power station workers, the American government stepped in to suppress the exercise. (Gayle Greene: Alice Stewart, the woman who knew too much.)
When the Swedish parliament voted her the alternative nobel prize, the British embassy didn’t even give her a car-lift from the air-port. She might as well have been a non-person, as far as the Establishment was concerned.
Despite the deficiencies of this after-word, as well as the previous essays, I hope I have said enough to justify the conclusion that nuclear power has been a stalking horse for nuclear proliferation, endangering the health and happiness of life on earth, thru subordinating civilian needs to military objectives. Nuclear power has not been possible on a commercial basis, independent of public funds insuring its disasters and catastrofes waiting to happen, and disposing of its chronic waste.
Nuclear power has been dishonesty personified. Its public relations, employing friends in high places, suppressing evidence on radiation sickness, and dumping its hyper-pollution on future generations to solve or suffer.
When extravagant promises of “atoms for peace” (perhaps the biggest threat to life on earth) were no longer remotely credible, the tyrants excuse of necessity (against climate change) is made.
No mature government by democratic consensus would inflict this bitterly opposed burden on the coming generations.
Today (21-09-2015) George Osborne announced the start of a nuclear deal with the Chinese government (as well as the French state-owned EDF). He claims that nuclear power is low carbon emitting, which, no matter how many times it is repeated, is still false.
As before mentioned, this fraud is exposed by a Total Energy Audit, specified by PR Rowland, of the whole production process from uranium mining to waste disposal, including all the facilitatory expenses.
British government is a typical Stalinist enslaver to white-elephant prestige projects, above all, the tarnished glamor of atomics. This unamiable mind-set has been characterised as “nuclear fascism.”
The biggest health benefit and energy economy would be thru really good insulation standards in buildings. A big coalition called the Energy Bill Revolution promotes this for every home. The Tories singularly neglect it, all the better to exploit their energy serfs, in a nuclear feudalism of centrally controlled power.
But disable the center and the whole is made helpless. That is why the US military developed the internet for decentralised communications.
Whereas decentralised energy, where everyone can get by in healthy insulated homes with their own renewable energy generators and storage, is the future that beckons to free democrats.
In the 1950s, a comprehensive expert energy report, to Pres Eisenhower, predicted correctly that nuclear power would never make more than a minor contribution, and that the future benefit to mankind lay with an aggressive research into solar power.
More than half a century later, David Attenborough spoke for a new Apollo project. The Kennedy presidency marshalled the nations resources to put a man on the moon within the decade. Likewise, an international fund could enable a doable project to collect and store enough solar energy for the worlds needs. Just a tiny proportion of all the radiation from the sun, that daily reaches the earth, would leave no need for fossil fuels and their climate-destabilising pollution.
He might have added, removing the risk of nuclear power contamination, rendering the planet more or less uninhabitable.
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Some women scientists who should have won nobel prizes.
Table of contents.
Links to sections:
Neither feminist nor “masculinist.”
Lise Meitner.
Madame Chien-Shiung Wu.
Rosalind Franklin.
Jocelyn Bell (Jocelyn Bell-Burnell).
Alice Stewart
Neither feminist nor “masculinist.”
I was told I was “condescending” about a woman being well-read in popular science. It was a reminder how sensitive women are, with regard to being treated as intellectual equals. My reply was that, in my own readings of popular science, I had come across three women scientists, who should have won nobel prizes.
I can think of four. (I came across more later, without looking or taking notes.) That being the case, it is probably only the tip of an iceberg of hidden injustice to the scientific abilities of women. This is not meant to be a comprehensive case for womens intellectual rights. It is just something I noticed a little of, without even looking for it. Nor is it meant to belittle the present attempts being made in education, to encourage girls and young women to become scientists.
If words are to mean what they say, I don’t believe in “feminism,” as a one-sided sexism, any more than “masculinism.” It is arguable that we now have a “feminist” culture in this one-sided sense. That is to say an excessive passivity towards restitution for people who are wronged by crime or civil injustice.
What, you may ask, is masculinism? Perhaps it is shown most blatantly in many old action movies. All the aggressive competition, all the fighting, racing, chasing, all the courageous acts are left to the men. The womans role is to stand in the wings, in a sort of agonised dither, while the men slug it out. She then falls into the arms of the victorious male in a swoon of admiration and adoration.
In my mis-spent youth, at the cinema, I remember being exasperated by this conventional view, which I now call masculinism. Sometimes, the script was enlivened by a spirited woman. But I learned to expect she would be the one with the tragic ending, while the passive or “womanly” woman was the one who lived happily ever after, with the hero.
Of course, not all the old action films were like that. One, that was not expected to become a classic, was a Western, which everyone has seen, High Noon. Here, all the town folk shun the sherriff, asking for help against a gang, gathering to gun him down. One man tries to make him change his mind. But only makes things worse, by fighting him.
Left on his own, the sheriff, Gary Cooper, playing a fine manly role, is surprised from being reduced to tears, by a boy bursting into the office. The boys offer of help, against hardened killers, has to be refused.
Meanwhile, the sheriffs fiancée is leaving on the train. A Spanish woman (with her nationalitys belief in family loyalty) tells her she would never leave her man. The fiancée returning, with shot-gun, is true to life, in that her presence both helps the sheriff and makes him vulnerable thru her.
Lise Meitner.
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Shortly before world war two, Lise Meitner worked out a process of nuclear “fission” leading to the possibility of a chain reaction and the unleashed energy of an atomic bomb, on the basis of the famous mass-energy equation.
She was sent to the United States with this information because it was too dangerous to send the news by post.
The secrecy, of research that would make lucrative Nobel dynamite seem inoffensive in comparison, prevented the nobel committee from hearing about it in a hurry. But that has never been a bar from scientists receiving eventual recognition.
In fact, Einsteins nobel prize was delayed, till the evidence for his revolutionary ideas was more assured. And he never did receive the prize for the theory of relativity, with which his name is associated.
When Niels Bohr first heard the Meitner explanation, he exclaimed, to the effect, what fools they had been. This seems a rather ungracious acknowledgement. After all, you could say about many discoveries that they are easy -- when you know how.
Meitner worked with Otto Hahn for thirty years. Like Bohr, Hahn was a nobel laureate. In his autobiography, he decried some ill-informed journalism, making extravagant claims for what Lise Meitner was working on. Again, it struck me as curious that Hahn should choose to refer to his female colleague, in this passing and negative way.
I make no claims to understand the states of mind of these foremost scientists with regard to Meitner. However, there is historical evidence of a male chauvinist attitude that women are no better than they should be. Boys may be brought up with the belief that they are going to be the ones who are going to do great things in the world.
In quite recent years, I heard a woman, of rural background, say her daughter was “only a girl.” And the same expression, recently, from a boy on tv. Tho, that was good-humoredly challenged.
However, we can safely assume that Bohr sparring partner, Albert Einstein thought Lise Meitner deserved a nobel prize, because he pointedly refered to her as “the German Curie.” He was trying to harness German national pride to her cause. Like her, he was a German Jew: he from Switzerland, she from Austria. And both were forced, by Nazi racism, to become emigrés.
Madame Wu.
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Implicit in the conduct of physical experiments are certain assumptions. When spelled out, they seem no more than common sense. It is thought not to matter to the out-come of experiments when or where they are conducted in space or time, as such. Likewise, it was thought that an experiment that was seen, as if in a mirror, could not be distinguished from direct observation of it.
This was a mirror image (or “parity”) conservation law of physical experiments. Up till the twentieth century, only two forces of nature were known, gravity and electro-magnetism. (Electric and magnetic forces received a unified treatment in the nineteenth century.) Tho, Isaac Newton anticipated there might be more forces of nature.
Two more were discovered, as it began to be realised that atoms were real but not the basic indivisible building blocks of matter. The “strong force” bound the constituents of the atom. The “weak force” was associated with the spontaneous disintegration of certain of the unstable heavy elements in radio-active decay.
By the middle of the twentieth century, examples of the weak force interactions, posed a dilemma, involving either one “strange” sub-atomic particle that violated parity conservation, or two such particles, with apparently identical properties.
The physicists Chen Ning Yang and Tsung Dao Lee proposed experiments “to determine whether weak interactions differentiate the right from the left.” The first team, to carry out these tests, was headed by their friend and fellow Chinese-born American, Madame Chien-Shiung Wu.
Martin Gardner described her, in The Ambidextrous Universe as:
widely regarded as the world’s leading woman physicist. She was already famous for her work on weak interactions and for the care and elegance with which her experiments were always designed.
This compliment reminds me of Elizabeth Barrett being called the worlds greatest woman poet.
In other words, she was very good -- for a woman!
Martin Gardner says:
Madame Wu’s experiment provided for the first time in the history of science a method of labelling the ends of a magnetic axis in a way that is not at all conventional. The south end is the end of a cobalt-60 nucleus that is most likely to fling out an electron!
It was pointed out that but for Yang and Lee telling the experimenters what to do, the experiments could never have been performed. This tacitly explained why the two theorists got the nobel prize but not the leading experimenter, who verified the violation of parity.
But experimental ability is also a gift. Ironically, Yang was legendary for his maladroitness anywhere near a physics lab. (Where there's a bang / There's Yang.) This is in no way meant to be disparaging of the man, who went onto further great things in mathematical physics -- the Yang-Mills gauge-field theory.
It is merely that experimental ability is equally to be respected as theoretical ability.
And the nobel committee recognised this, for example, with regard to the electro-weak theorists -- and their experimental demonstrators (as led by Carlo Rubbia, at the CERN laboratory, in Switzerland). This was the theory that gave a unified explanation of two of the four known natural forces, the electro-magnetic and the weak forces.
Madame Wu made just as epochal a result in the twentieth century history of physics. In justice, not to mention courtesy, a nobel prize should also have been hers.
Rosalind Franklin.
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James Watson, telling of the search for the genetic code, The Double Helix, starts by saying only five people in the world mattered, in its discovery. At any rate, one of these was the crystallographer, Rosalind Franklin.
Linus Pauling was known to be on the warpath for his third nobel prize. (He did eventually win another -- for peace, tho.) His son came over to Cambridge. With American generosity, he sided with Francis Crick and Watson, in their race to beat his father.
When Pauling came out with his model of a triple helix, it didn.t seem quite right. As Crick said, nature does things in pairs.
A recurring feature of the story was going round to take another look at what “Rosie” had done. They didn’t dare speak to her in such familiar terms, however. And Watson relates that she greeted their double helix idea with a womans fury and scorn.
In a later edition of his book, Watson sympathises with her for the difficulties she must have faced, as a woman in science. She was to die young of a painful illness, bravely continuing her work till the end. Her notebooks show she was moving towards the double helix explanation.
From Watsons account, one certainly gets the feeling that the superb quality of her X-ray diffraction studies were Crick and Watsons window on the problem.
In The Physicists, C P Snow said Rosalind Franklin got “a raw deal.” She should surely have shared that nobel prize.
Jocelyn Bell.
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Jocelyn Bell was the first to discover an astronomical object, that was to become known as a pulsar, short for pulsating star. This class of things were later to be identified as neutron stars. Like black holes, the possibility of their existence had been theorised, but few had believed in them.
Apart from black holes, neutron stars are stars in their most catastrophically collapsed state, occurring in super-nova explosions. This produces an enormously increased spin, the figure-skater effect, named after the increased spin of a skater after she draws in her arms. With it, goes a greatly increased magnetic field, whose poles may differ from the axis of spin. The former is whipped round eccentrically, by the latter, drawing in nearby charged particles to produce a rotating beam, a light-house effect.
The regularity of this pulsed radio signal made the Cambridge team, led by Anthony Hewish, think, at first, that their new large array radio telescope recording was artificial. However, only the worlds best atomic clocks could keep such accurate time, so it was no human interference.
An extra-terrestial contact was next thought of -- LGM or little green men. But then Bell found another such signal.
In his book, Perfect Symmetry, Heinz Pagels said of Bell:
It was (Hewish's) extreme good fortune to have Jocelyn Bell-Burnell, a twenty-four-year-old graduate student, on his team. Examining the output of the antenna which swept the sky as the earth rotated, she observed "a bit of scruff" -- a distinctive radio signal -- coming from a particular spot in the sky. It would be rather easy to disregard such a signal as nonsense noise. The actual output of the antenna was recorded as a line trace on a paper roll, and the "bit of scruff" was just some short jumps in the trace on hundreds of yards of paper, every inch of which was examined by Bell.
A month later, she saw the signal again and soon thereafter analysed the “scruff” in detail. She saw that it consisted of periodic pulses about one second long.
Some people thought Jocelyn Bell should also have had a nobel prize. After all, people think the idea is that the prize should go to the person who first makes a first class discovery. And it does seem to me that they are right that she should have had a share in the glory at Stockholm.
But for her, some other radio astronomy group could well have snatched the prize first. Also, it was a lost opportunity to write in the sky what a diligent young woman in science might achieve -- and be fully recognised and rewarded for.
In a recent interview, broadcast in 2015, Bell-Burnell regarded missing the big prize, as an advantage, because it would have put-off all the subsequent awards heaped upon her.
Alice Stewart
Post-script (2015)
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Other examples of disadvantaged women scientists, because they were women, impinged on my later attention. Emmy Noether has a theorem named after her on the necessary relation between symmetry principles and conservation laws. This concept is at the very foundation of modern physics. (While it is true that the nobel prize doesn’t include a mathematics category.) A German university only let her become part of the faculty, because David Hilbert backed her.
And then they didn’t pay her! What a practical comment on male condescension!
There were two pioneer women astronomers, one of whom Hubble wouldn’t let use the great telescope, to anticipate his findings. That would be Henrietta Swan Leavitt. A classic case of boys not sharing their toys with girls. Sharing is what man remains poor at.
The Hubble telescope? The Hubble-Leavitt telescope, methinks.
A century later, right into our own times, women scientists were still getting a raw deal. Alice Stewart should be a household name. I had never heard of her. Her curiously unfamous, and belatedly acted-upon discovery was that X-rays on unborn children are fatally cancer-inducing.
Adopting a low threhold against radiatioactivity could not guarantee against harm. This obviously raised questions about nuclear power. The US governmant was desperate to prevent Alice Stewart ruining the reputation, that nineteen fifties science propaganda had built-up for the nuclear utopia, just as she had spoiled the reputation of the medical professions favorite toy.
I once had the misfortune to come across a professional review of the biography by Gayle Greene: Alice Stewart. The woman who knew too much.
The critic started by saying that he wouldn’t dwell on her early career.
This was most convenient for his debunking exercise, because her diagnostic abilities were honored and recognised as out-standing.
He conceded that X-rays are pre-natally carcinogenic. There was no longer any use in trying to shut the stable door after that horse had bolted. The rest of his comments might be described as a war of attrition on her subsequent work. Her statistician colleague used dodgy techniques. (No demonstration of that conveniently floated claim.) Her biographer wasn’t a scientist. (Actually, Greene let her subject speak for herself.)
The article writer may have been a scientist but his objective knowledge was of how to follow the nuclear party line, not any desire to share in Stewarts attempts to know the precise extent of this undoubted menace.
The Swedish parliament awarded Alice Stewart the alternative nobel prize, since scientists couldn’t bring themselves to give credit where it was due. The British embassy didn’t even give her a lift from the air-port. She was not so much given the red carpet as brushed under the carpet, like her social health research.
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Murray Gell-Mann: The Quark and the Jaguar
Some themes illustrated from electoral methods.
Table of contents.
Complex systems.
Zipf’s law, self-similarity and fractals.
Borda method.
“Landslide” majorities.
Murray Gell-Mann (published by Little, Brown and Co in 1994) tells a little about his personal life, mostly his youth -- tho there are a few genial anecdotes about colleagues. This is just as well, because as CS Lewis said: He'd never read an autobiography yet in which the early years weren't by far the best. Lewis appears to have discovered a law of nature, or human nature. Gell-Mann is a student of both.
Gell-Mann, on “Quarks and all that” echoes “1066 and all that,” as if we would laugh-off his most famous discoveries as ancient history. In fact, he gives a typical account you might read in other popular physics books.
He is much better on current research to demystify quantum theory. The paradox of Schrodinger’s cat is laid to rest (tho attempts may be made to revive it).
Complex systems.
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Nevertheless, the focus has changed, from: what are the basic parts of the world? to: how does everything fit together so it works properly? Gell-Mann helped found the Santa Fe Institute concerned with the general properties of complex systems and their emergent features that make successive chemical, biotic and social systems irreducible wholes.
The Quark and the Jaguar sets out to define complexity. Complexity is in the observer and the observed. Observations are most complex when they are not so apparently random, that no rules can be abstracted by the observer, and when the observations are not so regular, that they can be summed-up in a simple rule.
Consequently, the skill of observers is most tried, as themselves “complex adaptive systems,” when they have to distinguish most carefully the essential patterns in the data – the sensory “signals,” if you like, from the random noise. (That is, if you consider ones perception of the real world rather like receiving a radio signal, so one has to fine tune out the interference to its message.)
The “noise” could be superstitions caused by ones conditioning to chance associations between events that have no rational connection. However, Malinowski anthropology and Jung psychology have impressed on us that apparently silly customs may have a ritual value for the integration of society and of the personality.
No doubt much of the paranormal is credulous. But I don’t agree with Gell-Mann, in his throwaway dismissal of “psychic detectives.” They are the subject of tv programs (writing in 2015) which sometimes offer no evident cause for the psychics “inside informatio.” The police are scientific investigators (in a democracy). If they find such people useful at times, that is surely being practical rather than dogmatic about things we don’t understand.
Anyway, Gell-Mann, on complexity, may be illustrated by voting methods. Candidates first past the post in marginal constituencies depend largely on chance factors to win. “There is no greater gamble than a British general election,” admitted one devotee of the simple majority system. An opposite fault applies in the safe seats, where results are too determined. An agent boasted he could put up a pint pot of beer in this constituency and still get it elected.
The random effects of marginal constituencies and the pre-determined effects of safe seats are both examples of low “effective complexity.”
The voters are caught between two extremes and have difficulty adapting to the system either way. If you are in a safe seat, you know your vote is unlikely to make any difference. That’s why party politicians tend to favor single member systems. A safe seat is a local monopoly for some party, whose candidate does not have to earn an elective proportion of the vote, in competition with candidates of his own party, as well as of other parties.
In a marginal constituency, you may have to vote tactically for the best chance to make your vote count. The information value of the X-vote is too low to register more than a single preference, unlike a ranked choice.
A combination of an elective proportion and a ranked choice (which exists as a voting system called the single transferable vote) therefore increases the effective complexity of a voting system in two ways. The ranked choice of a preference vote reduces all the “noise” from split votes that interferes with and frustrates the popular will. A proportional count prevents votes being wasted in predictable pile-ups that make safe seats.
In short, the voters have the best chance of adapting the political system with transferable voting.
That’s why the Establishment least wants that system, to disestablish its opposition to the worlds changing needs. Michels called this evident state of affairs “the iron law of oligarchy.” But government is supposed to be the cybernetic principle of the rulers responding to the (especially voting) information feedback of the ruled. The least effective government as cybernetic system has minimal feedback methods of voting.
Typically, these are partisan systems that only tell the rulers what they want to know from the ruled, namely that they follow their party lines. Indeed, the voters can do no other, as the likes of party list systems pre-define the terms of the popular vote.
Zipf’s law, self-similarity and fractals.
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I read somewhere that at a conference, Stephen Hawking had just quoted off the top of his head an equation about a mile long, when Murray Gell-Mann promptly stood up to point out a missed term. Yet The Quark And The Jaguar takes an interest in the simplest of arithmetic laws.
They may apply thru-out the sciences. Zipf’s law is one of many “scaling laws” or “power laws” about which “…we see what is going on but do not yet understand it.” For example, you can rank 1st, 2nd, 3rd etc the cities of a country by their population size, which turns out to be inversely proportional to that rank.
If the first city has about 10 million people, the second city turns out to have about half that number or around 5 million. The third largest city will have one-third the population of the biggest, or some three and one-third million people. And so on, down to, say, the hundredth city at about 100,000 citizens.
Similar relations hold for ranking countries by their volume of business in exports, or for ranking firms by their volume of business in sales.
Modified versions of the Zipf law may produce a formula that is a better fit of the data, but the point is that there is an underlying regularity. Gell-Mann says this is reminiscent of self-similarity found in nature. Trees from their largest branches to their smallest twigs, or rivers down to their smallest tributaries, have a characteristic shape at every scale. The same is true to some extent of clouds and mountains and many natural features.
Such features do not have regular dimensions, one, two or three. But they were found to have fractional dimensions. A screwed-up ball of paper is not a proper ball of three dimensions but is more than two dimensions. It may typically measure over 2.7 dimensions. Likewise, the squiggly lines, say, of rivers on maps, have a characteristic fractional dimension of slightly more than one dimension.
Hence, the term “fractals,” which relate to “chaos theory.” In Does God Play Dice? Ian Stewart says: “The same complexity of structure that lets fractals model the irregular geometry of the natural world is what leads to random behaviour in deterministic dynamics.”
Knowing the fractals of natural phenomena enables them to be modelled realistically as in computer “landscapes.”
You could also simulate a society and an economy, with the help of scaling laws like Zipf law.
Borda method
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The Santa Fe Institute includes political science in its array of systems studies. But it is possible Gell-Mann colleagues haven’t heard of Borda Method of counting votes for a single vacancy. This is actually an electoral version of Zipf law.
Voters can order their choice of candidates, 1st, 2nd, 3rd, 4th, etc. These preferences are given due weight in the count, as a measure of their order of importance. If there were five candidates, your first preference would get five points; your second would get four points, and so on to your last preference getting one point.
Laplace gave an involved proof of Borda Method.
In Elections and Electors, JFS Ross pointed out that the more candidates standing, the less important the first preference, using Borda method of weighting the count with an arithmetic series.
Ross suggested the preferences be weighted by a geometric series. The first preference would count as one vote, the second as half a vote, the third preference as one-quarter of a vote, the fourth as one-eighth of a vote…
A happy medium, between weighting by the arithmetic series and by the geometric series, would be to weight preferences with the harmonic series. Choice 1 counts as one vote; choice 2 counts as 1/2 a vote; choice 3 counts as 1/3 of a vote; choice 4 counts as 1/4 of a vote…
This modified version of Borda Method was once favored by Sir Robin Day. And it is Zipf law for an election, whereby the count is inversely proportional to the vote.
You could imagine Zipf law applied to cities as an “electoral” system of how people vote with their feet. The largest city attracts twice as many as the second largest, three times as many as the third largest, etc. Borda political justice turns out to be a case of art unconsciously imitating nature.
Borda method was designed to overcome an objection to the Second Ballot, which does not weight preferences to account for their order of importance. If three candidates contest one seat and none wins over half the votes, the candidate with least votes has to stand down. A second ballot decides between the two remaining candidates.
Condorcet pointed out that the eliminated candidate (say, a center candidate) might have won more votes from either a right or a left wing candidate than they would have won from each other. (By the way, this isn’t necessarily the case. Extremes may have more in common than moderates.)
Borda method, in turn, is open to the objection that the lesser weights given to lesser preferences, count to some extent against a voters first preference. That candidate has a better chance of winning if the voter refrains from adding further choices.
This problem is overcome by the transfer of votes, surplus to a quota or proportion of votes needed to elect the most prefered candidate, according to voters succeeding preferences for candidates, elected in multi-member constituencies.
The size of the most prefered candidates surplus vote determines how much weight to assign to the next preferences of the most popular candidates voters.
Borda method has to assume what value voters assign to their preferences. But with (the so-called Senatorial rules of) transferable voting, this is a real value based on the size of surplus votes, which does not count against more prefered candidates already elected.
Benoit Mandelbrot generalised Zipf law by adding a constant, c, to its inverse proportion. That is, 1/1, 1/2, 1/3,… becomes 1/(1+c), 1/(2+c), 1/(3+c),… Let that constant equal one, and you have the Droop quota, which gives the elective proportion of votes to become a representative in one, two, or three member constituencies etc.
The Droop quota is used with transferable voting. Candidates, winning more than their quota, have their surplus votes transfered to their voters next preferences.
Zipf law describes natural structures. Whereas Borda method is a similar structure, consciously imposed by the rules of an electoral system.
“Landslide” majorities.
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Murray Gell-Mann cites the work of Per Bak and associates on how structures arise naturally without imposed constraints.
Cone-like heaps of sand had more grains of sand piled on them. As their steepened slopes became more unstable, a critical value was passed for avalanches, which left the slope back at the critical value. This cycle was called self-organised criticality.
The single transferable vote is analgous to such “self-organised systems.” The surplus votes transfered to next prefered candidates are akin to the avalanche, a political “landslide majority,” caused by the piling of extra sand on a mound or cone, above the value for a stable heap. This critical value compares to the quota, or proportion of votes needed to elect the most prefered candidate (and in turn the next prefered candidates).
Proportional representation originally came about, in the early nineteenth century, when school children queued behind their favorite class-mates to form a committee. The most popular children had longer queues than they needed, so some supporters slid away to help their next prefered mate get elected. Children queueing behind least popular candidates, deserted them to help their next preferences, still in with a chance of forming a long enough queue.
The winning candidates would have queues of the same or proportionate length, that no other candidate could match. This original inter-active version of the modern single transferable vote form of proportional representation fits the Gell-Mann description of a self-organizing (voting) system.
The actual way that grains of sand tumble together is extremely complicated, just as is the way that thousands of voters preferences combine. But each scenario clearly follows a typical structural development. The contrast is that Per Bak and his colleagues evolved formulas from a phenomenum. Whereas the pioneers of electoral science, from Borda and Condorcet, Andrae and Hare, Clark, Droop and Gregory, onwards evolved a phenomenum from formulas.
The former is natural science, the latter is “moral science” but the two are complementary.
The introduction to The Quark And The Jaguar and the last chapter on a sustainable world is an admirable survey that perhaps speaks for many as to the kind of world they would like to work for.
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Paul Erdös: review of Paul Hoffmann, on The man who loved only numbers.
Table of contents.
Paul Erdös was the second most prolific mathematician in history, after the Swiss, Leonhard Euler. He is the most prolific collaborator with other mathematicians. Hence, an ordinal number system named after him, which grades how closely any mathematician came to work with him. If you have an “Erdös number one,” that means you actually did a mathematical paper with this prodigy. “Erdös number two” means you have done a joint paper with a mathematician who did a paper with Erdös.
I know someone whose bigger brother is an Erdös number two. She wonders if having her homework done for her, by said brother, makes her an Erdös number three!
At our book club, we gave little talks about a book we had read. She enjoyed my talk about the Paul Hoffmann title, some time ago. She said I was right: he did bring his mother along with him.
That much I had faithfully gleaned from Hoffmann. (Unfortunately, I can’t remember exactly what I said. The following account differs somewhat.)
Paul Erdös was of Jewish Hungarian extraction. The book says the name Erdös is pronounced “air-dish.” My faher said the Hungarian pronunciation is actually “err-desh.” That is err, as in “to err is human, to forgive is divine,” and desh, as in Bangladesh. (Scots would give “err” its traditional and fonetic pronunciation, which does sound like the normal pronouncing of “air,” but I mean the more typical pronunciation of “err,” as an unstressed vowel sound.)
The physics nobel prize winner, Abdus Salam also was a mathematical prodigy, which was how he came to be discovered in his home country of Bangladesh. He set up a foundation in Italy to help others like himself in under-privileged countries. Evidently, they put much back in their homelands, and do not constitute a “brain drain.”
You may think it is good to tell I am not a mathematician, because they do not work on irrational associations. The conclusion is correct but the inference false. Yes, I’m not a mathematician. No, they do. A versifying colleague (half) rhymed Erdös with “Kurdish,” as about the only people not to benefit from one of his math papers. Erdös was enthused to submit a paper to a Kurdish journal of mathematics. But he found there wasn’t one.
(Not being a mathematician, apologies for not knowing other mathematicians names from Hoffmanns book -- Erdöses own name being all I could master.)
Paul Erdöses parents were both mathematicians. Paul, himself, never married, as title of Hoffmann biography suggests. His genetic heritage went no further. He called children “epsilons,” the Greek letter mathematicians use for small quantities. He loved children and was good in their company. Earlier fotos show him beaming in their presence.
Paul never really grew up. He was always a mothers boy. War and conquest deprived him, as it has many, of a fathers influence. He was not allowed to tie his shoelaces, till his age was into double figures. His mother was so possessive, she once appeared out of an upstairs window to ask her son, in the street, what was he doing with that girl. She was the girl-friend accompanying Pauls friend.
By all accounts, his mother was likable, as well as dominating. Five years after she died, Paul was gloomily crossing some campus. A colleague asked him what was the matter. He replied: He missed his mother. Reminded this was five years ago, he said, he knew.
After her death, he threw himself even deeper into his vocation. More than one foto of him, in later life, show him asleep at formal group takes or just at dinner. Those in mathematical conversation with him would think he hadn’t heard. He was like a dolphin that sleeps with one half of his brain, while the other half stays alert. The mathematics went on even while he was dozing.
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Bernard Shaw wisely advised that anyone, who worked more than four hours a day on the most intellectually demanding work, such as research mathematics, was heading for an early grave. A mathematical collaboration with Erdös was a somewhat taxing seventeen-hour day. He would arrive at the door-step. If it was Christmas, he would say something like: Happy Christmas. Let n be the number…
At about four o'clock in the morning, the guest would start rummaging noisily about the kitchen, in his undomesticated way -- he was used to his mother having done everything for him and expected everyone else to. Moreover, this was the hint it was about time his colleague and host got up to do mathematics.
Erdös had no home. He lived from a suit-case, on a perpetual tour to tap other mathematicians brains. “Property is a nuisance” is one of his sayings. He kept himself solvent with earnings from mathematical journals and prizes. He never kept more than he needed to meet colleagues. He was constantly giving money away to the current needy cause, wherever in the world it was.
He might have seemed a sad case, had he not had this professional talent, because no-one might have known the extent of his goodness of character. Perhaps that is the moral of his life. He only needed mathematics but others had a most pressing need of money. He knew this and cared without fuss and without their asking.
One married couple of mathematicians built an extension for him to stay periods. He could be difficult to work with. He could drive the woman, of these hosting partners, to vow she would never work with him again, in her frustration at his bad conversational manners.
He would ask her to explain some maths and then interrupt her to try and re-formulate the problem in his own terms, which naturally stopped her in her tracks.
Yet Erdös was an incomparable ambassador of mathematics. He would set people questions with rewards, starting with a five dollar question, grading the prize according to the difficulty of the answer. He could judge ability, so he knew just what level of problem to set.
At the top end of ability, he once advised a graduate against a particular thesis. It was too difficult. The young man had cause to be grateful. The problem still wasn’t solved by the mathematical world twenty or thirty years later.
Erdös prefered tackling problems that didn't need a lot of specialist knowledge. He best liked solutions "straight from the book" -- Gods manual of creation, as it were -- that carried immediate conviction. In perhaps forty pages or so, two mathematicians, to their intense pride. had written a proof of a theorem. Erdös happened to notice it on a black-board. Asking the meaning of the notation, from a field of math he didn't know, he wrote straight down, in a couple of lines or so, a new proof -- straight from the book.
He wasn’t in the slightest interested in the practical value of his findings. He would be satisfied if no applications were found for another five hundred years. This was not just the doctrine of pure science but the freedom to enjoy mathematics for mathematics sake.
Paul Hoffmann is of interest for also discussing some recent mathematical milestones, such as the proof to Fermat last theorem. The solver happened to know the right branches of study and worked long alone to win the prize on offer. This way is in complete contrast to the co-operative Erdös. Ironically, a hole was found in the closet solvers proof and he was driven to seek help from another mathematician, to plug the leak in his proof.
However, I must admit to finding the cited problems, in pure math, neither practical nor interesting. There was one old brain-teaser I found appealing and remembered for a little talk to a non-mathematical group. A well-known tv hostess with a super IQ caused a storm of controversy with it:
Suppose you are on a quiz show. You may choose to open one of three doors. One has a prize behind it, an other a booby prize. Suppose you choose one door but before you open it, the show hostess, opens another door which reveals no prize. The hostess then allows you to stay with your first choice or to choose, instead, the other unopened door.
The question is: which is the best strategy?
The tv woman with the genius IQ said: change your choice. Letters at a rate of nine to one disagreed with her, including some academics, on the degenerate influence of tv, saying things like: you really blew it, this time! Their argument was that the move from one door to the other shouldn’t make any difference, because there was an equal probability that the prize would be behind either still unopened door.
Of the little group, this reviewer talked to, some guessed right some wrong. None really know. I had thought like the ignorant ninety per cent. The interesting thing is that the most prolific mathematician of the twentieth century couldn’t understand, either. Like a green student, he pestered his host and colleague for an explanation.
Erdös was shown a computer simulation of the quiz show given a large number of trials. On average, the probability of winning the prize was one-third, if one stayed at the door of ones first choice. If one changed ones choice, the probability of winning became two-thirds.
Erdös accepted the result but he still wanted a transparent explanation “straight from the book.”
Erdöses friend and colleague put it this way: You, as the quiz contestant, know you are going to be given the chance of making two choices for the prize.
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Julian Barbour: The End Of Time
- in classical physics.
Table of contents
(1) Triangle Land.
Links to sections:
All the world’s a kinema.
Time and motion
The shape space of triangle land.
Shortest paths in triangle land.
Deriving Newton laws from Mach principle.
Prominent role of time in Relativity.
A timeless geometric dynamics within general relativity.
Quantum gravitys conflict over time.
All the world’s a kinema.
Suppose the audience of a picture house are immortal souls and that the happenings on screen relate to this world of ours. A holographic movie would be even more like three-dimensional reality. The audience of souls become so absorbed in the goings-on, on screen, that they forget themselves and identify with the actors, some more than others, and perhaps one in particular, who becomes ones (mortal) self. Consciousness has shifted from timelessness to time.
Our holographic lives, in the kinema, are apparently kinematic or moving in time. But God, the great movie maker, knows better. A movie reel is made up of a lot of static images. The projector runs them too fast for us to see the jumps between them, giving an impression of flowing motion. Gods cuttings floor is strewn with rejected images, immensely more than left out by any human director, because the divine director works on a grand scale.
This stupendous totality of imaginary realities, from which a miniscule number are selected to become our conscious reality, is called Platonia by Julian Barbour.
This is after Plato, on an underlying reality of perfect forms, to which our world only approximates. We are likened to cave-dwellers, round a fire, who see only shadows of a real world outside.
Platonia is a timeless jumble of practically infinite possibilities. The images most likely to come together in an appearance of timely motion are the kinematic slides best matched to each other.
If you cut a movie reel into its individual slides, you could put them back together in sequence by comparing those which were best matched to run continuously. In platonia, tho, there is an embarrassment of choice from every conceivable possibility of image, tho the vast majority are so ill-matched that they are easily dismissed from any probable historical sequence.
The explanatory success of quantum theory has made physicists take seriously the notion that there is a graded potential for all logical possibilities of existence to become reality. Barbour makes the point that these possibilities could include unimaginable heaven, purgatory and hell. (He has a pantheistic belief in platonia, rather than in a personal god.)
Julian Barbour wrote The End Of Time (1999) about how such a timeless nature of reality might work. It is physics not the metaphysics of God and immortal souls. Tho not Barbours view, it is perhaps worth mentioning that karma, for example, is a sort of probability theory of reincarnation, whereby the moral conditions of souls best-matches them to a succession of mortal bodies.
Time and motion
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I attempt to explain Barbour to myself, but don’t pretend to fully understand his book, and apologise to readers for errors or infelicities. Barbour gives us passengers a privileged tour of the engine room to the ship of physics. This review is meant to be no more than a guide. My guides never managed to do without going back and obtaining the masterpiece in question.
Julian Barbour is a physicist, who forsook main-stream university life, to concentrate on the meaning of time. He decided, as a young man, that time could be reduced to terms of “movement.” This was the motive for, to quote the sub-title of his book: The next revolution in our understanding of the universe.
This is rather as if a clock was considered not as telling “the time” but as its “movement,” which is the name for the mechanics of a clock. Mechanical toy trains were described as (running on) “clock-work.”
The heavenly bodies regulate living bodies, so that they have their own “biological clocks.” Human beings even regulate themselves with an abstract concept of time. Where does this notion come from? And from what does time ultimately derive?
Astronomers used earth rotation for their clock. About the turn of the twentieth century, they found earth rotation rate was not quite regular enough, because of lunar gravitational effects. The earth may be put out of sync by the moon. The sun and its planets, as a whole, make a more regular clock-work, because they are isolated from any such intrusive influences.
This change to a newer heavenly time-piece may be compared to man making a more regular mechanism by designing a longer train of cog-wheels, that slows down the full force of the main-spring to unwind more gradually and uniformly. That way, the clock is less liable to gain too much when fully wound up and to lose too much as it runs down.
So, the solar system was adopted as a more regular natural clock. This is called ephemeris time. Ephemeris tables give the positions of the planets at given times. The most convenient planetary pointer for so telling the time, but not the most accurate, is the moon. Thus, time is a convention that depends on the most regular available time-piece.
Ephemeris time was soon superseded by a convention based on atomic periodicity. (On the very small scale of the atom, gravitational disturbances are negligible.) But it remains a pointer, in the Barbour quest for the true nature of time. The solar system is a little universe in itself. Indeed, by the start of the third millenium, mans technical ability scarcely reaches to its limits.
Barbour suggests that if time is more accurately measured by a graduation from diurnal time to ephemeris time, then the ultimate time-piece is nothing less than the universe itself.
The principle of Ernst Mach is that time only makes sense in terms of motion and motion is relative to the motions of all the masses of the universe, on which time, therefore, ultimately depends.
Universal gravity is its “main-spring” which spins whole galaxies in its train. (The actual train of the stellar clock-work mechanism, or which cogs connect with which, also might be used as a fanciful analogy to Barbour idea of “best matching.”)
By definition, the universe is everything that there is. It would be illogical to think of a god, outside it, timing its run, with a stop-watch. But that is essentially what the notion of “absolute time” is, in classical physics. Barbour wishes to promote Ernst Mach principle of a self-referential universe, from which time is a convenient derivation but no more than that. The very convenience of the concept of time may make it a most powerful illusion. But basically time is just an illusion, Barbour thinks. Hence, the title of his book.
The obvious precedent for this way of thinking is how physics over-turned the convenient fiction that the sun travels round the earth, rather than vice versa.
The shape space of triangle land.
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Barbour sets out to show how the notion of time may emerge from a universe considered merely as the relations between all its objects. To do this, he imagines a model of the simplest of universes, consisting of three objects. Three objects have a triangular relationship and he calls this universe “triangle land.” If these are massive objects, their relations will be governed by the law of gravity.
There is no-one running a stop-watch to gauge the time it takes for the three bodies to change their positions relative to each other. There is no absolute time. Nor is there an “absolute space”: these bodies are not spatially measured with respect to the sides of a box of co-ordinates, the length, breadth and height of some cube-shaped room.
Barbour triangle land is a universe sufficient to itself. So, it must fashion any co-ordinates, such as it may have, in its own terms. In practise, this means taking the three bodies in one configuration at a time. That is one differently shaped triangle at a time, with the bodies at its three corners. Barbour calls each of these triangles “time-instants.” They are like fotografs or snaps of the bodies at one instant of time.
All the possible configurations for three bodies can be represented by all the possible shapes of a triangle. The general geometry of triangles is, in effect, the structure of the space, or the “configuration space,” within which these three bodies must move.
For the possible relations between four bodies, the geometry of a tetrahedron land, in six dimensions, would apply, analgously. And so on, for greater numbers of bodies, in hugely multi-dimensional configuration spaces.
Happily, the simple “universe,” of positions for three bodies, can be visualised in three dimensions. Their three co-ordinates are AB, BC and CA. These represent the lengths of each side of a given triangle. Each triangle is then pin-pointed accordingly within this box of co-ordinates. One corner of this box is taken as the origin, meaning the point representing a triangle, all three of whose sides are zero. This is the unique point at which all three bodies, configuring a triangle, meet. Barbour suggestively calls this the alpha point.
From that corner, the three room-edges of length, breadth and height extend. A “triangle” pin-pointed exactly on co-ordinate AB, BC or CA is just a greater or shorter line AB, BC or CA, respectively.
The geometric properties of triangles limit the area of the box that may be filled with positions for possible triangle shapes. One triangle side cannot be longer than the sum of the other two. This limitation removes some points in the box as possible positions to represent triangles. The borders of this limit are described by a regular three-cornered pyramid, whose apex is at the origin of the box.
The pyramids three edges, to its base corners, extend at 45 degrees to the three co-ordinates AB, BC, CA. These pyramid-edges represent triangles, two of whose three corners coincide, one of their sides being zero length. This stands for two out of three bodies meeting. They are the next most unique positions to the origin or alpha point. (Diagrams are given in Barbours book!)
It does not matter how far the edges of the pyramid extend or how large is the base triangle of the pyramid. You can cut the pyramid into triangular cross-sections. Each cross-section is, in effect, a more or less broad base to the pyramid. These cross-sections all reveal the same pattern of information about the geometrical nature of the triangles that all the positions on their surfaces represent. These cross-sections are called “shape space.” Independently of scale, shape space contains positions for every possible shape of triangle.
Notice that the geometric meaning, of the pyramids edges, is retained in the cross-sections of shape space, considered as the base corners, that the edges extend-to from the pyramid apex. The base lines of the pyramid, which are also the sides of shape space, are the positions for all triangles, in which the length of one side equals the sum of the other two sides.
Also, the very centre of each cross-section is the one position in shape space marking an equilateral triangle. It is perpendicular to the pyramid apex, where three bodies are zero distance from each other. The apex or alfa point is like an equal-sided triangle, where all three sides are zero length.
A potential energy contour map can be drawn over shape space. Potential energy is inversely proportional to separation, so the potential energy rises like canyon walls over the shape spaces three corners, each representing two of the three masses coinciding. Indeed, the potential energy only depends on the relative configuration of masses. It is independent of a frame-work of absolute space and absolute time, and so is a suitable measure of changes from place to place, in platonia, as mentioned, below.
A fuller analysis of the shape space of triangle land revealed the following. Three lines, that cross at the center to make perpendicular bisections with the three sides of each cross-section, mark the only positions for isosceles triangles (where two of the sides are equal).
It turned-out that right-angled triangles positions were found only on three lines concave to the three sides of shape space. The three “lens” areas, in between, represented positions for triangles with an obtuse angle (greater than 90 degrees). The remaining central area of shape space designated acute-angled triangles: all their angles less than 90 degrees.
The moral is that even the simplest of platonias or relative configuration spaces, consisting of three bodies possible positions, has a geometric structure. Whereas, absolute space is treated as absolutely uniform. No point in absolute space is regarded as different from any other. It is essentially a transparent abstraction.
Julian Balfour believes that the extremely complex structure of a platonia of the real universe will be shown to guide the apparent “arrow of time,” which gives us the impression of being caught on a present flow of time out of the past into the future. The mathematical demonstration of Barbours conjecture is liable to be extremely difficult.
The notes of his book mention collaboration on a dynamic geometry to remove completely the concept of absolute distance. This would be analgous to the way shape space removes over-all scale from triangle land. But, as well as that, the ratios of lengths of sides would no longer be relevant. (Barbour web sites promised news: www.julianbarbour.com or platonia.com).
Shortest paths in triangle land.
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We must remember that triangle land has no outside observer timing a sequence that these triangles might make, as a means to determining the distances between them. The so-called “relative configuration space,” that makes up triangle land, consists of nothing more than a jumble of triangles. Barbour names “Platonia” all such relative configuration spaces for any given number of bodies, up to the totality of the universe itself.
It is still possible to measure a “distance,” in platonia, between neighboring triangles, without reference to an outside frame-work.
The distance measure for triangle land, used in The End Of Time, is like Pythogoras theorem in three dimensions, but with the hypotenuse transformed into the platonia “distance” and the three perpendicular sides of a three-dimensional triangle transformed into the distances, AA*, BB*, CC*, between the respective corners of two triangles in triangle land.
If the three bodies, at the corners, have different masses, this may be allowed-for by weighting this calculation.
In other words, if a, b, c, are the masses of three objects, in two different triangular configurations, ABC and A*B*C*, then the platonia distance, squared, equals a times (AA*) squared plus b times (BB*) squared plus c times (CC*) squared.
Any two such triangles can be moved relative to each other, so that their distance, represented by neighboring points in platonia, is at a minimum. This minimum distance is termed their “intrinsic difference” and, as such, is said to represent their best-matching position.
(The formula for intrinsic difference is modified by a function of potential energy, actually the square root of minus the potential, according to Barbour. This modification, “the action,” does not affect the argument.) It can be found for any two distributions of matter, such as astronomers might observe, at any time, in any arbitrary relation to each other.
These shortest paths, in platonia, are like the geodesics in general relativity. In that theory, light appears to bend or curve under gravitational attraction, because it is actually following a shortest path, determined by the geometrical curving of space around gravitational masses. Given that platonia can have its own geometrically determined “lines of least resistance” or geodesics, these are the most probable paths for a configuration of masses, plotted in platonia.
(2) geometric dynamics.
Deriving Newton laws from Mach principle.
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Bruno Bertotti and Julian Barbour constructed a theory of geodesics to determine the shortest path between any two fixed points in platonia. They found that the unique history this produced, from an initial point and direction, corresponds to one of many such histories that Newton framed in absolute space and time. This correspondence was to the special case of a Newtonian history, with zero energy and angular momentum, which solves in terms of “a simpler timeless and frameless theory.”
Barbour and Bertotti produced the mathematics that put paradigm Newton in the context of the Mach principle.
Assuming the universe is finite, the total energy and angular momentum of its sub-systems add up to zero. But this will not be true of most sub-systems, themselves. So, they will produce the much more common Newtonian solutions for galaxies or solar systems with non-zero energies and angular momentums, as if they were in absolute space and time.
Even to Isaac Newton contemporary, Leibniz, absolute space and time seemed a cumbersome frame of measurement. In determining the evolution of a configuration of, say, three masses, from its initial conditions and direction to a second configuration, fourteen dimensions (allowing fourteen “degrees of freedom”) are involved. But only four of them make any difference to the result.
The ten absolute dimensions that make no difference are: three spatial dimensions each for the first and second configurations, from the points of view of their centers of mass; the starting time, another dimension; and the three-dimensional orientation of the first triangle.
The four remaining dimensions that do matter are the orientation of the second triangular configuration and the absolute time elapsing between the two configurations positions, or to put it another way, the angular momentum and the kinetic energy, respectively. The angular momentum of the three-mass configuration is like a spinning top with an imaginary pivot, thru its center of mass, pointing skywards over two dimensions, with a third dimension from the axial rotation of the triangle perpendicular to the pivot.
Spiral galaxies and the rings of Saturn are spectacular astronomical examples of angular momentum.
In relative configuration space or platonia, the process of best matching, from one configuration to another, creates a determinate relation, of all the configurations to the first chosen configuration, which gives an appearance of a rigid frame-work, like absolute space and time.
Barbour completes a derivation of Newton, from Mach, world view, with respect to the “spacings in time”:
In the equations that describe how the objects move in the framework built up by best matching, it is very convenient to measure how far each body moves by making a comparison with a certain average of all the bodies in the universe. The choice of the average is obvious, and simplifies the equations dramatically…It is directly related to the quantities used to determine the geodesic paths in Platonia. To find how much it changes as the universe passes from one configuration to another slightly different one, it is necessary only to divide their intrinsic difference by the square root of minus the potential. (The action, by contrast, is found by multiplying it by the same quantity.) When this distinguished simplifier is used as “time,” it turns out that each object in the universe moves in the Mach framework described above exactly as Newton laws prescribe.
Prominent role of time in Relativity.
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Having derived a simulacrum of time from a dynamic geometry of gravitational masses, in Newtons system, the next step was to do likewise for the theory that superseded it, Einstein general theory of relativity. This was a daunting task, since time plays such a prominent role in both special and general relativity.
A first set-back is that Barbour platonia is a collection of time-instants or Nows. But special relativity seems to do away with the concept of (what time is) Now, or Simultaneity, as what we are agreed is the same time for all of us. Simultaneity turned out not to be a universal time but locally measured times, that differed as to what time is now, from their different frames of reference, in uniform relative motion to each other. Only observers at rest in relation to each other would agree when something happened, in high velocity or high energy physics.
Galileo relativity principle observed that the laws of motion hold within the cabin of a galley, whether it is moving or at rest. You couldn’t tell, say, from two people playing ping-pong on the captains table, whether the ship was cruising or at anchor. Therefore, we cannot say whether the galley is at rest or moving relative to some supposed “absolute space.”
As a so-called “inertial frame of reference,” the galley merely maintains the inertia of staying where it is or continuing to move steadily in a straight line, unless acted upon by outside forces.
Galileo relativity principle makes redundant so much of the afore-mentioned absolute space and time frame-work for measuring the lawful motion of bodies from known initial conditions.
In Einstein special relativity, Galileo relativity is combined, instead, with a new absolute, in the limiting speed of light. When the motions of bodies, being considered, are very slow compared with light, as they usually are on earth, light speed is effectively a stand-in for absolute space and time. And different observers can match each others frame of measurement, in a common-sense manner, called the Galilean transformations (of their respective co-ordinates, to agree with each other, as to what they’ve both seen).
But heavenly bodies, receding at astronomical speeds, or the basic constituents of matter, in particle accelerators, approach to a significant fraction of the speed of light. Observers in uniform relative motion need more general formulas to make their space and time measurements correspond to a given event. These are called the Lorentz transformations, which reduce to the Galilean transformations for motions not significant compared to light speed.
The Lorentz transformations adjust observers space and time measures, or rods and clocks, so that neither observer can send a light signal, in relative motion away from the other, that would move faster than the limiting speed of light. There follow apparent paradoxes of time and space. This includes an inability to agree on the Now that something happened: simultaneity is lost.
However, Hermann Minkowski came up with a quantity called “The Interval,” by which all observers, in uniform relative motion, agreed what they had measured, in terms of the same “space-time” event. All the observers noted their respective distances and times, from an event. But when you treated time as akin to a fourth dimension of space, the totality of each observers space and time readings always added up to the same amount, considered as a geometrically integrated space-time quantity.
Most physicists prize this formalism for its unexpected unification of two of classical physics basic concepts. But it seems at cross-purposes to the Barbour program, as he seeks to de-mystify time, rendered in terms purely of changing spatial relationships or a dynamic geometry.
Barbour goes back to Henri Poincaré, in an 1898 paper that posed two problems in the definition of time. One was simultaneity, which Einstein solved in 1905. The other was duration: how do we know a second today is the same as a second tomorrow? How do we know that the hands or pointers of different clocks will allow us literally to keep appointments?
An inertial clock of only three identified particles moving inertially could have four snapshots taken to show the distances between them. Peter Tait showed how this information of triangular positions provided enough known quantities, obeying the inertial law, to derive the unknown quantities of the times the snapshots were taken and their positions in an absolute space.
Using one of the particles as a reference point or origin, it turns out that space can be found in which the triangle corners move on mutually uniform straight lines. In other words, either of the other particles can serve as a clock “hand” for the motion of the other two particles, considered as the “movement” or mechanism of the inertial clock.
Duration is reduced to distance. If today or tomorrow any one of the “hands” of the inertial clock moves through the same distance, then we can say that the “same amount of time” has passed. The extra time dimension is redundant: everything we need to know about time can be read off from the distances.
The same applies for mechanisms of much larger numbers of bodies. The astronomers ephemeris time, using the fairly isolated solar system is an inertial clock. The universe is the ultimate inertial clock, because, by definition, there are no outside forces acting on the universe.
For more than five particles, three snapshots are enough to solve Tait problem but two are never enough to tell about relative orientations and separation in time. These variables require the four out of fourteen dimensions not redundant in Newton absolute space and time frame-work, that were mentioned above.
Angular momentum accounts for three of those four dimensions. Even if the rotation of a mass is uniform, the velocity of that rotation involves a (centripetal or normal) component of acceleration, due to the change in direction of the velocity. And Newton justified his absolute frame-work with regard to acceleration, not velocity.
A timeless geometric dynamics within general relativity.
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General relativity is usually considered in terms of Minkowski four-dimensional space-time generalised from Euclid flat surfaces to Riemann geometry of curved space. The special theory made use of Galileo relativity principle for the freedom it confered to observers irrespective of (uniform) relative velocity between their co-ordinate systems. Einstein general theory extended this relative freedom of observation with co-ordinate systems (of geometric curvature), irrespective of (uniform) relative acceleration between observers.
Albert Einstein equivalence principle was akin to Galileo relativity principle in that it imagined observing law-like physical effects, which were reproduced in either of two apparently different conditions. For instance, the apparent bending or curving of light, passing thru space-ship portals, would not tell the crew whether they were in tow under uniform acceleration in empty space or whether they were in a gravitational field.
This reviewer is no physics expert and Barbour account of relativity is mainly concerned to relate relativity to his own program that time does not exist. Suffice it to say that Barbour and Bertotti were told by Karel Kuchar that the math of their ideas of best matching and duration was implicit in a less well known dynamic treatment of general relativity.
Originally, Barbour and Bertotti didn’t know that a dynamic geometry (or geometro-dynamics) of curved three-dimensional spaces, based on Mach principle, is implicit in general relativity. (Einstein never knew it.) So, they set about creating one. One incentive was the work of Dirac and others:
They found that if general relativity is to be cast in dynamic form, then the “thing that changes” is not… the four-dimensional distances within space-time, but the distances within three-dimensional spaces nested in space-time. The dynamics of general relativity is about three-dimensional things: Riemannian spaces.
A platonia is defined as “any class of objects that differ intrinsically but are all constructed according to the same rule.” We saw this of sets of three particles according to the geometric rules of triangles. Likewise, Riemann spaces can be treated as time-instants, registering as points in a platonia, if one of infinitely many dimensions.
As mentioned (above, at end of section, the shape space of triangle land) the problem of infinite spaces might be removed. At any rate, it is convenient to consider finite Riemann spaces. How a three-dimensional space folds up on itself is hard to imagine. But a finite two-dimensional space is like the surface of the earth or an egg.
All the shapes, that cannot be transformed into each other, without stretching, count as a separate point in platonia, as is any number of superficial variations on these shapes. This platonia of possible empty spaces is itself vastly increased by “painted patterns” on the surfaces, to represent matter, electro-magnetic and other fields in the universe. Indeed, general relativity related gravitational mass to spatial curvature. And the platonia of three-dimensional Riemann spaces is known as “superspace” in the formalism, given to general relativity, by John Wheeler and colleagues.
Best matching in this Riemannian platonia is much more complicated than for triangle land. Barbour illustrated this at lectures, using two convoluted fungi, of rather different size and markings, which he labelled Tristan and Isolde. A first guess, at their corresponding positions, is marked by pins, with matching numbers 1, 2, 3, etc. This “trial pairing” serves as a basis to establish a “provisional difference,” an average of all the differences of curvature at each pair of points.
Keeping the pins fixed on Tristan, the pins are re-arranged on Isolde, as many ways as possible, in a continuous fashion. The best matching pairing and corresponding intrinsic difference is the transition, between any ever so slightly differing pairings, on this continuum, where the provisional difference remains unchanged. (That is the “stationary point.”)
“Now” appears even more arbitrary in general relativity space-time than Minkowski space-time. But general relativity exactly positions two three-spaces in four-dimensional space-time, whose geodesics are followed as the world-lines of bodies. Indeed, clocks, traveling the lines, measure the proper times. The world lines, like a series of “struts” compare to the “pin”-matching of pairs of three-spaces.
The Einstein equation states a best-matching condition between two three-spaces, which also feature as time-instants in platonia. As just these two “nows” are needed, general relativity turns out to justify Mach belief in the redundancy of a third “now,” thought necesary for measurement in the context of Newton absolute frame-work of space and time.
At the end of the above section, deriving Newton laws from Mach principle, Barbour was quoted on the distinguished simplifier, as creating the same “time separation” across all space. But, to quote him again:
In Einstein’s geometrodynamics, the separation between the 3-spaces varies from point to point, but the principle that determines it is a generalization, now applied locally, of the principle that works in the Newtonian case and explains how people can keep appointments…
Since the equivalence principle is essentially the condition that the law of inertia holds in small regions of space-time, and all clocks rely in one way or another on inertia, this is the ultimate explanation of why it is relatively easy (nowadays at least) to build clocks that all march in step. They all tick to the ephemeris time created by the universe through the best matching that fits it together.
Quantum gravitys conflict over time.
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Trying to give a description of general relativity in terms of quantum theory exposed their conflicting concepts of time. The theoretical likenesses between Maxwell on electro-magnetism and Einstein on space-time (especially when almost flat like Minkowski space-time) led physicists to believe that just as the photon is the quantum concept associated with the electro-magnetic wave field, an analgous massless particle, the graviton, could be conjectured for the field of gravitational waves, too weak to be detected by contemporary devices.
A relativistic effect of the masslessness of the photon nullifies (longitudinal) waves that would move along its direction. At right angles to the direction, two perpendicular (transverse) waves, giving two true degrees of freedom, constitute two independent polarisations of light. These are what bees can see to orient themselves.
Paul Dirac and the ADM team (Arnowitt, Deser and Misner) wanted a general quantum gravity, including for curved space. But the graviton and gravitational fields implied two true degrees of freedom did not match general relativitys three degrees of freedom of the three-spaces, whose “geometry – the way in which they are curved – is described by three numbers at each point of space.”
Physicists attempted to match the two theories differing degrees of freedom. Since time was thought to be needed for the quantum description of gravity, it was believed time might be identified with one of the three-spaces. But this would go against the equivalence of all co-ordinates in relativity theory, denying a definite time.
To apply the (Schrödinger) equation of quantum mechanics to geometric dynamics, the Wheeler-DeWitt equation fell back on Dirac method of defering a choice of time dimension from three spatial dimensions. Barbour advocates a “naive” interpretation, of this version, as the stationary state Schrödinger equation for a zero sum energy of the universe.
The text-book ball-and-strut models of molecules are only the most probable configurations of the structure of micro-scopic matter, this equation normally describes. The Wheeler-DeWitt equation is a telescopic version, with the universe as one “monster molecule” which also has its huge number of other possible configurations. These are the collections of “time-instants” or nows, that make up the points of a timeless landscape, that is Barbours platonia, at the other extreme of complexity to triangle land.
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Julian Barbour: The End Of Time
- in quantum mechanics:
Table of contents.
(1) making waves.
Links to sections:
Quantum energy waves of particles.
The double slit experiment.
Quantum entanglement of a two-particle system.
Shrödinger stationary wave equation.
Relativised Schrödinger equation of the cosmos.
Quantum theory of records.
Time dependence on configurations.
Quantum energy waves of particles.
Relativity is based on the limiting maximum velocity of the speed of light, c for celerity or constant. Quantum theory is based on a limiting minimum of energy transfer in “lumps” or “quanta” (or quantums, as I would say). Max Planck invoked this near infinitessimal quantity, h,[] the “quantum” in relation to a problem with infinitely continuous waves of radiation, thrown up by the theory of ovens or black boxes.
Light had been shown to have wave-like effects of interference patterns, such as are seen in water waves. Introducing “light quanta” suggested light waves are fundamentally made up of particles. It is as if one were to change ones belief, that water waves form a continuous flow, to considering them as really more like dunes or waves of sand, that on very close inspection are made up of tiny grains.
However, this simple analogy glosses over the deep puzzles encountered on an altogether smaller scale of physical phenomena.
Einstein took up Max Planck idea of the quantum, in the form of light quanta or photons, to explain the “photo-electric effect.” It was found that bombarding the surface of metal, with certain beams of light, dislodged electrons from its surface. The effect didn’t depend on the intensity or brightness of the light used. If the light was ultra-violet, no matter how dim the beam, it still succeeded in knocking off electrons.
Einstein explained that higher frequencies of light, or more generally electro-magnetic radiation, such as violet or ultra-violet light were, in effect, faster “bullets” or quanta (quantums) of light energy. Dimming this more energetic light only meant that fewer bullets were being fired. But when they hit an electron, bound to a metal atom on the surface of a metal plate, they would still dislodge it.
In contrast, it didn’t matter how bright you made longer wave, red or infra-red light; in other words, it didn’t matter how many red bullets were fired, they were all too low energy to ionise the metal surface.
An analogy to the photo-electric effect might be two walkers, walking into someone. They are both going at the same speed. One walker is taking long slow strides and so not using much energy. He hardly disturbs you, passing by. The other walker is taking short fast steps, and knocks you out of the way, bustling by.
These two walkers going the same speed compare to light speed as being constant. The velocity of light equals its wavelength times its frequency. (Or, v = l x f. Texts normally use Greek small letters lambda for wave-length and nu for frequency.) Red light is just as fast as violet but the former makes up for its lower frequency with longer wavelength. Red lights longer wave-length is like the longer strides of the walker making up for a lower frequency of steps.
Einstein 1905 paper on the photo-electric effect is summed up in the formula, E = hf. Energy, E, equals Planck constant, h, times the radiation frequency, f (usually denoted by the Greek small letter nu).
The formula, E = hf, suggests a compromise between the wave and particle theories, of matter in sub-atomic physics, measured as continuous and discrete quantities. The quantum, h, is a minimum discrete energy quantity, of which, wave-length quantities must always be an exact multiple. The frequency, f , is known in the classical physics of continuous circular motion as the time rate of change of an angle -- call it angle Q. (Again I've passed-by the typical Greek letter, in this case, theta used for an angle.)
The double slit experiment.
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Julian Barbour uses the standard introduction, to the many mysteries of wave mechanics or quantum mechanics, with the double slit experiment.
Shine a beam of light thru a single slit onto a wall. Most of the light will go straight thru the slit and form the densest target area on the wall. The rest of the light will get more or less deflected from the edges of the slit and scatter about the densest area.
Replacing one slit with two slits, close together, produces a surprisingly different picture. On the wall, a series of bars form, the middle bars being most densely lit. There is an absence of light strikes between the bars.
The single slit experiment could have been explained as either a particle or wave activity of light. But the double slit experiment has to be explained in terms of light being in the form of both particles and waves.
The bars of light are characteristic of interference patterns found in water waves. When two radiating circles of waves, such as from two stones dropped in a pond, collide, the crests of the two rings may reinforce each other creating higher crests. The trofs may reinforce each others depth. When one ring crest coincides with the other ring trof they neutralise each other to surface level.
The snag is that even when one photon at a time is sent at the double slit, the photon hits on the wall build up the same pattern, as if they were sent in a steady stream that waved into each other. The basic paradox of quantum theory wave-particle duality of light (and electrons etc) is that a single particle can interfere with itself like a wave.
The double slit experiment pattern of photon hits is given a probabilistic prediction by Schrödinger equation of the wave function (denoted by Greek letter psi). The wave function that gives the best interference effects, in the experiment are so-called momentum “eigenstates” (German for “proper” or “characteristic” states).
Eigenstates of position or momentum are the only ones that can be measured with complete or unit probability of matching prediction.
It turns out that the wave function of a particle with a definite momentum (the momentum eigenstate) is two super-posed plane waves out of phase by a quarter of a wave-length. By definition, sine and cosine waves are out of phase by 90 degrees or a quarter of a wave-length.
The quantum mechanical wave function is a complex function. Roger Penrose, in The Emperor’s New Mind, explains how complex numbers are used in this function.
Suffice it to say, a horizontal x-axis could represent the direction of the two waves. Backward or forward direction is related to which of the two waves comes first. A y-axis could give a back-ground dimension to the sine and cosine waves, turning them from just undulating lines into planar waves. A z-axis could measure the height or amplitude of these waves. But the y and z axes are treated as composite or complex numbers, which are ordered pairs of numbers. These represent two intensities. The sum of their squares is the “probability density” of the (complex) wave function, psi, of the x-variable. Notably, this gives the probability that a trial measurement will find a particle at x.
A particle has a definite momentum because its wave function has a regular and definite wave-length. At the same time, the particles position is completely indefinite. Its probability density is uniform thru-out space, because the sum of the squares of two sinusoidal waves, one-quarter wave-length out of phase, is always one, given that they have unit amplitude.
This comes from Pythagoras theorem in trigonometric form: sin²Q + cos²Q = 1.
Fourier showed that adding or super-posing harmonic waves of different wave-lengths can produce any curve, even down to a spike, that characterises the position eigenstate. (Mathematics imitates natures wave-particle duality. The same wave pattern can be regarded as super-posed waves of different wave-lengths or super-posed spikes, with different coefficients.)
These are the extremes between complete and null information we can have between complementary pairs of quantities, such as momentum and position, or energy and time. Heisenberg uncertainty relation measures the extent, that more accurately measuring one of these pairs of quantities, is always at the expense of precisely measuring the other.
The experimenter can measure one or other of the complementary pairs, all implicit in the wave function. So a lack of complete knowledge is offset by a range of choice as to what can be known.
The double slit experiment can be considered in terms of two similar plane light waves super-posed or merged at a slight angle (of five degrees) to each other. At right angles to the mid-line of the five-degree angle, Barbour shows, as a computer-generated probability density, the result, in a concertina-like series of light ridges, corresponding to the light “fringes” that show the interference effect on the wall.
This result relates to William Hamilton on optics. His wave theory showed that regular wave patterns reproduce light rays, without particles, yet corresponding to the older particle theory of light, and explaining more than it could.
Hamilton found an analogy to this in Newtonian dynamics, for only one value of energy allowed. Depending on an equation, his “principal function” has a varying intensity, like the “mist,” at each point of configuration space. This equation is like that for his wave optics, but in multi-dimensional configuration space, instead of the ordinary three dimensions. When the intensity forms regular wave patterns, their respective families of paths, at right angles to their crests, are Newtonian dynamics histories, having the same energy.
This path-making property of regular wave patterns has given them the name “semi-classical,” which is also the name of a physicists program, to show how apparent paths in time may “funnel” out of a timeless geometric structure underlying the universe.
Quantum entanglement of a two-particle system.
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The Einstein, Podolsky, Rosen paradox was a thought-experiment designed to reveal that quantum mechanics, compared to classical mechanics, was incomplete in the scope of its explanations. Two particles, such as photons sharing polarisation, or electrons whose total spin was conserved, would be correlated. Changing the state of one would have a correlative effect on the other.
For example, a system of two electrons, with a total of zero angular momentum, implies that if one electron has spin up, the other must have an equal and opposite spin down.
The EPR paradox was that quantum mechanics predicted that if you moved these correlated particles far apart, then changed the state of one, there would be an instant conservative response from the separated particle. But Einstein special relativity forbids any signal passing, at more than light speed, from one particle to influence the lawful adjustment of the other particle.
John Bell theorem showed how quantum correlations must surpass any relations attributable to classical causes. Roger Penrose has further refined these distinctions, especially in his second popular book, Shadows Of The Mind, on the “magic dodecahedra” (Penrose dodecahedrons).
The EPR team believed the law of local causes would be upheld against quantum correlation. But by the 1980s, the Alain Aspect experiment team were proving a super-luminal connection between correlated particles. This did not mean signals could be sent faster than light; it did not violate the foundations of special relativity. But it did mean the experimenter could bring about a known, faster-than-light change on a distant particle, by a certain change on its correlated particle.
With the help of half a dozen diagrams, Julian Barbour gives readers a feel for quantum correlations and “entanglement” in the simplest possible two-particle system. Two particles, moving on a line, combine their one-dimensional configuration spaces to make a two-dimensional “Q” space.
The wave function value, for a single particle or a duet, as here, varies with time at each point in Q, which carries information about both particles, as to their positions or other quantities. These predictions are comprehensive, if often mutually exclusive, and refer to the system rather than its parts.
To find the relative probability of configurations of the two-particle system at some point, representative averages are found from the mid-points of a grid on Q. The probability density gives the relative numbers of these representative configurations likely to be found by repeated trial measurements.
Barbour likened this process of prediction and measurement, to giving the predicted numbers of configurations a proportionate number of tags in a bag and then drawing them out at random, as a trial confirmation of the predicted proportions.
These configurations are, in effect, ranked by their greater or lesser probabilities, which is how the Schrödinger equation configures atomic and molecular structure from the configuration space of all possibilities.
(In the simplest "platonia" or relative configuration space of Triangle land -- discussed on the former chapter of this review -- a probability ranking may be established thru best matching all possible triangles, each represented by a point in their platonia.)
Measuring the two-particle quantum system for, say, the position of one of the particles reduces the two-dimensional grid on Q to one dimension. This so-called “collapse of the wave function” yields the only possible positions of the unmeasured particle, as relative probabilities of being somewhere on the remaining grid line.
Hugh Everett assumed that the wave function is “the basic physical entity.” Its unimaginably huge numbers of possibilities are taken to constitute “many worlds.” Tho we are only conscious of one world being realised, this does not necessarily mean that is all there is. Everett defended this possibility by the linearity or super-position principle of wave mechanics. Waves can split and combine, to create interference effects, but they remain themselves, essentially unaffected by it.
According to Barbour, “To save the appearances, we do not have to create a unique history: we need only explain why there seems to be a unique history. That was Everett’s insight.”
Barbour sees the essence of things in platonia, the relative configuration space, or a completely relativised version of Schrödinger Q space. This geometric landscape, of all possible configurations of reality, gathers, like a more or less dense mist, the wave function probability density. Following Boltzmann, Barbour assumes: only the probable is experienced.
(2) quantum cosmology.
Shrödinger stationary wave equation.
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Julian Barbour thought a timeless universe has to do with turning Scrödinger quantum mechanics into a quantum cosmology. To do that, he first has to relativise the remaining classical absolute time and space frame-work, within which the Schrödinger equation is expressed. Barbour tries to give non-physicists a new insight into these mysteries.
Barbour makes the point, that only the quantum mechanics of a single particle takes place in the ordinary three dimensions. Quantum phenomena create new puzzles in certain two-particle states, or more. These multi-particle effects take place in a configuration space, which Schrodinger called “Q.”
In Platonia or relative configuration space (described in the previous review chapter on classical physics) the simplest Platonia called Triangle Land consisted of each possible arrangement of three particles. This requires three dimensions for the lengths of the three sides of each triangular configuration, which has its own point in a “configuration space.”
But Schrödinger Q, for triangle land, would not merely rely on the relative positions between the three particles. Q also depends on an external or absolute frame-work. This locates the centre of mass of each triangle in absolute space, requiring three more numbers. Each triangle orientation in absolute space also requires three more numbers.
The Q, of triangle land is a nine-dimensional configuration space. In fact, for any number of particles, Q always has six more dimensions than Platonia.
The Schrödinger equation comes in a time-dependent and a time-independent form. Barbour suggests, contrary to conventional wisdom, that the latter is the more fundamental. The wave equation, that finds all the possible stationary states of a system, hints at a universal state of affairs in which super-positions of stationary waves create a variation in time of the probability density.
The stationary states, such as in Neils Bohr model of the atom, correspond to a fixed energy level, between quantum “jumps” with the emission or absorption of a photon. The probability density of finding the atom in these states is constant, while the complex or composite values of the wave function oscillate with a fixed frequency. But adding two such solutions, with their respective frequencies, makes them interfere. The oscillations cease to be regular. Adding these timeless solutions yet makes the probability density vary with time.
Barbour gives a pictorial description of the Schrödinger wave function in a steady state.
At each point of the configuration space Q, imagine a child swinging a ball in a vertical circle, on a string of constant length, the “amplitude,” denoted by Greek small letter phi. Its squared value stands for the constant probability density.
The swinging ball, continuously changing height, above or below the center, stands for one of the two ordered pairs of numbers, that make a complex variable. Its other component is the distance to the right (positive) and to the left (negative).
The stationary state is like swinging such balls at the same rate, every-where in Q, and all perfectly in phase or reaching the top of the circle together. In the momentum eigenstate, phi is the same every-where. But generally it varies according to a condition, imposed at each point of Q, by the equation of the stationary state. Barbour describes this condition as: Curvature number plus Potential number equals Energy number.
The curvature number is complicated. For a quantum system of three bodies, each point in Q corresponds to a configuration of the three bodies in absolute space. Holding two of the bodies fixed and moving the third, along a line in absolute space, moves on a line in Q.
Phi, the string length can be plotted as a curve more or less above the line.
A three-particle Q has three times three dimensions of movement. So there are nine such curvatures at each point of Q. The “curvature number” is “the sum of these nine curvatures after each has been multiplied by the mass of the particle for which it has been calculated.”
The second number, the Potential is derived by multiplying phi by the potential. The potential energy depends on a given configuration of bodies and their nature, such as their masses.
The third number, the Energy is found by multiplying phi by the previously mentioned quantum energy relation, E = hf. The frequency, f, is the number of rotations of the “balls” in a second.
Schrödinger compared the stationary state of the hydrogen atom to a vibrating string, which is fixed at either end and must always have a whole number of waves (counted in half-wave-lengths) like the harmonics of a musical instrument. The higher harmonics compare to an atoms higher energy levels. The fundamental note, when the string is just one over-arching and under-arching vibration (that is, one half wave-length) compares to the lowest energy level of the atom, its “ground state.”
This analogy supplies a boundary condition for the solution of Schrödinger stationary state equation, as an explanation of the discrete energy levels, posited in Bohr quantum model of the atom. This condition is that the ends of the vibrating string are fixed, therefore the amplitude of phi must tend to zero at large distances.
Where the energy, E, minus the potential, V, is more than zero, phi oscillates. Where E – V is less than zero, phi tends to zero, only in certain well-behaved solutions (the eigenfunctions) for special values of E (the energy eigenvalues). The eigenfunction of the system, with the lowest energy value, is the ground state. Higher energy states are called excited states.
Finally, if E is large enough for E – V to be positive everywhere, the eigenfunctions oscillate everywhere, though more rapidly where the potential is lowest. The negative eigenvalues E form the discrete spectrum, and the corresponding states are called bound states because for them phi has an appreciable value only over a finite region. The remaining states, with E greater than zero, are called unbound states, and their energy eigenvalues form the continuum spectrum.
Relativised Schrödinger equation of the cosmos.
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Barbour follows much the same plan, to dispense with the remaining Newtonian frame-work in quantum mechanics, as he did with classical physics. (See previous chapter.) The Schrödinger wave function, of a given system of particles, changes with their relative configuration, centre of mass, orientation and time.
Barbour dispenses with the latter three, as he did for classical dynamics, since the relative configuration of the whole universe is its own absolute space and time, deriving them independently of an external frame-work. This applies Mach principle to Schrödinger equation for a quantum cosmology.
In applying quantum rules to a classical theory of cosmology, Barbour says:
The central insight is this. A classical theory that treats time in a Machian manner can allow the universe only one value of its energy. But then its quantum theory is singular -- it can only have one energy eigenvalue. Since quantum dynamics of necessity has more than one energy eigenvalue, quantum dynamics of the universe is impossible. There can only be quantum statics. It's as simple as that!
In a timeless system, the over-all energy is zero. So, in the stationary Schrödinger equation, at every point of Q, the sum of the curvature number and the potential number is zero. As in classical physics, the potential already is derived from relative configurations of the bodies that make up a (Machian) system, independently of absolute space and time.
As for curvature, that is the rate at which a curve slope changes, with respect to a distance in absolute space, in ordinary quantum mechanics. Barbour suggests replacing these distances with the Machian best matching distances in relative configuration space, as he did to eliminate absolute space from classical physics.
We then add curvatures measured in as many mutually perpendicular directions as there are dimensions in that timeless arena, and set the sum equal to minus the potential number.
The “Machian” wave functions are the Schrödinger eigenfunctions, whose eigenvalues have zero angular momentum, which was the case for the Machian treatment of classical dynamics.
On platonia or relative configuration space, only the potential and best matching distance govern the static wave function variation from point to point. This timeless “topography” determines where the “mist” of the probability density gathers.
This predicts how probable all the inconceivably many permutations of atomic and molecular structures, and ultimately, Barbour seems to argue, how the most probable configurations of the universe best “resonate” each other, in a sort of competition for the appearance of historical reality.
Quantum theory of records.
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Barbour imagines how history emerges from what he sees as the essentially timeless arena of quantum mechanics. He relies on John Bell for analysis of how records are made, in the context of radio-active decay in a cloud chamber, where an alfa particle leaves a track of ionised atoms.
Bell gives two interpretations of this phenomenum depending on when it is assumed a measurement is taken, that supposedly “collapses the wave function” of possible places the particle will be found. The simpler interpretation assumes that atom ionisation is the “classical external measuring instrument” for revealing the alfa particle, and successively collapsing the wave function, with gradual loss of particle energy and increasing deflection, that can be statistically predicted.
Bell, on second thoughts, treats not only the alfa particle but the whole system in quantum mechanical terms, that is to say billions of potentially detachable electrons, from their hydrogen atoms, all given three dimensions each (together with the alfa particle in three dimensions).
Given time for the ionization of, say, a thousand atoms, a foto takes a measure of the complete system, collapsing the wave function onto a complete track, not onto one position of one particle.
In the second scenario, the wave function has a vastly increased configuration space to search-out. But this land-scape is also vastly more structured and the wave function, like a mist settles more densely, accordingly, determining the points most probably measured. Each point, in this bigger platonia path, “looks like a history of the three dimensional track up to some point along it.”
Despite the different view of when the wave function collapses, the results are much the same, because the experiment is a highly organised situation, whereby a highly regular Hamiltonian wave function produces a semi-classical solution that gives the appearance of a path taken in time or a history.
But, in quantum mechanical terms, it is the wave functions probabilistic search thru a timeless topography of all possible histories, that measurement, “collapsing the wave function,” realises as one history.
Time dependence on configurations.
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Barbour believes consistency demands that a consciousness of motion must be in one configuration. He guesses that the brain can take in several “snap-shots” at once and “play the movie.”
He follows Bell, that “the past” is inaccessible and therefore irrelevant. We have only records of an illusory history. Bell, however, didn’t deny the reality of time, whatever his beliefs about it.
This reviewer can’t help but think that records are of something. So, to deny that something is a contradiction. Suppose time has a comparable reality to that of space. It can be thought of in the same way. For instance, special relativity treats time, as well as space, as having speed.
Maps are a record of a topography, to a greater or lesser scale. As Charles Dodgson pointed out, the map can be increased to full scale, till the land-scape is a map of itself. Geological strata containing fossils might be considered naturally scaled-down time-maps of evolution. The strata are a spatial map of the ages. One could say they had lost most of their temporal dimension, rather like the Minkowski Interval makes for an inter-change of temporal with spatial dimensions, in space-time.
In denying the reality of time, Barbour finds himself at odds with other physicists wishing to develop the space-time frame-work of relativity theory.
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Brian Greene: The Elegant Universe.
Table of contents.
(1) Super-strings.
Links to sections:
Finite strings remove infinities.
String resonances as “elementary” particles.
Hidden dimensions
Beyond strings: M-theory.
Black holes as elementary particles;
super-string cosmology.
Postscript: Parallel universes.
Finite strings remove infinities.
This Brian Greene science book (published in 1999) is award-winning. He imagines familiar comparisons with the basic ideas of relativity and quantum mechanics. String theory makes general relativity and quantum theory compatible, if it is correct. We don’t know this yet because the supposed strings, that replace point elementary particles, are on far too small a scale (the Planck scale) to be reached by experiment, requiring the kind of energies to be found soon after the emergence of the universe in the big bang.
As Greene puts it, physicists will have to make the big bang itself do, as a cosmic accelerator experiment, and measure its results, in the laboratory of the universe.
In such extreme scenarios, as the big bang, or black holes, the conditions for the large scale physics of general relativity and small scales of quantum mechanics come together. General relativity predicted a singularity or infinitely dense point and infinite curvature at the origin of the universe, or within a black hole. This breaks down the law of space-time being geometrically curved with gravitational mass.
Quantum theory appeared to offer a way out of this impasse with Heisenberg uncertainty principle. A photon that throws light on an electron needs a short wave-length to determine its position accurately. But shorter wave-lengths have higher energy and give the electron a kick that creates an uncertainty in its momentum. A longer wave-length disturbs less its momentum but is a less precise observation of position. It’s not just a question of disturbing the momentum of a particle the more accurately it is measured for position, and vice versa.
Empty space is really a seething mass of energy eruptions, viewed on a sufficiently small scale. Tho, over-all, a vacuum has zero energy. The uncertainty relation allows energy to be borrowed in inverse proportion to the time taken. The more energetic a particle and anti-particle creation, the quicker they must annihilate each other, thus preserving the spirit of the conservation law of mass-energy, within Heisenberg terms.
At the incredibly small Planck length, to confine a particle, in so narrow a region, is to create (literally) massive uncertainty. Consequently, the curvature of the space and time dimensions will lose their continuity and become too grotesquely distorted to be meaningful as left or right etc. General relativity is inapplicable to the so-called “quantum foam.” A combined theory of quantum gravity is thus frustrated.
According to the string theorists, the cause of this difficulty is the treatment of elementary particles as infinitely small points of no dimension. Such points would be small enough to probe the quantum foam, below the Planck length. Suppose that elementary particles are one dimensional “strings,” so to speak, of about Planck length. Then they will be too “big” to probe the quantum foam, just as ones finger is too insensitive to feel the irregularities of a granite surface.
More exactly, a Feynman diagram, of particle interactions, has a new interpretation, owing to special relativity, if re-drawn in terms of looped strings, rather than point particles. Whoever observed the infinitessimal point particles would agree on their positions at a point of interaction. But for the finite-sized loops, coming together to form a different loop and therefore a different particle, described by a different vibration pattern, different observers would disagree when and where the interaction took place. For, in special relativity, observers in relative motion use space and time co-ordinate systems that disagree when “now” is.
The interaction location is smeared out along the observational indefiniteness, so the force of the particle need no longer be treated as of infinite strength at an infinitessimal point. Finite strings produce well-behaved finite answers due to the blurring over of the sub-Planck scale with its quantum foam.
Nor would it avail one to pump more energy, and therefore frequency, into a string, to give it a shorter wavelength, more probing of an objects position, as is done with photons. The string is merely magnified in size, rather than becoming a magnifier.
A combined theory of quantum gravity becomes possible, after all.
String resonances as “elementary” particles.
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A basic value of string theory is that all the supposedly elementary particles may be taken as various vibrational patterns, or resonances, of a single loop of Planck length “string.” (To give an idea of this measure, if an atom was expanded to the size of the known universe, the Planck length would scarcely reach the height of an average tree.)
Matter is made up of over a hundred or so atoms, depending on how many protons and neutrons they contain in their nuclei, until their number makes them too unstable to hold together. Their electric charges are neutralised by a cloud of electrons, with opposite electric charges. This accounts for the fact that the electro-magnetic force does not normally prevail over the extremely feeble, but purely attractive, gravitational force, that holds the galaxies together, the planets to the sun and things to the planets.
The electrons are elementary particles. They are mysteriously associated, in inter-actions, with neutrinos, a scarcely inter-acting particle, too light for any mass, it may possess, to be measured as yet. (After-note, 2002: the neutrino has been found to possess a small mass.) The protons and neutrons are made of combinations of three sub-atomic particles called quarks. (They can also pair to form “mesons.”) Quarks are held together by eight possible gluons, as the name suggests.
For some presently unknown reason, the electron, neutrino and a pair of quarks come in two further sets of more massive and ephemeral versions of themselves.
Particles classify into force particles and matter particles. There are four known forces of nature. The gluons are the interactive or force particles for the “strong force” that holds the nucleus of an atom together but does not extend beyond it.
A relatively “weak force” is responsible for radio-active decay of the nucleus. It has its own three inter-active particles, which have been compared to “heavy” photons, in the electro-weak theory that unites the electro-magnetic force with the weak force. The photons are the carriers or inter-active or “messenger” particles of the electro-magnetic force.
All the particles have anti-particles, which are the same but of opposite charge. Neutral particles, like the photon, are their own anti-particle.
If matter particles are hit with higher energies, they produce more massive versions of themselves, which quickly decay into their basic versions. These are called resonance particles, hundreds of which have been found. The name is by analogy with plucking a string to put more energy into it, producing higher resonances.
String theory, due to the extreme tension of strings, predicts an infinite number of higher resonances, just as there can be an infinite number of wave-lengths and correspondingly higher frequences and the higher energies that go with them.
A vibrating string has more energy with more and, therefore, shorter wave-lengths like choppy seas instead of gentle rollers. Also, there’s more energy if the “seas” are higher, that is if their crests and trofs mark higher amplitudes. Special relativity translates energy into mass. So, the mass of an elementary particle can be understood in terms of the vibration pattern of a string.
There is a hypothesised force-carrying particle for gravitational mass, called the graviton. General relativity predicted gravitational waves, too feeble to detect by present devices.
An early success of string theory of particles as resonances was to predict the properties of the graviton. It was also calculated that “the strength of the force transmitted by the proposed graviton pattern of string vibration is inversely proportional to the string’s tension.” Since gravity is so feeble, the tension worked out at 10, to the power of 39, tons (the Planck tension: enough perhaps to work the universe up into a light sweat)!
Not surprisingly, this tension contracts a string loop down to the afore-mentioned Planck length. The energy, for such stiff strings, must be extremely high for them to vibrate at all: on the Planck mass scale.
Greene says, suppose different people were only entrusted with one discrete monetary denomination, corresponding to energy being quantised or permitted only at certain discrete levels (as, in the Bohr atom, electron orbits are quantised). These people are only allowed to pay in whole number multiples of these denominations, as nearly as possible, up to the cost of a purchase (being let off, in so far as their denomination may not fully add up to the full price).
Likewise, strings have a minimum energy denomination proportional to the strings tension, itself proportional to the number of crests and troffs in a vibration pattern, whose energy is a whole number multiple, determined by its amplitude, of this quantised energy minimum. The typical mass-equivalent of some vibrating loop is 1, 2, 3,… times the Planck mass.
This is about the mass of a grain of dust, massively beyond the masses of elementary particles. Despite the tension of strings, quantum uncertainty ensures some vibration, which is associated with a negative energy that can cancel out the strings Planck energy, manifested in the lowest, or one times Planck energy, vibration levels. This can produce the tiny masses of elementary particles, tho not typically.
These cancellations worked perfectly for the vibration pattern hypothesised as the graviton, which is just as well, because the graviton, akin to the photon, as a force carrier, is reckoned to have zero mass.
(String pattern theory can also be related to natures other three force-carrying particles.)
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String theory is expected to incorporate the principle of super-symmetry. For the laws of nature to be truly general, they must apply in all manner of circumstances. It should not matter when or where an event happens, from what angle, or in what motion. The law should still be observed to hold.
Laws that respect these conditions are said to exhibit certain symmetries, such as thru translations in space and time.
Special relativity is symmetric with respect to observers in relative motion, who can all equally claim they are at rest, relative to any motion between themselves and other observers. General relativity goes further, as accelerating observers are, in effect, at rest in a gravitational field. This enforces a symmetry that ensures equality of all points of view.
The other three forces are also required to enforce other more abstract “gauge” symmetries. However, there was one further symmetry to do with space, time and motion, namely spin. This is rather as the earth rotates, as well as revolves. Elementary point particles would not seem to have a meaningful spin. But the magnetic properties of electrons, for instance, showed that in a quantum sense they had. Only, this spin did not vary like the skater pulling in her arms to spin faster.
The particle spin is a fixed quantity that goes to define its nature. Its quantum mechanical rate is spin one-half for all matter particles and spin one for three of the force carriers. The graviton would have spin two. String theory actually demands a vibration pattern that corresponds to a massless spin-two particle.
It turns out that spin invokes another symmetry principle of nature: “super-symmetry.” Brian Greene says no more about it than: supersymmetry can be associated with a change in observational vantage point in a “quantum-mechanical extension of space and time.”
Whatever that means, it implied that particles must come in pairs with spins differing by one half. This would naturally partner the matter and force particles. Unfortunately, the standard model, that unifies three of the four forces (leaving out gravity), matched none of the existing particles. Instead of effectively halving the number of particles, super-symmetry doubled them, by positing a complete new set of partners.
However, super-symmetry pairings of bosons (with whole number spins) and fermions (with half-number spins) give canceling contributions to particle interactions that the standard model can other-wise only make add-up by extreme fine tuning of its calculations.
The three non-gravitational forces, tho of greatly disparate strength, apparently diverged at an early stage of big bang evolution. The quantum flux of virtual particles were found to weaken the intrinsic force of an electrically charged particle they surrounded, until approach at very close distance. The opposite situation held for the strong force and to a lesser extent the weak force, so that at very short distance not greatly above the Planck length, the three forces strengths converge.
But, it was found that extra quantum fluctuations provided by super-symmetric particles would make the convergence perfect.
In the super-symmetric version of string theory, it emerged that the boson and fermion patterns of vibration came in pairs.
Part 2, hidden dimensions.
Hidden dimensions.
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Kaluza findings didn’t fit the experimental data about the electron mass and charge. Eventually, as more particles and the strong and weak forces became known, theorists wondered whether the fault with Kaluza-Klein theory had been too few dimensions rather than too many.
Just as an ordinary string may be allowed to vibrate in three independent directions, a theoretical string may vibrate in nine independent directions. The curled-up six dimensions, that fulfill the equations of string theory, are called Calabi-Yau spaces (or shapes).
These shapes may be likened to musical instruments that create particular vibration patterns. The testing question is: how well do these patterns match with the elementary particles found, or capable of being found, by experiment?
Calabi-Yau shapes contain various holes, which themselves have various dimensions, analgous to a do-nut and a double or triple do-nut. A family of lowest energy string patterns is associated with such holes. Multiple holes imply multiple families, like the three families of elementary particles. Just the right shapes are currently still being sought.
String theory predicts other fractional charges than those of the quarks. And experiments finding super-partners would also be relevant to super-strings.
Possible behavior of strings can be described in a simplified form of the large and the hidden dimensions, the afore-mentioned “garden hose” universe, with one familiar line dimension and a hidden, curled-up dimension. The universe may collapse back on itself, from a big bang to a big crunch. This depends on whether there is enough mass in the cosmos to pull it back. The big crunch may resemble the formation of a singularity at the heart of a black hole. All the cosmic mass may be crunching into a single linear stream. It looks to be of one dimension only but has a cylindrical dimension also, like a garden hose.
The difference from point particle physics is that strings can not only move about on this cylinder. They can also wrap around it: they have a winding mode. So, strings have two sources of energy, winding energy, as well as vibrational motion. The latter consist of uniform vibrations and ordinary vibrations. Ordinary vibrations are the kind of oscillations considered above, and are not decisive in this context. Uniform vibrations are “the overall motion of a string as it slides from one position to another without changing its shape.”
Uniform vibrations string energies are inversely proportional to the radius of the circular dimension. The uncertainty relation ensures that a constricting hose radius, confining the string, increases its energy.
But the winding energy is proportional to the radius. The greater radius and circumference, the longer the string and the greater its mass, when wrapped around the “hose,” and according to how many times wrapped round, giving the “winding number.” There are also multiple vibration numbers.
The units are on the Planck scale of length and energy.
The winding energies and vibration energies of the strings compensate each other. You could have a table of winding numbers and vibration numbers for a given radius, and another table of the same, for its inverse radius, giving an over-all correspondence of entries. You could have one universe with a small radius and large vibration energy that corresponded exactly in total energy with another universe, having a large radius and a small winding energy. The two universes are effectively the same, having the same allowed quantum particle energies and charges.
We do not know whether our own universe has a hidden curvature, in the sense that it is too large, rather than too small, for us to see. Space might be traversed as Magellan expedition circumnavigated the globe. If the universe has a 15 billion light year expansion age to put a radius to, say 10, to the power of 61, Planck lengths, then string theory provides an alternative inverse radius of the universe (10 to the power of minus 61), a radius that is miniscule and contracting, but just as valid in its own terms.
Measuring distance, the familiar way, by light amounts to using light (meaning not-heavy) string modes as probes. In principle, if they were technically able, astronomers might equally well measure distance by heavy wound-string modes. But such probes, being proportional to a cosmic radius would have to be incredibly massive.
Whichever string mode happened to be the light or “easy” mode, it never measures below Planck length. Even if the non-standard measure of distance were adopted, so the radius is below Planck length, the physics is the same as for the complementary table in which the radius is more than Planck length in the conventional measure of distance.
Having discovered that geometrical forms could differ in size, yet be physically indistinguishable, physicists, including Greene, found that the same could be true of different shapes, by orbifolding Calabi-Yau spaces. The number of odd-dimensional holes equaled the number of even ones, in the original, and vice versa. Their totals of holes is equal, implying the same number of particle families, tho their shapes and structures differ.
The shapes agreed on the rest of their physical properties. The beauty of these “mirror manifolds” was that one might be chosen as the possible hidden dimensions creating the sought-after particle masses and force charges. The calculations involved had often been impossible. This had also been the case for the pure mathematical study of Calabi-Yau spaces. But it turns out that the “mirror” partners, figuratively speaking, are often easy to calculate, a source of progress in string theory, and a return by physicists for what they’d learned from pure mathematics.
In 1987, Calabi-Yau spaces were found to be transformable into each other, according to a mathematical pattern of puncturing and sewing their surfaces. (This was a space-tearing “flop transition,” which is sometimes “topologically distinct.”) Considering such processes as possible physical tearings of space, mirror symmetry of Calabi-Yau spaces was used to give fuller grounds for the suspicion. An absence of catastrophe in the “mirror” partner would make the space-tearing original physically allowable.
Edward Witten showed that travelling strings, unlike point particles, could protectively encircle spatial tears, with relative possibilities (calculated from Feynman sum-over-paths) that would cancel out a “cosmic calamity.”
He and colleagues, including Brian Greene, also showed that spatial tears would leave types and families of particles unaffected. But the energies of the possible string vibration patterns, meaning the individual particle masses, could change. Experiment shows these to be stable. If there is any spatial tearing in the universe at large, it is too slow to be noticable.
Space tearing opens the way for the possibility of worm-holes, the creation of new space joining previously unconnected parts of the universe.
Beyond strings: M-theory.
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Up till 1995, five string theories seemed to be at odds with each other. Only approximate string equations could be found and each of the five theories differed from each other. Their difficulty meant that perturbation theory had to be used, that is a method of successive approximations. A classic example of this is how the gravitational interactions of the solar system are worked out. The sun is by far the most important gravitational mass. So, its effect in relation to the earth is calculated first. This result has to take into account the next most important effect, the moon in gravitational relation to the earth, and so on, until all the significant planetary masses have been allowed for.
The success of perturbation theory depends on being able to order the importance of the effects. Then, dealing with each in turn, one has some idea of how the margin of error should diminish in each successive approximation. Using Feynman diagrams, Richard Feynman, in popular lectures, QED, gives examples of this process of adding successively smaller corrections for all the possible ways a given particle inter-action might take place. Experiment confirmed this quantum electro-dynamics as the most accurate theory in history.
String theory has Feynman diagrams for strings instead of point particles. The Heisenberg uncertainty principles allowance for the creation and annihilation of virtual string pairs, in a string inter-action, is diagrammed as a series of loops between in-coming and out-going strings.
The likelihood of such temporary energy incursions is measured by the size of a “string coupling constant.” It would determine masses and charges of string vibrations. Strongly or weakly coupled values, above and below unity, respectively, determine whether it is increasingly likely or unlikely for more and more virtual particles to appear. Therefore, values, above one, for any of the five string theories, would invalidate the use of perturbation theory.
In 1995, Witten introduced “duality” to get beyond perturbation theory. Of the five string theories, two pairs of them get exchanged by the large/small radius duality, discussed in previous section.
Instead of assuming the five theories were independent competitors, all amenable to perturbation theory, by being weakly coupled, it was found that two of the theories could be transmuted into each other, because of a strong-weak duality. Their physics appeared the same, when one theory was weakly coupled and the other strongly coupled.
To this end, use was made of super-symmetry constraints and minimum mass constraints to give clues about particle states (BPS states), for the string theory with a strong coupling constant.
Another of the five theories appeared to correspond to itself when weakly and strongly coupled: it was self-dual.
To complete the link-up of the five theories required a further insight. Super-gravity theories attempted to use super-symmetry to unify quantum field theories with general relativity. It turns out that these point particle theories were approximations to various of the five string theories.
One of the super-gravity theories was in eleven dimensions, rather than ten, and didn’t fit in with the existing 10-D string theories. But a string theory, by gradually increasing its coupling constant, and with respect to its BPS states, showed 11-D super-gravity to be a low-energy approximation. The extra dimension emerges with the increasing coupling constant and a string loop turns into a two dimensional cylindrical membrane or a hoop, depending on the string theory.
Higher dimensional membranes, than two, are also possible. But with weak string coupling, all but the strings would be too massive to be produced without enormous energies.
Witten provisionally named the 11-D theory as M-theory, still something of a mystery, but the supposed under-lying theory to the five string theories, incorporating 11-D super-gravity. What was previously an embarrassment of theories, as to the truth, has become an inter-related variety of approaches to make the problems of theoretical prediction more tractable.
Witten demonstrated a primal emergence of the gravitational force from the other three forces, according to their varying strengths when the string coupling constant need not be small.
Black holes as elementary particles.
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Elementary particles and black holes have in common that they are distinguished only by their mass, force charges and spin. A black hole might be a huge elementary particle. A small enough black hole should resemble an elementary particle. But this brought into play the big versus small theory incompatibility between general relativity and quantum mechanics – until string theory, or M-theory.
In the context of space-tearing flop transitions (discussed above) string equations show three-dimensional surfaces, as well as beach-ball-like two-dimensional surfaces, embedded in a Calabi-Yau shape, are likely to vanishingly collapse. A one-dimensional string, moving in time, could “lasso” a 2-D sphere, preventing a cataclysmic spatial tear. In this respect, at any instant, a 1-D string (or one-brane) could only surround a circle; a 2-D string membrane, or “two-brane” wrap round a two-dimensional sphere (like an orange); and a three-brane wrap round a three-dimensional sphere.
Following-up the flop transition for the 3-D sphere (called a conifold transition), it was found that the sphere repairs and reinflates only as a 2-D sphere. This can only be imagined in lower dimensions. A two-dimensional sphere is “a collection of points in three-dimensional space that are the same distance from a chosen center.” Its reduction, to a one-dimensional sphere, would be to the points making up the circumference of a circle, which is in two spatial dimensions.
A further reduction would be to a zero-dimensional sphere, “the collection of points in a one-dimensional space (a line) that are the same distance from a chosen center.”
This is as if the Klein hidden extra dimensions of space transformed from the one curled-up shape to another, comparably to the normal extended three dimensions changing the shape of the universe from a torus to a ball.
Equations governing the “branes” showed that, from our limited three-dimensional view-point, the three-brane “smeared” around a three-dimensional sphere, within a (curled-up) Calabi-Yau space, sets up a gravitational field like a black hole.
The black hole is considered to have under-gone a phase transition to a massless elementary particle, like a photon. String theory has identified them as being made of the same “stringy material.” Much as ice under-goes a phase transition to water, they look different but their make-up is the same.
“Hawking radiation” established the “entropy” of black holes. To solve what this disorder was of, string theorists theoretically built
certain extremal black holes by starting with a particular collection of BPS branes (of certain specified dimensions) and binding them together according to a precise mathematical blueprint…
Strominger and Vafa could easily and directly count the number of rearrangements of the black hole's microscopic constituents that would leave its overall observable properties, its mass and force charges, unchanged. They could then compare this number with the area of the black hole's horizon -- the entropy predicted by Bekenstein and Hawking.
Hawking radiation implies the eventual evaporation of black holes. With the gradual shrinking of their areas, their entropy decreases. A current research question is whether order or “information,” lost to a black holes gravitational suction, could be recovered from the surrounding area that the shrinking event-horizon has given up.
If the answer is “no,” this would further take the edge off a deterministic physics. Quantum mechanics had made, only probabilistic, the totally determinist mechanics, conceived by Laplace.
[PS, sept. 2015: BBC news reported that Stephen Hawking answered “yes,” presumably accepting the holographic principle of Leonard Susskind.]
However, Greene doesn’t mention chaos theory, which requires infinite accuracy in initial conditions, to predict the oscillations of so simple a classical system as the force-driven pendulum. The Laplace school thought these information feed-ins, to apply physical laws to particular circumstances, would, in principle, determine the evolution of the universe.
Brian Greene discusses other questions, mainly to do with the new subject of super-string cosmology.
Already, some possible answers have been put forward. Pending the big bang, all the eleven dimensions of space and time were supposed to be curled up in a universe of Planck scale size. Why did only three dimensions of space extend (thru “inflation” and so forth)?
As related, above, strings can wrap round these dimensions. But there are anti-strings, wrapping round “the other way,” which annihilate them on contact, producing an unwrapped string, and releasing the dimension to expand. These releasing collisions are most likely in one dimension. At different speeds, two marbles, confined to a line, are sooner or later going to hit. This is less likely of two objects moving freely on a surface, and less likely still for objects moving freely in three dimensions. Thus, the chances were that the fourth and higher dimensions of space were not released from their string wrappings by string pair annihilations.
Alan Guth, in note to The Inflationary Universe, mentions there being about fifty versions of inflation theory, which explains several discrepancies in the earlier big bang model. Greene refers to a controversial pre-big bang version, derived from string theory, by Gasperini and Veneziano, which they hope presages a more inevitable development to inflation.
Closing remark.
Brian Greene, explaining string theory, may be likened to a series of beacon hills that trail off to who knows where. They give you an idea of the general direction string theory is going but they leave in darkness the maze of valleys below, which only whiz mathematicians and physicists can follow, to light up more beacons.
Postscript: Parallel universes.
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BBC tv Horizon (14 feb. 2002) featured an astonishing development in linking string theory to cosmology via the concept of parallel universes. The program followed the implication of a unified string theory or M-theory featuring an eleventh dimension and, beyond strings, the existence of membranes of various dimensions.
One of the scientists involved described the arrival of an Italian liner in New York, damaged by a rogue wave. It so happened that a study of the mathematical possibilities of what might happen when the membranes collide in their hyper-space also yielded catastrophic results of the order of the Big Bang itself, or innumerable big bangs.
Classical cosmology closes off possible events, before the big bang, with an infinitely small beginning, a singularity. But quantum theory of the Planck scale of events transcends the big bang, as the outcome of these colliding membranes. As they move, they ripple, so that collisions yield the clumps of matter after the big bang. That is the material universe.
This implies that time precedes the big bang, which is indeed one of an infinite number of different big bangs resulting in an infinite number of possible universes, with different laws of physics.
Hence, string theory has theoretically explained the origin of the big bang by implying parallel universes.
At the time of writing, this is a new theory, which the physics community has yet to decide whether to accept or not. As yet, parallel universes have not been the majority view.
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Brian Greene: The Hidden Reality.
Over the cosmic horizon to an infinite multiverse?
Table of contents
New Deal universes.
Matter over mind hypothesis.
String theory, membrane universes and a runaway multiverse.
Many worlds interpretation of quantum mechanics.
Parallel worlds of a holographic universe.
Reviewer comment: multiverse and multichoice.
New Deal universes.
To simplify the mathematics, the cosmological principle assumes that on a cosmic scale the distribution of matter and energy is approximately uniform. Einstein general theory of relativity replaces the Newtonian concept of gravitational force with matter exerting geometrical distortions on space and time. A universe which has a positive curvature of a sphere is finite in spatial extent. A universe which has the negative curvature of a saddle is infinite in extent. A flat universe like a tabletop may be either finite or infinite in extent.
Greene says a uniform presence of matter generally curves space-time but can leave zero space curvature. The curvature of space depends on the density of matter and energy. There did not appear to be enough for more than negative curvature of space. Then the behaviour of galaxies suggested there must be more energy than was visible, short of modifying the laws of motion.
Hence, a supposed “dark energy” suffusing space. This harks back to the idea of a cosmological constant. Einstein found that a positive constant produced a repulsive gravity, that counter-acted normal attractive gravity. Building on this principle, Greene later explains (to my confusion) why Inflationary theory of the Big Bang strongly implies an infinite multiverse.
Thru Inflation, the universe bears the signature of its quantum scale origins. Unlike the determinist measurements of classical physics, the quantum state has the jitters, being subject to random fluctuations. The consequences of a super-small quantum origin have been predicted and found to a surprising experimental accuracy in tiny ripples measured in the cosmic microwave background radiation, a sort of faded flash to the Big Bang.
Ultimately it is the random deviations from a completely uniform distribution of energy and matter, which gravity has gradually worked upon on the cosmic scale, to separate into larger clumps, the stars and clusters of stars.
From what little I could make out, Inflationary theory seems to be tied up with the idea of a compact region of differing energy potentials. At some points, in keeping with random quantum fluctuations, they seem to have realised their potential like stones rolling downhill.
These kinetic energies are likened to bubble universes, much like holes forming in a Swiss cheese.
This is the expanding Inflationary multi-verse.
Tho these are bubble universes, Greene explains how they may be likened to an infinite universe. It has to do with the relative measurement of time, familiar since Einstein special and general relativity, by which observers cease to share the same time, either when relative motion is significant compared to light speed or when gravity is strong, as in bending a light ray moving close to a solar mass.
Observers may co-ordinate their times, according to a shared measure of energy density or mass density. That is because, at any given time, the universe has a fairly uniform density, which becomes steadily more diffuse with the universal spatial expansion.
Reminiscent of Einsteins famous thought experiment of the two observers inside and outside an accelerating lift, there is now an observer outside the bubble universe as well as inside. According to Greene, “what appears as endless time to an outsider appears as endless space, at each moment of time, to one insider.
“Current evidence suggests a small positive cosmological constant, for a universe that seems to be increasing its rate of expansion. The astronomical evidence, taking into account the estimated dark energy, suggests the universe has zero curvature. It is not known whether this is finite or infinite.
Greene says that since the universe is extremely big any way, you may not think it matters -- but “you should.”
Einstein special theory of relativity postulates that nothing can move faster than light. Light bounds our observational universe in a cosmic horizon, analogous to the global horizon that we cannot see over.
In an expanding universe, different regions, with their own cosmic horizons, become so separated that they could not possibly influence each other.
All of these independent regions can consist of only a finite number of particles of matter or energy. And they are subject to only a finite number of possible re-configurations.
Greene stresses that “anything but measurements with perfect resolution reduces the number of possibilities from infinite to finite.” This is not merely a technical temporary limitation. This is a limitation in principle, according to the uncertainty principle, which specifies how much the gain in resolving the quantum scale measurement of one property is at the expense of another property.
To measure a particle position with complete precision would require infinite energy, which no particle can be given. In an infinite universe, the consequence, of an infinite number of re-shuffles of a finite number of particles, is that eventually all of those independent regions will undergo more or less exact repetition.
Matter over mind hypothesis.
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Greene postulates the reductionist view, common among physicists, that knowing the full arrangement of physical particles fully describes reality.
This is what Francis Crick called The Astonishing Hypothesis. Namely that consciousness is an emergent property of the evolution of the brain.
In other words, we cannot explain it, we just give it a mysterious characterisation, like emergence, and take it as given. But this attitude is what the philosopher, Robert Nozick, was getting at the young Brian Greene about, as Greene re-tells near the end of the book.
The assumption is one of materialistic determinism or that body determines mind. But what of mind over matter? What of the influence of faith, that medicine is pleased to call the placebo effect?
Treating a correlation as a one-way cause must be regarded as suspect. A critic has said, it is like assuming that when a television “dies” that is all there is to it, overlooking that it is just a receiver for a signal, not the signal itself.
Such an unsuspected consciousness “signal” might be labeled unhelpfully paranormal or psychic or spiritualist etc but is no more fantastic or implausible than – indeed seems rather complementary to – some physicists suggestion that the world is no more than the projection, as it were, of a computer simulation, discussed in the same chapter, saw Nozick philosophise.
The world as computer simulation re-opens the way for mind, if doing the computing from a hidden reality. This leads me to suspect that the physics of materialist determinism (matter over mind) is at odds with the physics of the world as computer simulation (mind over matter) short of some more balanced assessment of the inter-relation between mind and matter.
In the penultimate chapter on creating universes and simulating reality, Greene cites the research into computer simulation of the brain and the prospect for artificially realising emergent consciousness with the acceleration of computing power. While the objective creation of a new cosmos evidently needs energies beyond humanitys grasp for the forseeable future, the subjective creation of a new cosmos in any robotic consciousness seems a not too distant possibility.
Of course, every new birth into consciousness is a new subjective universe in the multiverse of human and other life forms. We cannot traverse individual universes in a multiverse any more than individual consciousnesses. But Greene is surely right in believing we may infer the existence of other universes in a conjectured multiverse, just as we avoid solipsism by infering individual consciousness, other than our own.
However that may be, physics may find that reductionism has severe limitations. Take for instance the conclusion of “infinite copies of you and everyone and everything.”
Well, we don’t have to imagine an infinite journey thru the cosmos for this. Twins are already exact copies. Despite their remarkable psychological affinities, as far as I know, they do not share the same consciousness. Nor, would I guess, any doppelgangers from here or the other ends of an infinite universe. Moving the same model of body, like driving the same model of car, does not make you the same person.
Are we not missing something here? Like a common quality of consciousness that different life-forms filter variously like so many specialised sensory instruments. I suppose I’m saying that it is a common consciousness that unites life but different bodies that divide it, even if they are physically identical.
String theory, membrane universes and a runaway multiverse.
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The Hidden Reality gives a summary update of The Elegant Universe (reviewed in previous chapter) and The Fabric Of The Cosmos, on string theory.
Nineteenth-century mathematics made an axiomatic modification that was the first revolution in geometry since Euclid. Riemann geometry of curvature made general relativity possible. In the late 20th century, physics led mathematics, when string theory generalised the classic geometry of zero dimensional dots, used to describe elementary particles, into a geometry of one-dimensional strings.
The over-coming of certain mathematical obstacles allowed and encouraged strings themselves to be generalised into higher dimensional membranes. Our own three dimensional universe could be considered as just one membrane among many floating in higher dimensional space.
It has been speculated that these membrane universes might more or less collide. A gentle collision with our membrane universe and another might leave an astronomically observable signature. Less happily, more violent collisions might induce what is being called facetiously Big Splat.
On this basis, a possible mechanism for a never-ending cyclic creation has been worked out, one avoiding the progressive disorder and collapse known as the second law of thermodynamics. The Calabi-Yau spaces of higher dimensions stitched in to the observable three dimensions lead to a stupendous number of possible universes, such that it would be much easier to locate a particular grain of sand on a beach than to find the particular higher dimensional space that would characterise our own universe.
String theorists have tried to show that this superfluity can be accounted for in terms of a theory of eternal Inflation. It seems to have something to do with extending the Swiss cheese model of the multi-verse, so that bubble universes cascade into ever more bubbles. This runaway multiverse involves a mountainous energy multiverse with different valleys, for the extra dimensions different forms, where quantum tunnelling can take place to lower energy levels indefinitely creating bubble universes within bubble universes.
This review is just a sketch caricature of admittedly extravagant speculation. To get the proper explanation, you have to read the book, which does indulge in some soul-searching about whether all this untested abstraction is really science.
Some hopes for theoretical hints are pinned on the large hadron collider experiments at CERN.
Many worlds interpretation of quantum mechanics.
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Not only string theory and Inflationary cosmology have invoked multiverses. Quantum theory has also resulted in a Many-Worlds scenario. Greene discusses the Copenhagen interpretation of a collapse of probabilities, generated by Schrodinger equation of the evolution of a particle property, like position in space and time, into a certain measurement, which when repeated confirms the equations predicted odds.
Neils Bohr drew a line between microscopic and macroscopic objects. But experiments have shown the Schrodinger equations probabilistic (and increasingly difficult) measure to hold for ever increasing collections of particles, without any supposed collapse into one definitive measure of where the object actually is.
Everett sought to get round the ad hoc nature of the Copenhagen interpretation. His proposal raised ad hoc problems of its own. Suppose the Schrodinger equation simply describes the evolution of many worlds represented by the different peaks of a wave-function, with one observer or measurer becoming many, for each peak or spike, tho each of the experimenters proliferating selves are unaware of the others in their many worlds.
The probability of observing a given particle position measurement, say, is determined by the height of the wave spike. This probability weighting of some observations over others undermines the Many Worlds view that all the possibilities are equally real, and that our worlds observer is no more real than other observers in other worlds of possibility.
An Oxford UK suggestion is that the wave-function probabilities are just the odds that one out of any number of possible worlds will be the one you turn-up, for any given measurement.
Gary Zukav, in The Dancing Wu Li Masters, notes that the basic quandry of quantum mechanics is that a single foton may interfere with itself. This is a reference to the double-slit experiment. Here one foton at a time can be fired to pass thru either of two closely placed slits such that they arrive at a foto-sensitive target.
If one of the slits is blocked, you’ll get a fog of strikes on the side in line with the open slit. And conversely if only the other slit is left open. If both slits are left open, there is not an undifferentiated mass of strikes on both sides of the target. Instead, there is a series of bright vertical bars alternating with vertical dark regions. These resemble the crests and trofs in waves, in this case light waves.
A member of the above-mentioned Oxford group, David Deutsch, in The Fabric Of Reality, argues that the interference effect is evidence of the existence of another foton, we cannot see, and therefore indicating another world impinging on our own. The single foton, that the experimenter fires and that goes thru one of the slits, is accompanied by a sort of ghost foton that goes thru the other slit. But the interference effect is reckoned to be just as sure as when two stones are dropped into a pond and their radiating ripples bump into each other.
I also recommend “Quantum Enigma” by Bruce Rosenblum and Fred Kuttner. It is an ever so polite renunciation by veteran physicists (one of them met Einstein as a grand old man) of the doctrine of “Shut up and calculate,” in order to more fully expose the outrageous implications of quantum theory.
I must admit I hadn’t realised just how extra-ordinary it really is.
Parallel worlds of a holographic universe.
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Greenes chapter nine, on Black Holes and Holograms, starts with a brilliantly simple introduction on the relation of entropy to information, before going on to the conservation of information apparently lost in black holes. Hawking radiation, from the black holes surface area, is where the information is encoded from an object passing the event-horizons point of no return. That is from an outsiders point of view, seeing a gravitationally attracted observer sizzled on the event horizon.
The faller inside would not notice any such spontaneous combustion but the outside observer would only observe this too late to catch up with the inside observer, to confront him with a paradox. The holographic principle expresses this reduction of information from three to two dimensions.
This principle was couched in terms of Maldecana revealing a duality between string theory (as bulk physics in three dimensions) and a specific kind of quantum field theory (as the boundary physics in two dimensions). From Maldecana, it turns out that the math of string theory can facilitate intractable calculations in the quantum theory, which have a direct bearing on experimental observations.
Indirectly, at least, string theory has come of age, as experimental science. The Maldecana holographic result has led to speculation about a whole universe of three dimensional space having a parallel universe on its two-dimensional boundary – whatever that means, as Leonard Susskind would say.
Reviewer comment: multiverse and multichoice.
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Brian Greene says that he avoids the free will debate. The physicists emphasis on multiverse at the expense of multi-choice follows David Hume, who dualised facts and values, rather than Immanuel Kant, whose response identified a unifying graduation between the natural sciences and the moral sciences.
Since the 1970s, I have been more or less vaguely aware that theories of physics, most obviously special and general relativity and quantum theory, seem to have corresponding methods of observational choice. The more general the method of choice, the more comprehensive the tests that physical theory may be subjected to.
Election science is a mathematics and experiments of ethics. Mathematically, I guess “electics” would be another facilitative duality for physics. And experimentally, if the method of choice, the electoral method, is generalised to provide broader observational frames of reference, then their operation is in effect creating new universes of observation, by which this universe is more multiverse than it otherwise would have been.
(That argument reminds me of the reasoning from fuzzy logic.) At least as important would be the strict standards of honesty imparted to electoral method, so sadly lacking in politics – and so destabilising of society, because of the greater power for disaster from an honest science in the hands of a dishonest politics.
CP Snow partly highlighted this, when he talked of The Two Cultures, tho HG Wells anticipated the problem more starkly. In case it be thought that I am being merely moralistic, consider the crudely inefficient voting systems used by most so-called democracies. Consider the current referendum for the alternative vote in the United Kingdom and the stupefying level of debate, especially from the opposition to progress from the illiterate least choice of an x-vote to counting 1, 2, 3, etc order of choice for candidates.
The explosion of scientific exploration, characterised in this book, stupefying in its potential, is in grotesque contrast to the stupefying atavism of politics. This imbalance is likely to capsize society, if not corrected. The Establishment reaction against democratic progress, from its most crude and primitive form, is inevitably an attack on the progressive mandate of science from taking its nine-to-five honesty outside office hours.
20 April 2011.
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Lee Smolin: Three roads to quantum gravity.
Table of contents.
Links to sections:
Four principles.
Black hole thermo-dynamics.
Loop quantum gravity.
Four principles.
Quantum theory changed the assumptions about the relation between observer and observed but retained how Newton viewed space and time. General relativity changed the latter but not the former. So, Lee Smolin said, in the year 2000, of the search for the unifying theory of quantum gravity.
Three main groups research this, by way of string theory, which is mainly a development of quantum theory; or loop quantum gravity, based on general relativity with quantum modifications; and a third small group of originals. Smolin is hopeful that the three groups are converging to enhance each others understanding.
String theory is introduced as the third road to quantum gravity. Having reviewed Brian Greene, on The Elegant Universe, I’ve said no more about it here.
Smolin outlines four principles as a basis for progress. First, consistency, with the definition of a universe, requires that “there is nothing outside the universe.” So far, he agrees with his friend and colleague, Julian Barbour (also reviewed in this book). He agrees that time only makes sense in terms of change. But Smolin doesn’t treat time as an illusion.
When we look into space we are looking back, also, in time. The light from further away comes from further back in the history of the universe.
Hence, Smolin principle two: in the future we shall know more. Nothing can travel faster than light. But as time passes by, the spot-light, we are in, grows bigger. However, the spot-light is the limit to which we can see. This spot-light is different for different parts of the universe, depending on the time light has had to reach a given spot.
(The spot-lights are another word for the “light-cones” familiar to readers of popular books on general relativity.)
No-one can have access to total knowledge about events in the universe. So, we cannot always say whether a thing is true or false, as Aristotle assumed for classical logic. New systems of logic, acknowledging only partial information, dependent on the observers situation, reflect the nature of society. One of these systems, topos theory was found, by Fotini Markopoulou-Kalamara, to suit cosmology.
The quantum theory paradox of Schrödingers cat, and so forth, make no sense in terms of classical logic or common sense. This is the “super-position principle” that a cat in a box, subject to the chance of a fatal accident, is in superposed states of being alive and dead, until an observer opens the box. Then, the conventional interpretation goes, there is a “collapse of the wave function” of superposed states, resulting in either of the definite states: dead or alive – but not both! (Tho, this paradox begs the question of being “half dead.”)
The paradox gives a vivid idea of the observer being outside the observed system. But combining quantum theory with cosmology means that the observer cannot be conceived as existing outside the system, when it is the whole universe.
The Wheeler-DeWitt equations suppose the quantum constraints on the universe. The author played a part in hitting upon their exact solutions, saying it took another ten years to find out what they meant.
Later, Smolin adds, like Douglas Adams, the galactic hitch-hiker seeking the meaning of life:
He goes on that this is not surprising, since the whole universe is not within our purview like a quantum experiment in the laboratory. Context-dependent theories, such as Markopoulou cosmological logic, applied to quantum theory, provide a reason for observers different points of view, from which the super-position paradox follows. One may observe a system, that includes another observer in a super-position of states. But that observer never so describes himself, remaining outside the system he describes.
And this is never precluded: the system observed can never be the totality of the universe, because of the light-speed limits on the size of the observable universe.
A slogan for this point of view is: “One universe, seen by many observers, rather than many universes, seen by one mythical observer outside the universe.” And this is Smolin principle three.
His fourth principle is: The universe is made of processes not things. Here he clearly differs from Julian Barbour. The world is not made up of a lot of static snap-shots put together like a movie. Taking the analogy further, he points out that real snap-shots decompose. Everything we observe is always changing more or less. In direct contrast to Barbour, Smolin speaks of “the illusion of the frozen moment.”
Smolin says we learn about things, just as we do about people, from their stories, which are essentially about causes.
The fundamental idea in general relativity is that the causal structure of events can itself be influenced by those events…The laws that determine how the causal structure of the universe grows in time are called the Einstein equations. They are very complicated, but when there are big, slow-moving klutzes of matter around, like stars and planets, they become much simpler. Basically, what happens then is that the light cones tilt towards the matter… (This is what is often described as the curvature, or distortion of the geometry of space and time.) As a result matter tends to fall towards massive objects. This is…the gravitational force. If matter moves around, then waves travel through the causal structure and the light cones oscillate back and forth…These are the gravitational waves.
Nevertheless, Smolin says physicists tend to think there is a limit to the number of events in a process. And that space and time are not continuous but form into fundamental discrete units (rather like the quantum, h, is such). (Barbour “snap-shot” reality may yet get a look in.)
Black hole thermo-dynamics.
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This is the first road to quantum gravity.
In accord with Einstein equivalence principle, a space-ship can maintain a position out-side the event horizon of a black hole, with a force of acceleration matching the holes force of gravitational attraction. The horizon is a “curtain” of unseen photons that are just unable to escape the black hole. Hence the name: this curtain and everything behind it is a hidden region.
In general, wherever a light source cannot reach an observer, that is in a hidden region to that observer. Even far away from any black holes, a space-ships acceleration will create a hidden region behind a horizon of photons that cannot be seen from the ship, because its acceleration has been enough to put them out of reach, even tho the ship itself cannot reach light speed.
Bill Unruh predicted that the energy source provided by the acceleration will activate the ships particle detectors to register quantum fluctuations, in the vacuum of space, between electric and magnetic fields. By Heisenberg uncertainty relation, both fields cannot be measured, in a region, as zero.
Another principle, that of quantum correlation, predicts the fluctuations will be random, which implies heat, detectable as a temperature proportional to the ships acceleration. A pair of spontaneously created particles, like photons, within the limits allowed by the uncertainty relation, are, in effect a system, which can only be properly understood as a whole. A change in the condition of one photon, such as its polarisation, will affect the polarisation of the other, conserving the pair as a system, even tho they may have moved too far apart for a light signal to have been quick enough to effect this correlation.
The accelerating space-ship detects photons correlated with photons in its hidden region, denying their systemic information and, in effect, surrounding the ship with a random “gas” of photons.
This “Unruh law” is the study of quantum gravitys first prediction.
The entropy of the gas is the measure of all the positions and motions of molecules in the gas. This measure is made in terms of information theory that counts in sequences of bits, as the number of answers to yes/no questions, like a digital computer. This information is missed when taking only temperature and density averages used in statistical mechanics or thermo-dynamics.
The photon gas randomness results from the missing information, in the accelerating space-ships hidden region, which the entropy measures as exactly proportional to the area of the horizon boundary between the ship and its hidden region.
This is Bekenstein law, the second prediction of quantum gravity.
The “Bekenstein bound” is a limit on the information that can be contained in any region. This finite capacity for information implies space is discrete, on the Planck scale.
Thermo-dynamics states an entropy law of the over-all increasing disorder of things, which gives a sense of time being irreversible. Black holes have this character, because nothing falling in one can ever get out. Consequently, Stephen Hawking showed, that, like entropy, the area of a black hole can never decrease.
In the case of black holes, the random photon gas is known as Hawking radiation, when pairs of virtual particles, created from the quantum fluctuations of space, are split near the event horizon. One partner may fall in the black hole, the other be shot off into space. The random radiation meant a black hole would give off (very minimal) heat, the result of missing information from black-holed partner particles, which showed a black hole could have entropy.
The Hawking law is a third prediction, that the temperature of a black hole is inversely proportional to its mass. Hawking radiation means a black hole will lose mass, therefore lose area, and lose entropy. The outside world should gain the entropy, so there is no over-all loss, contrary to thermo-dynamic law.
A black hole of the suns mass would take ten, to the power of 57, times the fourteen billion year-age of our universe, to evaporate. So, the nature of the information trapped in the black hole, and possibly released by evaporation, is of decidedly theoretical interest.
The information lost in a black hole, is measured in discrete units of atoms and photons. But the measure of the black hole entropy is in terms of the continuous area of its horizon. The three roads to quantum gravity are converging on an atomised or quantised concept of space and time having fundamental units.
The Bekenstein surprise that information capacity of space is proportional to a regions area, and not its volume, makes one think of a holograph. This is a two dimensional picture that encodes three dimensions depending on which angle you look at it.
The weak holographic principle treats the surfaces of things as screens with finite capacities to channel information from observer to observer.
Finally, Smolin discusses frontiers of knowledge, including several versions of a holographic principle, for which there are great hopes, as a new founding principle of quantum gravity, as the uncertainty relation is for quantum mechanics, and the equivalence principle for general relativity.
Loop quantum gravity.
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The atomic nature of thermo-dynamics was not accepted in the nineteenth century. Einstein 1905 paper explained Brownian motion in terms of collisions from the random motions of atoms or molecules. Another of his 1905 papers also explained light atomically in terms of light-quanta, carrying a unit of energy proportional to the light frequency.
A theory of quantum gravity will likewise quantise space and time.
Chapters nine and ten are the core of Smolins book, because they describe him working with many colleagues, the world over, to come to loop quantum gravity -- the second road. Because he is telling a story, this reader was given an illusion of understanding their progress, which clearly depends so much on professional co-operation.
With apologies, this amateur reviewer merely makes a few notes, by way of memoranda, hoping they are not too misleading.
Of the four known forces of nature, the strong force binds the three quark constituents of particles, like protons and neutrons, that themselves make up an atomic nucleus, whose cloud of electrons, in turn makes up an atom. Electrons are more or less easily stripped from atoms.
But energy directed at freeing quarks from protons only seems to add to the length of an apparent string, joining the quarks, without diminishing its strength. This “quark confinement” has an analog in super-conductive metals at temperatures a few degrees above absolute zero. Normally, magnetic lines of force are continuous, tho the size of iron filings, put on paper around a magnet, show discrete lines. Only in a super-conductor are magnetic field lines quantised, carrying a whole number multiple of a basic unit of magnetic flux.
The electric force is closely related to the magnetic force. Together they comprise one of the four forces of nature. And the theory of the strong force was based on an analogy with electric charge, except that quarks are distinguished by having three distinct charges (three “colors”).
(Quantum Chromo-dynamics, QCD, is the analgous theory to Quantum Electro-dynamics, QED.)
The color-electric lines of force, holding the color charges of quarks together, could become discrete like a line of magnetic flux in a super-conductor. The guess is that “empty space is a color-electric superconductor.” The complete lack of electrical resistance found in super-conductivity is as if the temporary quantum fluctuations of energy in a vacuum also had large-scale effects.
The stretched “strings,” between quarks, have been thought basic entities, rather than just lines from force fields. Other physicists thought both points of view valid. One of the latter was Smolin after hearing of “Wilson loops.” Ken Wilson assumed a discrete space based on a grid or lattice, of units far smaller than a proton diameter. Quarks could only be on the nodes and strings on the edges of the lattice.
Using simple rules, the three-color-electric field was described by the movement of discrete field lines along the discrete space. Given only one charge, like normal electricity, the field lines tended to lose their discreteness by joining to behave like continuous electric field lines. But, given three charges, as with quarks, the field lines always stayed discrete, no matter how big they got.
The next step would be to dispense with the grid, as a fixed back-ground, leaving only the “quantised loops of electric flux” to characterise a discrete space. Building on the work of many colleagues, as always, the authors work included using Polyakov expressions for the quantised loops of electric fields as the quantum states for a geometry of space-time, given in a simplified version of the Wheeler-DeWitt equations.
It would not matter where these back-ground independent loops were in space. That would have no meaning, because space itself would be defined by the inter-relations of the loops, their intersections, knots, links and kinks.
The idea of discrete lines of force, taken from that in a super-conductors magnetic field, quantised areas into discrete units, on the Planck scale, each carrying finite amounts of area; likewise for volume.
Loop states could be arranged in “spin networks,” previously derived by Roger Penrose, one of the originals, amongst the groups researching quantum gravity. The various lengths, of the joined lines in the net, are integers coming from quantum theorys allowed spin states of particles.
Arduous translation, of loop quantum gravity into spin networks, revealed:
…each spin network gives a possible quantum state for the geometry of space. The integers on each edge of a network correspond to units of area carried by that edge. Rather than carrying a certain amount of electric or magnetic flux, the lines of a spin network carry units of area. The nodes…correspond to quantised units of volume. The volume contained in a simple spin network, when measured in Planck units, is basically equal to the number of nodes of the network.
Theorems show “that the spin network picture of quantum geometry… follows directly from combining the basic principles of quantum theory with those of relativity.”
“Connections have been discovered to… such as Alain Connes’ non-commutative approach to geometry, Roger Penrose’s twistor theory and string theory.”
Giovanni Amelino-Camelia suggested a test whether the geometry of space is discrete on the Planck scale. A photons path should be deviated, from its expected classical path, by interference effects of its associated wave being scattered by the discrete nodes of the quantum geometry. Altho the effect is extremely small, it is cumulative and might be detectable over large fractions of the observable universe.
How probable would it be that this atomic structure of space yielded the Euclidean space we see? The universe under-went a phase transition, like a gas turning to liquid. The early plasma of photons “froze” into matter. A smoothly featureless three-dimensional geometry resembles the crystalline atomic structure of a metal with its smooth surface. For the atoms of space to organise themselves over the cosmos, so highly, seems fantastically improbable.
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Lee Smolin: the trouble with physics.
Table of contents
“we have failed.”
making research more effective.
“we have failed.”
“We have failed” is Smolin verdict on his generation of physicists to sustain the momentum of scientific progress, at a fundamental level.
Lee Smolin has written a good book on scientific method in current physics. He doesn’t explain the theories in detail but he gives you a good insight into the difficulties of giving coherent explanations of nature in all its depth. It doesn’t matter that research will soon render much of this explanation out of date.
There is the Large Hadron Collider coming online at CERN. And there is an increasing ability to see events at energy scales, previously thought unreachable, by the use of astronomical data, as if from cosmic experiments. But the beauty of the book is in understanding physicists decisions with the information currently at hand.
I wouldn’t recommend Smolins book as a primer on scientific method, because the physics of string theory, and the like, is so far out. I would recommend it as essential reading for someone who has progressed in learning scientific method, with respect to more familiar fields of knowledge.
Smolins work is based on a methodological decision that it is high time for physicists to diversify from string theory. One of the controversial ideas, he had a hand in, was so-called Doubly (or Deformed) Special Relativity (DSR). This was to over-come the problem that Special Relativity, in its current formulation, does not allow different observers to see the fundamental quantum unit of length, the Planck length, as the same length. Observers moving ever closer to light speed, relatively to the Planck measure can always measure it as shorter.
The Planck length is a universal quantity of length as light speed is a universal quantity of speed. Observers do agree when they measure light speed and when something is moving less than light, they agree to the extent that it is moving less. DSR re-formulates SR so that observers agree on the Planck length and when anything is more than the Planck length.
In a later version of DSR, the cosmic creation began with a concentration of energy that started fotons with infinite speed, which decreased in the expansion called the Big Bang. Only the low energy fotons, we know in our much-expanded universe, are reckoned to have constant speed. This modification allowed theorists to do away with the SR paradox of a varying Planck length with respect to constant light speed.
DSR with a variable speed of light is a possible replacement for Inflation theory (an enormously accelerated initial expansion of the universe beyond that first thought in the Big Bang theory) as an explanation of why the universe is causally connected at the same temperature. This wouldn’t be possible if nothing could move faster than the currently observed constant speed of light.
The standard model of theoretical physics has held up well in experiments. But in 30 years, there has been no new theory to replace it. String theory, which mathematicly derived the known particles, as well as the yet unobserved graviton, as modes of minute vibration, and its developments in extra dimensions as “branes,” have not so far fulfilled the high hopes for it, as a “theory of everything.”
Smolin repeats his objection to string theory that it has been slow to become a background-independent theory, like general relativity, the physics of the very large. It would be sensible for string theory to have this property, if it is indeed to reconcile general relativity with quantum theory, the physics of the very small.
Smolin calls this problem, of quantum gravity, the first great problem of theoretical physics.
The lesson of general relativity is that space and time are not a passive back-ground according to any number of assumed geometries. Instead, space-time geometry changes its shape in relation to the presence of matter. Having an infinite number of possible back-grounds has weakened the predictive power of string theory, because when anything doesn’t fit, the back-ground can always be changed.
In this respect, Smolin talks about doing science the old-fashioned way. This means abiding by the rules of scientific method (a term he denigrates), such as that theories should come up with testable predictions.
Smolin believes the second great problem of physics is to make sense of quantum mechanics or replace it with a more sensible theory. Neils Bohr said that if anyone thinks they understand it, they don’t.
Problem three is to unify the particles (leptons and quarks are at present the two known kinds) with the four known forces (gravitational and electro-magnetic forces, weak and strong nuclear forces). Theoretical developments, like super-symmetry, have depended on creating a lot of unknown partner particles. The LHC at CERN might turn up new particles of theoretical relevance. But this is only a hope, not solidly backed-up predictions of the kind made by the Salam-Weinberg theory of the unified electro-weak force.
Murray Gell-Mann used group theory to explain the abundance of new particles observed with higher energy accelerators. They produced more unstable versions of the proton and neutron. There was high hopes of building on Gell-Manns work with symmetry groups to predict rare decays of the proton. Even if an average proton was so stable as to last longer than the age of the universe, a few of their multitude might be detected to decay.
By 1990 (in a Physics anthology essay, edited by Paul Davies) Abdus Salam was already wanting a moon base for more suitable conditions to measure the rate of proton decay and give a clue to what class of symmetry group applied to the real world.
Since 1975, the standard model has explained known particles and forces (except gravity) but has to adjust their relations with the help of about twenty constants, which are just given experimently. Their theoretical justification is big problem four.
Problem five is to explain dark matter and dark energy, or whatever is the correct explanation of the discrepancy in astronomical mass measurements.
There is a suspicion of dark matter, as the galactic orbits of stars give a greater mass than the number of stars and light matter observable would warrant, according to gravitational law. Likewise, a uniform dispersal of dark energy would account for the acceleration rate of the universe-expanding dispersal of galaxies.
“Dark” matter or energy doesn’t interact with the electromagnetic force, the medium of light.
If dark matter accounts for about 26% and dark energy, 70%, this leaves 4% for the matter physicists understand by their standard model. This unknown matter seems to be the only alternative to abandoning Newton laws of gravity and their modification by Einstein theory of general relativity.
Besides dark matter and energy, the discovery that neutrinos have masses, are the only two recent finds Smolin credits as major. According to him, at no time for two hundred years has physics had such a dearth of major advances, as in the last thirty years.
[Since this review, CERN have discovered the Higgs particle, crucial for confirming the Standard Model.]
Making research more effective.
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Never mind, the biggest achievment of physics for science, in their so-called failed period was the democratic inauguration at CERN of the World Wide Web, the most revolutionary advance for both the acquisition and dissemination of knowledge, since printing was invented.
If you want to publish ideas, you don’t have to under-go the censorship of peer review. Tho not an academic, I have, on rare occasions, fallen foul of this closed shop. I don’t mean to say my work was necessarily deserving of appearing in some journal. I mean that the decision is held in secret, so you cannot appeal to out-side opinion in defense of your views against arbitrary decisions of the editors expert. It can be like talking to a brick wall or the Inquisition. When anonymous, peer review can be a hooded Inquisition as well. It’s not science, it’s authority.
Admittedly, the odd Guardian journalist says: Sorry the Web hasn’t made a difference. And she has some grounds for saying so. Authoritarian politics continue to defy human rights and, in so-called democracies, hold out against electoral justice.
The last part, of “the trouble with physics,” is concerned with reforms to academic life to make research more effective. One chapter is called: How do you fight sociology?
He is not talking about the discipline of that name. He is really talking about tribalism. He reckons there is a string theory tribe.
The work of physics is so hard and the competition is so fierce that students have to resort to other methods than sheer brilliance to keep their job. Following the intellectual fashion, heeding the seniors, knowing how to pitch for grants are also useful talents in the scramble.
One sees this in the popular arts, when even geniuses are not averse from looking over their shoulders to see what fashion is coming next. Alright, I’ll tell you the instance I’m thinking of. In the mid-1960s, The Beatles had gone all flower power and transcendentalism. Then there was just a hint of a few rock ‘n’ roll songs coming back into the charts. Suddenly, John Lennon was saying: We are all rockers really. We’re just rockers.
And their songs went back to a simpler style more like the 1950s.
Another critical chapter, How science really works, was not an expected exposition of scientific procedure but a description of how scientists assess each other. The time-consuming system of committees and informers reminded of the Soviet arts bureaucracy, which usually didn’t get round to publishing works suspect as to ideological purity.
Solzhenitsyn gives an irreverant version of the ordeal, in The Oak and the Calf. Elsewhere, more credit is given to the daring of an editor like Alexander Tvardovsky, he portrays so affably.
In the chapter on What is science? Smolin sets out a scientific ethic, which he hopes might help to keep research on course. This involves a common adherence to reason and evidence as a means of arriving at decisions as to the truth.
Thus the scientific community is a democracy not beholden to authority. At least this is the theory. In practise, status and track-record weigh in the balance.
Smolin says that science is not like a democracy in that it doesn’t abide by majority rule.
Mill said: democracy is not majority rule or maiorocracy. This is the undeveloped notion of democracy that still prevails in the world. It is not Smolins fault that he shares this view. When he says science tries to achieve a consensus, that is what the developed conception of democracy does.
The difference between majority rule and consensual rule can be re-stated precisely in terms of the difference between single majority rule and multi-majority rule. The latter is a rationalisation of the former, basicly because choice, like motion, is relative, which means in practise a transferable vote.
(I explain this in my book, Scientific Method of Elections.)
It is common sense that government must be by democratic consensus if it is to work. This is most obviously shown by considering what would happen if world government were to be run on a majority basis. It would be impossible for either the East or the West to agree to let the other monopolise government, even if they could trust each other to do so for only a limited period.
A few mature democracies, like Switzerland, recognise this and have power-sharing governments. Or in the case of Northern Ireland, the long-suffering people of Ulster have had to endure a drawn-out civil war, while the consensual principle was being accepted. (And little wonder, when Britains two-party state still won’t accept it.) Consensus couldn’t work well, without the most effective electoral machinery to make this possible: the single transferable vote (STV). (Tho, Ulster still does not have STV in UK general elections.)
My main criticism of Smolins book is his discounting of scientific method for his scientific ethic. Reducing scientific method to a bare scientific ethic might be compared to reducing relativity theory to the meaning that all motion is relative. Scientific method is really the study of what reason and evidence amount to. Knowing how to think is an art, which improves with practice, like any other art. And scientific method is a guide or a set of instructions to mastering this art.
Scientific method, as the study of what reasoning and evidence collecting entail, should be the defense of the community against arbitrary authority that is not answerable for its decisions and can over-rule popular opinion, because of the unscientific disregard for genuine democracy.
Currently some governments are trying to push the dangerous, expensive and obsolete vested interest in nuclear power. The physicists utopian promises, that massively over-subsidised fission energy for over 50 years, were a disasterous blunder, which the world may yet suffer more grievously.
[This was written before the Fukushima disaster. Just how disasterous this was, or could be, has not been well broadcast. Further nuclear disasters are likely, given unteachable governments.]
The International Panel on Climate Change has properly down-played nuclear power. Why was this voice of the scientific community not heard loud and clear before the peoples and their parliaments?
Smolin repeats Feyerabend anathema that there is no such thing as scientific method. (Notably in an art catalog on art, science and democracy.) This is about as sensible as saying there is no such thing as physics because it hasn’t a theory of everything. Who on earth would expect the discipline of scientific method to provide a definitive guide for scientific discovery?
Of course, Smolin heroes, Einstein and the rest of them, had to use what methods or devices they could to make their discoveries. If they had no such need for ingenious improvisation, then scientific method itself would be the theory of everything. That doesn’t mean to say that scientific method or the philosophy of science is non-existent, any more than is physics or natural philosophy.
Smolin believes that the lack of theoretical progress in physics is because they are missing something. One thing I can tell them for sure is that they are too dualistic with regard to science and ethics. Immanuel Kant refuted David Hume, for radical scepticism against deriving values from facts.
The academic community, both in the natural sciences and the social sciences, or as Kant called them, the moral sciences, have divided the world in two, which naturally prevents them from understanding the whole world.
The Smolin “scientific ethic” reminds of HG Wells, who promoted a Charter of Scientific Fellowship, in 1942, which recognises “the democracy of science.”
Wells also saw that democracy is scientific, in the work started by John Stuart Mill to promote the transferable vote as the scientific method of elections, beyond all the rival and ruinously wrong electoral methods, as theories of choice.
Smolin may think physicists have “failed” in the past 30 years. If true, it is still as nothing compared to the academic communitys 150 year failure to recognise the scientific conception of democracy. Within six months, John Stuart Mill had noticed how its opponents projected the faults in their own beliefs onto the Hare system, fore-runner of the single transferable vote.
Mill never belonged to the conservative academic community, tho British universities used System Of Logic as their standard text on scientific method, for fifty years.
Right on Smolins door-step, at the Perimeter Institute, the Ontario Citizens Assembly on Electoral Reform gave an unedifying example of what happens when so-called world experts – the Perimeter Institute would call them, like themselves, “global souls” – flown in to coach the Assembly, abandon all cognisance of scientific method or the search for truth. They merely followed an official line that no voting system is, on balance, better than another. Ontarians should just choose the one that suited them best.
Such an attitude is the end of science as well as of democracy.
Not for a moment would Smolin tolerate the idea, in physics, that no theory is decisively better than another and you just believe what theory suits you subjectively and arbitrarily “as Ontarians,” without the need to put theories to the test of a principle, in this case, democracy. Yet that is what the Ontario Citizens Assembly was told by politicians and academics supervising them.
A British Columbia physicist, John Huntley, from Simon Fraser University, was curious about the BC Citizens Assembly choice of a system called the single transferable vote. On finding out what it was about, he thought everyone would see that it was easily the best method. When he found they didn’t, he tried to promote it. He and a colleague wrote a good submission to Ontario Citizens Assembly.
18 april 2008.
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A plea to automate and test Binomial STV.
Table of contents
A three-day course, while our lecturers were absent, was all the education in computer programming, that we received as college students back in the 1960s. Since then, the basic skills have become part of the new maths, or so I fondly believe, that is taught to school-children at as early an age as possible.
At any rate, you occasionally hear of teenage computer geniuses.
All that passed me by. If I was starting off today, in my line of study, it would be an essential attainment. Anyway, there are plenty of people about, that are much better at it, than I ever would have been.
Here, I’m trying to draw their attention to writing a program for Binomial STV. If, by any chance, you happen to be familiar with the program for Meek method of STV, then you would be in an ideal position to automate Binomial STV.
Meek method is already used for some official elections in New Zealand. It is also the election method of such expert bodies as the London mathematical society, the Royal statistical society, and the British computer society.
It is possible that someone amongst any number of skilled individuals and expert organisations might conceivably take an interest in programming Binomial STV. So, it is always advisable, when answering such calls for help, as this, to make sure that the call has not been answered, since the call was written. I probably shall go on approaching possible candidates.
One of the main drawbacks of the traditional Single Transferable Vote, and also Meek method, is that it does not fully apply to single seat elections.
There are no surplus votes to transfer, from a candidate already elected on a quota, to a next prefered candidate, taking a second seat, because there is only one seat to be taken!
Thus, single seat elections rob STV of a principal advantage over other voting methods. In effect, traditional STV is reduced to the so-called Alternative Vote, sometimes known as Instant Run-off Voting. If no one candidate wins over half the votes, then the candidate with least votes has to step down, in favor of his voters next preferences, till some candidate gets an overall majority.
The advantage of Binomial STV is that it never finally excludes or disqualifies any candidates, even in single-member constituencies, until the final count, which is an average of systematic recounts of preferences, in an election count, and reversed preferences, in an exclusion count.
For over a decade since inventing Binomial STV, I didn’t ask anyone to write the program. Now I think it is ready, for two reasons.
The first reason.
The bi-, in binomial, refers to a reverse preference count, as well as the normal preference count.
I recently overcame the problem of establishing the relative importance of the preference count and the reverse preference count. This is done by counting all the preference abstentions.
In a typical election, voters will abstain on later preferences. A reverse preference count, that does not take this into account, would effectively give the same importance to the first reversed preference, in an exclusion count, as is normally given to the first preference in an election count. This is even when the first reversed preference is most probably not the least possible preference that a voter could make among all the candidates.
In a Binomial STV count, returning a blank ballot paper is equivalent to the “none of the above” option. But a preference abstentions-inclusive Binomial STV count is actually more discriminating, as to lack of popularity among the candidates, because every preference, not filled-in on the ballot paper, may go towards the quota for an unfilled seat, in the contest.
My second reason for this formal request to programmers or coders is that Binomial STV may have research applications beyond political elections. Binomial STV need not stop at a simple combination or averaging of a preference count and a reverse preference or un-preference count. That is only a first order Binomial STV. (Traditional STV is a zero order or uninomial STV, perfectly adequate for practically all political elections.)
My free e-book, Scientific Method of Elections (which belongs to a series of works referenced at the end of this book) explains how Binomial STV can conduct a second order count of preferences and un-preferences, or a third or any higher order count, according to the binomial formula. I realised that this progression made for the systematic mining of preferential information.
This was before becoming aware, recently, that there actually is a data-mining community, experimenting with more satisfactory voting systems, than simple plurality, for extracting information from data-bases.
(Again with reference to my above-cited e-book) Binomial STV is a continuation of Meek method. Unlike traditional STV, Meek method takes into account extra preferences gained by candidates already elected, by readjusting their keep values. Binomial STV calculates the keep values for every candidate, whether in surplus or deficit of the quota.
Being in deficit of the quota, in an exclusion count, may help a candidate become elected, when the election and exclusion counts are averaged.
Binomial STV would leave out Meek method of reducing the quota as preferences run out. By contrast, Binomial STV counts preference abstentions.
A greater and more rational use of (preferential) information sums up the case for computer automating Binomial STV.
It would also be interesting to compare results, using the Harmonic Mean quota (which I have also advocated, as explained in above-mentioned book) as distinct from the Droop quota, or even the Hare quota.
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Guide to three book series by the author.
Table of contents.
The Commentaries series
Commentaries book one:
Literary Liberties
Literary Liberties with reality allow us to do the impossible of being other people, from all over the world. Our imagined other lives make the many worlds theory a fact thru fiction.
This book of books or illustrated reviews span fiction, faction and non-fiction.
In promoting others writings, I hoped to promote my own, any-way, the liberal values that inform my writings. It took a lot more preparation than I had anticipated. This is usually the case with my books.
Commentaries book two:
Science and Democracy reviews
Works reviewed and studied here include the following.
The physicist, John Davidson under-took an epic investigation into the mystic meaning of Jesuses teachings, as for our other-worldly salvation, supplemented by a revelation in non-canonic texts of the gnostics.
The Life and Struggles of William Lovett, 1876 autobiography of the “moral force” Chartist and author of the famous six points for equal representation.
Organiser who anticipated the peace and cultural initiatives of the UN, such as UNESCO.
Jill Liddington: Rebel Girls. Largely new historical evidence for the role especially of working women in Yorkshire campaigning for the suffrage.
“How the banks robbed the world” is an abridged description of the BBC2 program explanation of the fraud in corporate finance, that destroys public investments.
David Craig and Matthew Elliott: Fleeced!
The political system fails the eco-system.
Green warnings, over the years, by campaigners and the media, and the hope for grass roots reforms.
From Paul Harrison, how expensively professionalised services deprive the poor of even their most essential needs. And the developed countries are over-strained, on this account, drawing-in trained people from deprived countries.
Why society should deprofessionalise basic skills important for peoples most essential needs, whether in the third world or the “over-developed” countries.
The sixth extinction
Richard Leakey and other experts on how mankind is the agent of destruction for countless life forms including possibly itself, in the sixth mass extinction, that planet earth has endured in its history. Why world politicians must work together to counter the effects of global warming.
On a topic where science and democracy have not harmonised, a few essays from 2006 to 2010, after “nuclear croneyism” infested New Labour and before Japans tsunami-induced chronic nuclear pollution. There’s a 2015 after-word.
Some women scientists who should have won nobel prizes.
Lise Meitner, Madame Wu, Rosalind Franklin and Jocelyn Bell, Alice Stewart, to name some. Reading of their work in popular science accounts led me, by chance, to think they deserved nobel prizes; no feminist program at work here.
Julian Barbour: The End Of Time.
Applying the Mach principle, to an external frame-work of Newtonian absolute space and time, both in classical physics and to Schrödinger wave equation of quantum mechanics, by which the universe is made properly self-referential, as a timeless “relative configuration space” or Platonia.
Murray Gell-Mann: The Quark and the Jaguar.
Themes, including complex systems analysis, which the reviewer illustrates by voting methods.
Brian Greene: The Elegant Universe.
Beyond point particle physics to a theory of “strings” that may under-lie the four known forces of nature, and its material constituents, thru super-symmetry, given that the “super-strings,” as such, are allowed to vibrate, their characteristic particle patterns, in extra hidden dimensions of space.
Brian Greene: The Hidden Reality.
A survey of the more extravagant physics theories that have invoked many worlds or a multiverse..
Lee Smolin: Three roads to quantum gravity.
Reviewing the other two roads (besides string theory) namely black hole cosmology and loop quantum gravity. All three approaches are converging on a discrete view of space and time, in basic units, on the Planck scale. General relativitys space-time continuum is being quantised, rather as nineteenth century thermo-dynamics of continuous radiation was quantised.
Lee Smolin: the trouble with physics.
Impatience with the remoteness of string theory and hope for progress from theories with more experimental predictions. How to make research more effective. Smolin on a scientific ethic. Reviewer criticises the artificial divide academics make between science and ethics.
Commentaries book three.
If and when time allows, it is intended to gather a final note-book, consisting largely of tables, graphs and diagrams, too large to conveniently include for e-book readers…
The Democracy Science series.
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The Democracy Science series of books, by Richard Lung, also is edited and renovated from this authors material on the Democracy Science web-site.
Book 1: Peace-making Power-sharing.
The first, of two books on voting method, has more to do with electoral reform. (The second is more about electoral research.)
“Peace-making Power-sharing” features new approaches to electoral reform, like the Canadian Citizens Assemblies and referendums. I followed and took part in the Canadian debate from before the assemblies were set-up, right thru the referendums.
Some developments in America are reviewed.
The anarchy of voting methods, from the power struggle in Britain, is investigated over a century of ruling class resistance to electoral reform.
Peace-making Power-sharing
from Shakespir in epub format: here free.
[It is also available for Amazon kindle, here.
(Amazon charge a nominal amount, currently.)]
Book 2: Scientific Method of Elections.
The previous book had a last chapter in French, which is the earliest surviving version of the foundation of this sequel, Scientific Method of Elections. I base voting method on a widely accepted logic of measurement, to be found in the sciences. This is supported by reflections on the philosophy of science.
The more familiar approach, of judging voting methods by (questionable) selections of basic rules or criteria, is critically examined.
This author is a researcher, as well as a reformer, and my innovations of Binomial STV and the Harmonic Mean quota are explained.
This second book has more emphasis on electoral research, to progress freedom thru knowledge.
Two great pioneers of electoral reform are represented here, in speeches (also letters) of John Stuart Mill on parliamentary reform (obtained from Hansard on-line). And there is commentary and bibliography of HG Wells on proportional representation (mainly).
Official reports of British commissions on election systems are assessed. These reports are of Plant, Jenkins, Kerley, Sunderland, Arbuthnott, Richard, and (Helena Kennedy) Power report.
The work begins with a short history on the sheer difficulty of genuine electoral reform. The defeat of democracy is also a defeat for science. Freedom and knowledge depend on each other.
Therein is the remedy.
Book 3: Science is ethics as electics.
Political elections, that absorbed the first two books in this series, are only the tip of the iceberg, where choice is concerned. Book three, in preparation, intends to take an electoral perspective on the social sciences and natural sciences, from physics to metaphysics of a free universe within limits of determinism and chance.
[]Collected Verse in five books by Richard Lung
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The Valesman.
[_Published, 3 august 2014, _
with ten per cent free sample, and available at Amazon here.]
Dates and Dorothy.
[Published on 2nd september 2014.
And is available here for the Kindle version.]
_Also available from Shakespir here, in epub format _
He’s a good dog. (He just doesnt like you to laf.)
[Published on 14 november 2014.
And is available from Amazon here.]
In the meadow of night.
[_Published on 26 january 2015. _
_And is available from Amazon here. _]
[_Published on 3 march 2015. _
_And is available from Amazon here. _
Also available from Shakespir in epub format here.]
If you read and enjoy any of these books of collected verse, please post on-line a review of why you liked the work.
_While preparing this series, I made minor changes to arrangement and content of the material, so the descriptions of companion volumes, at the end of each book, might not always quite tally. _
The Valesman
The first volume is mainly traditional nature poetry.
(160 poems, including longer narrative verse in section three.)
The nature poet Dorothy Cowlin re-connected me with my rural origins. Many of the poems, about animals and birds and the environs, could never have been written without her companionship.
The unity of themes, especially across the first two sections, as well as within the third section, makes this volume my most strongly constructed collection. I guess most people would think it my best. Moreover, there is something for all ages here.
1. How we lived for thousands of years.
Dorothy thought my best poems were those of the farming grand-father, the Valesman.
2. Flash-backs from the early train.
More memories of early childhood on the farm and first year at the village school.
3. Trickster.
Narrative verse about boyish pranks and prat-falls.
4. Oyh! Old Yorkshire Holidays.
Features playtime aspects of old rural and sea-side Yorkshire.
Dates and Dorothy
Book two begins with eight-chapter review of works, plus list of publications & prizes by Dorothy Cowlin.
(Seven of these chapters are currently freely available as web pages.)
This second volume continues with the second instalment of my own poems, classed as life and love poetry.
The Dates are historical and romantic plus the friendship of Dorothy and the romance of religion.
169 poems plus two short essays.
Prelude: review of Dorothy Cowlin.
Dates, historical and romantic, and Dorothy:
1. dates.
2. the Dorothy poems.
3. loves loneliness loves company.
4. the romance of religion.
The hidden influence of Dorothy, in the first volume, shows in this second volume. The first two sections were written mostly after she died. Thus, the first section, Dates, reads like a count-down before meeting her, in the second section, as prentice poet.
She was warmly responsive to the romantic lyrics of the third section. This was reassuring because some originated in my twenties. (I gave-up writing formal poetry during my thirties, to all practical purposes. There were only about three exceptions.) These surviving early poems, like most of my out-put, under-went intensive revision.
The fourth section probably stems from the importance attached to religion at primary school. Here humanitarian Dorothys influence only slightly made itself felt by her liking to visit churches.
The prelude review of Dorothy as a professional writer is freely available, at present, on my website: Poetry and novels of Dorothy Cowlin.
Nearly all the text is there, except a preface and last section, which I didnt up-load before losing access to the site in 2007.
The fotos, I took of Dorothy, are published for the first time.
The continued availability of my Dorothy Cowlin website is not guaranteed, so I welcome this opportunity to publish my literary review of her work, as an extra to volume 2.
The third book is a miscellaneous collection of 163 poems/pieces, with the arts and politics the strongest themes, as well as themes found in the companion books. There is also a story in section one, and a final short essay.
1. with children
2. or animals
3. never act
4. the political malaise
5. the lost
6. short essay:
Proportional Representation for peace-making power-sharing.
“A boot boy in the Great War,” in the first section, is a sort of verse novela and dramatic poem with an eye on the centenary of the First World War. The idea stemmed from an incident related by Dorothy Cowlin (yet again). Her uncle was stopped flying a kite on the beach, because he might be signaling to the enemy battle fleet.
No kidding!
In this miscellany, previous themes appear, such as children, animals and birds. Verse on the arts comes in. I’ve organised these poems on the WC Fields principle: Never act with children or animals.
The fourth section collects political satires from over the years. The fifth section reflects on loneliness.
This volume is classed as of “presentatives” because largely about politics and the arts, with politicians acting like performing artists or representatives degenerating into presentatives on behalf of the few rather than the many.
However, the title poem, He’s a good dog…, hints how eccentric and resistent to classification is this third volume. This title poem is based on a true war-time air incident. The good dog is also derived from a true dog, whose own story is told in the poem, the bleat dog (part of the free sample in volume 1).
In the meadow of night
The fourth volume is of 160 poems and two short stories on the theme of progress or lack of it.
part one: allure.
The allure of astronomy and the glamor of the stars.
part two: endeavor.
The romance and the terror of the onset of the space age and the cold war.
part three: fate.
An uncertain future of technologies and possible dystopias. Ultimate questions of reality.
This fourth volume is of SF poetry. SF stands for science fiction, or, more recently, speculative fiction. The verse ranges from hard science to fantasy.
The literary tradition of HG Wells and other futurists exert a strong influence.
Otherwise, I have followed my own star, neither of my nature poet friends, Dorothy and Nikki, having a regard for SF poetry.
Yet science fiction poetry is a continuation of nature poetry by other means.
This may be my most imaginative collection. Its very diversity discourages summary.
Volume 5 opens with a play about the most radical of us all, Mother Teresa: If the poor are on the moon…
This is freely available, for the time being, on my website: Poetry and novels of Dorothy Cowlin. (Performers are asked to give author royalties to the Mother Teresa Mission of Charity.)
The previously unpublished content consists largely of fairly long verse monologs, starting with artistic radicals, in “Symfonic Dreams,” which is a sequence of The Impresario Berlioz, and The Senses of Sibelius.
Next, the intellectual radical, Sigmund Freud, followed by short poems on a sprinkling of more great names, who no doubt deserved longer. (Art is long, life is short.)
The title sequence, Radical! is made-up of verse about John Stuart Mill, Arthur Conan Doyle, George Bernard Shaw, HG Wells, George Orwell and JB Priestley.
Volume five ends with an environmental collection, largely available on my website: Poetry and novels of Dorothy Cowlin. However, those available verses have been more or less revised.
Should that website close down, I hope the green verses and the Mother Teresa play can still be obtained in this volume five.
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Science and democracy reviews.
I have drawn on my experience of life and books, in discussing the following works. The physicist, John Davidson under-took an epic investigation into the mystic meaning of Jesuses teachings, as for our other-worldly salvation, supplemented by a revelation in non-canonic texts of the gnostics. The Life and Struggles of William Lovett, 1876 autobiography of the "moral force" Chartist and author of the famous six points for equal representation. Organiser who anticipated the peace and cultural initiatives of the UN, such as UNESCO. Jill Liddington: Rebel Girls. Largely new historical evidence for the role especially of working women in Yorkshire campaigning for the suffrage. "How the banks robbed the world" is an abridged description of the BBC2 program explanation of the fraud in corporate finance, that destroys public investments. David Craig and Matthew Elliott: Fleeced! How we've been betrayed by the politicians, bureaucrats and bankers and how much they've cost us. The political system fails the eco-system. Green warnings, over the years, by campaigners and the media, and the hope for grass roots reforms. From Paul Harrison, how expensively professionalised services deprive the poor of even their most essential needs. And how even the developed countries are over-strained on this account and draw in trained people from deprived countries. Why society should deprofessionalise basic skills important for peoples most essential needs, whether in the third world or the "over-developed" countries. The sixth extinction Richard Leakey and other experts on how mankind is the agent of destruction for countless life forms including possibly himself, in the sixth mass extinction planet earth has endured in its history. Why world politicians must work together to counter the effects of global warming. Some women scientists who should have won nobel prizes. Lise Meitner, Madame Wu, Rosalind Franklin and Jocelyn Bell, Alice Stewart, to name some. Reading of their work in popular science accounts led me, by chance, to think they deserved nobel prizes: no feminist program at work here. Julian Barbour: The End Of Time. Applying Mach's principle, to Newton's external frame-work of absolute space and time, both in classical physics and to Schrödinger's wave equation of quantum mechanics, by which the universe is made properly self-referential, as a timeless "relative configuration space" or Platonia. Murray Gell-Mann: The Quark and the Jaguar. Themes, including complex systems analysis, which the reviewer illustrates by voting methods. Brian Greene: The Elegant Universe. Beyond point particle physics to a theory of "strings" that may under-lie the four known forces of nature, and its material constituents, thru super-symmetry, given that the "super-strings," as such, are allowed to vibrate, their characteristic particle patterns, in extra hidden dimensions of space. Brian Greene: The Hidden Reality. A survey of the more extravagant physics theories that have invoked many worlds or a multiverse.. Lee Smolin: Three roads to quantum gravity. Reviewing the other two roads (besides string theory) namely black hole cosmology and loop quantum gravity. All three approaches are converging on a discrete view of space and time, in basic units, on the Planck scale. General relativity's space-time continuum is being quantised, rather as nineteenth century thermo-dynamics of continuous radiation was quantised. Lee Smolin: the trouble with physics. Impatience with the remoteness of string theory and hope for progress from theories with more experimental predictions. How to make research more effective. Smolin's scientific ethic. Reviewer criticises artificial divide between science and ethics.
• Author: Richard Lung
• Published: 2015-10-12 22:20:22
• Words: 93431
Science and democracy reviews. Science and democracy reviews. |
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splitting method; semilinear evolution equations; error analysis
We consider a Strang-type splitting method for an abstract semilinear evolution equation $$ \partial _t u = Au+F(u). $$ Roughly speaking, the splitting method is a time-discretization approximation based on the decomposition of the operators $A$ and $F.$ Particularly, the Strang method is a popular splitting method and is known to be convergent at a second order rate for some particular ODEs and PDEs. Moreover, such estimates usually address the case of splitting the operator into two parts. In this paper, we consider the splitting method which is split into three parts and prove that our proposed method is convergent at a second order rate.
[1] Besse, C., Bidégaray, B., Descombes, S.: Order estimates in the time of splitting methods for the nonlinear Schrödinger equation. SIAM J. Numer. Anal. 40 (2002), 26-40. DOI 10.1137/S0036142900381497 | MR 1921908 | Zbl 1026.65073
[2] Borgna, J. P., Leo, M. De, Rial, D., Vega, C. Sánchez de la: General splitting methods for abstract semilinear evolution equations. Commun. Math. Sci. 13 (2015), 83-101. DOI 10.4310/CMS.2015.v13.n1.a4 | MR 3238139 | Zbl 1311.65106
[3] Cazenave, T., Haraux, A.: An Introduction to Semilinear Evolution Equations. Oxford Lecture Series in Mathematics and Its Applications 13, Clarendon Press, Oxford (1998). MR 1691574 | Zbl 0926.35049
[4] Descombes, S., Thalhammer, M.: An exact local error representation of exponential operator splitting methods for evolutionary problems and applications to linear Schrödinger equations in the semi-classical regime. BIT 50 (2010), 729-749. DOI 10.1007/s10543-010-0282-4 | MR 2739463 | Zbl 1205.65250
[5] Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations. Springer Series in Computational Mathematics 31, Springer, Berlin (2006). DOI 10.1007/3-540-30666-8 | MR 2221614 | Zbl 1094.65125
[6] Jahnke, T., Lubich, C.: Error bounds for exponential operator splittings. BIT 40 (2000), 735-744. DOI 10.1023/A:1022396519656 | MR 1799313 | Zbl 0972.65061
[7] Lubich, C.: On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations. Math. Comput. 77 (2008), 2141-2153. DOI 10.1090/S0025-5718-08-02101-7 | MR 2429878 | Zbl 1198.65186
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More on funding
by Michael Nielsen on August 15, 2007
Chad Orzel has some thoughtful comments on my earlier questions about research funding. Here’s a few excerpts and some further thoughts:
… a good deal of the image problems that science in general has at the moment can be traced to a failure to grapple more directly with issues of funding and the justification of funding… In the latter half of the 20th century, we probably worked out the quantum details of 1000 times as many physical systems as in the first half, but that sort of thing feels a little like stamp collecting– adding one new element to a mixture and then re-measuring the band structure of the resulting solid doesn’t really seem to be on the same level as, say, the Schrödinger equation, but I’m at a loss for how to quantify the difference… The more important question, though, is should we really expect or demand that learning be proportional to funding?
This really gets to the nub of it. In research, as in so many other things, funding may hit a point of diminishing returns beyond which what we learn becomes more and more marginal. However, it is by no means obvious where the threshold is beyond which society as a whole would be better off allocating its resources to other more worthy causes.
And what, exactly, do we as a society expect to get out of fundamental research?
For years, the argument has been based on technology– that fundamental research is necessary to understand how to build the technologies of the future, and put a flying car in every garage. This has worked well for a long time, and it’s still true in a lot of fields, but I think it’s starting to break down in the really big-ticket areas. You can make a decent case that, say, a major neutron diffraction facility will provide materials science information that will allow better understanding of high-temperature superconductors, and make life better for everyone. It’s a little harder to make that case for the Higgs boson, and you’re sort of left with the Tang and Velcro argument– that working on making the next generation of whopping huge accelerators will lead to spin-off technologies that benefit large numbers of people. It’s not clear to me that this is a winning argument– we’ve gotten some nice things out of CERN, the Web among them, but I don’t know that the return on investment really justifies the expense.
The spinoff argument also has the problem that it’s hard to argue that these things wouldn’t have happened anyway. No disrespect to Tim Berners-Lee’s wonderful work, but it’s hard to believe that if he hadn’t started the web, some MIT student in a dorm room wouldn’t have done so shortly thereafter.
Of course, it’s not like I have a sure-fire argument. Like most scientists, I think that research is inherently worth funding– it’s practically axiomatic. Science is, at a fundamental level, what sets us apart from other animals. We don’t just accept the world around us as inscrutable and unchangeable, we poke at it until we figure out how it works, and we use that knowledge to our advantage. No matter what poets and musicians say, it’s science that makes us human, and that’s worth a few bucks to keep going. And if it takes millions or billions of dollars, well, we’re a wealthy society, and we can afford it.
We really ought to have a better argument than that, though.
As for the appropriate level of funding, I’m not sure I have a concrete number in mind. If we’ve got half a trillion to piss away on misguided military adventures, though, I think we can throw a few billion to the sciences without demanding anything particular in return.
One could attempt to frame this in purely economic terms: what’s the optimal rate at which to invest in research in order to maximize utility, under reasonable assumptions? This framing misses some of the other social benefits that Chad alludes to – all other things being equal, I’d rather live in a world where we understand general relativity, just because – but has the benefit of being at less passably well posed. I don’t know a lot about their conclusions, but I believe this kind of question has recently come under a lot of scrutiny from economists like Paul Romer, under the name endogeneous growth theory.
From → Science
1. Travis permalink
I think the particular social benefit you describe in your conclusion (knowing “just because”) is rapidly diminishing for fundamental science.
Let’s consider the LHC, which is expected to cost around $2 billion USD. If we assume that the cost is evenly distributed over the roughly one billion people living in developed countries, that’s $2 USD per person. If you explained to the average person what the LHC was for and what knowledge it might discover, do you think you could convince them it was worth $2 of their money? If the answer is no, we have a problem.
I know that, as a physicist, I derive a lot of “intrinsic value” from simply understanding something like quantum mechanics or general relativity. It’s not clear to me that the general public derives much–if any–value from knowing that there are a few people who understand GR. Fortunately with GR, interested laymen can at least understand what it’s about, which certainly boosts the value to the public. Once things get more specialized (string theory, I’m looking at you), that public benefit goes away.
Put another way, would it mean anything to you if you knew that some person somewhere knew the True Meaning of Life, but that you could never understand it? Would you pay money to help that person discover the TMoL, knowing that you could never know it yourself?
As for the technological benefits, it’s also not clear to me that that is a politically viable justification for funding basic science. A good deal of basic physics research that is being done will take decades to generate practical applications (if it does at all). Our society isn’t good at making investments on that timescale; as John Maynard Keynes said, “In the long run, we’re all dead.” Perhaps the focus on biomedical research is a good strategy–if people live longer, they’re more likely to care about the distant future. Funding research so that you can have flying cars five decades from now becomes a lot more compelling if you expect to still be alive to fly one fifty years hence.
2. Michael Nielsen permalink
I don’t think people look at it like that. So far as I can see, many people in the public at large are proud of the Einsteins and Cricks of the world, even while they don’t necessarily fully understand what those people have done.
To put it a different way, an awful lot of people bought Roger Penrose’s recent book, and I don’t think it’s because they wanted to read up on holomorphic functions; I think it’s because they enjoy and appreciate contact with great minds, even if they don’t fully understand what those minds do.
On the LHC, I have some of the same concerns as you. However, I don’t think it’s fair to generalize from this to all of fundamental science as you have done. There is a lot of excellent fundamental science being done that is many orders of magnitude cheaper than the LHC.
3. Travis permalink
People have some idea of what Einstein did– E=mc^2 is one of the most famous equations in the world, after all. Penrose is still somewhat understandable, too. That said, book sales should be taken with a grain of salt: Deepak Chopra’s “The Book of Secrets” was a NYT Bestseller.
The problem is the trend; as it stands, we’re heading towards things being less comprehensible to the public, even as we ask for more and more funding. I picked on the LHC because, as someone pointed out at DAMOP (to subsequently be quoted on one of the other blogs discussing this topic), BEC experiments and many other areas of physics are asymptotically approaching particle physics in complexity, cost, and inaccessibility. One can also see parallels between topological quantum computing and string theory. If high energy physics is the model for the future of science, we’re in trouble.
Either we need to get much, much better at communicating, or the public is eventually going to lose faith that what we’re doing means anything. The less accessible science becomes, the more it starts to look from the outside like religion.
4. aram permalink
Romer says in part that all per-capita economic growth ultimately comes from technological improvement, and that R&D from the private sector will always be far less than the optimal. This doesn’t say how much gov’t-supported science is too much, but we should remember that it doesn’t take much to have an enormous spillover.
For example, maybe some MIT kids would have made the web a year later, but even that difference of a year probably means an enormous amount to the global economy.
5. Michael Nielsen permalink
Travi: “The less accessible science becomes, the more it starts to look from the outside like religion.”
I agree, although I’d probably say “look meaningless” instead of making the comparison to religion, which is a bit loaded. This is a big problem. I don’t know what the solution is, although I suspect that serious efforts at outreach by the scientific community would be a good start.
6. Michael Nielsen permalink
Aram: do you know enough about Romer et al’s work to know whether this question (what’s the optimal investment in basic research) could even be given an answer within their framework?
Your point about the year delay is well taken, although I’m not sure it isn’t a red herring. There are many contingencies here: maybe the web would have taken off faster if it had been invented a year later in an MIT dorm room.
(Of course, I’m not seriously trying to make this argument. All I’m trying to say is that evaluating the economic impact of basic research is pretty darn complicated, and one can’t just ascribe all or even much of the value of something like the web to the investment in CERN.)
7. John Novak permalink
You must know the answer to that question already, Michael.
If not, answer me this: Given a billion dollars, what is the optimal partition in basic research between, say, string theory, quantum gravity, and ultracold physics? If you can’t answer that, why can’t you answer it, and what does that say about your ability to determine an optimal lump sum to invest?
If you can answer it, how did you arrive at your answer?
8. Travis permalink
Michael: I agree that we need much better science outreach, starting with vastly improved science education. Unfortunately, this is a chicken-and-egg problem: voters don’t know enough about science to know why improved science education should be a priority.
On a different note, let’s think about where the point of optimal investment might lie. If return on investment in basic science is linear, then either we reach an optimum only when we’re spending so much on science that we’re significantly neglecting other things (which have non-linear returns), or the optimum is essentially zero. Since science is currently a relatively small portion of total government spending, this suggests that we’re either spending way too much or way too little on science (or perhaps we’re in a transition between the two regimes).
Could the return on science be substantially non-linear? I can really only think of two ways this could happen. The first is if progress in any given area is inherently speed-limited, no matter how many people are thrown at the problem. In other words, there’s a limit to how many people can productively work on the same problem before you end up with too much duplication of effort and overhead. The second way is if science progress is limited by the number of capable researchers.
Given the relatively low percentage of grant applications that get funded, I’d guess the latter isn’t currently the case–we have plenty of people who could be doing quality research who currently aren’t getting funded. As for the former possibility, I suspect it’s probably also not true, except in a few very crowded fields.
To (partially) answer John, if the return on science is high, then the I know the optimal sum to invest is very large, even though I may know nothing about how much of that sum to invest in ultracold physics versus quantum gravity. It’s not necessary to solve the funding allocation problem at the micro level to solve it at the macro level.
9. Michael Nielsen permalink
Travis: I certainly think that the return on science is incredibly non-linear. Empirically, we spent a huge amount more on science in the second half of the twentieth century, yet the return (comparatively) didn’t match the investment. Don’t get me wrong, I still think the return was worth it, it just wasn’t as exceptional as earlier.
10. Michael Nielsen permalink
John: I have lots of ideas about how to decide this. But none that I feel all that confident of. A quote from Robin Hanson seems apt:
My core politics is “I don’t know”; most people seem far too confident in their political opinions.
With that said, I tend to agree with Travis that at the larger scale we can make some plausible guesses based on past performance. But at the individual scale, where science changes very rapidly, this becomes much less useful as a predictive mechanism. String theory may have no economic impact; it may spawn multi-trillion dollar industries. I don’t know which, and I suspect no-one else does either. How should I compare that to AMO?
Robin Hanson’s ideas about idea futures may have some relevance here.
11. Travis permalink
Michael: Were you implying that, in the first part of the 20th century, physics picked all the low-hanging fruit, and we are now left “mining the low-grade ore” (sorry for mixing metaphors)? That’s not incompatible with what I mean by “linear”. To better define it, what I mean that is that the return we get depends only on the amount we spend, and not on the rate. In other words, $1 billion spent over 1 year gets us the same return as $1 billion over 5 years. It’s still possible that the second billion we spend will get a lower rate of return than the first billion.
Taking the mining metaphor further, if we throw more money into science now, we’ll run out of high- and medium-grade “scientific ore” sooner, and have to start “mining” the low-grade stuff with its poor returns. This still fits within what I meant by “linear”, though. In such a scenario, we should mine as fast as we can, until we run out of “ore” that’s economical to extract. At that point, science is over, at least from an economically-rational viewpoint.
The question is–have we already hit that point? Have we passed “peak science”? At least in physics, we seem to be out of the high-grade ore (barring a strike of a “new vein” of good stuff).
12. Michael Nielsen permalink
Travis: I certainly don’t think the return is linear. The best funded fields in science are often very crowded, and this tends to diminish people, who will (often) narrow their ambition.
In regard to your question about “peak science” and physics: Physics has historically gone through many long dry spells, only to return to rapid major progress. There are lots of really big unsolved problems in physics, and some of those may turn out to have solutions as revolutionary as quantum mechanics or relativity; it’s also possible that for some of them a major advance may come in the very near future.
13. John Novak permalink
I meant to answer my own not-quite-rhetorical question sooner, but it’s been a hectic week. In any event, I respect your answer for its honesty, but as most “I don’t know” answers, it’s not the most helpful one.
Here’s the key insight I was driving at with my questions: You can’t allocate between three broad research areas precisely because they’re research. If you could allocate properly, you would have to have some good foreknowledge of the results of the research, in which case it wouldn’t be research, would it? The same principle applies to trying to allocate an amount of funds in general to research in general.
Any time you allocate funds like that, you’re effectively placing a wager, which makes the reference to Robin Hansen’s “Idea Futures” an interesting one… although the pragmatist here must point out that we already have an idea futures market in effect by way of research univiersities, large technical corporations, and research arms of the various governments.
It’s worth pointing out another insight, related to the above– not all research is created equal, in that regard. Qualitatively, and not at all comprehensively, there’s a continuum ranging from basic, paradigm-shifting fundamental scientific research (say, reconciling GR and QM) at one end, down through more “technical” but still science-related research that expands and maps out semi-known territory (say, bioinformatics research), into more engineering-research fields (say, the first few decades after the development of the transistor, or some of the things my company does in defense applications), and finally down into more easily product-oriented engineering (say, cell phones and road construction techniques.)
That’s probably not an entirely single-axis continuum, but let that go for a moment. The thing is, the farther toward the product end of that scale you are, the more certain you are of both your results and of the cost of your goals. (And, I suppose, of the achievability of your goals in the first place.) The near term consequences of a road construction technique that slashes costs by half and slashes maintenance time by half are pretty easy to predict, in brad strokes– companies make more money, state governments either save more or provide more services, etc. You can generate a sophisticated model, plug in some assumptions (Which is more important– more roads, or smaller budget? Where will new roads go?) and get a range of sophisticated and plausible results.
The closer you move toward the basic science regime, the more inherent uncertainty you crank into the equation, and the less smooth the results are. The results, in fact, tend to be all or nothing, with the “all” being completely unpredictable even aside from that. I hate to be the naysayer, but I cannot imagine even the shape of a better solution than the market. Perhaps there are tweaks or moderate experiments to be made, though:
Perhaps an experimental fund of, say, $10,000,000 per year to be allocated more directly by an idea futures market subject to some constraints (e.g., only people with a post graduate degree in science, engineering, or related technical fields are allowed to wager.)
14. John Novak permalink
I take that back, actually.
Two ideas futures market funds, with the same amount of money in each, administered by the same rules, with one difference: One open only to holders of the proper credentials, the other open to anyone.
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1eaeaa695df4e7e5 | I'm on the very beginning of learning quantum mechanics. When we solve the time independent Schrödinger Equation as far as I understand we will get the general solution:
$$\Psi(r,t)=\sum c_n\cdot \psi(r)\cdot \exp(-iE_nt/\hbar)$$
But I have learnt that $\Psi$ has no physical meaning, and that we have to use $\mid \Psi \mid^2$ for a physical interpretation. Describing the probability of finding a particle at a given neighbourhood. We know that $(\exp(-iE_nt/\hbar))^2=1$
Does that mean that the factor $\exp(-iE_nt/\hbar)$ is totally useless, and that for any system we may change our solution to $\Psi(r,t)=\sum c_n\cdot \psi(r)\cdot "1"$ and still have an equivalent physical system?
I clearly see that this is a mathematical sin, but would it be okay from a purely physical perspective?
• $\begingroup$ $A^2\neq |A|^2$. And if you change that factor to 1, $\Psi$ is no longer a solution of the Schrödinger equation. $\endgroup$ – Demosthene Jan 23 '17 at 9:41
You are mistakenly assuming that $|\sum_k a_k|^2 = \sum_k |a_k|^2$.
Make sure you understand why this relation does not hold (find a counterexample), and you should be able to apply it to your problem.
It's not quite correct that the wavefunction lacks physical meaning. If you left out the imaginary rotation term, you couldn't get interference effects. It'd be kinda okay if your system was in an energy eigenstate, as those do have time symmetry when you look at the probability density with respect to time, but the same would not be true for a superposition of two energy eigenstates, which should more or less oscillate between the two.
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f473c3c5ab8623c2 | The Coulomb's force is given by $$ F = {k q^2 \over r^2} $$ When $ r \rightarrow 0 $, $ F \rightarrow \infty $ Does this mean two electrons never touch each other?
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$\begingroup$ Define touch. Electrons are standing waves and are known to show interference patterns. I don't know what you mean by them 'touching'. $\endgroup$ – Gerard Aug 4 '13 at 12:05
• $\begingroup$ @user1305192 can there not be free electron in space at some given postion? $\endgroup$ – hasExams Aug 5 '13 at 15:57
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$\begingroup$ Pauli Exclusion . . . $\endgroup$ – Abhimanyu Pallavi Sudhir Aug 9 '13 at 11:55
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$\begingroup$ Pauli Exclusion won't save them from touching if they have different spins $\endgroup$ – Ruslan May 22 '14 at 6:17
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$\begingroup$ @Ruslan so 50% of the time it works every time $\endgroup$ – Jim May 22 '14 at 14:28
Yes, two electrons or two charged particles are not allowed to overlap – the interaction energy goes like $1/r$ so it would diverge in the strict limit and the world doesn't have the infinite energy to do that. Two electrons are forbidden to overlap for another reason, the Pauli exclusion principle.
On the other hand, two particles of opposite signs attract by a force that goes to infinity in the $r\to 0$ limit, too. Nevertheless, even in this case, they won't end up overlapping due to the uncertainty principle of quantum mechanics.
For example, take the hydrogen atom (or any atom). The electron can't sit exactly at the nucleus even though it would save an infinite amount of energy. The reason is that a very low $r$ implies a very low $\Delta x$ of the same order and, therefore, a high $\Delta p \geq \hbar / \Delta x$. A high $\Delta p$ means a high average $\Delta p^2$, and therefore high kinetic energy, and this one actually wins if the proton-electron distance is too short. If the electrostatic force increased faster than $1/r^2$ at short distances, one could actually beat the kinetic energy and oppositely charged particles would choose to sit on top of each other.
• $\begingroup$ @ Lubos Motl what dou you mean by electrostatic force increased faster than 1/r^2 .please clarify on that. $\endgroup$ – Ufomammut Mar 31 '13 at 12:42
• $\begingroup$ The answer is wrong. Correct one is the opposite - see my answer. $\endgroup$ – Ruslan May 22 '14 at 7:40
• $\begingroup$ @AaKASH - I mean if the potential energy were $K/r^n$ for $n\gt 2$ or as any function that is greater than that for all $r\lt \varepsilon$. For example $K/r^3$. This is so huge near $r=0$ that even the huge kinetic energy you get from the inevitable $\Delta p$ is smaller and it's energetically favored for the electron to sit strictly at $r=0$. Of course, it's just a mathematical limit - you won't find those things exactly in Nature. $\endgroup$ – Luboš Motl May 30 '14 at 4:52
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$\begingroup$ Lubos is nothing but a dick. $\endgroup$ – Les Adieux Aug 26 '16 at 17:23
• $\begingroup$ Thanks, you're far from the first person who considers me a reincarnation of dick feynman. $\endgroup$ – Luboš Motl Aug 27 '16 at 15:10
Although this is quite an old question, I have to disagree with answer by Luboš.
First, Pauli exclusion principle says that no two fermions can share the same state. But, if the electrons have different spins (i.e. are in so called spin-singlet state), then they can be in the same positional state.
Next, indeed, in classical case, two charged particles with equal signs cannot touch, because their potential energy $\sim1/r$ will go to infinity at the collision point. But, as electrons are quantum particles, they obey uncertainty principle, which allows them, in particular, to tunnel to classically restricted locations. To more precisely describe their motion, one has to use Schrödinger equation.
Suppose two electrons with different spins are in a parabolic 3D potential well, while interacting by Coulomb force between each other. Schrödinger equation for them can be simplified by separation of variables*. Then the part for inter-electron wave function would look like (neglecting dimensional constants):
The boundary conditions for $\phi(r)$ are boundedness at $r=0$ and $r\to\infty$.
Solving** this equation with $l=0$ (otherwise electrons won't touch because of centrifugal force, which grows faster than Coulomb force), we get the wavefunction for ground state:
enter image description here
As you can see, the wavefunction doesn't go to zero at $r=0$. Neither will it vanish for excited $S$-states (i.e. excited states with $l=0$). Instead, there's a cusp with a local minimum of probability density at the point of collision. Were the potential to go to infinity at a higher rate, like $r^{-2}$, the wavefunction would indeed vanish at that point. This is what happens for $P, D$ and other states with $l>0$.
Note that the reason for this is very similar to reason why electron doesn't have infinite binding energy in atom in $S$ states: the wavefunction there has a local maximum cusp, but it still is bounded. For potentials $\sim -r^{-2}$ the electron would fall onto the nucleus and have infinite binding energy.
* See the (paywalled) article for details on separation of variables.
** I solved it via Frobenius method, limiting to 1000 series terms. I'm not sure if there's a closed-form solution for this BVP.
• $\begingroup$ An interesting answer, @Ruslan, but I don't know why it implies that they will touch. The probability of strictly $r=0$ is still zero, right? Would you say that the electron touches the nucleus at the 1s state? $\endgroup$ – Luboš Motl May 30 '14 at 4:54
• $\begingroup$ @LubošMotl Of course, probability at $(r,\theta,\phi)=(0,?,?)$ is zero, but so it is for any other $(r,\theta,\phi)$ (because of integration over zero volume). The difference with centrifugal potential is that in the Coulomb case relative probability $P(0,?,?)/P(r,\theta,\phi)$ is non-zero for all $r$, $\theta$ and $\phi$. So, yes, I'd say the electron does touch the nucleus in 1s state. Using point-likeness of particles as a reason for them to not touch would be somewhat vacuous (and then your answer taking potential into account would be redundant). $\endgroup$ – Ruslan May 30 '14 at 8:34
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$\begingroup$ It is not redundant! For attractive potentials stronger than $1/r^2$, like $-1/r^3$, the electron would touch the nucleus even in my, strong sense, as the wave function would be proportional to the delta-function! So the probability for the electron to be strictly at the origin would be positive, finite. I would agree with you that the probability density for the $1s$ state is parametrically higher than for $l\neq 0$ states but it isn't enough to call these things "touching" (in the approximation of a pointlike nucleus; for a finite realistic nucleus, the touching occurs for any $l$). $\endgroup$ – Luboš Motl Jun 3 '14 at 11:26
• $\begingroup$ Indeed. Although, I would't call it a touch, instead it'd be a (permanent and irreversible) fusion :) Also it seems wavefunction won't be proportional to delta function per se, although it would be something qualitatively very similar to it. $\endgroup$ – Ruslan Jun 3 '14 at 11:41
• $\begingroup$ @user104 collision of quantum particles is a fuzzy concept. Usually it just means scattering (elastic or inelastic), and doesn't imply actual touch. $\endgroup$ – Ruslan Feb 14 '16 at 11:59
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8d213c5c9ab13937 | Search This Blog
Monday, August 31, 2015
Quantum Aether Electromagnetism and Gravity
Saturday, August 22, 2015
Spacetime Emerges from Quantum Aether
Sunday, August 9, 2015
Quantumology is the belief that quantum action describes all force and that gravity is a discrete quantum force. Quantumology necessarily begins with some kind of universal particle, like a discrete aether, and the decay of discrete aether then defines all force. What this means is that the photon diploe exchange that defines charge force also then defines a photon pair as the monopole-quadrupole force of gravity. Photon pairs as monopole-quadrupoles then bond neutral matter particles for quantum gravity and are scaled versions of the photon dipole emissions that bond charged particles.
A very simple way to scale gravity force from charge dipole force is to wrap the universe onto itself and let the ratio of the time delay of the atom to the time delay of the universe scale gravity. In other words, the charge dipole force acts locally between the charges of an atom as well as globally as a monopole-quadrupole force when the universe wraps onto itself in time. Gravity force is simply charge force scaled by the ratio of the time delay of an atom with the time delay of the universe as a pulse in time. This simple statement of unification is completely consistent with mass-energy equivalence, Lorentz invariance, gravitational radiation, and many of the other precepts of general relativity. This simple way of unifying gravity and charge force is not yet accepted by mainstream science.
However, the notions of discrete aether, matter exchange, and time delay are much more general that the notions of continuous space, motion, and time as axioms. Continuous space and motion are not congruent between gravity and charge forces and that incongruence precludes unification within the limits of continuous space and time. Instead of continuous space and motion, unification necessitates a pair of conjugates that are congruent and compatible for both charge and gravity forces.
Even though continuous space and motion are very intuitive and deeply embedded into our consciousness, the notions of continuous space and motion are not a priori axioms for all action. Discrete matter and time delay as the proper conjugate quantum operators apply even beyond the current limits of continuous space and motion, which bound more typical conjugates of space and momentum. Space and momentum still have the same meanings and utility for many predictions of action, but for both very large and very small scales, there are no expectation values for space and momentum. Time, for example, has a fundamental two dimensional representation instead of a single continuous dimension of spacetime and time reflects the nature of the boson aether pulse that is the universe.
Things happen to objects of matter in the universe because of the actions of both gravity and charge and we think of gravity and charge as being very different, but in fact they are simply different manifestations of the same force of aether decay at much different scale. The scale ranges from the time delay of the atom to the overall time delay the universe aether pulse. While charge force is a result of the boson matter decay of the universe, gravity force is a result of the fermion decay of microscopic matter. While the universe is mostly boson aether, it is fermion matter that makes up common objects.
The action of the earth's gravity creates stone from cooling inner molten magma and it is the microscopic charges of stone's atoms and molecules that hold those stones together. The much weaker action of gravity is only evident in holding those stones and us to earth's surface, but gravity is what makes earth earth. Someone building a stone wall depends on gravity not only to keep them and the stone wall bound to earth, that gravity also compresses and slightly heats stones in the actions of building a stone wall. That very slight heating of the stone is part of the gravity force of earth and leads to much greater heating of the inner earth.
Action is both what forms objects like stones from atoms and action is how we form objects like stone walls from stone. In both cases, smaller moments of matter come together to form larger objects. The heat and pressure of earth's gravity makes stone while people gather those stones and make stone walls on earth’s surface for some purpose. The gravitational bond between the stones in the wall and the earth heats the stones up very slightly on earth's surface and it is that radiative and conductive cooling that results in the bonding that we call gravitational compression.
Gravity describes how most things of common experience happen and simply depends on mass action, like the action of a deterministic path of an apple falling from a tree. Gravity results in a very deterministic cause and effect universe where it appears that all action results in only local effects. Our notions of space and momentum emerge from the actions of gravity on objects that we sense.
Charge describes how the microscopic actions of atoms and molecules of matter objects happen with quantum matter with both phase and amplitude. Quantum charge is how the apple grew on the tree in the first place and quantum charge released the apple from the tree into gravity mass action. Charge results in a wavelike and probabilistic universe that allows the matter wave amplitude of one object to affect the matter wave amplitude of another object instantaneously across the universe. As a result, both philosophy and science therefore have very different interpretations of the very different natures of gravity and charge actions.
Quantumology is the belief that gravity is just a scaled version of charge force and that quantum of gravity force is a coherent photon pair as a monopole-quadrupole. Although mainstream science and general relativity are not consistent with this view of quantum gravity, the decay of discrete aether and time delay are consistent with quantum gravity.
Charge bonds involve matter exchange between objects while gravity bonds also involve matter exchange between objects and the universe. Motion in the universe emerges from a change in an object’s inertial mass as equivalent energy and it is that exchange of aether that we call object momentum. Changes in an object’s inertial mass or kinetic energy define an object’s action for a given frame of reference while gains and losses of mass as impulse change object momentum. Although motion is a very common way to define momentum in space, the dimensionless ratio of velocity squared to the speed of light squared in ppb is embodied in the dimensionless Lorentz factor.
The equivalence of matter and energy means that velocity and acceleration are equivalent to changes in inertial mass. The dimensionless Lorentz factor impacts space, matter, and time even while most object action involves gains and loses of ordinary matter as impulsive momentum, which typically overwhelm changes in inertial mass. What we call the fields of charge or gravity force are actually matter exchanges among objects that result in acceleration and changes in object velocities. Charge and gravity fields are potential matter, which is the rate of change of inertial matter in time and is that proper matter that comes into existence as velocity or kinetic matter from an inertial frame. In matter time, fields in space are simply a manifestation of the exchange of matter between objects and those matter exchanges are the forces or accelerations of potential matter.
The decay of all universe matter with time, mdot, is in fact a fundamental principle of matter time and is the determinant of both gravity and charge actions, just at very different scales. This decay constant is simply a restatement of charge and gravity forces as cross sections and is equivalent to the dimensionless universal decay of all matter, αdot, at 0.255 ppb/yr. For charge force, αdot applies to the electron mass as the fundamental fermion while for gravity force, αdot applies to the gaechron mass as the fundamental boson, which is some 1e-39 times less than the electron mass.
Currently science uses two somewhat inconsistent theorets to separately predict the gravity and quantum futures of objects in time. This patchwork approach actually works very well for predictions of action within certain scales, but mainstream science yearns to describe gravity as part of a unified quantum action that includes both charge and gravity.
Gravity action is what holds us to the earth as well as what holds the earth in orbit around the sun and gravity action holds the rest of the greater universe together as well. So, gravity action is the way that we predict how objects move for much of our very deterministic and causal and chaotic reality here on earth and gravity action is how we measure the billions of years of our universe time delay. We have come to know gravity action as general relativity but still gravity action scales with the mass distribution of objects and gravity does not depend on exactly what the matter is.
Gravity action in matter time is very simply related to the binding of objects to the boson matter of the universe. Just like the quantum bonds of electrons to nuclei, the quantum bonds of atoms to the universe boson matter result in the attraction between neutral objects that we call gravity. Gravity is a quantum excitation that involves correlated pairs of photons as a mono-quadrupole time and for most common gravity action, quadrupole time is equivalent to proper time, τ. This approximation does not account for any quantum exchange effects, where the exchange of identical particles leads to an additional quantum gravity binding energy.
Our microscopic reality, though, is bound with charge and quantum action and, unless an object is very massive, gravity action is not much of a factor at all. In contrast to gravity action, quantum action is very dependent on the exact nature of matter amplitude and phase. Matter amplitude and phase are part of the quantum action that determines the nature of the bonds that hold an object’s matter together.
For example, an atom of hydrogen bonds much differently with another hydrogen atom as compared to a different element like oxygen. Oxygen bonds to two hydrogens and forms the water of our earth and comets. In contrast to charge action, the predictions of gravity action do not really need the details of atoms and bonds and amplitude and phase as long as we know a object's density and mass. However, at larger and smaller scale, the natures of quantum amplitude and phase do indeed impact gravity bonds.
Gravity and quantum actions represent somewhat inconsistent theorets or realities for science, but somehow we know that there is a relationship. General relativity is basically the gravity action that is what holds us to the earth and holds the sun in the galaxy and all galaxies to the universe and is very intuitive and deterministic. Each effect of gravity has a cause and that cause is local to that effect. In contrast to gravity action, quantum action depends on both matter amplitude and phase and not just mass. An extra phase coherence between objects links not only local object actions, but also correlates nonlocal object actions as well.
One of the more notable aspects of relativity is the statement of equivalence of energy and mass, E = mc2, with the proportionality of the speed of light squared and indeed quantum action has adopted that same principle as well. Just this simple matter-energy equivalence (MEE) explains much about both gravity and quantum action since all motion increases the inertial mass for each object proportional to its velocity squared, which is the kinetic energy of motion. Somehow an object gains and loses extremely small amounts of matter simply by changing its velocity.
Another notable result of relativity is the fact that the speed of light for an object does not depend on object velocity, which is a direct result of the equivalence of mass and energy and further results in a dilations of space and time associated with any motion as velocity and acceleration. When it comes to explaining the anomalous precession of Mercury about the sun or the bending of starlight by sol, the proportionality of energy and matter explains about one-half of such observations and the dilation of space and time explains the other half.
While the mass-energy equivalence principle is completely consistent with the formulation of a quantum gravity in matter time, the distortion of a continuous space by velocity and acceleration represents a little bit of a problem for any discrete quantum gravity. This is because dilation of continuous space is a result of gravity and so a particle that carries gravity force would therefore dilate space and alter the particle, which further dilates space, and so on. With discrete matter and time delay, spatial dilation is the result of action in discrete matter and time delay and not a result of gravity per se. While the distortion of continuous space and time with motion is definitely a part of our reality, this distortion is where there is a strain between gravity and quantum actions.
The question comes down to whether or not there is a continuous deterministic and predictable path for an object through space time. In general relativity, gravity distorts space and time and that is what results in a continuous deterministic path as a straight line in continuous 4-D space time. However, it is possible with mass-energy equivalence to have the same dilation of time and along with discrete changes in inertial matter, have the same path emerge for that object. In this reinterpretation, spatial dilation then emerges from the action of discrete matter and time delay and the result is what we call motion. What we imagine as action in space is really first of all an action or change of discrete matter with time delays and then only secondarily do continuous motions and dilations of continuous space emerge.
With discrete matter and time delay, a continuous spatial dilation emerges from the gravity action of an object in discrete matter time and spatial dilation therefore does not therefore cause action or motion in space. With this approach, quantum gravity becomes a straightforward result of action in matter time. While charge force is the exchange of photon dipoles between electrons and nuclei, gravity force is exchange of complementary photon pairs as mono-quadrupoles between the neutral matter and the boson matter of the universe.
The stress-energy tensor of GR then more properly emerges from a mono-quadrupole time and is not an a priori axiom. In quantum gravity, it is the mono-quadrupole time operator and its tensors that provide a proper time for each action from the two time dipoles of the rest and moving frames. For most common actions, the quantum time quadrupole is largely identical to proper time. However, for certain very massive and very small objects, there is a quantum exchange that enhances the gravitational bond. Gravity objects bind to each other by means of exchange of time quadrupoles.
Quantum action is largely about the behavior of coherent microscopic matter and is much less intuitive than gravity action at all scales. Quantum action depends on matter or mass just like gravity but quantum action also depends on something called phase and coherence and charge amplitude, properties of matter that have no relevance in general relativity. The interference effects of light are due to light’s phase and amplitude and so light shows polarization and partial reflection as a result. Yet these coherent effects occur for all objects of matter, not just for light. Neutral matter can show polarization and neutral matter can show partial reflection as well.
The basic equation of motion for quantum action is the Schrödinger equation for discrete matter,
which is a proportionality between the amplitude and phase for a matter wave of the future,
and the amplitude and phase of a matter wave of the present,
This is what is called a differential equation in time and is an action equation that describes how a matter wave changes over time, both in mass and phase. In this equation, mR represents the photon exchange energy that binds an electron to a proton to make hydrogen and is the mass equivalent of the Rydberg energy. There is an infinity of excited states for hydrogen whose energies emerge as spectral lines that converge to a finite ionization energy, which is called the Rydberg energy.
The integral form of the Schrödinger equation for discrete matter is
and shows that matter waves are also proportional to their integration over time, which is their action over time. That proportionality is the ratio of a binding energy, mR, and Planck’s constant and of course a phase factor, -i, which means that the action of an object is somehow orthogonal to its matter in time.
There is an infinity of excited states for hydrogen whose energies as discrete spectral lines converge to a finite value that is the hydrogen ionization or Rydberg energy. There are two solutions to each Schrödinger equation; an inner charge solution involving the charged electron along with an outer gravity solution involving discrete aether. The inner solution has photon dipole exchange that binds electrons to the nuclei of atoms and the outer solution involves pairs of complementary emitted photons that bind neutral atoms to the outer boson aether of the universe.
Matter waves scale with the square root of mass in matter time while the more typical wavefunctions of quantum mechanics are just dimensionless phase as probability amplitude. This means that the integral of a matter wave over all time is an action that results in the measurable property that we call mass. Matter waves are the moments of matter that make up all objects and sensation is the exchange of the matter waves of our senses with the matter waves of an object being sensed. In the parlance of quantum action, a matter wave or wavefunction collapses as a product of each exchange between us and an object and that collapse is the sensation that we imagine as the mass or some other property of an object.
We might see light from an object, feel the object, hear it, smell it, or even taste it. What we sense of an object alone is not the matter wave itself, but the product of the object matter wave with our own sensory matter waves. Sensation is an exchange of both amplitude and phase with objects in a bonding action that we imagine as reality. The discrete exchange of matter actually bonds us to objects with a quantum action that necessarily occurs in discrete quantum steps with discrete quantum states. This bonding action involves our whole body and not just our sensory organs.
A journey from point A to point B involves a series of steps or quantum jumps as an object exchanges discrete aether with other objects in order to get around the universe, successively bonding and conflicting with the matter waves of objects in order to move. Matter waves show action under the influence of operators and those actions result in discrete changes in object matter over time.
Time delay waves also show action, but now as a function of a quasi-continuum of matter. A journey from matter state A to matter state B involves a series of quantum jumps as an object exchanges time delays with other objects. While objects exists with discrete time delays, time is a quasi-continuum that depends on the very large number of quantum jumps of matter particles.
A continuum force like gravity in general relativity does not show the discrete states of quantum gravity but rather shows continuous motion from point A to point B. Continuous motion in space is a very natural and intuitive concept that is not how objects move in discrete matter and time delay. In fact, motion in continuous space results in serious conundrums like the Zeno’s paradox of an infinity of points and quantum action of whole particles resolves Zeno’s paradox but at the expense of a different interpretation for continuous macroscopic gravity action in the universe.
Gravity in matter time is a quantum action that binds atom pairs to the boson aether of the universe, which is discrete gaechron. The complementary photon pairs emitted from the charge actions of electron bonds for two atoms are the light that objects emit from charge and are the gravity force bonds between atoms and molecules as well. Emitted light represents the complementary outer state for the inner binding states of each atom and molecule and emitted light is the exchange that binds the matter waves of atoms and molecules with each other as the matter waves of the universe. Because we see light, we imagine emitted photons on trajectories through the void of space. In fact, emitted photons represent complementary changes in matter states that we call charge and gravity action.
There is a photon dipole exchange that binds an electron to a proton to form a hydrogen atom and such a mass defect is the Rydberg energy for hydrogen as well as binding atoms to each other with further energies and further shared electrons. That same charge force defect represents an equivalent photon pair exchange with the boson aether of the universe that is the gravity force that binds the hydrogen atom to the universe. The dephasing of discrete aether results in what we call gravity force and by scaling discrete aether exchange by the ratio of electron mass to discrete aeither, discrete aether decay is then what we call charge force as well. The light that we see from the stars at night represents a discrete aether exchange that binds the electrons and protons as well as atoms into stars and stars into the galaxy as well as the galaxy into the very fabric of the cosmos.
Although science expects a new particle called a graviton to be the exchange particle of gravity force, with the scaling of photon pairs in discrete matter, there is no new gravity particle. Rather, it is the universal dephasing of discrete boson aether that determines both gravity and charge forces and the photon is the basic exchange particle for both gravity and charge forces. Whereas photon exchange between the electron and proton represents charge force, photon pairs exchange between the electron and discrete aether represents gravity force. Thus, the ratio of the gaechron particle of discrete aether to the electron mass represents the 1e39 scaling between gravity and charge force cross sections.
Quantum action is often called odd although quantum action has been extraordinarily successful for virtually all predictions of action. However, quantum predictions are always probabilistic and uncertain and sometimes matter waves show correlated and coherent effects that entangle different locations in space. Even for a highly local matter wave action there is still some quantum uncertainty, which bothers many people. Since quantum phase can persist between two objects across the universe, the observation of one object phase seems to determine the other object phase instantaneously. So when that quantum uncertainty involves locations across the universe, people get even more uncomfortable and bothered.
And yet quantum action does not violate any causal principles, rather quantum action simply refines those causal principles to include matter wave phase, amplitude, and coherence as well as mass as the product of two matter waves. The phase or coherence of a matter wave is a property of an object that we do not directly experience and so it is less intuitive than just the mass of an object, which is the square of its amplitude and does not carry phase information.
There are many different ways of describing the issues of quantum nonlocality and entanglement, but basically it comes down to a set of fundamental differences between quantum and gravity notions of space and motion. Quantum motion involves both the wave amplitude and phase of an object, while gravity motion involves only the mass of an object, i.e. the product of two matter waves, and so gravity action for mainstream science does not involve or entangle matter wave phase and amplitudes between objects at all.
Objects follow certain action principles where action is the integral or sum total of an object’s matter over time. Any macroscopic object is the product of a very large number of actions over time and objects continually gain and lose discrete aether as a part of their existence in the universe. Our intuition typically represents action as some kind of spatial displacement of an object, but it is the discrete aether exchanges of an object in time that better represent quantum action instead of motion. Discrete matter exchanges occur as quantum action and are the action we see as motion for an object in space.
Einstein first recognized that both event and action times are equivalent to spatial displacements and his general relativity shows how gravity action dilates matter, space, and time in a continuous four dimensional spacetime. Objects that gain inertial mass from their potential matter we interpret as a relative motion in space and that mass gain affects the space and action time between objects as well.
There are, however, different ways to interpret the dilation of matter, space, and time, with quantum gravity and therefore with a pure quantum action. Objects are in constant discrete aether exchange with other objects and it is from the gained inertial mass from other objects that object motion in space emerges. However, in general relativity the trajectory of an object follows a determinate geodesic path determined by gravity. If rather the distortion of space is a result of the gravity actions of that object, the same principles apply but now with a complementary quantum action for both gravity and charge.
An object like a rocket ship gains velocity and momentum by ejecting matter with the mass impulse of some kind of burning fuel and the action of the burning fuel propels the rocket in the opposite direction by its equivalent momentum. However, the relative motions of both ship and fuel actually are a result of much smaller gains in inertial masses, discrete aether, as equivalent kinetic energy by the matter-energy equivalence principle.
In other words, even while we imagine that the total rest mass of rocket and fuel does not change due to exchange of equivalent and opposite momentum, in fact, it is the the very small changes in the inertial masses of both rocket and ejected fuel that results in their respective motions. In a strict sense, then, what causes motion in space is the increase in inertial masses of two objects with equal and opposite momentum by exchange of discrete aether. Both objects increase in mass proportionately with their velocities squared relative to a rest frame and this matter increase comes from the potential matter as energy that was embedded into the chemical and gravity and nuclear bonds of the fuel.
The quantum action of discrete matter and time delay, which along with action, are the three axioms that close our universe. An action equation predicts the future of an object as discrete exchanges of matter with other objects over time. Quantum gravity predicts a large number of possible futures for macroscopic objects, but quantum action for macroscopic objects involves much greater scale than the local actions of gravity.
While there are a large number of possible futures for an object undergoing quantum action, including nonlocal futures, under gravity action of mainstream science, there is only one possible future for an object. This difference of action principles goes for the same object and the same reality and leads to interminable scientific and philosophical discourse about which action actually better describes an object’s possible future. Gravity and quantum actions are largely consistent with each other in common experience, but the two actions can represent irreconcilable futures for certain very large or very small objects. For example, until science reconciles gravity with quantum action, there is simply no way to definitively address the mystery of quantum gravity nonlocality.
The single future of gravity action in GR is consistent with a reality that is deterministic and local. Local effects always have local causes and this is the reality that we normally experience with gravity. Gravity is a continuous and infinitesimal force with a pesky microscopic singularity centered on each particle of matter and so there is no coherence for an object between two different locations in space. Since gravity action is the basis of our intuition for macroscopic objects in everyday life, we therefore have a very strong expectation that local actions only correlate to other local effects. We know that two ballistic particles from a source can arrive simultaneously at very different locations along separate paths A and B. However, a single matter wave can propagate along both paths A and B and yet only appear as a single particle at A or B. Note the appearance at A is coherent and correlated with no appearance at B, but neither causes one nor the other to occur.
Our intuition and experience, after all, are both largely based on an intuition of gravity action and so we greatly favor gravity action and mass as bases for predictions. Gravity action is usually very predictable since after all, what goes up, must come down. For gravity force, there is no allowance for the coherence of a single matter wave across the time delay of the universe.
Phase coherence can make it seem like the appearance of an object in one place causes its absence in another place, or that the absence of an object in another place causes the object appearance in the one place. Coherence has many effects, but quantum action does not violate any causal principle. Quantum action simply includes phase along with amplitude and a source and so better represents the actions of the entire universe, including actions at very small and very large scales.
A quantum universe consists of objects simultaneously located everywhere in the universe as amplitudes of matter waves. What provides us with the sensation of an object in one place and on one path is the time and phase that separates that object from other objects. It is an object’s incoherence with all of its other possibilities as a matter wave that we sense as a local object in time and space. While some of the many possible futures of an object from quantum action are nonlocal, the issues with quantum nonlocality and entanglement are fundamentally related to the many very different possible futures or phases for quantum action.
Quantum action is perfectly causal, but unfortunately quantum action is just sometimes not very intuitive since quantum can involve phase and coherence among objects in different places. We find it hard to accept how a perfectly real and observable ballistic object could ever be a matter wave that has both an amplitude and phase and magically disappears from one place due to destructive interference and then equally magically reappears in a completely different place due to constructive interference of those same amplitudes.
Worse yet, objects as matter waves can actually exist as a possibility in more than one Cartesian location until it finally interacts with another object at one place or the other, i.e., the matter wave collapses or dephases. And yet our quantum reality shows that matter has both amplitude and phase and therefore matter will show the many nonintuitive effects of coherency and interference.
It is particularly confusing when explanations of quantum action give macroscopic objects like people and cats the coherent attributes of microscopic matter. Coherent matter behaves so differently from incoherent matter that comparisons between coherent and incoherent macroscopic matter can result in very confusing allegories. Although it is possible for macroscopic matter to show coherence, the dephasing times for any macroscopic object are typically very short unless the objects are very massive neutron stars or black holes.
Until science unites charge and gravity into a common quantum action for all objects, there will continue to be confusion and strong differences of opinion about the nature of quantum action versus gravity action. For example, given similar charge and gravity forces for a coherent object, quantum action shows interference effects due to superposition but gravity only predicts ballistic collisions between objects. We have an intuition and life experience with macroscopic matter and gravity action that is very difficult to reconcile with the reality of microscopic matter and quantum action.
Light is a rather unusual form of matter and a photon of light on a trajectory in space is also the exchange particle that binds charged particles together. An exchange of a photon dipole between an electron and proton represents the dipolar charge force that stabilizes a hydrogen atom dipole, which is the basis of quantum electrodynamics and is well accepted by science.
That emitted photon pair is then the binding force for gravity, but this is not a common understanding. For one thing, charge is a dipole force while gravity is a mono-quadrupole force and so it is not clear how a dipolar photon with spin = 1 and plus/minus amplitudes can result in mono-quadrupole gravity with spin = {2, 0, -2} and quadrupolar amplitudes. The radiative cooling of hydrogen at the CMB created photon pairs that are a quadrupole attractive force called gravity. Since there is a pair of photons for every two neutral atoms to the universe, it is that mono-quadrupole pair that is responsible for gravity force.
In order for a neutral atom to form from charged electrons and protons, the neutral atom must emit or otherwise radiate its dipole charge binding energy as a complementary photon. That emitted photon is equal to the atom’s binding energy, which is the Rydberg energy for hydrogen, for example. There actually can be and are many photon emissions and absorptions of various energies and so this description just simplifies that complexity into one single event pair.
Each pair of neutral atoms emits a pair of photons at creation and those photon matter waves have complementary spin and polarization. While the dipole force between these particles and the photons progressively cancels out over time, the mono/quadrupole force persists as a tensor. Thus gravity force behaves as the quadrupole tensor of a coherent photon pair with spin = 0 and is a single particle with physical dimensions that literally define the age of the universe.
There is just one future for gravity action in general relativity and that one future is still consistent with our deterministic intuition. General relativity dilates or distorts continuous matter, space, and time with gravity action and there are many strange results of general relativity having to do with time dilation, simultaneity, and frames of reference. But while distant objects far away from a gravity action do not affect a local gravity action very much, the ratio of hydrogen’s time dipole to the time dipole of the universe is the scaling between gravity and charge forces.
In contrast to the determinism of gravity action in GR, there are actually a large number of possible futures for the same action as a quantum time quadrupole. The Rydberg photon emitted from hydrogen at creation is the exchange with the universe that binds each hydrogen atom to the boson matter of the universe. The time delay of that bond is coherent with that of the electron around the proton. The photon exchange between the universe and each pair of such atoms binds each atom to the universe matter and therefore to each other as well. It is then the shrinkage of the universe about those atom’s center of mass that represents what we interpret as the binding force of gravity between these two hydrogens.
Therefore the binding energy for hydrogen is the sum of the binding energy of the electron and proton along with a second term that is the binding energy of the atom with the discrete aether of the universe. In a strict sense, the binding matter of the electron and proton of an atom scales to the binding matter of that atom to the universe. Since they are equal and opposite in sign, their sum is zero and that result is an example of the Taylor-DeWitt equation. Even though their energies are equal and opposite, charge and gravity matter waves are quite different.
Whatever future actions occur for atoms in their many possible futures, their center’s of action and the gravity action that goes along with these centers persists. As matter evolves into heavier elements in star fusion engines, there are additional light and energy exchanges between those heavier elements and the universe and this additional action matter means that matter bonds in more complex ways to the universe just as matter bonds in more complex ways with different elements. The nature of gravity force actually increases over time just as the universe of matter shrinks or dephases and it is the overall shrinkage of the universe that is the origin of all force.
Quantum mechanics represents matter as the two dimensions of amplitude and phase. Thus a particle on a trajectory in space represents the matter of an object as a wave in a spectrum of matter waves across all space and time. A classic example of the wave nature of light is a series of strong and weak intensities, fringes, that is an interference pattern. An equally classic example of the particle nature of light as photons is the photoelectric effect where a photon of some minimum energy results in ejection of an electron from a metal surface.
The wave nature of light results in a pattern of light and dark fringes due to a coherent action from a single source between two or more possible paths for a source’s photons. This coherence can be the result of any number of means but the typical experiment is with two slits and the resultant diffraction of a light source.
However, each peak of intensity of the fringe pattern comprises a large number of measurable single photon events from the source. We want very badly that each of those photons journeyed ballistically along straight line paths from the source to the pattern and are disappointed to learn that there is not a single ballistic path for any single photon. Rather, each photon journeys as a matter wave with a wavelike trajectory on multiple paths to the interference pattern.
We are further disappointed to learn that this fringe pattern could persist over the dimensions of the universe. That is, the photon that we detect right here right now that come to us from a source may have also possibly been on a different path, somewhere very far away connecting some other object to the same source at the same time distance away. Since the photon wave journeyed across the universe somehow on its way to us right here we presume that its journey was ballistic as a particle.
When we record the photon right here, right now, we know for certain that the photon was here now and therefore not ever anywhere else. But the moment before we measured the photon here now, there had been a possibility that that same photon as a wave would have occurred somewhere else in the universe and therefore not here. Our intuition, though, tells us that photons that emanate from a source do so in a continuous ballistic manner and those photons are on continuous ballistic paths. The quantum truth is that it is photon matter waves that emanate from a source, and a photon matter wave is not yet a ballistic photon localized in space.
This seems like a funny result since when we see a photon, we know that the photon came from the image of a source that we imagine behind the photon and so we imagine a ballistic Cartesian journey in a more or less straight line from the source to our eye. If the source is incoherent, we imagine that it shines equivalently in all directions, but still imagine each light wave as a ballistic photon particle.
This is how we imagine objects in our Cartesian minds and a quantum action as a wave goes against the deterministic intuition of our ballistic gravity action. This does not mean that the photon did not exist before its wave dephased from the source, rather it means that the photon existed as a matter wave with both amplitude and phase and not as a ballistic particle.
What gives? Why can an object appear to be in more than one place as a matter wave prior to its interaction with another object at a different location? And what about the recoil momentum of the source? The ballistic action of a photon leaving a source means a recoil of equal and opposite momentum of the source since that is our experience with the ballistics of firing a bullet from a gun. A gun immediately recoils with the bullet momentum and does not wait until the bullet hits a target. In other words, the bullet does not remain coherent with the gun from which it discharged for very long and so the ballistic path of the bullet is a single path from the source.
However, a bullet is really not an apt analogy for a photon as a matter wave.
A different perspective provides different information about an object and while that information from a different perspective is in principle knowable, we cannot ever know about an object from every possible perspective. We can never observe all of the different perspectives of an object, but still that lack of knowledge does not represent anything that is fundamentally unknowable.
The path of a photon through space, however, can represent information that is fundamentally unknowable. A matter wave is necessarily a superposition of states and so we can only know the result from say two possibilities, A and B, by seeing the photon along path A. However, we can only then conclude that the photon’s amplitude wave included path A and we cannot know that the photon was ballistic on path A. The photon may or may not have existed as a matter wave superposition on A and B even though we can still use the photon location at A or B to know the direction of the source.
A single photon event does not tell us very much about a source and we typically depend on many more than thousands of photons to locate a source image with any precision.
A photon and its source can remain coherent with each other and that coherency will persist until some kind of dephasing action occurs with another object. An action with another object can dephase either the photon or the source and if that happens, the photon becomes ballistic. A subsequent action between an object and the photon, such as reflection, polarization, diffraction, refraction, etc., in effect creates a new source and a new phase relationship with the photon.
Actually we readily accept some degree of time and spatial uncertainty for events as long as the uncertainites are local to an object or action. But it really distresses our causal nature when there are large spatial gaps between an object’s possibilities, i.e., when the fringe patterns of quantum interference are really large. It simply is not possible to assign ballistic trajectories to photons with anything more than a probability.
We as quantum beings are in a quantum universe and only have relational experiences with objects by exchange of matter. Yet we imagine from those limited relations a ballistic Cartesian existence outside of our quantum mind with well-defined objects that we recognize from past experience. While a Cartesian object has a single ballistic trajectory in space and time, there are many possible futures for a relational object with which we are in direct contact and so we exchange our own matter waves with that of an object. Quantum events and actions reveal that there is a relational dimension in our quantum existence, even though we normally only imagine a Cartesian world of objects from our relational experiences with those objects.
It is from our relational experience with an object that we project its Cartesian or ballistic reality and so that is the dilemma of existence. It is only possible for us to experience an object through our relations with an object’s matter waves, but we then imagine a ballistic Cartesian existence in our mind that represents that object on a trajectory in the space outside of our mind.
We can prepare a coherent state that represents a particle’s matter wave amplitude at two places across the universe from each other with different phases. However, once the particle interacts with an object in one place or the other, that action can dephase or collapse that matter wave and therefore localize the matter wave to a particle in that one place.
The background matter of the universe, whatever you want to call it, is mostly what defines the universe and there is necessarily a coherence in time for any matter action. The phase of an action of a particle defines the location and direction of the particle journey and so a particle reality occurs in just one location. A particle amplitude, though, goes into and out of existence as its matter wave oscillates in time, in principle for the whole time of the universe. And a particle as a matter wave at a given moment also varies in the matter spectrum of the universe, in principle involving all of the matter in the universe.
One way to unite gravity and charge force is by the principles of discrete matter and time delay. In discrete matter time, light is the exchange particle that is responsible for both charge and gravity forces. Light binds charges together into an atom with a single photon and light also binds atoms to the universe with photon pairs as an exchange that binds atoms to each other with gravity.
In much of our experience, particles are well localized and that means particles are dephased and incoherent and ballistic in both the time and matter of the universe. In quantum parlance, this is what we know as our Cartesian reality, where particles and objects all seem to behave ballistically and independently. If a particle is on a trajectory through space, that trajectory represents a continuum of displacements along that trajectory.
However, a particle as a coherent matter wave manifests itself with additional possible futures in both proper and action times of the universe. While charge force is a local exchange on the dimensions of an atom, gravity force is the stabilization of that atom with a photon exchange that occurs on the dimensions of the universe. A coherent charge state binds each atom with a coherent gravity state due to an emitted photon wave, a wave that has 2π symmetry. Gravity force, though, is a result of two complementary photon waves, which are the exchanges of photons on the much larger time and matter dimensions of the universe and therefore have a 4π symmetry.
In effect, gravity force is therefore coherent with charge force and the action of light scales both gravity and charge forces by the matter and time dimensions of the universe. The photon, electron, and proton of each atom are in an action that binds the atom together while a complementary emitted photon wave exchanges with discrete aether and binds atoms to each other through the universe of matter.
Coherent gravitational states are therefore possible, but only with very simple gravitational matter. The boson accretion that we call a black hole, for example, is an example of highly coherent gravitational matter.
In principle, a gravity beamsplitter as shown in the figure at right prepares small objects like atoms or molecules into a superposition of coherent gravity states. Two identical massive bodies like the earth and moon orbit each other around a center of mass as in the figure. Two much smaller and identical objects, A and B, are in orbits that intersect at a gravitational Lagrange point between the earth and moon.
It appears that any gravitational Lagrange point can result in generating coherent gravity matter states for small objects on different orbits. Moreover, two stars that are equidistant from a third star result in a similar degeneracy that results in a coherent matter wave resonance that affects all three stars. Such matter waves perturb the underlying discrete boson aether of the universe and so matter waves affect both charge and gravity actions in complementary ways.
Coherent matter states in the universe have the same proper times relative to a source event, even though they are widely separated in action time. While a matter wave can remain coherent with a source for a very long time, that does not mean that a particle’s existence is uncertain; it does mean that a particle’s state or future is uncertain.
There is a conflict between the ballistic Cartesian existence for an object that we typically project with our mind and the relational existence that actually binds us to the matter waves of objects with matter exchange. These two dimensions of existence represent the dual aspects of our quantum reality as well as the duality of Descartes’ and other philosophies. In our ballistic Cartesian experience, existence has one meaning; an object that exists does so right here and right now as part of a proper existence.
In our relational experience, the matter waves we exchange with objects only represent possible futures. When we exchange discrete aether waves, we in essence share or exchange both matter and phase with objects in the wavelike realm of quantum exchange, and existence of quantum matter waves means something more than Cartesian ballistic existence. The relational aether wave exchange that binds us to an object means that the object becomes a part of us and we become a part of the object, even though we only sense some small fraction of that matter wave exchange.
When we exchange matter waves with an object, we call that experience, and there is always a period of both matter exchange as well as phase coherence between two objects. Any residual coherence between us and the object can result in a further relational component beyond a mass change and is a quantum entanglement that is beyond the typical ballistic Cartesian experience of action and reaction that we imagine. Note that Cartesian and relational dimensions of experience are really both part of a dual quantum reality.
We can and do imagine and know that there are other possible futures for any event that we experience. In particular, an action can dephase a photon from its source in which case the photon becomes ballistic. But as long as a photon remains coherent with its source, a matter wave binds not only the photon to the source, but to other objects as well at the same time distance from the source.
The photon could have a single ballistic future or it could have the many possible matter wave futures that entangle it with other objects. It is the other possible nonlocal and unknowable futures that somehow bother our causal ballistic natures. We want to place each object that we experience on a single ballistic Cartesian trajectory that is continuous from an origin to a destiny. Our intuition does not have much patience for the seemingly endless waves of quantum coherency that entangle local aether waves with other aether waves on other trajectories in the universe.
A photon that remains coherent with the action of its source has different possible futures from a photon that has dephased from its source. A photon that has dephased from its source has a single ballistic future much like any macroscopic object. All macroscopic objects, though, continually emit and absorb light and particles with incoherent phases and so a macroscopic objects’ decoherence times can be quite short. Simple quantum objects like photons, though, can retain coherence with their sources across the universe.
We are very comfortable with the causal notion of directional coherence and expect that a single point of an object emits photons in a single direction. When we see a photon from such a point on an object, we know the direction from which it came and our quantum logic does not change that truth. Where we have trouble is in imagining a single photon event that also has a transverse phase coherence as a matter wave that is perpendicular to the photon direction from a source. Transverse phase coherence means that a photon amplitude travels as a coherent wave in different possible directions at the same time even though the photon will only be absorbed by another matter wave in one particular location or phase.
There are actually two dimensions to time and our two dimensional time along with two dimensional matter represents a total of four dimensions in matter time. Given a π/2 or perpendicular phase relationship between matter and time, these four matter time dimensions reduce to three; matter, time, and phase. Time’s two dimensions include a proper time and an action time and matter’s two dimensions likewise include proper matter and action matter.
Our proper time is relative to the CMB in our 371 km/s velocity inertial frame. Action time is that associated with velocities of common experience, perhaps all of several meters per second and so action time represents displacements that are orders of magnitude less than the displacement of proper time.
Proper matter describes our galaxy as it moves at 550 km/s with respect to the CMB and rotates at 200 km/s, while our sun moves at 220 km/s, about 20 km/s faster than the galaxy rotates. These actions all make up the proper matter that results from our 371 km/s proper motion with respect to the CMB while our action matter is what occurs at lower scale.
Earth rotates about the sun at 30 km/s and spins about its axis at 0.47 km/s while we travel down the freeway at 0.027 km/s and walk around at about 0.001 km/s. Matter is likewise two dimensional with one dimension being the proper matter of our comoving frame of reference in the universe. The second matter dimension is the action matter of common experience that we call kinetic and potential energies.
Each atom of the universe forms as bound charges in a quantum exchange of light and other bosons that complements a gravitational quantum exchange orbit of that atom with the gaechron matter of the universe. We like to imagine a ballistic orbit for gaechron around an atom through space just as we like to imagine an electron in a ballistic orbit around a proton. But the atom-gaechron orbit is through time and quantum phase and not through space just as the electron orbit is through time and quantum phase as well.
While continuous space and motion are very useful ways to imagine the universe, continuous space and motion do not always represent either electron-proton states or atom-aether states very well. In addition to the time of this atom-aether orbit, there is a quantum phase angle between time and matter and for typical action, and it is from matter and time and from that phase angle that we project what we call space.
For any pair of atom-universe bonds, the shrinkage of the universe aether is the gravity force by which atoms appear to attract each other. In fact, the shrinkage of the universe is responsible for both charge and gravity force, just at very different scale. Eventually, these gravitational accretions of fermionic matter evolve from hydrogen into other elements in stars and that nucleosynthesis releases more action matter. A portion of the total energy and luminosity or action matter of each galaxy derives from nucleosynthesis and that action matter eventually ends up as large boson accretions known as black holes.
The formation of protons and electrons from the aether of the early universe results in a light that is the integrated CMB luminosity at 2.7 K, very much colder than the 70-80 F that people prefer. Once stars begin to fuse hydrogen into other elements, there is enough action matter to reionize hydrogen as well as to begin to fuse matter into excited states of the universe. And this reionization is an additional source of energy that then contributes to an overall universe energy balance.
Suppose you see an object along path A, if the object was at some incremental displacement as A – ds the previous moment, then the object was ballistic and its action was local. There are objects that exist in a superposition of quantum states, {A, B, C, …} and such an object can distribute around the universe according to some prior coherent quantum action. Note that a ballistic object actually also follows that same quantum logic, but a ballistic object has dephased and no longer coherent with its source.
The action of a beamsplitter creates coherency between the two paths A and B and some kind of magic occurs at the beamsplitter that makes 50% of photons disappear by destructive interference at both A and B. The ballistic Cartesian interpretation is that the beamsplitter reflects 50% of the photons as particles to A and transmits 50% to B and although this answer is technically wrong, it is good enough for many applications. If all you need is a one-way mirror or a grayed window or sunglasses to block sunlight, you really do not need to know much about single photon coherence. Thus our ballistic Cartesian reality does work fairly well for most predictions of action, even for those quantum actions with quantum devices like sunglasses.
We often lack knowledge about the appearance of an object even though that object exists as a single state and its appearance is in principle knowable. We can also lack knowledge about the state of an object, but if the object does exist in a single state, that single state is in principle knowable as well and not subject to quantum entanglement. When an object or image is a superposition of two coherent amplitudes, though, a single state is not yet realized and therefore not even knowable in principle. The object or image will not appear until we or other objects dephase the amplitudes from each other and a single state occurs.
Using logic to test quantumology tries to get a more graphic description of nonlocality. Remember, though, that quantum logic is already quite rigorous since it is based on math. It is rather the word descriptions of quantum logic that somehow fail to convince our common ballistic intuition of the principle of coherency. Our language is full of loopholes and conundrums and logic itself is often thwarted by the words that confuse meaning.
You say A is B or A is not B, but of course, we have a lot of examples of words that provide ambiguous meaning even to simple logic statements. Nothing is true, but if that is correct, it means that nothing is not true as well. The universe is finite, and if that is true, it would mean that the universe is not finite as well. Everything is finite and if that is true, nothing is finite since nothing is a part of everything. If there is anything that is really true, it is that nothing is really true. But if nothing is true, then anything is not true as well.
Is matter real? Is time real? Is action real? What is matter and why is matter the way that it is? What is time and why is time the way that it is? What is action and why is action the way that it is? Why does the world exist?
Thinking is being, but thinking is in our mind and being is not in our mind, and if that is all true, thinking is not being.
One very significant issue with quantum versus gravity actions is in the definition of consciousness. Unless there is a way to express conscious choice in the context of quantum action, there will always be those who believe that conscious choice is an illusion of the chaos of a ballistic determinism. Usually the reasoning goes that all action in the world is actually deterministic, but the world it is also just really, really very complicated and so we can never hope to know all of that complexity and chaos.
In a world of chaotic determinism, while it seems like we have free choice, this is just an illusion and the truth is that we just have more choices that we can ever possibly know about. However, philosophers who take this position then need to stipulate that there is still a need for personal responsibility and morality. In a deterministic universe, it is not clear that anyone is really responsible for their actions. After all, action and behavior are simply the some total of their genes and experiences up until that point.
All choice comes down to a binary decision between action and inaction at some threshold of a neural action potential and since quantum probability determines the neural action potential as it does all action of the universe, quantum probability also governs choice. Circumstances at the time of a choice predetermine most choices that we make and so in that sense, even binary decisions are not random. Each set of circumstances determines the threshold of action, but at the threshold of each action/inaction there is a distribution of quantum possibilities and a superposition of action and inaction states. In particular, there are a number of even odds choices that we make that may still substantially change the path of our lives. Every action, then, is a quantum action and involves some superposition of states for some period after the action.
An aware matter algorithm is part of our consciousness is therefore an important part of what makes us us. While most actions have fairly predictable results, there are no perfectly predictable results of action, especially for the results of human actions. Given the free choice that is quantum action, we do have a responsibility for choosing moral action since we freely choose our path in life as part of our purpose.
What we know of as right and wrong and just and unjust is part of the purpose with which we journey in life from our origin to a destiny. We are not programmed to be good or evil, but we are free to choose our destiny despite any experience of our past.
Some of what happened in the past involved objects that persisted as amplitudes and never collapsed into intensities. What this means is not that these objects do not exist as one phase, rather it means that the objects persist with more than one possibility as matter amplitudes that still project into more than one spatial location in the present moment.
Continuous space and motion are really just the results of discrete matter and action and so space exists only as a result of discrete matter, time delay, and the action of matter exchange. What this means is that while space is a convenient and necessary way to imagine discrete matter and action, the notions of continuous space and motion are limited. Although we find it useful to remember space as an object of the past that contains objects of action, the universe exists as an object of matter and its matter spectrum is what actually exists. While we get confused by objects that appear to simultaneously exist in different places in space, the state of the universe matter spectrum at any past time is knowable. |
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Stanislas Dehaene
Max Delbrück
Abraham de Moivre
Paul Dirac
Hans Driesch
John Eccles
Arthur Stanley Eddington
Gerald Edelman
Paul Ehrenfest
Manfred Eigen
Albert Einstein
George F. R. Ellis
Hugh Everett, III
Franz Exner
Richard Feynman
R. A. Fisher
David Foster
Joseph Fourier
Philipp Frank
Steven Frautschi
Edward Fredkin
Lila Gatlin
Michael Gazzaniga
Nicholas Georgescu-Roegen
GianCarlo Ghirardi
J. Willard Gibbs
Nicolas Gisin
Paul Glimcher
Thomas Gold
A. O. Gomes
Brian Goodwin
Joshua Greene
Dirk ter Haar
Jacques Hadamard
Mark Hadley
Patrick Haggard
J. B. S. Haldane
Stuart Hameroff
Augustin Hamon
Sam Harris
Ralph Hartley
Hyman Hartman
John-Dylan Haynes
Donald Hebb
Martin Heisenberg
Werner Heisenberg
John Herschel
Basil Hiley
Art Hobson
Jesper Hoffmeyer
Don Howard
William Stanley Jevons
Roman Jakobson
E. T. Jaynes
Pascual Jordan
Ruth E. Kastner
Stuart Kauffman
Martin J. Klein
William R. Klemm
Christof Koch
Simon Kochen
Hans Kornhuber
Stephen Kosslyn
Daniel Koshland
Ladislav Kovàč
Leopold Kronecker
Rolf Landauer
Alfred Landé
Pierre-Simon Laplace
David Layzer
Joseph LeDoux
Gilbert Lewis
Benjamin Libet
David Lindley
Seth Lloyd
Hendrik Lorentz
Josef Loschmidt
Ernst Mach
Donald MacKay
Henry Margenau
Owen Maroney
Humberto Maturana
James Clerk Maxwell
Ernst Mayr
John McCarthy
Warren McCulloch
N. David Mermin
George Miller
Stanley Miller
Ulrich Mohrhoff
Jacques Monod
Emmy Noether
Alexander Oparin
Abraham Pais
Howard Pattee
Wolfgang Pauli
Massimo Pauri
Roger Penrose
Steven Pinker
Colin Pittendrigh
Max Planck
Susan Pockett
Henri Poincaré
Daniel Pollen
Ilya Prigogine
Hans Primas
Henry Quastler
Adolphe Quételet
Lord Rayleigh
Jürgen Renn
Juan Roederer
Jerome Rothstein
David Ruelle
Tilman Sauer
Jürgen Schmidhuber
Erwin Schrödinger
Aaron Schurger
Sebastian Seung
Thomas Sebeok
Claude Shannon
David Shiang
Abner Shimony
Herbert Simon
Dean Keith Simonton
B. F. Skinner
Lee Smolin
Ray Solomonoff
Roger Sperry
John Stachel
Henry Stapp
Tom Stonier
Antoine Suarez
Leo Szilard
Max Tegmark
Libb Thims
William Thomson (Kelvin)
Giulio Tononi
Peter Tse
Francisco Varela
Vlatko Vedral
Mikhail Volkenstein
Heinz von Foerster
Richard von Mises
John von Neumann
Jakob von Uexküll
John B. Watson
Daniel Wegner
Steven Weinberg
Paul A. Weiss
Herman Weyl
John Wheeler
Wilhelm Wien
Norbert Wiener
Eugene Wigner
E. O. Wilson
Stephen Wolfram
H. Dieter Zeh
Ernst Zermelo
Wojciech Zurek
Konrad Zuse
Fritz Zwicky
Free Will
Mental Causation
James Symposium
Probability and Uncertainty
- the Quantum Mechanical View of Nature
Chapter 6 of The Character of Physical Law (annotated)
In the beginning of the history of experimental observation, or any other kind of observation on scientific things, it is intuition, which is really based on simple experience with everyday objects, that suggests reasonable explanations for things. But as we try to widen and make more consistent our description of what we see, as it gets wider and wider and we see a greater range of phenomena, the explanations become what we call laws instead of simple explanations. One odd characteristic is that they often seem to become more and more unreasonable and more and more intuitively far from obvious. To take an example, in the relativity theory the proposition is that if you think two things occur at the same time that is just your opinion, someone else could conclude that of those events one was before the other, and that therefore simultaneity is merely a subjective impression.
There is no reason why we should expect things to be otherwise, because the things of everyday experience involve large numbers of particles, or involve things moving very slowly, or involve other conditions that are special and represent in fact a limited experience with nature. It is a small section only of natural phenomena that one gets from direct experience. It is only through refined measurements and careful experimentation that we can have a wider vision. And then we see unexpected things: we see things that are far from what we would guess — far from what we could have imagined. Our imagination is stretched to the utmost, not, as in fiction, to imagine things which are not really there, but just to comprehend those things which are there. It is this kind of situation that I want to discuss.
Let us start with the history of light. At first light was assumed to behave very much like a shower of particles, of corpuscles, like rain, or like bullets from a gun. Then with further research it was clear that this was not right, that the light actually behaved like waves, like water waves for instance. Then in the twentieth century, on further research, it appeared again that light actually behaved in many ways like particles. In the photo-electric effect you could count these particles — they are called photons now. Electrons, when they were first discovered, behaved exactly like particles or bullets, very simply. Further research showed, from electron diffraction experiments for example, that they behaved like waves. As time went on there was a growing confusion about how these things really behaved — waves or particles, particles or waves? Everything looked like both.
In 1925 quantum mechanics discovered the equations that let us calculate physical properties to extraordinary accuracy, but the founders did not provide us with an intuitive picture of what is going on at the quantum level
This growing confusion was resolved in 1925 or 1926 with the advent of the correct equations for quantum mechanics. Now we know how the electrons and light behave. But what can I call it? If I say they behave like particles I give the wrong impression; also if I say they behave like waves. They behave in their own inimitable way, which technically could be called a quantum mechanical way. They behave in a way that is like nothing that you have ever seen before. Your experience with things that you have seen before is incomplete. The behaviour of things on a very tiny scale is simply different. An atom does not behave like a weight hanging on a spring and oscillating. Nor does it behave like a miniature representation of the solar system with little planets going around in orbits. Nor does it appear to be somewhat like a cloud or fog of some sort surrounding the nucleus. It behaves like nothing you have ever seen before.
How they behave, therefore, takes a great deal of imagination to appreciate, because we are going to describe something which is different from anything you know about. In that respect at least this is perhaps the most difficult lecture of the series, in the sense that it is abstract, in the sense that it is not close to experience. I cannot avoid that. Were I to give a series of lectures on the character of physical law, and to leave out from this series the description of the actual behaviour of particles on a small scale, I would certainly not be doing the job. This thing is completely characteristic of all of the particles of nature, and of a universal character, so if you want to hear about the character of physical law it is essential to talk about this particular aspect.
It will be difficult. But the difficulty really is psychological and exists in the perpetual torment that results from your saying to yourself, 'But how can it be like that?' which is a reflection of uncontrolled but utterly vain desire to see it in terms of something familiar. I will not describe it in terms of an analogy with something familiar; I will simply describe it. There was a time when the newspapers said that only twelve men understood the theory of relativity. I do not believe there ever was such a time. There might have been a time when only one man did, because he was the only guy who caught on, before he wrote his paper. But after people read the paper a lot of people understood the theory of relativity in some way or other, certainly more than twelve.
"nobody understands quantum mechanics"
Watch this famous Feynman quote
On the other hand, I think I can safely say that nobody understands quantum mechanics. So do not take the lecture too seriously, feeling that you really have to understand in terms of some model what I am going to describe, but just relax and enjoy it. I am going to tell you what nature behaves like. If you will simply admit that maybe she does behave like this, you will find her a delightful, entrancing thing. Do not keep saying to yourself, if you can possibly avoid it, 'But how can it be like that?' because you will get 'down the drain', into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.
So then, let me describe to you the behaviour of electrons or of photons in their typical quantum mechanical way. I am going to do this by a mixture of analogy and contrast.
If I made it pure analogy we would fail; it must be by analogy and contrast with things which are familiar to you. So I make it by analogy and contrast, first to the behaviour of particles, for which I will use bullets, and second to the behaviour of waves, for which I will use water waves. What I am going to do is to invent a particular experiment and first tell you what the situation would be in that experiment using particles, then what you would expect to happen if waves were involved, and finally what happens when there are actually electrons or photons in the system.
We will show that the (one) mystery of quantum mechanics is how mere "probabilities" (immaterial information) can causally control (statistically) the positions of material particles
I will take just this one experiment, which has been designed to contain all of the mystery of quantum mechanics, to put you up against the paradoxes and mysteries and peculiarities of nature one hundred per cent. Any other situation in quantum mechanics, it turns out, can always be explained by saying, 'You remember the case of the experiment with the two holes ? It's the same thing'. I am going to tell you about the experiment with the two holes. It does contain the general mystery; I am avoiding nothing; I am baring nature in her most elegant and difficult form.
We start with bullets (fig. 28). Suppose that we have some source of bullets, a machine gun, and in front of it a plate with a hole for the bullets to come through, and this plate is armour plate. A long distance away we have a second plate which has two holes in it — that is the famous two-hole business. I am going to talk a lot about these holes, so I will call them hole No. 1 and hole No. 2. You can imagine round holes in three dimensions — the drawing is just a cross section. A long distance away again we have another screen which is just a backstop of some sort on which we can put in various places a detector, which in the case of bullets is a box of sand into which the bullets will be caught so that we can count them. I am going to do experiments in which I count how many bullets come into this detector or box of sand when the box is in different positions, and to describe that I will measure the distance of the box from somewhere, and call that distance 'x', and I will talk about what happens when you change 'x', which means only that you move the detector box up and down. First I would like to make a few modifications from real bullets, in three idealizations. The first is that the machine gun is very shaky and wobbly and the bullets go in various directions, not just exactly straight on; they can ricochet off the edges of the holes in the armour plate. Secondly, we should say, although this is not very important, that the bullets have all the same speed or energy. The most important idealization in which this situation differs from real bullets is that I want these bullets to be absolutely indestructible, so that what we find in the box is not pieces of lead, of some bullet that broke in half, but we get the whole bullet. Imagine indestructible bullets, or hard bullets and soft armour plate.
The first thing that we shall notice about bullets is that the things that arrive come in lumps. When the energy comes it is all in one bulletful, one bang. If you count the bullets, there are one, two, three, four bullets; the things come in lumps. They are of equal size, you suppose, in this case, and when a thing comes into the box it is either all in the box or it is not in the box. Moreover, if I put up two boxes I never get two bullets in the boxes at the same time, presuming that the gun is not going off too fast and I have enough time between them to see. Slow down the gun so it goes off very slowly, then look very quickly in the two boxes, and you will never get two bullets at the same time in the two boxes, because a bullet is a single identifiable lump.
Now what I am going to measure is how many bullets arrive on the average over a period of time. Say we wait an hour, and we count how many bullets are in the sand and average that. We take the number of bullets that arrive per hour, and we can call that the probability of arrival, because it just gives the chance that a bullet going through a slit arrives in the particular box. The number of bullets that arrive in the box will vary of course as I vary 'x'. On the diagram I have plotted horizontally the number of bullets that I get if I hold the box in each position for an hour. I shall get a curve that will look more or less like curve N12 because when the box is behind one of the holes it gets a lot of bullets, and if it is a little out of line it does not get as many, they have to bounce off the edges of the holes, and eventually the curve disappears. The curve looks like curve N12, and the number that we get in an hour when both holes are open I will call N12 which merely means the number which arrive through hole No. 1 and hole No. 2.
I must remind you that the number that I have plotted does not come in lumps. It can have any size it wants. It can be two and a half bullets in an hour, in spite of the fact that bullets come in lumps. All I mean by two and a half bullets per hour is that if you run for ten hours you will get twenty-five bullets, so on the average it is two and a half bullets. I am sure you are all familiar with the joke about the average family in the United States seeming to have two and a half children. It does not mean that there is a half child in any family — children come in lumps. Nevertheless, when you take the average number per family it can be any number whatsoever, and in the same way this number N12, which is the number of bullets that arrive in the container per hour, on the average, need not be an integer. What we measure is the probability of arrival, which is a technical term for the average number that arrive in a given length of time.
Finally, if we analyse the curve N12 we can interpret it very nicely as the sum of two curves, one which will represent what I will call N1, the number which will come if hole No. 2 is closed by another piece of armour plate in front, and N2, the number which will come through hole No. 2 alone, if hole No. 1 is closed. We discover now a very important law, which is that the number that arrive with both holes open is the number that arrive by coming through hole No. 1, plus the number that come through hole No. 2. This proposition, the fact that all you have to do is to add these together, I call `no interference'.
N12 = N1 + N2 (no interference).
That is for bullets, and now we have done with bullets we begin again, this time with water waves (fig. 29).
The source is now a big mass of stuff which is being shaken up and down in the water. The armour plate becomes a long line of barges or jetties with a gap in the water in between. Perhaps it would be better to do it with ripples than with big ocean waves; it sounds more sensible. I wiggle my finger up and down to make waves, and I have a little piece of wood as a barrier with a hole for the ripples to come through. Then I have a second barrier with two holes, and finally a detector. What do I do with the detector? What the detector detects is how much the water is jiggling. For instance, I put a cork in the water and measure how it moves up and down, and what I am going to measure in fact is the energy of the agitation of the cork, which is exactly proportional to the energy carried by the waves. One other thing: the jiggling is made very regular and perfect so that the waves are all the same space from one another. One thing that is important for water waves is that the thing we are measuring can have any size at all. We are measuring the intensity of the waves, or the energy in the cork, and if the waves are very quiet, if my finger is only jiggling a little, then there will be very little motion of the cork. No matter how much it is, it is proportional. It can have any size; it does not come in lumps; it is not all there or nothing.
What we are going to measure is the intensity of the waves, or, to be precise, the energy generated by the waves at a point. What happens if we measure this intensity, which I will call 'I' to remind you that it is an intensity and not a number of particles of any kind? The curve I12, that is when both holes are open, is shown in the diagram (fig. 29). It is an interesting, complicated looking curve. If we put the detector in different places we get an intensity which varies very rapidly in a peculiar manner. You are probably familiar with the reason for that. The reason is that the ripples as they come have crests and troughs spreading from hole No. 1, and they have crests and troughs spreading from hole No. 2. If we are at a place which is exactly in between the two holes, so that the two waves arrive at the same time, the crests will come on top of each other and there will be plenty of jiggling. We have a lot of jiggling right in dead centre. On the other hand if I move the detector to some point further from hole No. 2 than hole No. 1, it takes a little longer for the waves to come from 2 than from 1, and when a crest is arriving from 1 the crest has not quite reached there yet from hole 2, in fact it is a trough from 2, so that the water tries to move up and it tries to move down, from the influences of the waves coming from the two holes, and the net result is that it does not move at all, or practically not at all. So we have a low bump at that place. Then if it moves still further over we get enough delay so that crests come together from both holes, although one crest is in fact a whole wave behind, and so you get a big one again, then a small one, a big one, a small one... depending upon the way the crests and troughs 'interfere'. The word interference again is used in science in a funny way. We can have what we call constructive interference, as when both waves interfere to make the intensity stronger. The important thing is that I12 is not the same as I1 plus I2, and we say it shows constructive and destructive interference. We can find out what I1 and I2 look like by closing hole No. 2 to find I1, and closing hole No. 1 to find I2. The intensity that we get if one hole is closed is simply the waves from one hole, with no interference, and the curves are shown in fig. 2. You will notice that I1 is the same as N1, and I2 the same as N2 and yet I12 is quite different from N12.
As a matter of fact, the mathematics of the curve I12 is rather interesting. What is true is that the height of the water, which we will call h, when both holes are open is equal to the height that you would get from No. 1 open, plus the height that you would get from No. 2 open. Thus, if it is a trough the height from No. 2 is negative and cancels out the height from No. 1. You can represent it by talking about the height of the water, but it turns out that the intensity in any case, for instance when both holes are open, is not the same as the height but is proportional to the square of the height. It is because of the fact that we are dealing with squares that we get these very interesting curves.
h12 = h1 + h2
I12 ≠ I1 + I2 (Interference)
I12 = (h12)2
I1 = (h1)2
I2 = (h2)2
That was water. Now we start again, this time with electrons (fig. 30).
The source is a filament, the barriers tungsten plates, these are holes in the tungsten plate, and for a detector we have any electrical system which is sufficiently sensitive to pick up the charge of an electron arriving with whatever energy the source has. If you would prefer it, we could use photons with black paper instead of the tungsten plate — in fact black paper is not very good because the fibres do not make sharp holes, so we would have to have something better — and for a detector a photo-multiplier capable of detecting the individual photons arriving. What happens with either case? I will discuss only the electron case, since the case with photons is exactly the same.
First, what we receive in the electrical detector, with a sufficiently powerful amplifier behind it, are clicks, lumps, absolute lumps. When the click comes it is a certain size, and the size is always the same. If you turn the source weaker the clicks come further apart, but it is the same sized click. If you turn it up they come so fast that they jam the amplifier. You have to turn it down enough so, that there are not too many clicks for the machinery that you are using for the detector. Next, if you put another detector in a different place and listen to both of them you will never get two clicks at the same time, at least if the source is weak enough and the precision with which you measure the time is good enough. If you cut down the intensity of the source so that the electrons come few and far between, they never give a click in both detectors at once. That means that the thing which is coming comes in lumps — it has a definite size, and it only comes to one place at a time. Right, so electrons, or photons, come in lumps. Therefore what we can do is the same thing as we did for bullets: we can measure the probability of arrival. What we do is hold the detector in various places — actually if we wanted to although it is expensive, we could put detectors all over at the same time and make the whole curve simultaneously — but we hold the detector in each place, say for an hour, and we measure at the end of the hour how many electrons came, and we average it. What do we get for the number of electrons that arrive? The same kind of N12 as with bullets? Figure 30 shows what we get for N12, that is what we get with both holes open. That is the phenomenon of nature, that she produces the curve which is the same as you would get for the interference of waves. She produces this curve for what? Not for the energy in a wave but for the probability of arrival of one of these lumps.
The mathematics is simple. You change I to N, so you have to change h to something else, which is new — it is not the height of anything — so we invent an 'a', which we call a probability amplitude, because we do not know what it means. In this case a1 is the probability amplitude to arrive from hole No. 1, and a2 the probability amplitude to arrive from hole No. 2. To get the total probability amplitude to arrive you add the two together and square it. This is a direct imitation of what happens with the waves, because we have to get the same curve out so we use the same mathematics. I should check on one point though, about the interference. I did not say what happens if we close one of the holes. Let us try to analyse this interesting curve by presuming that the electrons came through one hole or through the other. We close one hole, and measure how many come through hole No. 1, and we get the simple curve N1. Or we can close the other hole and measure how many come through hole No. 2, and we get the N2 curve. But these two added together do not give the same as N1 + N2; it does show interference. In fact the mathematics is given by this funny formula that the probability of arrival is the square of an amplitude which itself is the sum of two pieces, N12 = (a1+ a2)2. The question is how it can come about that when the electrons go through hole No. 1 they will be distributed one way, when they go through hole No. 2 they will be distributed another way, and yet when both holes are open you do not get the sum of the two. For instance, if I hold the detector at the point q with both holes open I get practically nothing, yet if I close one of the holes I get plenty, and if I close the other hole I get something. I leave both holes open and I get nothing; I let them come through both holes and they do not come any more. Or take the point at the centre; you can show that that is higher than the sum of the two single hole curves. You might think that if you were clever enough you could argue that they have some way of going around through the holes back and forth, or they do something complicated, or one splits in half and goes through the two holes, or something similar, in order to explain this phenomenon. Nobody, however, has succeeded in producing an explanation that is satisfactory, because the mathematics in the end are so very simple, the curve is so very simple (fig. 30).
Feynman only adds to the mystery by saying a particle is both a wave and a particle. The wave is just abstract information (a theoretical and statistical prediction) about the distribution of paths and positions of particles over large numbers of experiments.
There is no "it" in the wave.
I will summarize, then, by saying that electrons arrive in lumps, like particles, but the probability of arrival of these lumps is determined as the intensity of waves would be. It is in this sense that the electron behaves sometimes like a particle and sometimes like a wave. It behaves in two different ways at the same time (fig. 31).
That is all there is to say. I could give a mathematical description to figure out the probability of arrival of electrons under any circumstances, and that would in principle be the end of the lecture — except that there are a number of subtleties involved in the fact that nature works this way. There are a number of peculiar things, and I would like to discuss those peculiarities because they may not be self-evident at this point.
To discuss the subtleties, we begin by discussing a proposition which we would have thought reasonable, since these things are lumps. Since what comes is always one complete lump, in this case an electron, it is obviously reasonable to assume that either an electron goes through hole No. 1 or it goes through hole No. 2. It seems very obvious that it cannot do anything else if it is a lump. I am going to discuss this proposition, so I have to give it a name; I will call it `proposition A'.
Now we have already discussed a little what happens with proposition A. If it were true that an electron either goes through hole No. 1 or through hole No. 2, then the total number to arrive would have to be analysable as the sum of two contributions. The total number which arrive will be the number that come via hole 1, plus the number that come via hole 2. Since the resulting curve cannot be easily analysed as the sum of two pieces in such a nice manner, and since the experiments which determine how many would arrive if only one hole or the other were open do not give the result that the total is the sum of the two parts, it is obvious that we should conclude that this proposition is false.
We can show that the electron can go through just one hole and yet proposition A is not false, because Feynman has ignored something very important - the wave function that determines the probabilities of finding particles is different when both holes are open. The information that generates interference comes from the surrounding environment.
If it is not true that the electron either comes through hole No. 1 or hole No. 2, maybe it divides itself in half temporarily or something. So proposition A is false. That is logic. Unfortunately, or otherwise, we can test logic by experiment. We have to find out whether it is true or not that the electrons come through either hole 1 or hole 2, or maybe they go round through both holes and get temporarily split up, or something.
Why interference patterns show up when both holes are open, even when particles go through just one hole, though we cannot know which hole or we lose the interference
When there is only one slit open (here the left slit), the probabilities pattern has one large maximum (directly behind the slit) and small side fringes. If only the right slit were open, this pattern would move behind the right slit. And the combination of some experiments with the left open and others with the right open resembles Feynman's Figure 28 (no interference).
When both slits are open, the maximum is now at the center between the two slits, there are more interference fringes, and these probabilities apply whichever slit the particle enters. The solution of the Schrödinger equation depends on the boundary conditions - different when two holes are open. The "one mystery" remains - how these "probabilities" can exercise causal control (statistically) over material particles.
Feynman's path integral formulation of quantum mechanics suggests the answer. His "virtual particles" explore all space (the "sum over paths") as they determine the variational minimum for least action, thus the resulting probability amplitude wave function can "know" which holes are open.
All we have to do is watch them. And to watch them we need light. So we put behind the holes a source of very intense light. Light is scattered by electrons, bounced off them, so if the light is strong enough you can see electrons as they go by. We stand back, then, and we look to see whether when an electron is counted we see, or have seen the moment before the electron is counted, a flash behind hole 1 or a flash behind hole 2, or maybe a sort of half flash in each place at the same time. We are going to find out now how it goes, by looking. We turn on the light and look, and lo, we discover that every time there is a count at the detector we see either a flash behind No. 1 or a flash behind No. 2. What we find is that the electron comes one hundred per cent, complete, through hole 1 or through hole 2 — when we look. A paradox!
Let us squeeze nature into some kind of a difficulty here. I will show you what we are going to do. We are going to keep the light on and we are going to watch and count how many electrons come through. We will make two columns, one for hole No. 1 and one for hole No. 2, and as each electron arrives at the detector we will note in the appropriate column which hole it came through. What does the column for hole No. 1 look like when we add it all together for different positions of the detector? If I watch behind hole No. 1 what do I see? I see the curve N1 (fig. 30). That column is distributed just as we thought when we closed hole 2, much the same way whether we are looking or not. If we close hole 2 we get the same distribution in those that arrive as if we were watching hole No. 1; likewise the number that have arrived via hole No. 2 is also a simple curve N2. Now look, the total number which arrive has to be the total number. It has to be the sum of the number N1 plus the number N2; because each one that comes through has been checked off in either column 1 or column 2. The total number which arrive absolutely has to be the sum of these two. It has to be distributed as N1 + N2. But I said it was distributed as the curve N12. No, it is distributed as N1 + N2. It really is, of course; it has to be and it is. If we mark with a prime the results when a light is lit, then we find that N1', is practically the same as N1, without the light, and N2' is almost the same as N2. But the number N12', that we see when the light is on and both holes are open is equal to the number that we see through hole 1 plus the number that we see through hole 2. This is the result that we get when the light is on. We get a different answer whether I turn on the light or not. If I have the light turned on, the distribution is the curve N1 + N2. If I turn off the light, the distribution is N12. Turn on the light and it is N1 + N2 again. So you see, nature has squeezed out! We could say, then, that the light affects the result. If the light is on you get a different answer from that when the light is off. You can say too that light affects the behaviour of electrons. If you talk about the motion of the electrons through the experiment, which is a little inaccurate, you can say that the light affects the motion, so that those which might have arrived at the maximum have somehow been deviated or kicked by the light and arrive at the minimum instead, thus smoothing the curve to produce the simple N1 + N2 curve.
Electrons are very delicate. When you are looking at a baseball and you shine a light on it, it does not make any difference, the baseball still goes the same way. But when you shine a light on an electron it knocks him about a bit, and instead of doing one thing he does another, because you have turned the light on and it is so strong. Suppose we try turning it weaker and weaker, until it is very dim, then use very careful detectors that can see very dim lights, and look with a dim light. As the light gets dimmer and dimmer you cannot expect the very very weak light to affect the electron so completely as to change the pattern a hundred per cent from N12 to N1 + N2. As the light gets weaker and weaker, somehow it should get more and more like no light at all. How then does one curve turn into another? But of course light is not like a wave of water. Light also comes in particle-like character, called photons, and as you turn down the intensity of the light you are not turning down the effect, you are turning down the number of photons that are coming out of the source. As I turn down the light I am getting fewer and fewer photons. The least I can scatter from an electron is one photon, and if I have too few photons sometimes the electron will get through when there is no photon coming by, in which case I will not see it. A very weak light, therefore, does not mean a small disturbance, it just means a few photons. The result is that with a very weak light I have to invent a third column under the title 'didn't see'. When the light is very strong there are few in there, and when the light is very weak most of them end in there. So there are three columns, hole 1, hole 2, and didn't see. You can guess what happens. The ones I do see are distributed according to the curve N1 + N2. The ones I do not see are distributed as the curve N12. As I turn the light weaker and weaker I see less and less and a greater and greater fraction are not seen. The actual curve in any case is a mixture of the two curves, so as the light gets weaker it gets more and more like N12 in a continuous fashion.
I am not able here to discuss a large number of different ways which you might suggest to find out which hole the electron went through. It always turns out, however, that it is impossible to arrange the light in any way so that you can tell through which hole the thing is going without disturbing the pattern of arrival of the electrons, without destroying the interference. Not only light, but anything else — whatever you use, in principle it is impossible to do it. You can, if you want, invent many ways to tell which hole the electron is going through, and then it turns out that it is going through one or the other. But if you try to make that instrument so that at the same time it does not disturb the motion of the electron, then what happens is that you can no longer tell which hole it goes through and you get the complicated result again.
Heisenberg noticed, when he discovered the laws of quantum mechanics, that the new laws of nature that he had discovered could only be consistent if there were some basic limitation to our experimental abilities that had not been previously recognized. In other words, you cannot experimentally be as delicate as you wish. Heisenberg proposed his uncertainty principle which, stated in terms of our own experiment, is the following. (He stated it in another way, but they are exactly equivalent, and you can get from one to the other.) 'It is impossible to design any apparatus whatsoever to determine through which hole the electron passes that will not at the same time disturb the electron enough to destroy the interference pattern'. No one has found a way around this. I am sure you are itching with inventions of methods of detecting which hole the electron went through; but if each one of them is analysed carefully you will find out that there is something the matter with it. You may think you could do it without disturbing the electron, but it turns out there is always something the matter, and you can always account for the difference in the patterns by the disturbance of the instruments used to determine through which hole the electron comes.
This is a basic characteristic of nature, and tells us something about everything. If a new particle is found tomorrow, the kaon — actually the kaon has already been found, but to give it a name let us call it that — and I use kaons to interact with electrons to determine which hole the electron is going through, I already know, ahead of time — I hope — enough about the behaviour of a new particle to say that it cannot be of such a type that I could tell through which hole the electron would go without at the same time producing a disturbance on the electron and changing the pattern from interference to no interference. The uncertainty principle can therefore be used as a general principle to guess ahead at many of the characteristics of unknown objects. They are limited in their likely character.
The Copenhagen interpretation insists we know nothing about a path when not looking (measuring). Our measurements create the path, they said. But Einstein said that goes too far.
We can say, and know, things like the particle is conserving its mass, momentum, energy, spin, etc. along its path. It cannot divide in two and go through both holes!
Let us return to our proposition A — 'Electrons must go either through one hole or another'. Is it true or not? Physicists have a way of avoiding the pitfalls which exist. They make their rules of thinking as follows. If you have an apparatus which is capable of telling which hole the electron goes through (and you can have such an apparatus), then you can say that it either goes through one hole or the other. It does; it always is going through one hole or the other — when you look. But when you have no apparatus to determine through which hole the thing goes, then you cannot say that it either goes through one hole or the other. (You can always say it — provided you stop thinking immediately and make no deductions from it. Physicists prefer not to say it, rather than to stop thinking at the moment.) To conclude that it goes either through one hole or the other when you are not looking is to produce an error in prediction. That is the logical tight-rope on which we have to walk if we wish to interpret nature.
This proposition that I am talking about is general. It is not just for two holes, but is a general proposition which can be stated this way. The probability of any event in an ideal experiment — that is just an experiment in which everything is specified as well as it can be — is the square of something, which in this case I have called 'a', the probability amplitude. When an event can occur in several alternative ways, the probability amplitude, this 'a' number, is the sum of the 'a's for each of the various alternatives. If an experiment is performed which is capable of determining which alternative is taken, the probability of the event is changed; it is then the sum of the probabilities for each alternative. That is, you lose the interference.
The question now is, how does it really work ? What machinery is actually producing this thing? Nobody knows any machinery. Nobody can give you a deeper explanation of this phenomenon than I have given; that is, a description of it. They can give you a wider explanation, in the sense that they can do more examples to show how it is impossible to tell which hole the electron goes through and not at the same time destroy the interference pattern. They can give a wider class of experiments than just the two slit interference experiment. But that is just repeating the same thing to drive it in. It is not any deeper; it is only wider. The mathematics can be made more precise; you can mention that they are complex numbers instead of real numbers, and a couple of other minor points which have nothing to do with the main idea. But the deep mystery is what I have described, and no one can go any deeper today.
We can not know the path or the position of an individual particle. If we do measure it to learn its path, the experimental results change. There is no longer interference.
What we have calculated so far is the probability of arrival of an electron. The question is whether there is any way to determine where an individual electron really arrives? Of course we are not averse to using the theory of probability, that is calculating odds, when a situation is very complicated. We throw up a dice into the air, and with the various resistances, and atoms, and all the complicated business, we are perfectly willing to allow that we do not know enough details to make a definite prediction; so we calculate the odds that the thing will come this way or that way. But here what we are proposing, is it not, is that there is probability all the way back: that in the fundamental laws of physics there are odds.
Suppose that I have an experiment so set up that with the light out I get the interference situation. Then I say that even with the light on I cannot predict through which hole an electron will go. I only know that each time I look it will be one hole or the other; there is no way to predict ahead of time which hole it will be. The future, in other words, is unpredictable. It is impossible to predict in any way, from any information ahead of time, through which hole the thing will go, or which hole it will be seen behind. That means that physics has, in a way, given up, if the original purpose was — and everybody thought it was — to know enough so that given the circumstances we can predict what will happen next. Here are the circumstances: electron source, strong light source, tungsten plate with two holes: tell me, behind which hole shall I see the electron? One theory is that the reason you cannot tell through which hole you are going to see the electron is that it is determined by some very complicated things back at the source: it has internal wheels, internal gears, and so forth, to determine which hole it goes through; it is fifty-fifty probability, because, like a die, it is set at random; physics is incomplete, and if we get a complete enough physics then we shall be able to predict through which hole it goes. That is called the hidden variable theory. That theory cannot be true; it is not due to lack of detailed knowledge that we cannot make a prediction.
Feynman is wrong. We can say it went through either hole 1 or hole 2. We just cannot say which hole without destroying the interference pattern!
I said that if I did not turn on the light I should get the interference pattern. If I have a circumstance in which I get that interference pattern, then it is impossible to analyse it in terms of saying it goes through hole 1 or hole 2, because that interference curve is so simple, mathematically a completely different thing from the contribution of the two other curves as probabilities. If it had been possible for us to determine through which hole the electron was going to go if we had the light on, then whether we have the light on or off is nothing to do with it. Whatever gears there are at the source, which we observed, and which permitted us to tell whether the thing was going to go through 1 or 2, we could have observed with the light off, and therefore we could have told with the light off through which hole each electron was going to go. But if we could do this, the resulting curve would have to be represented as the sum of those that go through hole 1 and those that go through hole 2, and it is not. It must then be impossible to have any information ahead of time about which hole the electron is going to go through, whether the light is on or off, in any circumstance when the experiment is set up so that it can produce the interference with the light off. It is not our ignorance of the internal gears, of the internal complications, that makes nature appear to have probability in it. It seems to be somehow intrinsic. Someone has said it this way — 'Nature herself does not even know which way the electron is going to go'.
The same conditions do not always produce the same results. It is this quantum indeterminism that breaks the causal chain of physical determinism.
A philosopher once said 'It is necessary for the very existence of science that the same conditions always produce the same results'. Well, they do not. You set up the circumstances, with the same conditions every time, and you cannot predict behind which hole you will see the electron. Yet science goes on in spite of it — although the same conditions do not always produce the same results. That makes us unhappy, that we cannot predict exactly what will happen. Incidentally, you could think up a circumstance in which it is very dangerous and serious, and man must know, and still you cannot predict.
Sad these tragic examples scientists imagine, like Schrödinger's Cat
For instance we could cook up — we'd better not, but we could — a scheme by which we set up a photo cell, and one electron to go through, and if we see it behind hole No. 1 we set off the atomic bomb and start World War III, whereas if we see it behind hole No. 2 we make peace feelers and delay the war a little longer.
The future is unpredictable
Then the future of man would be dependent on something which no amount of science can predict. The future is unpredictable.
What is necessary 'for the very existence of science', and what the characteristics of nature are, are not to be determined by pompous preconditions, they are determined always by the material with which we work, by nature herself. We look, and we see what we find, and we cannot say ahead of time successfully what it is going to look like. The most reasonable possibilities often turn out not to be the situation. If science is to progress, what we need is the ability to experiment, honesty in reporting results — the results must be reported without somebody saying what they would like the results to have been — and finally — an important thing — the intelligence to interpret the results. An important point about this intelligence is that it should not be sure ahead of time what must be. It can be prejudiced and say 'That is very unlikely; I don't like that'. Prejudice is different from absolute certainty. I do not mean absolute prejudice — just bias. As long as you are only biased it does not make any difference, because if your bias is wrong a perpetual accumulation of experiments will perpetually annoy you until they cannot be disregarded any longer. They can only be disregarded if you are absolutely sure ahead of time of some precondition that science has to have. In fact it is necessary for the very existence of science that minds exist which do not allow that nature must satisfy some preconceived conditions, like those of our philosopher.
See also The Distinction of Past and Future
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Moving gapless indirect excitons in monolayer graphene
The existence of moving indirect excitons in monolayer graphene is theoretically evidenced in the envelope-function approximation. The excitons are formed from electrons and holes near the opposite conic points. The electron-hole binding is conditioned by the trigonal warping of the electron spectrum. It is stated that the exciton exists in some sectors of the exciton momentum space and has the strong trigonal warping of the spectrum.
An exciton is a usual two-particle state of semiconductors. The electron-hole attraction decreases the excitation energy compared to independent particles producing the bound states in the bandgap of a semiconductor. The absence of the gap makes this picture inapplicable to graphene, and the immobile exciton becomes impossible in a material with zero gap. However, at a finite total momentum, the gap opens that makes the binding of the moving pair allowable.
The purpose of the present paper is an envelope-approximation study of the possibility of the Wannier-Mott exciton formation near the conic point in a neutral graphene. In the present paper, we use the term ‘exciton’ in its direct meaning, unlike other papers where this term is referred to as many-body (‘excitonic’) effects[1, 2], exciton insulator with full spectrum reconstruction, or exciton-like singularities originating from saddle points (van Hove singularity) of the single-particle spectrum[3]. On the contrary, our goal is the pair bound states of electrons and holes. There is a widely accepted opinion that zero gap in graphene forbids the Mott exciton states (see, e.g.,[4]). This statement which is valid in the conic approximation proves to be incorrect beyond this approximation. Our aim is to demonstrate that the excitons exist if one takes the deviations from the conic spectrum into consideration.
We consider the envelope tight-binding Hamiltonian of monolayer graphene as follows:
H ex =( p e )+( p h )+V( r e r h ),
(p)= γ 0 1 + 4 cos a p x 2 cos 3 a p y 2 + 4 cos 2 a p x 2 ,
is the single-electron energy, a = 0.246 nm is the lattice constant, = 1 , V(r )= −e2/(χr) is the potential energy of the electron-hole interaction. The electron spectrum has conic points ν K,ν = ±1, K = (4Π/3a,0), where (p)≈s|pν K|, s = γ 0 a 3 / 2 is the electron velocity in the conic approximation.
The electron and hole momenta pe,hcan be expressed via pair q=p e + p h and relative p=p e p h momenta. The momenta pe,h can be situated near the same (qk 2K) or near the opposite conic points (q = 2K + k ,k 2K).
We assumed that graphene is embedded into the insulator with a relatively large dielectric constant χ so that the effective dimensionless constant of interaction g = e 2 / ( sχℏ ) 2 / χ 1 and the many-body complications are inessential. In the conic approximation, the classical electron and hole with the same direction of momentum have the same velocities s. The interaction changes their momenta, but not their velocities. The two-particle Hamiltonian contains no terms quadratic in the component of the relative momentum p along k. In a quantum language, such attraction does not result in binding. Thus, the problem of binding demands accounting for the corrections to the conic spectrum.
Two kinds of excitons are potentially allowed in graphene: a direct exciton with k 1/a(when the pair belongs to the same extremum) and an indirect exciton with q = 2K + k. Assuming pk (this results from the smallness of g), we get to the quadratic Hamiltonian
H ex =sk+ p 1 2 2 m 1 + p 2 2 2 m 2 e 2 χr ,
where the coordinate system with the basis vectors e1k/k and e2e1 is chosen, r = (x1,x2). In the conic approximation, we have m2 = k/s, m1 = . Thus, this approximation is not sufficient to find m1. Beyond the conic approximation (but near the conic point), we should expand the spectrum (2) with respect to k up to the square terms, which results in the trigonal spectrum warping. As a result, we have for the indirect exciton,
1 m 1 = ν sa 4 3 cos 3 ϕ k ,
where ϕ k is an angle between k and K.
The effective mass m1 m2is directly determined by the trigonal spectrum warping, and the large value of m1 follows from the warping smallness. The sign of m1is determined by ν cos3 ϕ k . If ν cos3 ϕ k > 0, electrons and holes tend to bind, or else to run away from each other. Thus, the binding of an indirect pair is permitted for ν cos3ϕ k >0. Apart from the conic point, this condition transforms to
( 1 + u + v ) < 0 ( 1 + u + v + ) < 0 ( 1 + u + v ) < 0 ( 1 + v + v + ) < 0 ( 1 + u + v + ) < 0 ( 1 + v + v + ) < 0 ,
where u = cos a k x , v ± = cos ( ( k x ± 3 k y ) a / 2 ) .
To find the indirect exciton states analytically, we solved the Schrödinger equation with the Hamiltonian (3) using the large ratio of effective masses. This parameter can be utilized by the adiabatic approximation similar with the problem of molecular levels. Coordinates 1 and 2 play a role of heavy ‘ion’ and ‘electron’ coordinates. At the first stage, the ion term in the Hamiltonian is omitted, and the Schrödinger equation is solved with respect to the electron wave function at a fixed ion position. The resulting electron terms then are used to solve the ion equation. This gives the approximate ground level of exciton ε(k)=skε ex (k), where the binding energy of the exciton is ε ex (k) = Π−1sk g2 log2(m1/m2) (the coefficient 1/Π here is found by a variational method).
A similar reasoning for the direct exciton gives negative mass m1=−32/(ks a2(7−cos6ϕ k )). As a result, the direct exciton kinetic energy of the electron-hole relative motion is not positively determined and that means the impossibility of binding of electrons with holes from the same cone point.
Results and discussion
Figure1 shows the domain of indirect exciton existence in the momentum space. This domain covers a small part of the Brillouin zone.
Figure 1
Relief of the single-electron spectrum. Domains where exciton states exist are bounded by a thick line.
The quantity ε ex (k) essentially depends on the momentum via the ratio of effective masses m1/m2. Within the accepted assumptions, ε ex is less than the energy of unbound pair sk. However, at a small-enough dielectric constant χ, the ratio of both quantities is not too small. Although we have no right to consider the problem with a large g in the two-particle approach, it is obvious that the increase of the parameter g can only result in the binding energy growth.
Besides, we have studied the problem of the exciton numerically in the same approximation and by means of a variational approach. Figure2 represents the dependence of the exciton binding energy on its momentum for χ=10. Figure3 shows the radial sections of the two-dimensional plot. The characteristic exciton binding energies have the order of 0.2 eV.
Figure 2
Relief map of indirect exciton ground-state binding energy. The map shows ε ex (in eV) as a function of the wave vector in units of reciprocal lattice constant. The exciton exists in the colored sectors.
Figure 3
Radial sections of Figure 2 at fixed angles in degrees (marked). Curves run up to the ends of exciton spectrum.
All results for embedded graphene are applicable to the free-suspended layer if the interaction constant g is replaced with a smaller quantity g ~ , which is renormalized by many-body effects. In this case, the exciton binding energy becomes essentially larger and comparable to kinetic energy sk.
We discuss the possibility of observation of the indirect excitons in graphene. As we saw, their energies are distributed between zero and some tenth of eV that smears up the exciton resonance. The large exciton momentum blocks both direct optical excitation and recombination. However, a slow recombination and an intervalley relaxation preserve the excitons (when generated someway) from recombination or the decay. On the other hand, the absence of a low-energy threshold results in the contribution of excitons in the specific heat and the thermal conductivity even at low temperature.
It is found that the exciton contribution to the specific heat at low temperatures in the Dirac point is proportional to (gT/s)2log2(aT/s)). It is essentially lower than the electron specific heat (T/s)2 and the acoustic phonon contribution (T/c)2, where c is the phonon velocity. Nevertheless, the exciton contribution to the electron-hole plasma specific heat is essential for experiments with hot electrons.
In conclusion, the exciton states in graphene are gapless and possess strong angular dependence. This behavior coheres with the angular selectivity of the electron-hole scattering rate[5]. In our opinion, it is reasonable to observe the excitons by means of high-resolution electron energy loss spectroscopy of the free-suspended graphene in vacuum. Such energy and angle-resolving measurements can reproduce the indirect exciton spectrum.
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This research has been supported in part by the grants of RFBR nos. 11-02-00730 and 11-02-12142.
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Correspondence to Mahmood Mahmoodian.
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All results were obtained by the collective work of MM and ME. Both authors read and approved the final manuscript.
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Mahmoodian, M., Entin, M. Moving gapless indirect excitons in monolayer graphene. Nanoscale Res Lett 7, 599 (2012).
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• Monolayer graphene
• Exciton
• Energy spectrum
• Optical absorption
• Specific heat
• 71.35.-y; 73.22.Lp; 73.22.Pr; 78.67.Wj; 65.80.Ck |
706d79f8c3906381 | Quantum Dynamics in Open Quantum-Classical Systems
Quantum Dynamics in Open Quantum-Classical Systems
Raymond Kapral Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, ON, M5S 3H6 Canada
Often quantum systems are not isolated and interactions with their environments must be taken into account. In such open quantum systems these environmental interactions can lead to decoherence and dissipation, which have a marked influence on the properties of the quantum system. In many instances the environment is well-approximated by classical mechanics, so that one is led to consider the dynamics of open quantum-classical systems. Since a full quantum dynamical description of large many-body systems is not currently feasible, mixed quantum-classical methods can provide accurate and computationally tractable ways to follow the dynamics of both the system and its environment. This review focuses on quantum-classical Liouville dynamics, one of several quantum-classical descriptions, and discusses the problems that arise when one attempts to combine quantum and classical mechanics, coherence and decoherence in quantum-classical systems, nonadiabatic dynamics, surface-hopping and mean-field theories and their relation to quantum-classical Liouville dynamics, as well as methods for simulating the dynamics.
I Introduction
It is difficult to follow the dynamics of quantum processes that occur in large and complex systems. Yet, often the quantum phenomena we wish to understand and study take place in such systems. Both naturally-occurring and man-made systems provide examples: excitation energy transfer from light harvesting antenna molecules to the reaction center in photosynthetic bacteria and plants, electronic energy transfer processes in the semiconductor materials used in solar cells, proton transfer processes in some molecular machines that operate in the cell, and the interactions of the qbits in quantum computers with their environment. Although the systems in which these processes take place are complicated and large, it is often the properties that pertain to only a small part of the entire system that are of interest; for example, the electrons or protons that are transferred in a biomolecule. This subsystem of the entire system can then be viewed as an open quantum system that interacts with its environment. In open quantum systems the dynamics of the environment can influence the behavior of the quantum subsystem in significant ways. In particular, it can lead to decoherence and dissipation which can play central roles in the rates and mechanisms of physical processes. This partition of the entire system into two parts has motivated the standard system-bath picture where one of these subsystems (henceforth called the subsystem) is of primary interest while the remainder of the degrees of freedom constitute the environment or bath.
Most system-bath descriptions focus on the dynamics of the subsystem density matrix, which is obtained by tracing over the bath degrees of freedom: . If such a program were carried out fully an exact equation of motion for could be derived and no information about the bath would be lost in this process. Of course, for problems of most interest where the bath is very large with complicated interactions this is not feasible and would defeat the motivation behind the system-bath partition. Consequently, the influence of the bath on the dynamics of the subsystem is embodied in dissipative and other coupling terms in the subsystem evolution equation.
There are many instances where more detailed information about the bath dynamics and its coupling to the subsystem is important. Examples are provided by proton and electron transfer processes in condensed phases or biological systems. As a specific example, consider the proton transfer reaction in a phenol-amine complex, , when the complex is solvated by polar molecules (see Fig. 1). The proton transfer events are strongly correlated with local solvent collective polarization changes. Subtle changes in the orientations of neighboring solvent molecules can induce proton transfers within the complex, which, in turn, influence the polarization of the solvent. The treatment of the dynamics in such cases requires detailed information about the dynamics of the environment and its coupling to the quantum process. It is difficult to capture such subtle effects without fully accounting for dynamics of individual solvent molecules in the bath.
Figure 1: Schematic representation showing the local solvent structure around the phenol-triethylamine complex. The covalent form of the phenol-amine complex (left) is unfavorably solvated by the polar solvent molecules. This induces a proton transfer giving rise to the ionic form (right). Subsequent solvent dynamics can lead to solvent polarization that favors the covalent form and the reverse proton transfer.
When investigating the dynamics of a quantum system it is often useful and appropriate to take into account the characteristics of the different degrees of freedom that comprise the system. The fact that electronic and nuclear motions occur on very different time scales, as a result of the disparity in their masses, forms the basis for the Born-Oppenheimer approximation where the nuclear-configuration-dependent electronic energy is used as the potential energy for the evolution of the nuclear degrees of freedom. This distinction between electronic and nuclear degrees of freedom is an example of the more general partition of a quantum system into subsystems with different characteristics.
Since the scale separation in the Born-Oppenheimer approximation is approximate, it can break down and its breakdown leads to coupling among many electronic energy surfaces. When this occurs, the evolution is no longer described by adiabatic dynamics on a single potential energy surface and nonadiabatic effects become important. Nonadiabatic dynamics plays an essential role in the description of many physical phenomena, such as photochemical processes where transitions among various electronic states occur as a result of avoided crossings of adiabatic states or conical intersections between potential energy surfaces.
In the examples presented above the molecules comprising the bath are often much more massive than those in the subsystem (). This fact motivates the construction of a quantum-classical description where the bath, in the absence of interactions with the quantum subsystem, is described by classical mechanics. Mixed quantum-classical methods provide a means to investigate quantum dynamics in large complex systems, since fully quantum treatments of the dynamics of such systems are not feasible. The study of such open quantum-classical systems is the main topic of this review. Since quantum and classical mechanics do not easily mix, one must consider the properties of schemes that combine these two types of mechanics. One such scheme, quantum-classical Liouville dynamics, will be discussed in detail and its features will be compared to other quantum-classical and full quantum methods.
Ii Open Quantum Systems
Since the quantum systems we study are rarely isolated and interact with the environments within which they reside, the investigation of the dynamics of such open quantum systems is a worthy endeavor. The full description of the time evolution of a composite quantum system comprising a subsystem and bath is given by the quantum Liouville equation,
where is the density matrix at time , is the total Hamiltonian, and the square brackets denote the commutator.
Introducing some of the notation that will be used in this paper, we denote by the coordinate operators for the subsystem degrees of freedom with mass , while the remaining bath degrees of freedom with mass have coordinate operators . (The formalism is easily generalized to situations where the masses and depend on the particle index.) The total Hamiltonian takes the form
where the momentum operators for the subsystem and bath are and , respectively. It is convenient to write the potential energy operator, as a sum of subsystem, bath and coupling contributions: . In this case the Hamiltonian operator can be written as a sum of contributions,
where is the quantum subsystem Hamiltonian, is the quantum bath Hamiltonian and is the coupling between these two subsystems.
Most often in considering the dynamics of such open quantum systems one traces over the bath since it is the dynamics of the subsystem that is of interest. As noted in the Introduction, a considerable research effort has been devoted to the construction of equations of motion for the reduced density matrix, . The Redfield equation Redfield (1965) describes the dynamics of a subsystem weakly coupled to a bath with suitably fast bath relaxation time scales, since a Born-Markov approximation is made in its derivation. In a basis of eigenstates of , , it has the form,
where the summation convention has been used. This convention will be used throughout the paper when confusion is unlikely. Here , while the second term on the right accounts for dissipative effects due to the bath. Remaining within the Born-Markov approximation, the general form of the equation of motion for a reduced density matrix that guarantees its positivity is given by the Lindblad equation Lindblad (1976),
where the are operators that account for interactions with the bath. In addition to these equations, a number of other expressions for the evolution of the reduced density matrix have been derived. These include various master equations and generalized quantum master equations. There is a large literature dealing with open quantum systems, which is described and surveyed in books on this topic. Davies (1976); Weiss (1999); Breuer and Petruccione (2006) In such reduced descriptions information about the bath is contained in parameters that enter in the operators that describe the coupling between the subsystem and bath. Also, quantum-classical versions of the Redfield Toutounji (2005) and Lindblad Toutounji and Kapral (2001) equations have been derived.
There are many applications, such as those mentioned in the Introduction, where a more detailed treatment of the bath dynamics and its interactions with the subsystem is required, even though one’s primary interest is in the dynamics of the subsystem. If, as we suppose here, the systems we wish to study are large and may involve complex molecular constituents, a full quantum mechanical treatment is beyond the scope of existing computational power and algorithms. Currently, the only viable way to simulate the dynamics of such systems is by using mixed quantum-classical schemes. Quantum-classical methods in a variety of forms and derived in a variety of ways have been used to simulate the dynamics. Herman (1994); Tully (1998); Ben-Nun and Martińez (1998); Kapral (2006); Tully (2012) Mean field and surface-hopping methods are widely employed and will be discussed in some detail below. Mixed quantum-classical dynamics Agostini et al. (2013) based the on the exact time-dependent potential energy surfaces derived from the exact decomposition of electronic and nuclear motions Abedi, Maitra, and Gross (2010) has been constructed. In addition, semiclassical path integral formulations of quantum mechanics Sun and Miller (1997); Sun, Wang, and Miller (1998); Makri and Thompson (1998); Miller (2009); Lambert and Makri (2012) and ring polymer dynamics methods Habershon et al. (2013) have been developed to approximate the dynamics of open quantum systems.
In the next section the specific version of mixed quantum-classical dynamics that is the subject of this review, quantum-classical Liouville dynamics, will be described. The passage from quantum to classical dynamics is itself a difficult problem with an extensive literature, and decoherence is often invoked to effect this passage. Joos et al. (2003); Zurek (1991) Considerations based on decoherence can also be used motivate the use of mixed quantum-classical descriptions. Shiokawa and Kapral (2002) Mean-field and surface-hopping methods suffer from difficulties related to the treatment of coherence and decoherence, and these methods will be discussed in the context of the quantum-classical Liouville equation, which is derived and discussed in the next two sections. Some applications of the theory to specific systems will be presented in order to test the accuracy of this equation description and the algorithms used to simulate its dynamics.
Iii Quantum-Classical Liouville Dynamics
The first step in constructing a quantum-classical Liouville description is to introduce a phase space representation of the bath degrees of freedom in preparation for the passage to the classical bath limit. This is conveniently done by introducing a partial Wigner transform Wigner (1932) over the bath degrees of freedom defined by
with an analogous expression for the partial Wigner transform of an operator in which the prefactor is absent. We let to simplify the notation. The quantum Liouville equation then takes the form,
To obtain this equation the formula for the Wigner transform of a product of operators Imre et al. (1967),
was used. Here the operator , where the arrows denote the directions in which the derivatives act, is the negative of the Poisson bracket operator,
The partial Wigner transform of the total Hamiltonian is,
We have dropped the subscript W on the potential energy operator to simplify the notation; when the argument contains the partial Wigner transform is implied.
Derivation of the QCLE
The quantum-classical Liouville equation (QCLE) can be derived by formally expanding the exponential operators on the right side of Eq. (III) to Aleksandrov (1981); Gerasimenko (1982) The truncation of the series expansion can be justified for systems where the masses of particles in the environment are much greater than those of the subsystem, Kapral and Ciccotti (1999) Scaling similar to that in the microscopic derivation of the Langevin equation for Brownian motion from the classical Liouville equation Mazur and Oppenheim (1970) can be used for this purpose, and we may write the equations in terms of the reduced bath momenta, where . In this variable the kinetic energies of the light and heavy particle systems are comparable so that is of order . To see this more explicitly we introduce scaled units where energy is expressed in the unit , time in and length in units of . Using these length and time units, the scaling factor for the momentum is . Thus, in terms of the scaled variables and we have
where the prime on indicates that it is expressed in the primed variables. Note that for a system characterized by a temperature the small parameter can be written as the ratio of the thermal de Broglie wavelengths and of the heavy bath and light subsystem particles, respectively, , and truncation of the dynamics to terms of effectively averages out the quantum bath oscillations on the longer quantum length scale of the light subsystem.
Inserting the expression for the exponential Poisson bracket operator, valid to , into the scaled version of Eq. (III) and returning to unscaled variables we obtain the quantum-classical Liouville equation Kapral and Ciccotti (1999),
Additional discussion of this equation can be found in the literature Kapral (2006); Kapral and Ciccotti (1999); Aleksandrov (1981); Gerasimenko (1982); Boucher and Traschen (1988); Zhang and Balescu (1988); Donoso and Martens (1998); Horenko et al. (2002); Shi and Geva (2004); Thorndyke and Micha (2005); Bousquet et al. (2011). Comparison of the second and third equalities in this equation defines the QCL operator , and given this definition the formal solution of the QCLE is
where we let here and in the following to simplify the notation. The QCLE (12) may also be written in the form Nielsen, Kapral, and Ciccotti (2001),
which resembles the quantum Liouville equation but the quantum Hamiltonian operator is replaced by the forward and backward operators,
This form of the evolution equation has been used to discuss the statistical mechanical properties of QCL dynamics Nielsen, Kapral, and Ciccotti (2001), and will be used later to derive approximate solutions to the QCLE.
In applications it is often more convenient to evolve an operator rather than the density matrix and we may easily write the evolution equations for operators. Starting from the Heisenberg equation of motion for an operator ,
one can carry out an analogous calculation to find the QCLE for the partial Wigner transform of this operator:
whose formal solution can be written as
QCLE from linearization
The QCLE, when expressed in the adiabatic or subsystem bases, has been derived from linearization of the path integral expression for the density matrix by Shi and Geva Shi and Geva (2004). It can also be derived in a basis-free form by linearization Bonella, Ciccotti, and Kapral. (2010) and it is instructive to sketch this derivation here to see how the QCLE can be obtained from a perspective that differs from that discussed in the previous subsection.
The time evolution of the quantum density operator from time to a short later time is given by
Writing the Hamiltonian in the form , for this short time interval, a Trotter factorization of the propagators can be made:
For simplicity, we have suppressed the dependence in but kept the dependence since it is required in the derivation. Working in the representation for the bath, inserting resolutions of the identity, and evaluating the contributions coming from the kinetic energy operators that appear in the resulting expression, we obtain
Next, we make the change of variables and , along with similar variable changes for the momenta and . In the new variables, the density matrix element is
We may now make use of the definition of the partial Wigner transform (see Eq. (6)) in the expression for the matrix element of the density operator in Eq. (22) to derive an equation of motion for . To do this we first expand the exponentials that depend on to first order in this parameter; e.g., . We may then use this expansion to compute the finite difference expression . Finally we multiply the equation by , integrate the result over and take the limit . The result of these operations is
where . This integro-differential equation describes the full quantum evolution of the density matrix element; however, it is not a closed equation for because of the dependence of on . If we make use of the expansion of this operator to linear order in , when performing the integrals in the right side of Eq. (23), we obtain the QCLE in Eq. (12). The linearization approximation can be justified for systems where Bonella, Ciccotti, and Kapral. (2010) The same scaled variables introduced above in the first derivation may also be used to re-express Eq. (23) in scaled form. In this scaled form one may show that the expansion in is equivalent to an expansion in the mass ratio parameter .
QCLE in a dissipative environment
At times it may be convenient to further partition the bath into two subsets of degrees of freedom, , where the variables are directly coupled to the quantum subsystem and the remainder of the (usually large number of) degrees of freedom denoted by only participate in the subsystem dynamics indirectly through their coupling to . In such a case we can project these degrees of freedom out of the QCLE to derive a dissipative evolution equation for the quantum subsystem and the directly coupled variables Kapral (2001). For example, such a description could be useful in studies of proton or electron transfer in biomolecules where remote portions of the biomolecule and solvent need not be treated in detail but, nevertheless, these remote degrees of freedom do provide a source of decoherence and dissipation on the relevant degrees of freedom.
For a system of this type the partially Wigner transformed total Hamiltonian of the system is,
The potential energy operator, , includes all of the coupling contributions discussed above, namely, the potential energy operator for the quantum subsystem and directly coupled degrees of freedom, the potential energy of the outer bath and the coupling between the two bath subsystems, .
An evolution equation for the reduced density matrix of the quantum subsystem and directly coupled degrees of freedom,
can be obtained by using projection operator methods. Nakajima (1958); Zwanzig (1961) The result of this calculation is a dissipative QCLE, which takes the form Kapral (2001),
where is the QCL operator introduced earlier but now only for the quantum subsystem and bath degrees of freedom. The effects of the less relevant bath degrees of freedom are accounted for by the mean force defined by , where the average is over a canonical equilibrium distribution involving the Hamiltonian . The Fokker-Planck-like operator in Eq. (26) depends on the fixed particle friction tensor, , defined by
where , and its time evolution is given by the classical dynamics of the degrees of freedom in the field of the fixed coordinates. The quantum-classical limit of the multi-state Fokker-Planck equation introduced by Tanimura and Mukamel Tanimura and Mukamel (1994) is similar to the dissipative QCLE (26) when expressed in the subsystem basis.
Iv Some properties of the QCLE
The QCLE specifies the time evolution the density matrix of the entire system comprising the subsystem and bath and conserves the energy of the system. If the coupling potential in the Hamiltonian is zero, the density matrix factors into a product of subsystem and bath density matrices, . In this limit the subsystem density matrix satisfies the quantum Liouville equation,
and bath phase space density satisfies the classical Liouville equation,
While the bath evolves by classical mechanics when it is not coupled to the quantum subsystem, its evolution is no longer classical when coupling is present. As we shall see in more detail below, not only does the bath serve to account for the effects of decoherence and dissipation in the subsystem, it is also responsible for the creation of coherence. Conversely, the subsystem can interact with the bath to modify its dynamics. This leads to a very complicated evolution, but one which incorporates many of the features that are essential for the description of physical systems.
Often, when considering the dynamics of a quantum system coupled to a bath, the bath is modeled by a collection of harmonic oscillators which are bilinearly coupled to the quantum subsystem. In this case we may write the coupling potential as . The partially Wigner transformed Hamiltonian then takes the form,
where is the harmonic oscillator bath Hamiltonian. When the Hamiltonian has this form one may show easily that . Consequently, when the exponential Poisson bracket operators in Eq. (III) are expanded in a power series, the series truncates at linear order and we obtain the QCLE in the form given in Eq. (14); thus, the QCLE is exact for general quantum subsystems which are bilinearly coupled to harmonic baths. For more general Hamiltonian operators the series does not truncate and QCL dynamics is an approximation to full quantum dynamics.
Quantum and classical mechanics do not like to mix. The coupling between the smooth classical phase space evolution of the bath and the quantum subsystem dynamics with quantum fluctuations on small scales presents challenges for any quantum-classical description. The QCLE, being an approximation to full quantum dynamics, is not without defects. One of its features that requires consideration is its lack of a Lie algebraic structure. The quantum commutator bracket and Poisson bracket for quantum and classical mechanics, respectively, are bilinear, skew symmetric, and satisfy the Jacobi identity, so that these brackets have Lie algebraic structures. The quantum-classical bracket, , which is the combination of the commutator and the Poisson bracket terms does not have such a Lie algebraic structure. While this bracket is bilinear and skew symmetric, it does not exactly satisfy the Jacobi identity. Instead, the Jacobi identity is satisfied only to order (or if scaled variables are considered): . The lack of a Lie algebraic structure, its implications for the dynamics, and the construction of the statistical mechanics of quantum-classical systems were discussed earlier Nielsen, Kapral, and Ciccotti (2001); Kapral and Ciccotti (2002) where full details may be found. For example, the standard linear response derivations of quantum transport properties have to be modified, and in quantum-classical dynamics the evolution of a product of operators is not the product of the evolved operators; this is true only to order . These feature are not unique to QCL dynamics and almost all mixed quantum-classical methods used in simulations suffer from such defects, although they are rarely discussed. Mixed quantum-classical dynamics and its algebraic structure continue to attract the attention of researchers. Salcedo (1996, 2012); Prezhdo and Kisil (1997); Sergi (2005); Prezhdo (2006); Salcedo (2007); F. Agostini and Ciccotti (2007); Hall and Reginatto (2005); Hall (2008)
One way to bypass some of the difficulties in the formulation of the statistical mechanics of quantum-classical systems that are associated with a lack of a Lie algebraic structure is to derive expressions for average values and transport property using full quantum statistical mechanics. Then, starting with these exact quantum expressions, one may approximate the quantum dynamics by quantum-classical dynamics. Sergi and Kapral (2004); Kim and Kapral (2005a, b); Hsieh and Kapral (2014) In this framework the expectation value of an observable is given by,
where is the partial Wigner transform of the initial quantum density operator and the evolution of is given by the QCLE. Similarly, the expressions for transport coefficients involve time integrals of correlation functions of the form,
where is the partial Wigner transform of the product of the quantum canonical density operator and the operator , and the time evolution of is again given by the QCLE. Such formulations preserve the full quantum equilibrium structure which, while difficult to compute, is computationally much more tractable than full quantum dynamics. Poulsen, Nyman, and Rossky (2003); Ananth and Miller (2010) The importance of quantum versus classical equilibrium sampling on reactive-flux correlation functions, whose time integrals are reaction rate coefficients, has been investigated in the context of quantum-classical Liouville dynamics. Kim and Kapral (2005b) In this review we shall focus on dynamics but, when applications are considered, the above equations that contain the quantum initial or equilibrium density matrices will be used.
V Surface hopping, coherence and decoherence
Surface-hopping methods are commonly used to simulate the nonadiabatic dynamics of quantum-classical systems. In such schemes the bath phase space variables follow Newtonian trajectories on single adiabatic surfaces. Nonadiabatic effects are taken into account by hops between different adiabatic surfaces that are governed by probabilistic rules.
One of the most widely used schemes is Tully’s fewest-switches surface hopping. Tully (1990, 1991, 1998) In this method one assumes that the electronic wave function depends on the time-dependent nuclear positions , whose evolution is governed by a stochastic algorithm. More specifically, choosing to work in a basis of the instantaneous adiabatic eigenfunctions of the Hamiltonian , , we may expand the wave function as . An expression for the time evolution of the subsystem density matrix can be obtained by substitution into the Schrödinger equation. The equations of motion for its matrix elements, are given by,
In this equation is the nonadiabatic coupling matrix element, . From this expression the rate of change of the population in state may be written as
where, for simplicity, we have suppressed the time dependence in the variables, and stands for the real part. This rate has contributions from transitions to and from all other states . Consider a single specific state . Then transitions into from and out of to will determine the rate of change of the population due to transitions involving this state. In fewest-switches surface hopping the transitions are dropped and the transition rate for , , is adjusted to give the correct weighting of populations:
This transition rate is used to construct surface-hopping trajectories that specify the evolution of the phase space variables as follows: When the system is in state , the coordinates evolve by Newtonian trajectories on the adiabatic surface. Transitions to other states occur with probabilities per unit time, . Since the rates may take negative values, the Heaviside function sets the probability zero for negative values of the rate. If the transition to state occurs, the momentum of the system is adjusted to conserve energy and the system then propagates on the adiabatic surface. The momentum adjustment is taken to occur along the direction of the nonadiabatic coupling vector and is given by , with
The form that the stochastic evolution takes can be seen from an examination of Fig. 2, which schematically shows the evolution of a wave packet that starts on the upper adiabatic surface of a two level system with a simple avoided crossing. (This is Tully’s simple avoided crossing model. Tully (1990)) When the system enters the region of strong nonadiabatic coupling near the avoided crossing, nonadiabatic transitions to the lower state are likely, a surface hop occurs and the system then continues to evolve on the lower surface after momentum adjustment.
Figure 2: Schematic representation of the evolution of a wave packet in a two-level system with a simple avoided crossing. The diabatic (crossing curves) and adiabatic (avoided crossing curves) are shown. Following the nonadiabatic transition from the upper to lower adiabatic surfaces, the system continues to evolve on the lower surface until the next nonadiabatic transition.
For upward transitions it may happen that there is insufficient energy in the environment to insure energy conservation. In this case the transition rule needs to be modified, usually by setting the transition probability to zero. This scheme is very easy to simulate and captures much of the essential physics of the nonadiabatic dynamics.
Fewest-switches surface hopping does suffer from some defects associated with the fact that decoherence is not properly treated. The transition probability depends on the off-diagonal elements of the density matrix but no mechanism for their decay is included in the model. As a result, the fewest-switches surface hopping model overestimates coherence effects and retains memory which can influence the probabilities of subsequent hops. Several methods have been proposed to incorporate the effects of decoherence in mixed quantum-classical theories and, in particular, in surface-hopping schemes. Neria and Nitzan (1993); Hammes-Schiffer and Tully (1994); Bittner and Rossky (1995); Bittner, Schwartz, and Rossky (1997); Schwartz et al. (1996); Bedard-Hearn, Larsen, and Schwartz (2005); Subotnik and Shenvi (2011); Shenvi, Subotnik, and Yang (2011); Landry, J.Falk, and Subotnik (2013); Subotnik, Ouyang, and Landry (2013); Subotnik (2011); Jaeger, Fischer, and Prezhdo (2012) In many of these methods a term of the form, , is appended to the equation of motion for the off-diagonal elements of the subsystem density matrix to account for the decay of coherence. The decoherence rate is estimated using perturbation theory or from physical considerations involving the overlap of nuclear wave functions. In the remainder of this section we discuss how the QCLE accounts for decoherence and comment on its links to surface-hopping methods.
QCL dynamics in the adiabatic basis and decoherence
Since surface-hopping methods are often formulated in the adiabatic basis, it is instructive to discuss the dynamical picture that emerges when the QCLE is expressed in this basis. Adopting an Eulerian description, the adiabatic energies, , and the adiabatic states, , depend parametrically on the coordinates of the bath. We may then take matrix elements of Eq. (12),
to find an evolution equation for the density matrix elements, . Evaluation of the matrix elements on the right side of this equation yields an expression for the QCL superoperator Kapral and Ciccotti (1999),
Here the frequency (now in the adiabatic basis), and is the classical Liouville operator
and involves the Hellmann-Feynman forces, . The superoperator, , whose matrix elements are
couples the dynamics on the individual and mean adiabatic surfaces so that the evolution is no longer described by Newtonian dynamics.
The resulting QCLE in the adiabatic representation reads,
To simplify we shall often use a formal notation and write Eq. (41) as
where and (without “hats”) are understood to be a matrix and superoperator, respectively, in the adiabatic basis.
Insight into the nature of QCL dynamics can be obtained as follows. If the operator is dropped the resulting equation of motion for the diagonal elements of the density matrix is
which implies that the phase space density is constant along trajectories on the adiabatic surface,
with the notation . The off-diagonal density matrix elements satisfy
whose solution is
where and the evolution of the phase space coordinates of the bath is given by
The off-diagonal elements accumulate a phase in the course of their evolution on the mean of the two and adiabatic surfaces.
The momentum derivative terms in are responsible for the energy transfers that occur to and from the bath when the subsystem density matrix changes its quantum state. Consequently the subsystem and bath interact with each other and the dynamics of both the subsystem and bath are modified in the course of the evolution. Further, we can see from the structure of the QCLE that there are continuous changes to the subsystem quantum state and bath momenta during the evolution, as opposed to the jumps that appear in surface-hopping schemes. Nonetheless, links to surface-hopping methods can be made.
Subotnik, Ouyang and Landry Subotnik, Ouyang, and Landry (2013) established a connection between fewest-switches surface-hopping and the QCLE. They investigated what must be done to the equations describing fewest-switches surface hopping in order to obtain the QCL dynamics. Since there are continuous bath momentum changes in QCL dynamics and discontinuous changes in fewest-switches surface hopping, there are limitations on the nuclear momenta. An important element in their analysis is the fact that terms of the form, , that account for decoherence must be added to the fewest-switches approach. The specific form of the decoherence rate in their analysis is
The superscript indicates that evolution is on the adiabatic surface and all quantities on the right are taken to evolve on this surface. An analogous expression can be written for .
Recall that surface-hopping schemes assume that the dynamics occurs on single adiabatic surfaces between hops. Given this fact, we can understand the need for such a term by viewing QCL dynamics in a frame of reference corresponding to motion along single adiabatic surfaces. To see this consider the equation of motion for an off-diagonal element of the density matrix as given by the QCLE. From Eqs. (38)-(41) we have
Defining the material derivative for the flow on the adiabatic surface as
we obtain
We see that the second term on the right side of this equation is just the decoherence factor that appears in Eq. (49). The fact that decoherence depends on the difference between the forces is a common factor in many of the models for decoherence mentioned above. The decoherence contribution is difficult to compute in its current form because of the bath momentum derivative and it is usually approximated in applications. Subotnik, Ouyang, and Landry (2013)
Surface-hopping solution of the QCLE
As discussed above, the dynamics prescribed by the QCLE is not in the form of surface hopping since quantum state and bath momentum changes as embodied in the superoperator occur continuously throughout the evolution. The effects of can be seen by considering the formal solution of Eq. (41), |
98e8a16094e90af5 | What does Quantum Mechanics Mean?
Patrice Ayme gave a long comment to my previous post that effectively asked me to explain in some detail the significance of some of my comments on my conference talk involving quantum mechanics. But before that, I should explain why there is even a problem, and I apologise if the following potted history seems a little turgid. Unfortuately, the background situation is important.
First, we are familiar with classical mechanics, where, given all necessary conditions, exact values of the position and momentum of something can be calculated for any future time, and thanks to Newtom and Leibniz, we do this through differential equations involving familiar concepts such as force, time, position, etc. Thus suppose we shot an arrow into the air and ignored friction and we wanted to know where it was, when. Velocity is the differential of position with respect to time, so we take the velocity and integrate it. However, to get an answer, because there are two degrees of freedom (assuming we know which direction it was shot) we get two constants to the two integrations. In classical mechanics these are easily assigned: the horizontal constant depends on where it was fired from, and the other constant comes from the angle of elevation.
Classical mechanics reached a mathematical peak through Lagrange and Hamilton. Lagrange introduced a term that is usually the difference between the potential and kinetic energy, and thus converted the problem from forces to one of energy. Hamilton and Jacobi converted the problem to one involving action, which is the time integral of the Lagrangian. The significance of this is that in one sense action summarises all that is involved in our particle going from A to B. All of these variations are equivalent, and merely reflect alternative ways of going about the problem, however the Hamilton Jacobi equation is of special significance because it can be mathematically transformed into a mathematical wave expression. When Hamilton did this, there were undoubtedly a lot of yawns. Only an abstract mathematician would want to represent a cannonball as a wave.
So what is a wave? While energy can be transmitted by particles moving (like a cannon ball) waves transmit energy without moving matter, apart from a small local oscillation. Thus if you place a cork on the sea far from land, the cork basically goes around in a circle, but on average stays in the same place. If there is an ocean current, that will be superimposed on the circular motion without affecting it. The wave has two terms required to describe it: an amplitude (how big is the oscillation?) and a phase (where on the circle is it?).
Then at the end of the 19th century, suddenly classical mechanics gave wrong answers for what was occurring at the atomic level. As a hot body cools, it should give radiation from all possible oscillators and it does not. To explain this, Planck assumed radiation was given off in discrete packets, and introduced the quantum of action h. Einstein, recognizing the Principle of Microscopic Reversibility should apply, argued that light should be absorbed in discrete packages as well, which solved the problem of the photoelectric effect. A big problem arose with atoms, which have positively charged nuclei and electrons moving around it. To move, electrons must accelerate, and hence should radiate energy and spiral into the nucleus. They don’t. Bohr “solved” this problem with the ad hoc assumption that angular momentum was quantised, nevertheless his circular orbits (like planetary orbits) are wrong. For example, if they occurred, hydrogen would be a powerful magnet and it isn’t. Oops. Undeterred, Sommerfeld recognised that angular momentum is dimensionally equivalent to action, and he explained the theory in terms of action integrals. So near, but so far.
The next step involved the French physicist de Broglie. With a little algebra and a bit more inspiration, he represented the motion in terms of momentum and a wavelength, linked by the quantum of action. At this point, it was noted that if you fired very few electrons through two slits at an appropriate distance apart and let them travel to a screen, each electron was registered as a point, but if you kept going, the points started to form a diffraction pattern, the characteristic of waves. The way to solve this was if you take Hamilton’s wave approach, do a couple of pages of algebra and quantise the period by making the phase complex and proportional to the action divided by (to be dimensionally correct bcause the phase must be a number), you arrive at the Schrödinger equation, which is a partial differential equation, and thus is fiendishly difficult to solve. About the same time, Heisenberg introduced what we call the Uncertainty Principle, which usually states that you cannot know the product of the position and the momentum to better than h/2π. Mathematicians then formulated the Schrödinger equation into what we call the state vector formalism, in part to ensure that there are no cunning tricks to get around the Uncertainty Principle.
The Schrödinger equation expresses the energy in terms of a wave function ψ. That immediately raised the question, what does ψ mean? The square of a wave amplitude usually indicats the energy transmitted by the wave. Because ψ is complex, Born interpreted ψ.ψ* as indicating the probability that you would find the particle at the nominated point. The state vector formalism then proposed that ψ.ψ* indicates the probability that a state will have probabilities of certain properties at that point. There was an immediate problem that no experiment could detect the wave. Either there is a wave or there is not. De Broglie and Bohm assumed there was and developed what we call the pilot wave theory, but almost all physicists assume, because you cannot detect it, there is no actual wave.
What do we know happens? First, the particle is always detected as a point, and it is the sum of the points that gives the diffraction pattern characteristic of waves. You never see half a particle. This becomes significant because you can get this diffraction pattern using molecules made from 60 carbon atoms. In the two-slit experiment, what are called weak measurements have shown that the particle always goes through only one slit, and not only that, they do so with exactly the pattern predicted by David Bohm. That triumph appears to be ignored. Another odd feature is that while momentum and energy are part of uncertainty relationships, unlike random variation in something like Brownian motion, the uncertainty never grows
Now for the problems. The state vector formalism considers ψ to represent states. Further, because waves add linearly, the state may be a linear superposition of possibilities. If this merely meant that the probabilities merely represented what you do not know, then there would be no problem, but instead there is a near mystical assertion that all probabilities are present until the subject is observed, at which point the state collapses to what you see. Schrödinger could not tolerate this, not the least because the derivation of his equation is incompatible with this interpretation, and he presented his famous cat paradox, in which a cat is neither dead nor alive but in some sort of quantum superposition until observed. The result was the opposite of what he expected: this ridiculous outcome was asserted to be true, and we have the peculiar logic applied in that you cannot prove it is not true (because the state collapses if you try to observe the cat). Equally, you cannot prove it is true, but that does not deter the mystics. However, there is worse. Recall I noted when we integrate we have to assign necessary constants. When all positions are uncertain, and when we are merely dealing with probabilities in superposition, how do you do this? As John Pople stated in his Nobel lecture, for the chemical bonds of hydrocarbons, he assigned values to the constants by validating them with over two hundred reference compounds. But suppose there is something fundamentally wrong? You can always get the right answer if you have enough assignable constants.The same logic applies to the two-slit experiment. Because the particle could go through either slit and the wave must go through both to get the diffraction pattern, when you assume there is no wave it is argued that the particle goes through both slits as a superposition of the possibilities. This is asserted even though it has clearly been demonstrated that it does not. There is another problem. The assertion that the wave function collapses on observation, and all other probabilities are lost actually lies outside the theory. How does that actually happen? That is called the measurement problem, and as far as I am aware, nobody has an answer, although the obvious answer, the probabilities merely reflected possibilities and the system was always just one but we did not know it is always rejected. Confused? You should be. Next week I shall get around to some from my conference talk that caused stunned concern with the audience. |
03c8f04b6d57df2f | A Breather Solution in the Causal Interpretation of Quantum Mechanics
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A breather solution appears often in nonlinear wave mechanics, in which a nonlinear wave has energy concentrated in a localized oscillatory manner. This Demonstration studies a breather solution with a hyperbolic secant envelope of the focusing nonlinear Schrödinger (NLS) equation , with and so on, also known as the Gross–Pitaevskii equation in the causal interpretation, developed by Louis de Broglie and David Bohm.
The NLS equation could be interpreted as the Schrödinger equation (SE) with the nonlinear potential term with , although for most situations it has no relationship with the quantum Schrödinger equation other than in name. In the studied breather, there are large density amplitudes at certain times, which could be interpreted as rogue waves.
The graphic on the left shows the density (blue), the quantum potential (red), and the velocity (green). On the middle and on the right, you can see the density and the trajectories in space and the quantum potential and the trajectories in space. The velocity and the quantum potential on the left side are scaled to fit.
Contributed by: Klaus von Bloh (July 2014)
Open content licensed under CC BY-NC-SA
One of the exact breather solutions for is a complex field as a function of and which is periodic in space .
The complex-valued wavefunction
, with ,
has several remarkable features (see [1]), including periodicity of the squared wavefunction with period implicit analytic solutions of quantum trajectories for some values a time-independent solution for and , and a symmetry according to , where , with . For the case , the wavefunction becomes , where an implicit analytic solution for the quantum motion is given by the gradient of the phase function of the wavefunction in the eikonal form (often called polar form). For this special case, the implicit function for is , with the integration constant .
Exploiting to the symmetry of the wave density, only one half of the trajectories were calculated numerically.
On YouTube there are some videos by the author, which show additional breather solutions with a hyperbolic secant envelope in the de Broglie–Bohm interpretation for the Gross–Pitavevskii equation.
[1] D. Schrader, Asymptotisch Auftretende Solitonen-Lösungen der Nichtlinearen Schrödinger-Gleichung zu Beliebigen Secans-Hyperbolicus-Förmigen Anregungen, Aachen, Germany: Shaker Verlag, 1998.
[2] Bohmian-Mechanics.net. (Jul 11, 2014) www.bohmian-mechanics.net/index.html.
[3] S. Goldstein. "Bohmian Mechanics." The Stanford Encyclopedia of Philosophy. (Jul 11, 2014)plato.stanford.edu/entries/qm-bohm.
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99979f2e21439426 | Strong electron correlations
Strong electron correlations
Many properties of materials are the result of correlated behaviour of electrons. Magnetic order, metallic conductivity, and superconductivity are just a few examples of different states of matter, each characterized by different types of correlation between the charge and spin of the electrons in a solid.
Theoretical work faces the tremendous challenge posed by the difficulty to define a minimal model for such a large group of materials. Clearly, from a reductionist point of view, the full Schrödinger equation, including all electrons and details of the various different stoichiometry and crystal structures, is not a satisfactory starting point; moreover, exact solutions including the many-body aspects are not a realistic option with the available computational techniques.
Experimentally, one faces the challenge as to how to identify quantities that (i) can be determined experimentally and (ii) provide insights independent from theoretical bias. Apart from direct phenomenological quantities such as resistivity, magnetization, entropy, our understanding is rooted in fundamental quantities related to symmetry, conservation laws and topology. Particularly powerful examples are Mott-insulators (Sr2VO4), Fermi-liquid-like phases (Sr2RuO4), correlation-induced metal-insulator transitions (SmNiO3), hidden order (URu2Si2), topological states of matter, unconventional pairing, and so forth.
In our group we obtain reliable insights in all of these subjects, using various kind of optical spectroscopies, as well as other spectroscopies using synchrotron radiation and neutron sources.
(a) Statistical analysis of the relaxation rate in strontium ruthanate, extracted from optical measurements, which shows universal Fermi liquid behaviour below a temperature T ∼ 40 K; the specific statistical factor p=2 relates the energy and temperature dependence of relaxation processes, and is typical of Fermi liquids. (b) Collapse of the relaxation rate data, measured from optics, on a universal scaling curve for T ≤ 40 K. [D. Stricker et al. (2014)]
Top image: real part of the dielectric function (top) and optical conductivity (bottom) of NdNiO3 on a NdGaO3 (110) substrate (a), NdNiO3 on a NdGaO3 (101) substrate (b), and for SmNiO3 on a LaAlO3 (001) substrate. Bottom image: real part of the optical conductivity for some temperatures, and energy/temperature color maps of samples (a) NNO/NGO-110, (b) NNO/NGO-101, and (c) SNO/LAO-001. Arrows on the color map mark metal-insulator phase transitions. A and B designate two peaks in the insulating phase. Data at 0 eV come from DC measurements. [J Ruppen et al (2015)]
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branes orbit Branes in the Bulk of The BRANE
November 11, 2016
Severe Tropical Storm Parma and Typhoon Melor on October 6, 2009.
Fujiwhara Effect
When cyclones are in proximity of one another, their centers will begin orbiting cyclonically (counter-clockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere)[1] about a point[barycenter] between the two systems due to their cyclonic wind circulations. The two vortices will be attracted to each other, and eventually spiral into the center point and merge. It has not been agreed upon whether this is due to the divergent portion of the wind or vorticity advection.[2] When the two vortices are of unequal size, the larger vortex will tend to dominate the interaction, and the smaller vortex will orbit around it. The effect is named after Sakuhei Fujiwhara, the Japanese meteorologist who initially described it in a 1921 paper about the motion of vortices in water.[3][4]
Quantum superposition is a fundamental principle of quantum mechanics. It states that, much like waves in classical physics, any two (or more) quantum states [for instance a black hole] can be added together (“superposed”) and the result will be another valid quantum state; and conversely, that every quantum state can be represented as a sum of two or more other distinct states. Mathematically, it refers to a property of solutions to the Schrödinger equation; since the Schrödinger equation is linear, any linear combination of solutions will also be a solution.
There is a probability of finding that black hole anywhere along that disturbance. He is triggering condensing in the atmosphere as he pops in/out at various locations. That is why there is an increasing probability of rain and electomagnetic discharge (lightning) as he approaches and also why he produces gravity waves. He will interact with other branes quantum gravitationally.
Go catch you one, but be ready for a wrestling match, there is a whole lot of energy there to tussle with…
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d83ada7d32c84ccd | Blog Archives
It was my distinct pleasure for me to participate in the organization of the latest edition of the Mexican Meeting on Theoretical Physical Chemistry, RMFQT which took place last week here in Toluca. With the help of the School of Chemistry from the Universidad Autónoma del Estado de México.
This year the national committee created a Lifetime Achievement Award for Dr. Annik Vivier, Dr. Carlos Bunge, and Dr. José Luis Gázquez. This recognition from our community is awarded to these fine scientists for their contributions to theoretical chemistry but also for their pioneering work in the field in Mexico. The three of them were invited to talk about any topic of their choosing, particularly, Dr. Vivier stirred the imagination of younger students by showing her pictures of the times when she used to hangout with Slater, Roothan, Löwdin, etc., it is always nice to put faces onto equations.
Continuing with a recent tradition we also had the pleasure to host three invited plenary lectures by great scientists and good friends of our community: Prof. William Tiznado (Chile), Prof. Samuel B. Trickey (USA), and Prof. Julia Contreras (France) who shared their progress on their recent work.
As I’ve abundantly pointed out in the past, the RMFQT is a joyous occasion for the Mexican theoretical community to get together with old friends and discuss very exciting research being done in our country and by our colleagues abroad. I’d like to add a big shoutout to Dr. Jacinto Sandoval-Lira for his valuable help with the organization of our event.
All you wanted to know about Hybrid Orbitals…
… but were afraid to ask
How I learned to stop worrying and not caring that much about hybridization.
The math behind orbital hybridization is fairly simple as I’ll try to show below, but first let me give my praise once again to the formidable Linus Pauling, whose creation of this model built a bridge between quantum mechanics and chemistry; I often say Pauling was the first Quantum Chemist (Gilbert N. Lewis’ fans, please settle down). Hybrid orbitals are therefore a way to create a basis that better suits the geometry formed by the bonds around a given atom and not the result of a process in which atomic orbitals transform themselves for better sterical fitting, or like I’ve said before, the C atom in CH4 is sp3 hybridized because CH4 is tetrahedral and not the other way around. Jack Simmons put it better in his book:
2017-08-09 20.29.45
Taken from “Quantum Mechanics in Chemistry” by Jack Simmons
The atomic orbitals we all know and love are the set of solutions to the Schrödinger equation for the Hydrogen atom and more generally they are solutions to the hydrogen-like atoms for which the value of Z in the potential term of the Hamiltonian changes according to each element’s atomic number.
Since the Hamiltonian, and any other quantum mechanical operator for that matter, is a Hermitian operator, any given linear combination of wave functions that are solutions to it, will also be an acceptable solution. Therefore, since the 2s and 2p valence orbitals of Carbon do not point towards the edges of a tetrahedron they don’t offer a suitable basis for explaining the geometry of methane; even more so these atomic orbitals are not degenerate and there is no reason to assume all C-H bonds in methane aren’t equal. However we can come up with a linear combination of them that might and at the same time will be a solution to the Schrödinger equation of the hydrogen-like atom.
Ok, so we need four degenerate orbitals which we’ll name ζi and formulate them as linear combinations of the C atom valence orbitals:
ζ1a12s + b12px + c12py + d12pz
ζ2a22s + b22px + c22py + d22pz
ζ3a32s + b32px + c32py + d32pz
ζ4a42s + b42px + c42py + d42pz
to comply with equivalency lets set a1 = a2 = a3 = a4 and normalize them:
a12 + a22 + a32 + a42 = 1 ∴ ai = 1/√4
Lets take ζ1 to be directed along the z axis so b1 = c1 = 0
ζ= 1/√4(2s) + d12pz
since ζ1 must be normalized the sum of the squares of the coefficients is equal to 1:
1/4 + d12 = 1;
d1 = √3/2
Therefore the first hybrid orbital looks like:
ζ1 = 1/√4(2s) +√3/2(2pz)
We now set the second hybrid orbital on the xz plane, therefore c2 = 0
ζ2 = 1/√4(2s) + b22px + d22pz
since these hybrid orbitals must comply with all the conditions of atomic orbitals they should also be orthonormal:
ζ1|ζ2〉 = δ1,2 = 0
1/4 + d2√3/2 = 0
d2 = –1/2√3
our second hybrid orbital is almost complete, we are only missing the value of b2:
ζ2 = 1/√4(2s) +b22px +-1/2√3(2pz)
again we make use of the normalization condition:
1/4 + b22 + 1/12 = 1; b2 = √2/√3
Finally, our second hybrid orbital takes the following form:
ζ2 = 1/√4(2s) +√2/√3(2px) –1/√12(2pz)
The procedure to obtain the remaining two hybrid orbitals is the same but I’d like to stop here and analyze the relative direction ζ1 and ζ2 take from each other. To that end, we take the angular part of the hydrogen-like atomic orbitals involved in the linear combinations we just found. Let us remember the canonical form of atomic orbitals and explicitly show the spherical harmonic functions to which the 2s, 2px, and 2pz atomic orbitals correspond:
ψ2s = (1/4π)½R(r)
ψ2px = (3/4π)½sinθcosφR(r)
ψ2pz = (3/4π)½cosθR(r)
we substitute these in ζ2 and factorize R(r) and 1/√(4π)
ζ2 = (R(r)/√(4π))[1/√4 + √2 sinθcosφ –√3/√12cosθ]
We differentiate ζ2 respect to θ, and set it to zero to find the maximum value of θ respect to the z axis we get the angle between the first to hybrid orbitals ζ1 and ζ2 (remember that ζ1 is projected entirely over the z axis)
dζ2/dθ = (R(r)/√(4π))[√2 cosθ –√3/√12sinθ] = 0
sinθ/cosθ = tanθ = -√8
θ = -70.53°,
but since θ is measured from the z axis towards the xy plane this result is equivalent to the complementary angle 180.0° – 70.53° = 109.47° which is exactly the angle between the C-H bonds in methane we all know! and we didn’t need to invoke the unpairing of electrons in full orbitals, their promotion of any electron into empty orbitals nor the ‘reorganization‘ of said orbitals into new ones. Orbital hybridization is nothing but a mathematical tool to find a set of orbitals which comply with the experimental observation and that is the important thing here!
To summarize, you can take any number of orbitals and build any linear combination you want, in order to comply with the observed geometry. Furthermore, no matter what hybridization scheme you follow, you still take the entire orbital, you cannot take half of it because they are basis functions. That is why you should never believe that any atom exhibits something like an sp2.5 hybridization just because their bond angles lie between 109 and 120°. Take a vector v = xi+yj+zk, even if you specify it to be v = 1/2i that means x = 1/2, not that you took half of the unit vector i, and it doesn’t mean you took nothing of j and k but rather than y = z = 0.
This was a very lengthy post so please let me know if you read it all the way through by commenting, liking, or sharing. Thanks for reading.
No, seriously, why can’t orbitals be observed?
The concept of electronic orbital has become such a useful and engraved tool in understanding chemical structure and reactivity that it has almost become one of those things whose original meaning has been lost and replaced for a utilitarian concept, one which is not bad in itself but that may lead to some wrong conclusions when certain fundamental facts are overlooked.
Last week a wrote -what I thought was- a humorous post on this topic because a couple of weeks ago a viewpoint in JPC-A was published by Pham and Gordon on the possibility of observing molecular orbitals through microscopy methods, which elicited a ‘seriously? again?‘ reaction from me, since I distinctly remember the Nature article by Zuo from the year 2000 when I just had entered graduate school. The article is titled “direct observation of d-orbital holes.” We discussed this paper in class and the discussion it prompted was very interesting at various levels: for starters, the allegedly observed d-orbital was strikingly similar to a dz2, which we had learned in class (thanks, prof. Carlos Amador!) that is actually a linear combination of d(z2-x2) and d(z2-y2) orbitals, a mathematical -lets say- trick to conform to spectroscopic observations.
Pham and Gordon are pretty clear in their first paragraph: “The wave function amplitude Ψ*Ψ is interpreted as the probability density. All observable atomic or molecular properties are determined by the probability and a corresponding quantum mechanical operator, not by the wave function itself. Wave functions, even exact wave functions, are not observables.” There is even another problem, about which I wrote a post long time ago: orbitals are non-unique, this means that I could get a set of orbitals by solving the Schrödinger equation for any given molecule and then perform a unit transformation on them (such as renormalizing them, re-orthonormalizing them to get a localized version, or even hybridizing them) and the electronic density derived from them would be the same! In quantum mechanical terms this means that the probability density associated with the wave function internal product, Ψ*Ψ, is not changed upon unit transformations; why then would a specific version be “observed” under a microscope? As Pham and Gordon state more eloquently it has to do with the Density of States (DOS) rather than with the orbitals. Furthermore, an orbital, or more precisely a spinorbital, is conveniently (in math terms) separated into a radial, an angular and a spin component R(r)Ylm(θ,φ)σ(α,β) with the angular part given by the spherical harmonic functions Ylm(θ,φ), which in turn -when plotted in spherical coordinates- create the famous lobes we all chemists know and love. Zuo’s observation claim was based on the resemblance of the observed density to the angular part of an atomic orbital. Another thing, orbitals have phases, no experimental observation claims to have resolved those.
Now, I may be entering a dangerous comparison but, can you observe a 2? If you say you just did, well, that “2” is just a symbol used to represent a quantity: two, the cardinality of a set containing two elements. You might as well depict such quantity as “II” or “⋅⋅” but still cannot observe “a two”. (If any mathematician is reading this, please, be gentle.) I know a number and a function are different, sorry if I’m just rambling here and overextending a metaphor.
Pretending to having observed an orbital through direct experimental methods is to neglect the Born interpretation of the wave function, Heisenberg’s uncertainty principle and even Schrödinger’s cat! (I know, I know, Schrödinger came up with this gedankenexperiment in order to refute the Copenhagen interpretation of quantum mechanics, but it seems like after all the cat is still not out of the box!)
So, the take home message from the viewpoint in JPC is that molecular properties are defined by the expected values of a given wave function for a specific quantum mechanical operator of the property under investigation and not from the wave function itself. Wave functions are not observables and although some imaging techniques seem to accomplish a formidable task the physical impossibility hints to a misinterpretation of facts.
I think I’ll write more about this in a future post but for now, my take home message is to keep in mind that orbitals are wave functions and therefore are not more observable (as in imaging) than a partition function is in statistical mechanics.
Dealing with Spin Contamination
Most organic chemistry deals with closed shell calculations, but every once in a while you want to calculate carbenes, free radicals or radical transition states coming from a homolytic bond break, which means your structure is now open shell.
Closed shell systems are characterized by having doubly occupied molecular orbitals, that is to say the calculation is ‘restricted’: Two electrons with opposite spin occupy the same orbital. In open shell systems, unrestricted calculations have a complete set of orbitals for the electrons with alpha spin and another set for those with beta spin. Spin contamination arises from the fact that wavefunctions obtained from unrestricted calculations are no longer eigenfunctions of the total spin operator <S^2>. In other words, one obtains an artificial mixture of spin states; up until now we’re dealing only with single reference methods. With each step of the SCF procedure the value of <S^2> is calculated and compared to s(s+1) where s is half the number of unpaired electrons (0.75 for a radical and 2.0 for triplets, and so on); if a large deviation between these two numbers is found, the then calculation stops.
Gaussian includes an annihilation step during SCF to reduce the amount of spin contamination but it’s not 100% reliable. Spin contaminated wavefunctions aren’t reliable and lead to errors in geometries, energies and population analyses.
One solution to overcome spin contamination is using Restricted Open Shell calculations (ROHF, ROMP2, etc.) for which singly occupied orbitals is used for the unpaired electrons and doubly occupied ones for the rest. These calculations are far more expensive than the unrestricted ones and energies for the unpaired electrons (the interesting ones) are unreliable, specially spin polarization is lost since dynamical correlation is hardly accounted for. The IOP(5/14=2) in Gaussian uses the annihilated wavefunction for the population analysis if acceptable but since Mulliken’s method is not reliable either I don’t advice it anyway.
The case of DFT is different since rho.alpha and rho.beta can be separated (similarly to the case of unrestricted ab initio calculations), but the fact that both densities are built of Kohn-Sham orbitals and not true canonical orbitals, compensates the contamination somehow. That is not to say that it never shows up in DFT calculations but it is usually less severe, of course for the case of hybrid functional the more HF exchange is included the more important spin contamination may become.
So, in short, for spin contaminated wavefunctions you want to change from restricted to unrestricted and if that doesn’t work then move to Restricted Open Shell; if using DFT you can use the same scheme and also try changing from hybrid to pure orbitals at the cost of CPU time. There is a last option which is using spin projection methods but I’ll discuss that in a following post.
Rank your QM knowledge according to Pauli’s Exclusion Principle
QM Evolutionary tree!
QM Evolutionary tree!
LOL just feeling a little humorous this morning!
New paper in JPC-A
As we approach to the end of another year, and with that the time where my office becomes covered with post-it notes so as to find my way back into work after the holidays, we celebrate another paper published! This time at the Journal of Physical Chemistry A as a follow up to this other paper published last year on JPC-C. Back then we reported the development of a selective sensor for Hg(II); this sensor consisted on 1-amino-8-naphthol-3,6-disulphonic acid (H-Acid) covalently bound to a modified silica SBA-15 surface. H-Acid is fluorescent and we took advantage of the fact that, when in the presence of Hg(II) in aqueous media, its fluorescence is quenched but not with other ions, even with closely related ions such as Zn(II) and Cd(II). In this new report we delve into the electronic reasons behind the quenching process by calculating the most important electronic transitions with the framework laid by the Time Dependent Density Functional Theory (TD-DFT) at the PBE0/cc-pVQZ level of theory (we also included an electron core potential on the heavy metal atoms in order to decrease the time of each calculation). One of the things I personally liked about this work is the combination of different techniques that were used to assess the photochemical phenomenon at hand; some of those techniques included calculation of various bond orders (Mayer, Fuzzy, Wiberg, delocalization indexes), time dependent DFT and charge transfer delocalizations. Although we calculated all these various different descriptors to account for changes in the electronic structure of the ligand which lead to the fluorescence quenching, only delocalization indexes as calculated with QTAIM were used to draw conclusion, while the rest are collected in the SI section.
Thanks a lot to my good friend and collaborator Dr. Pezhman Zarabadi-Poor for all his work, interest and insight into the rationalization of this phenomenon. This is our second paper published together. By the way, if any of you readers is aware of a way to finance a postdoc stay for Pezhman here at our lab, please send us a message because right now funding is scarce and we’d love to keep bringing you many more interesting papers.
For our research group this was the fourth paper published during 2014. We can only hope (and work hard) to have at least five next year without compromising their quality. I’m setting the goal to be 6 papers; we’ll see in a year if we delivered or not.
I’d like to also take this opportunity to thank all the readers of this little blog of mine for your visits and your live demonstrations of appreciation at various local and global meetings such as the ACS meeting in San Francisco and WATOC14 in Chile, it means a lot to me to know that the things I write are read; if I were to make any New Year’s resolutions it would be to reply quicker to questions posted because if you took the time to write I should take the time to reply.
I wish you all the best for 2015 in and out of the lab!
XIth Mexican Reunion on Theoretical Physical Chemistry
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6daee91f854d753e | Wolfram Blog
Michael Trott
An Exact Value for the Planck Constant: Why Reaching It Took 100 Years
May 19, 2016 — Michael Trott, Chief Scientist, Wolfram|Alpha Scientific Content
Blog communicated on behalf of Jean-Charles de Borda.
Some thoughts for World Metrology Day 2016
Please allow me to introduce myself
I’m a man of precision and science
I’ve been around for a long, long time
Stole many a man’s pound and toise
And I was around when Louis XVI
Had his moment of doubt and pain
Made damn sure that metric rules
Through platinum standards made forever
Pleased to meet you
Hope you guess my name
Introduction and about me
In case you can’t guess: I am Jean-Charles de Borda, sailor, mathematician, scientist, and member of the Académie des Sciences, born on May 4, 1733, in Dax, France. Two weeks ago would have been my 283rd birthday. This is me:
Jean-Charles de Borda
In my hometown of Dax there is a statue of me. Please stop by when you visit. In case you do not know where Dax is, here is a map:
Map of Dax and statue of Jean-Charles de Borda
In Europe when I was a boy, France looked basically like it does today. We had a bit less territory on our eastern border. On the American continent, my country owned a good fraction of land:
France and French territory in America in 1733
I led a diverse earthly life. At 32 years old I carried out a lot of military and scientific work at sea. As a result, in my forties I commanded several ships in the Seven Years’ War. Most of the rest of my life I devoted to the sciences.
But today nobody even knows where my grave is, as my physical body died on February 19, 1799, in Paris, France, in the upheaval of the French Revolution. (Of course, I know where it is, but I can’t communicate it anymore.) My name is the twelfth listed on the northeast side of the Eiffel Tower:
Borda listed on the northeast side of the Eiffel Tower
Over the centuries many of my fellow Frenchman who joined me up here told me that I deserved a place in the Panthéon. But you will not find me there, nor at the Père Lachaise, Montparnasse, or Montmartre cemeteries.
But this is not why I still cannot rest in peace. I am a humble man; it is the kilogram that keeps me up at night. But soon I will be able to rest in peace at night for all time and approach new scientific challenges.
Let me tell you why I will soon find a good night’s sleep.
All my life, I was into mathematics, geometry, physics, and hydrology. And overall, I loved to measure things. You might have heard of substitution weighing (also called Borda’s method)—yes, this was my invention, as was the Borda count method. I also substantially improved the repeating circle. Here is where the story starts. The repeating circle was crucial in making a high-precision determination of the size of the Earth, which in turn defined the meter. (A good discussion of my circle can be found here.)
Repeating circle
I lived in France when it was still a monarchy. Times were difficult for many people—especially peasants—partially because trade and commerce were difficult due to the lack of measures all over the country. If you enjoy reading about history, I highly recommend Kula’s Measures and Men to understand the weights and measurements situation in France in 1790. The state of the weights and measures were similar in other countries; see for instance Johann Georg Trallesreport about the situation in Switzerland.
In August 1790, I was made the chairman of the Commission of Weights and Measures as a result of a 1789 initiative from Louis XVI. (I still find it quite miraculous that 1,000 years after Charlemagne’s initiative to unify weights and measures, the next big initiative in this direction would be started.) Our commission created the metric system that today is the International System of Units, often abbreviated as SI (le Système international d’unités in French).
In the commission were, among others, Pierre-Simon Laplace (think the Laplace equation), Adrien-Marie Legendre (Legendre polynomials), Joseph-Louis Lagrange (think Lagrangian), Antoine Lavoisier (conservation of mass), and the Marquis de Condorcet. (I always told Adrien-Marie that he should have some proper portrait made of him, but he always said he was too busy calculating. But for 10 years now, the politician Louis Legendre’s portrait has not been used in math books instead of Adrien-Marie’s. Over the last decades, Adrien-Marie befriended Jacques-Louis David, and Jacques-Louis has made a whole collection of paintings of Adrien-Marie; unfortunately, mortals will never see them.) Lagrange, Laplace, Monge, Condorcet, and I were on the original team. (And, in the very beginning, Jérôme Lalande was also involved; later, some others were as well, such as Louis Lefèvre‑Gineau.)
Portraits of Pierre-Simon Laplace, Adrien-Marie Legendre, Joseph-Louis Lagrange, Antoine Lavoisier, and Marquis de Condorcet
Three of us (Monge, Lagrange, and Condorcet) are today interred or commemorated at the Panthéon. It is my strong hope that Pierre-Simon is one day added; he really deserves it.
As I said before, things were difficult for French citizens in this era. Laplace wrote:
The prodigious number of measures in use, not only among different people, but in the same nation; their whimsical divisions, inconvenient for calculation, and the difficulty of knowing and comparing them; finally, the embarrassments and frauds which they produce in commerce, cannot be observed without acknowledging that the adoption of a system of measures, of which the uniform divisions are easily subjected to calculation, and which are derived in a manner the least arbitrary, from a fundamental measure, indicated by nature itself, would be one of the most important services which any government could confer on society. A nation which would originate such a system of measures, would combine the advantage of gathering the first fruits of it with that of seeing its example followed by other nations, of which it would thus become the benefactor; for the slow but irresistible empire of reason predominates at length over all national jealousies, and surmounts all the obstacles which oppose themselves to an advantage, which would be universally felt.
All five of the mathematicians (Monge, Lagrange, Laplace, Legendre, and Condorcet) have made historic contributions to mathematics. Their names are still used for many mathematical theorems, structures, and operations:
Monge, Lagrange, Laplace, Legendre, and Condorcet's contributions to mathematics
In 1979, Ruth Inez Champagne wrote a detailed thesis about the influence of my five fellow citizens on the creation of the metric system. For Legendre’s contribution especially, see C. Doris Hellman’s paper. Today it seems to me that most mathematicians no longer care much about units and measures and that physicists are the driving force behind advancements in units and measures. But I did like Theodore P. Hill’s arXiv paper about the method of conflations of probability distributions that allows one to consolidate knowledge from various experiments. (Yes, before you ask, we do have instant access to arXiv up here. Actually, I would say that the direct arXiv connection has been the greatest improvement here in the last millennium.)
Our task was to make standardized units of measure for time, length, volume, and mass. We needed measures that were easily extensible, and could be useful for both tiny things and astronomic scales. The principles of our approach were nicely summarized by John Quincy Adams, Secretary of State of the United States, in his 1821 book Report upon the Weights and Measures.
Excerpt from John Quincy Adams' Report upon Weights and Measures
Originally we (we being the metric men, as we call ourselves up here) had suggested just a few prefixes: kilo-, deca-, hecto-, deci-, centi-, milli-, and the no-longer-used myria-. In some old books you can find the myria- units.
We had the idea of using prefixes quite early in the process of developing the new measurements. Here are our original proposals from 1794:
Excerpts of original proposals from 1794
Side note: in my time, we also used the demis and the doubles, such as a demi-hectoliter (=50 liters) or a double dekaliter (=20 liters).
As inhabitants of the twenty-first century know, times, lengths, and masses are measured in physics, chemistry, and astronomy over ranges spanning more than 50 orders of magnitude. And the units we created in the tumultuous era of the French Revolution stood the test of time:
Orders of magnitude plots for length and area
Orders of magnitude plots for length Orders of magnitude plot for area
In the future, the SI might need some more prefixes. In a recent LIGO discovery, the length of the interferometer arms changed on the order of 10 yoctometers. Yoctogram resolution mass sensors exist. One yoctometer equals 10–24 meter. Mankind can already measure tiny forces on the order of zeptonewtons.
On the other hand, astronomy needs prefixes larger than 1024. One day, these prefixes might become official.
Proposed prefixes larger than 10^24
I am a man of strict rules, and it drives me nuts when I see people in the twenty-first century not obeying the rules for using SI prefixes. Recently I saw somebody writing on a whiteboard that a year is pretty much exactly 𝜋 dekamegaseconds (𝜋 daMs):
1 year approximately pi dekamegaseconds
While it’s a good approximation (only 0.4% off), when will this person learn that one shouldn’t concatenate prefixes?
The technological progress of mankind has occurred quickly in the last two centuries. And mega-, giga-, tera- or nano-, pico-, and femto- are common prefixes in the twenty-first century. Measured in meters per second, here is the probability distribution of speed values used by people. Some speeds (like speed limits, the speed of sound, or the speed of light) are much more common than others, but many local maxima can be found in the distribution function:
Probability distribution of speed values used by people
Here is the report we delivered in March of 1791 that started the metric system and gave the conceptual meaning of the meter and the kilogram, signed by myself, Lagrange, Laplace, Monge, and Concordet (now even available through what the modern world calls a “digital object identifier,” or DOI, like 10.3931/e-rara-28950):
Report from 1791 that started the metric system and gave conceptual meaning of the meter and kilogram
Today most people think that base 10 and the meter, second, and kilogram units are intimately related. But only on October 27, 1790, did we decide to use base 10 for subdividing the units. We were seriously considering a base-12 subdivision, because the divisibility by 2, 3, 4, and 6 is a nice feature for trading objects. It is clear today, though, that we made the right choice. Lagrange’s insistence on base 10 was the right thing. At the time of the French Revolution, we made no compromises. On November 5, 1792, I even suggested changing clocks to a decimal system. (D’Alambert had suggested this in 1754; for the detailed history of decimal time, see this paper.) Mankind was not ready yet; maybe in the twenty-first century decimal clocks and clock readings would finally be recognized as much better than 24 hours, 60 minutes, and 60 seconds. I loved our decimal clocks—they were so beautiful. So it’s a real surprise to me today that mankind still divides the angle into 90 degrees. In my repeating circle, I was dividing the right angle into 100 grades.
We wanted to make the new (metric) units truly equal for all people, not base them, for instance, on the length of the forearm of a king. Rather, “For all time, for all people” (“À tous les temps, à tous les peuples”). Now, in just a few years, this dream will be achieved.
And I am sure there will come the day where Mendeleev’s prediction (“Let us facilitate the universal spreading of the metric system and thus assist the common welfare and the desired future rapprochement of the peoples. It will come not yet, slowly, but surely.”) will come true even in the three remaining countries of the world that have not yet gone metric:
Countries that have not gone metric
The SI units have been legal for trade in the USA since the mid-twentieth century, when United States customary units became derived from the SI definitions of the base units. Citizens can choose which units they want for trade.
We also introduced the decimal subdivision of money, and our franc was in use from 1793 to 2002. At least today all countries divide their money on the basis of base 10—no coins with label 12 are in use anymore. Here is the coin label breakdown by country:
Coin label breakdown by country
We took the “all” in “all people” quite seriously, and worked with our archenemy Britain and the new United States (through Thomas Jefferson personally) together to make a new system of units for all the major countries in my time. But, as is still so often the case today, politics won over reason.
I died on February 19, 1799, just a few months before the our group’s efforts. On June 22, 1799, my dear friend Laplace gave a speech about the finished efforts to build new units of length and mass before the new prototypes were delivered to the Archives of the Republic (where they are still today).
In case the reader is interested in my eventful life, Jean Mascart wrote a nice biography about me in 1919, and it is now available as a reprint from the Sorbonne.
From the beginnings of the metric system to today
Two of my friends, Jean Baptiste Joseph Delambre and Pierre Méchain, were sent out to measure distances in France and Spain from mountain to mountain to define the meter as one ten-millionth of the distance from the North Pole to the equator of the Earth. Historically, I am glad the mission was approved. Louis XVI was already under arrest when he approved the financing of the mission. My dear friend Lavoisier called their task “the most important mission that any man has ever been charged with.”
Pierre Méchain and Jean Baptiste Joseph Delambre
If you haven’t done so, you must read the book The Measure of All Things by Ken Alder. There is even a German movie about the adventures of my two old friends. Equipped with a special instrument that I had built for them, they did the work that resulted in the meter. Although we wanted the length of the meter to be one ten-millionth of the length of the half-meridian through Paris from pole to equator, I think today this is a beautiful definition conceptually. That the Earth isn’t quite as round as we had hoped for we did not know at the time, and this resulted in a small, regrettable error of 0.2 mm due to a miscalculation of the flattening of the Earth. Here is the length of the half-meridian through Paris, expressed through meters along an ellipsoid that approximates the Earth:
If they had elevation taken into account (which they did not do—Delambre and Méchain would have had to travel the whole meridian to catch every mountain and hill!), and had used 3D coordinates (meaning including the elevation of the terrain) every few kilometers, they would have ended up with a meter that was 0.4 mm too short:
Length of the meridian meter when taking elevation into account
Here is the elevation profile along the Paris meridian:
Elevation along the Paris meridian
And the meter would be another 0.9 mm longer if measured with a yardstick the length of a few hundred meters:
Length of the meridian meter when taking detailed elevation into account
Because of the fractality of the Earth’s surface, an even smaller yardstick would have given an even longer half-meridian.
It’s more realistic to follow the sea-level height. The difference between the length of the sea-level meridian meter and the ellipsoid approximation meter is just a few micrometers:
Difference between the length of the sea-level meridian and the ellipsoid approximation meter
But at least the meridian had to go through Paris (not London, as some British scientists of my time proposed). But anyway, the meridian length was only a stepping stone to make a meter prototype. Once we had the meter prototype, we didn’t have to refer to the meridian anymore.
Here is a sketch of the triangulation carried out by Pierre and Jean Baptiste in their adventurous six-year expedition. Thanks to the internet and various French digitization projects, the French-speaking reader interested in metrology and history can now read the original results online and reproduce our calculations:
Reproducing the triangulation carried out by Pierre and Jean Baptiste
The part of the meridian through Paris (and especially through the Paris Observatory, marked in red) is today marked with the Arago markers—do not miss them during your next visit to Paris! François Arago remeasured the Paris meridian. After Méchain joined me up here in 1804, Laplace got the go-ahead (and the money) from Napoléon to remeasure the meridian and to verify and improve our work:
Plotting the meridian through Paris and the Arago markers
Plotting the meridian through Paris
The second we derived from the length of a year. And the kilogram as a unit of mass we wanted to (and did) derive from a liter of water. If any liquid is special, it is surely water. Lavoisier and I had many discussions about the ideal temperature. The two temperatures that stand out are 0 °C and
4 °C. Originally we were thinking about 0 °C, as with ice water it is easy to see. But because of the maximal density of water at 4 °C, we later thought that would be the better choice. The switch to
4 °C was suggested by Louis Lefèvre-Gineau. The liter as a volume in turn we defined as one-tenth of a meter cubed. As it turns out, compared with high-precision measurements of distilled water,
1 kg equals the mass of 1.000028 dm3 of water. The interested reader can find many more details of the process of the water measurements here and about making the original metric system here. A shorter history in English can be found in the recent book by Williams and the ten-part series by Chisholm.
I don’t want to brag, but we also came up with the name “meter” (derived from the Greek metron and the Latin metrum), which we suggested on July 11 of 1792 as the name of the new unit of length. And then we had the area (=100 m2) and the stere (=1 m3).
And I have to mention this for historical accuracy: until I entered the heavenly spheres, I always thought our group was the first to carry out such an undertaking. How amazed and impressed I was when shortly after my arrival up here, I-Hsing and Nankung Yiieh introduced themselves to me and told me about their expedition from the years 721 to 725, more than 1,000 years before ours, to define a unit of length.
I am so glad we defined the meter this way. Originally the idea was to define a meter through a pendulum of proper length as a period of one second. But I didn’t want any potential change in the second to affect the length of the meter. While dependencies will be unavoidable in a complete unit system, they should be minimized.
Basing the meter on the Earth’s shape and the second on the Earth’s movement around the Sun seemed like a good idea at the time. Actually, it was the best idea that we could technologically realize at this time. We did not know how tides and time changed the shape of the Earth, or how continents drift apart. But we believed in the future of mankind, in ever-increasing measurement precision, but we did not know what concretely would change. But it was our initial steps for precisely measuring distances in France that were carried out. Today we have high-precision geo potential maps as high-order series of Legendre polynomials:
GeogravityModelData for the astronomical observatory in Paris
With great care, the finest craftsmen of my time melted platinum, and we forged a meter bar and a kilogram. It was an exciting time. Twice a week I would stop by Janety’s place when he was forging our first kilograms. Melting and forming platinum was still a very new process. And Janety, Louis XVI’s goldsmith, was a true master of forming platinum—to be precise, a spongelike eutectic made of platinum and arsenic. Just a few years earlier, on June 6, 1782, Lavoisier showed the melting of platinum in a hydrogen-oxygen flame to (the future) Tsar Paul I at a garden party at Versailles; Tsar Paul I was visiting Marie Antoinette and Loius XVI. And Étienne Lenoir made our platinum meter, and Jean Nicolas Fortin our platinum kilogram. For the reader interested in the history of platinum, I recommend McDonald’s and Hunt’s book.
Platinum is a very special metal; it has a high density and is chemically very inert. It is also not as soft as gold. The best kilogram realizations today are made from a platinum-iridium mixture (10% iridium), as adding iridium to platinum does improve its mechanical properties. Here is a comparison of some physical characteristics of platinum, gold, and iridium:
Comparison of physical characteristics of platinum, gold, and iridium
This sounds easy, but at the time the best scientists spent countless hours calculating and experimenting to find the best materials, the best shapes, and the best conditions to define the new units. But both the new meter bar and the new kilogram cylinder were macroscopic bodies. And the meter has two markings of finite width. All macroscopic artifacts are difficult to transport (we developed special travel cases); they change by very small amounts over a hundred years through usage, absorption, desorption, heating, and cooling. In the amazing technological progress of the nineteenth and twentieth centuries, measuring time, mass, and length with precisions better than one in a billion has become possible. And measuring time can even be done a billion times better.
I still vividly remember when, after we had made and delivered the new meter and the mass prototypes, Lavoisier said, “Never has anything grander and simpler and more coherent in all its parts come from the hands of man.” And I still feel so today.
Our goal was to make units that truly belonged to everyone. “For all time, for all people” was our motto. We put copies of the meter all over Paris to let everybody know how long it was. (If you have not done so, next time you visit Paris, make sure to visit the mètre étalon near to the Luxembourg Palace.) Here is a picture I recently found, showing an interested German tourist studying the history of one of the few remaining mètres étalons:
It was an exciting time (even if I was no longer around when the committee’s work was done). Our units served many European countries well into the nineteenth and large parts of the twentieth century. We made the meter, the second, and the kilogram. Four more base units (the ampere, the candela, the mole, and the kelvin) have been added since our work. And with these extensions, the metric system has served mankind very well for 200+ years.
How the metric system took off after 1875, the year of the Metre Convention, can be seen by plotting how often the words kilogram, kilometer, and kilohertz appear in books:
How often the words kilogram, kilometer, and kilohertz appear in books
We defined only the meter, the seond, the liter, and the kilogram. Today many more name units belong to the SI: becquerel, coulomb, farad, gray, henry, hertz, joule, katal, lumen, lux, newton, ohm, pascal, siemen, sievert, tesla, volt, watt, and weber. Here is a list of the dimensional relations (no physical meaning implied) between the derived units:
List of the dimensional relations between the derived units
List of the dimensional relations between the derived units
Many new named units have been added since my death, often related to electrical and magnetic phenomena that were not yet known when I was alive. And although I am a serious person in general, I am often open to a joke or a pun—I just don’t like when fun is made of units. Like Don Knuth’s Potrzebie system of units, with units such as the potrzebie, ngogn, blintz, whatmeworry, cowznofski, vreeble, hoo, and hah. Not only are their names nonsensical, but so are their values:
Portzerbies and blintz units
Or look at Max Pettersson’s proposal for units for biology. The names of the units and the prefixes might sound funny, but for me units are too serious a subject to make fun of:
Max Pettersson's proposal for units for biology
These unit names do not even rhyme with any of the proper names:
Words that rhyme with meter
Words that rhyme with mile
To reiterate, I am all in favor of having fun, even with units, but it must be clear that it is not meant seriously:
Converting humorous units of measurement
Or explicitly nonscientific units, such as helens for beauty, puppies for happiness, or darwins for fame are fine with me:
Measuring beauty in helens
Measuring happiness in puppies
Measuring fame in darwins
I am so proud that the SI units are not just dead paper symbols, but tools that govern the modern world in an ever-increasing way. Although I am not a comics guy, I love the recent promotion of the base units to superheroes by the National Institute of Standards and Technology:
Base units to superheroes
Base units to superheroes
Note that, to honor the contributions of the five great mathematicians to the metric system, the curves in the rightmost column of the unit-representing characters are given as mathematical formulas, e.g. for Dr. Kelvin we have the following purely trigonometric parametrization:
Purely trigonometric parametrization of Dr. Kelvin
So we can plot Dr. Kelvin:
Plotting Dr. Kelvin
Having the characters in parametric form is handy: when my family has reunions, the little ones’ favorite activity is coloring SI superheroes. I just print the curves, and then the kids can go crazy with the crayons. (I got this idea a couple years ago from a coloring book by the NCSA.)
Printing randomly colored curves
And whenever a new episode comes out, all us “measure men” (George Clooney, if you see this: hint, hint for an exciting movie set in the 1790s!) come together to watch it. As you can imagine, the last episode is our all-time favorite. Rumor has it up here that there will be a forthcoming book The Return of the Metrologists (2018 would be a perfect year) complementing the current book.
And I am glad to see that the importance of measuring and the underlying metric system is in modern times honored through the World Metrology Day on May 20, which is today.
In my lifetime, most of what people measured were goods: corn, potatoes, and other foods, wine, fabric, and firewood, etc. So all my country really needed were length, area, volume, angles, and, of course, time units. I always knew that the importance of measuring would increase over time. But I find it quite remarkable that only 200 years after I entered the heavenly spheres, hundreds and hundreds of different physical quantities are measured. Today even the International Organization for Standardization (ISO) lists, defines, and describes what physical quantities to use. Below is an image of an interactive Demonstration (download the notebook at the bottom of this post to interact with it) showing graphically the dimensions of physical quantities for subsets of selectable dimensions. First select two or three dimensions (base units). Then the resulting graphics show spheres with sizes proportional to the number of different physical quantities with these dimensions. Mouse over the spheres in the notebook to see the dimensions. For example, with “meter”, “second”, and “kilogram” checked, the diagram shows the units of physical quantities like momentum (kg1 m1 s–1) or energy (kg2 m1 s–2):
Physical quantities of given dimensions
Here is a an excerpt of the code that I used to make these graphics. These are all physical quantities that have dimensions L2 M1 T–1. The last one is the slightly exotic electrodynamic observable
Excerpt of code from physical quantities of given dimensions demonstration
Today with smart phones and wearable devices, a large number of physical quantities are measured all the time by ordinary people. “Measuring rules,” as I like to say. Or, as my (since 1907) dear friend William Thomson liked to say:
Here is a graphical visualization of the physical quantities that are measured by the most common measurement devices:
Graphical visualization of the physical quantities that are measured by the most common measurement devices
Electrical and magnetic phenomena were just starting to become popular when I was around. Electromagnetic effects related to physical quantities that are expressed through the electric current only become popular much later:
Electrical and magnetic phenomena timeline
Electrical and magnetic phenomena timeline
I remember how excited I was when in the second half of the nineteenth century and the beginning of the twentieth century the various physical quantities of electromagnetism were discovered and their connections were understood. (And, not to be forgotten: the recent addition of memristance.) Here is a diagram showing the most important electric/magnetic physical quantities qk that have a relation of the form qk=qi qj with each other:
Diagram showing the most important electric/magnetic physical quantities q sub k, with relation of the form q subk = q sub i, q sub j, with each other
On the other hand, I was sure that temperature-related phenomena would soon be fully understood after my death. And indeed just 25 years later, Carnot proved that heat and mechanical work are equivalent. Now I also know about time dilation and length contraction due to Einstein’s theories. But mankind still does not know if a moving body is colder or warmer than a stationary body (or if they have the same temperature). I hear every week from Josiah Willard about the related topic of negative temperatures. And recently, he was so excited about a value for a maximal temperature for a given volume V expressed through fundamental constants:
Maximal temperature for a given volume V expressed through fundamental constants
For one cubic centimeter, the maximal temperature is about 5PK:
Maximal temperature for once cubic centimeter
The rise of the constants
Long after my physical death, some of the giants of physics of the nineteenth century and early twentieth century, foremost among them James Clerk Maxwell, George Johnstone Stoney, and Max Planck (and Gilbert Lewis) were considering units for time, length, and mass that were built from unchanging properties of microscopic particles and the associated fundamental constants of physics (speed of light, gravitational constant, electron charge, Planck constant, etc.):
James Clerk Maxwell, George Johnstone Stoney, and Max Planck
Maxwell wrote in 1870:
Yet, after all, the dimensions of our Earth and its time of rotation, though, relative to our present means of comparison, very permanent, are not so by any physical necessity. The earth might contract by cooling, or it might be enlarged by a layer of meteorites falling on it, or its rate of revolution might slowly slacken, and yet it would continue to be as much a planet as before.
But a molecule, say of hydrogen, if either its mass or its time of vibration were to be altered in the least, would no longer be a molecule of hydrogen.
If, then, we wish to obtain standards of length, time, and mass which shall be absolutely permanent, we must seek them not in the dimensions, or the motion, or the mass of our planet, but in the wavelength, the period of vibration, and the absolute mass of these imperishable and unalterable and perfectly similar molecules.
When we find that here, and in the starry heavens, there are innumerable multitudes of little bodies of exactly the same mass, so many, and no more, to the grain, and vibrating in exactly the same time, so many times, and no more, in a second, and when we reflect that no power in nature can now alter in the least either the mass or the period of any one of them, we seem to have advanced along the path of natural knowledge to one of those points at which we must accept the guidance of that faith by which we understand that “that which is seen was not made of things which do appear.’
At the time when Maxwell wrote this, I was already a man’s lifetime up here, and when I read it I applauded him (although at this time I still had some skepticism toward all ideas coming from Britain). I knew that this was the path forward to immortalize the units we forged in the French Revolution.
There are many physical constants. And they are not all known to the same precision. Here are some examples:
Examples of physical constants
Converting the values of constants with uncertainties into arbitrary precision numbers is convenient for the following computations. The connection between the intervals and the number of digits is given as follows. The arbitrary precision number that corresponds to v ± δ is the number v with precision –log10(2 δ/v) Conversely, given an arbitrary precision number (numbers are always convenient for computations), we can recover the v ± δ form:
Converting arbitrary precision numbers to intervals
After the exactly defined constants, the Rydberg constant with 11 known digits stands out for a very precisely known constant. On the end of the spectrum is G, the gravitational constant. At least once a month Henry Cavendish stops at my place with yet another idea on how to build a tabletop device to measure G. Sometimes his ideas are based on cold atoms, sometimes on superconductors, and sometimes on high-precision spheres. If he could still communicate with the living, he would write a comment to Nature every week. A little over a year ago Henry was worried that he should have done his measurements in winter as well in summer, but he was relieved to see that no seasonal dependence of G’s value seems to exist. The preliminary proposal deadline for the NSF’s Big G Challenge was just four days ago. I think sometime next week I will take a heavenly peek at the program officer’s preselected experiments.
There are more physical constants, and they are not all equal. Some are more fundamental than others, but for reasons of length I don’t want to get into a detailed discussion about this topic now. A good start for interested readers is Lévy-Leblond’s papers (also here), as well as this paper, this paper, and the now-classic Duff–Okun–Veneziano paper. For the purpose of making units from physical constants, the distinction of the various classes of physical constants is not so relevant.
The absolute values of the constants and their relations to heaven, hell, and Earth is an interesting subject on its own. It is a hot topic of discussion for mortals (also see this paper), as well as up here. Some numerical coincidences (?) are just too puzzling:
Absolute values of the constants and their relations to heaven, hell, and Earth
Of course, using modern mathematical algorithms, such as lattice reduction, we can indulge in the numerology of the numerical part of physical constants:
Numerology of the numerical part of physical constants
For instance, how can we form 𝜋 out of fundamental constant products?
Forming pi out of fundamental constant products
Or let’s look at my favorite number, 10, the mathematical basis of the metric system:
Forming 10 out of fundamental constant products
And given a set of constants, there are many ways to form a unit of a given unit. There are so many physical constants in use today, you have to be really interested to keep up on them. Here are some of the lesser-known constants:
Some of the lesser-known physical constants
Physical constants appear in so many equations of modern physics. Here is a selection of 100 simple physics formulas that contain the fundamental constants:
100 simple physics formulas that contain the fundamental constants
Of course, more complicated formulas also contain the physical constants. For instance, the gravitational constant appears (of course!) in the formula of the gravitational potentials of various objects, e.g. for the potential of a line segment and of a triangle:
Gravitational constant appears in formula of gravitational potentials of various objects
My friend Maurits Cornelis Escher loves these kinds of formulas. He recently showed me some variations of a few of his 3D pictures that show the equipotential surfaces of all objects in the pictures by triangulating all surfaces, then using the above formula—like his Escher solid. The graphic shows a cut version of two equipotential surfaces:
Equipotential surfaces of all objects in the pictures by triangulating all surfaces
I frequently stop by at Maurits Cornelis’, and often he has company—usually, it is Albrecht Dürer. The two love to play with shapes, surfaces, and polyhedra. They deform them, Kelvin-invert them, everse them, and more. Albrecht also likes the technique of smoothing with gravitational potentials, but he often does this with just the edges. Here is what a Dürer solid’s equipotential surfaces look like:
Dürer solid's equipotential surfaces
And here is a visualization of formulas that contain cα–hβ–Gγ in the exponent space αγβγγ. The size of the spheres is proportional to the number of formulas containing cα·hβ·Gγ; mousing over the balls in the attached notebook shows the actual formulas. We treat positive and negative exponents identically:
Visualization of formulas that contain c^alpha-h^beta-G^gamma in the exponant space of alpha-beta-gamma
One of my all-time favorite formulas is for the quantum-corrected gravitational force between two bodies, which contains my three favorite constants: the speed of light, the gravitational constants, and the Planck constant:
Quantum-corrected gravitational force between two bodies
Another of my favorite formulas is the one for the entropy of a black hole. It contains the Boltzmann constant in addition to c, h, and G:
Entropy of a black hole
And, of course, the second-order correction to the speed of light in a vacuum in the presence of an electric or magnetic field due to photon-photon scattering (ignoring a polarization-dependent constant). Even in very large electric and magnetic fields, the changes in the speed of light are very small:
In my lifetime, we did not yet understand the physical world enough to have come up with the idea of natural units. That took until 1874, when Stoney proposed for the first time natural units in his lecture to the British Science Association. And then, in his 1906–07 lectures, Planck made use of the now-called Planck units extensively, already introduced in his famous 1900 article in Annalen der Physik. Unfortunately, both these unit systems use the gravitational constant G prominently. It is a constant that we today cannot measure very accurately. As a result, also the values of the Planck units in the SI have only about four digits:
Use of Planck units
These units were never intended for daily use because they are either far too small or far too large compared to the typical lengths, areas, volumes, and masses that humans deal with on a daily basis. But why not base the units of daily use on such unchanging microscopic properties?
(Side note: The funny thing is that in the last 20 years Max Planck again doubts if his constant h is truly fundamental. He had hoped in 1900 to derive its value from a semi-classical theory. Now he hopes to derive it from some holographic arguments. Or at least he thinks he can derive the value of h/kB from first principles. I don’t know if he will succeed, but who knows? He is a smart guy and just might be able to.)
Many exact and approximate relations between fundamental constants are known today. Some more might be discovered in the future. One of my favorites is the following identity—within a small integer factor, is the value of the Planck constant potentially related to the size of the universe?
Is the value of the Planck constant potentially related to the size of the universe?
Another one is Beck’s formula, showing a remarkable coincidence (?):
Beck's formula
But nevertheless, in my time we never thought it would be possible to express the height of a giraffe through the fundamental constants. But how amazed I was nearly ten years ago, when looking through the newly arrived arXiv preprints to find a closed form for the height of the tallest running, breathing organism derived by Don Page. Within a factor of two he got the height of a giraffe (Brachiosaurus and Sauroposeidon don’t count because they can’t run) derived in terms of fundamental constants—I find this just amazing:
Typical height of a giraffe
I should not have been surprised, as in 1983 Press, Lightman, Peierls, and Gold expressed the maximal running speed of a human (see also Press’ earlier paper):
Maximal running speed of a human
In the same spirit, I really liked Burrows’ and Ostriker’s work on expressing the sizes of a variety of astronomical objects through fundamental constants only. For instance, for a typical galaxy mass we obtain the following expression:
Expression for a typical galaxy mass
This value is within a small factor from the mass of the Milky Way:
Mass of the Milky Way
But back to units, and fast forward another 100+ years to the second half of the twentieth century: the idea of basing units on microscopic properties of objects gained more and more ground.
Since 1967, the second has been defined through 9,192,631,770 periods of the light from the transition between the two hyperfine levels of the ground state of the cesium 133, and the meter has been defined since 1983 as the distance light travels in one second when we define the speed of light as the exact quantity 299,792,458 meters per second. To be precise, this definition is to be realized at rest, at a temperature of 0 K, and at sea level, as motion, temperature, and the gravitational potential influence the oscillation period and (proper) time. Ignoring the sea-level condition can lead to significant measurement errors; the center of the Earth is about 2.5 years younger than its surface due to differences in the gravitational potential.
Now, these definitions for the unit second and meter are truly equal for all people. Equal not just for people on Earth right now, but also for in the future and far, far away from Earth for any alien. (One day, the 9,192,631,770 periods of cesium might be replaced by a larger number of periods of another element, but that will not change its universal character.)
But if we wanted to ground all units in physical constants, which ones should we choose? There are often many, many ways to express a base unit through a set of constants. Using the constants from the table above, there are thirty (thirty!) ways to combine them to make a mass dimension:
Thirty ways to combine constants to make a mass dimension
Because of the varying precision of the constants, the combinations are also of varying precision (and of course, of different numerical values):
Combinations are of varying precision
Now the question is which constants should be selected to define the units of the metric system? Many aspects, from precision to practicality to the overall coherence (meaning there is no need for various prefactors in equations to compensate for unit factors) must be kept in mind. We want our formulas to look like F = m a, rather than containing explicit numbers such as in the Thanksgiving turkey cooking time formulas (assuming a spherical turkey):
Turkey cooking time formulas
Or in the PLANK formula (Max hates this name) for the calculation of indicated horsepower:
Calculation of indicated horsepower
Here in the clouds of heaven, we can’t use physical computers, so I am glad that I can use the more virtual Wolfram Open Cloud to do my calculations and mathematical experimentation. I have played for many hours with the interactive units-constants explorer below, and agree fully with the choices made by the International Bureau of Weights and Measures (BIPM), meaning the speed of light, the Planck constant, the elementary charge, the Avogadro constant, and the Boltzmann constant. I showed a preliminary version of this blog to Edgar, and he was very pleased to see this table based on his old paper:
Tables based on Edgar's paper
I want to mention that the most popular physical constant, the fine-structure constant, is not really useful for building units. Just by its special status as a unitless physical quantity, it can’t be directly connected to a unit. But it is, of course, one of the most important physical constants in our universe (and is probably only surpassed by the simple integer constant describing how many spatial dimensions our universe has). Often various dimensionless combinations can be found from a given set of physical constants because of relations between the constants, such as c2=1/(ε0 μ0). Here are some examples:
Various dimensionless combinations found from a given set of physical constants
But there is probably no other constant that Paul Adrien Maurice Dirac and I have discussed more over the last 32 years than the fine-structure constant α=e2/(4 𝜋 ε0 ħ c). Although up here we meet with the Lord regularly in a friendly and productive atmosphere, he still refuses to tell us a closed form of α . And he will not even tell us if he selected the same value for all times and all places. For the related topic of the values of the constants chosen, he also refuses to discuss fine tuning and alternative values. He says that he chose a beautiful expression, and one day we will find out. He gave some bounds, but they were not much sharper than the ones we know from the Earth’s existence. So, like living mortals, for now we must just guess mathematical formulas:
Conjectured exact forms of the fine-structure constant
Or guess combinations of constants:
Guessing combinations of constants
And here is one of my favorite coincidences:
Favorite coincidence
And a few more:
A few more coincidences
The rise in importance and usage of the physical constants is nicely reflected in the scientific literature. Here is a plot of how often (in publications per year) the most common constants appear in scientific publications from the publishing company Springer. The logarithmic vertical axis shows the exponential increase in how often physical constants are mentioned:
How often the most common constants appear in scientific publications from the publishing company Springer
While the fundamental constants are everywhere in physics and chemistry, one does not see them so much in newspapers, movies, or advertisements, as they deserve. I was very pleased to see the introduction of the Measures for Measure column in Nature recently.
Fundamental constants in Measures for Measure column
To give the physical constants the presence they deserve, I hope that before (or at least not long after) the redefinition we will see some interesting video games released that allow players to change the values of at least c, G, and h to see how the world around us would change if the constants had different values. It makes me want to play such a video game right now. With large values of h, not only could one build a world with macroscopic Schrödinger cats, but interpersonal correlations would also become much stronger. This could make the constants known to children at a young age. Such a video game would be a kind of twenty-first-century Mr. Tompkins adventure:
Mr. Tompkins
It will be interesting to see how quickly and efficiently the human brain will adapt to a possible life in a different universe. Initial research seems to be pretty encouraging. But maybe our world and our heaven are really especially fine-tuned.
The current SI and the issue with the kilogram
The modern system of units, the current SI has, in addition to the second, the meter, and the kilogram, other units. The ampere is defined as the force between two infinitely long wires, the kelvin through the triple point of water, the mole through the kilogram and carbon-12, and the candela through blackbody radiation. If you have never read the SI brochure, I strongly encourage you to do so.
Two infinitely long wires are surely macroscopic and do not fulfill Maxwell’s demand (but it is at least an idealized system), and de facto it defines the magnetic constant. And the triple point of water needs a macroscopic amount of water. This is not perfect, but it’s OK. Carbon-12 atoms are already microscopic objects. Blackbody radiation is again an ensemble of microscopic objects, but a very reproducible one. So some of the current SI fulfills in some sense Maxwell’s goals.
But most of my insomnia over the last 50 years has been caused by the kilogram. It caused me real headaches, and sometimes even nightmares, when we could not put it on the same level as the second and the meter.
In the year of my physical death (1799), the first prototype of a kilogram, a little platinum cylinder, was made. About 39.7 mm in height and 39.4 mm in diameter, this was for 75 years “the” kilogram. It was made from the forged platinum sponge made by Janety. Miller gives a lot of the details of this kilogram. It is today in the Archives nationales. In 1879, Johnson Matthey (in Britain—the country I fought with my ships!), using new melting techniques, made the material for three new kilogram prototypes. Because of a slightly higher density, these kilograms were slightly smaller in size, at 39.14 mm in height. The cylinder was called KIII and became the current international prototype kilogram K. Here is the last sentence from the preface of the mass determination of the the international prototype kilogram from 1885, introducing K:
A few kilograms were selected and carefully compared to our original kilogram; for the detailed measurements, see this book. All three kilograms had a mass less than 1 mg different from the original kilogram. But one stood out: it had a mass difference of less than 0.01 mg compared to the original kilogram. For a detailed history of the making of K, see Quinn. And so, still today, per definition, a kilogram is the mass of a small metal cylinder sitting in a safe at the International Bureau of Weights and Measures near Paris. (It’s technically actually not on French soil, but this is another issue.) In the safe, which needs three keys to be opened, under three glass domes, is a small platinum-iridium cylinder that defines what a kilogram is. For the reader’s geographical orientation, here is a map of Paris with the current kilogram prototype (in the southwest), our original one (in the northeast), both with a yellow border, and some other Paris visitor essentials:
Map of Paris with current kilogram prototype (in the southwest) and our original one (in the northeast)
In addition to being an artifact, it was so difficult to get access to the kilogram (which made me unhappy). Once a year, a small group of people checks if it is still there, and every few years its weight (mass) is measured. Of course, the result is, per definition and the agreement made at the first General Conference on Weights and Measures in 1889, exactly one kilogram.
Over the years the original kilogram prototype gained dozens of siblings in the form of other countries’ national prototypes, all of the same size, material, and weight (up to a few micrograms, which are carefully recorded). (I wish the internet had been invented earlier, so that I had a communication path to tell what happened with the stolen Argentine prototype 45; since then, it has been melted down.) At least, when they were made they had the same weight. Same material, same size, similarly stored—one would expect that all these cylinders would keep their weight. But this is not what history showed. Rather than all staying at the same weight, repeated measurements showed that virtually all other prototypes got heavier and heavier over the years. Or, more probable, the international prototype has gotten lighter.
From my place here in heaven I have watched many of these the comparisons with both great interest and concern. Comparing their weights (a.k.a. masses) is a big ordeal. First you must get the national prototypes to Paris. I have silently listened in on long discussions with TSA members (and other countries’ equivalents) when a metrologist comes with a kilogram of platinum, worth north of $50k in materials—and add another $20k for the making (in its cute, golden, shiny, special travel container that should only be opened in a clean room with gloves and mouth guard, and never ever touched by a human hand)—and explains all of this to the TSA. An official letter is of great help here. The instances that I have watched from up here were even funnier than the scene in the movie 1001 Grams.
Then comes a complicated cleaning procedure with hot water, alcohol, and UV light. The kilograms all lose weight in this process. And they are all carefully compared with each other. And the result is that with very high probability, “the” kilogram, our beloved international prototype kilogram (IPK), loses weight. This fact steals my sleep.
Here are the results from the third periodic verification (1988 to 1992). The graphic shows the weight difference compared to the international prototype:
Weight difference between countries' national kilograms versus the international prototype
For some newer measurements from the last two years, see this paper.
What I mean by “the” kilogram losing weight is the following. Per definition (independent of its “real objective” mass), the international prototype has a mass of exactly 1 kg. Compared with this mass, most other kilogram prototypes of the world seem to gain weight. As the other prototypes were made, using different techniques over more than 100 years, very likely the real issue is that the international prototype is losing weight. (And no, it is not because of Ceaușescu’s greed and theft of platinum that Romania’s prototype is so much lighter; in 1889 the Romanian prototype was already 953 μg lighter than the international prototype kilogram.)
Josiah Willard Gibbs, who has been my friend up here for more than 110 years, always mentions that his home country is still using the pound rather than the kilogram. His vote in this year’s election would clearly go to Bernie. But at least the pound is an exact fraction of the kilogram, so anything that will happen to the kilogram will affect the pound the same way:
The pound is an exact fraction of the kilogram
The new SI
But soon all my dreams and centuries-long hopes will come true and I can find sleep again. In 2018, two years from now, the greatest change in the history of units and measures since my work with my friend Laplace and the others will happen.
All units will be based on things that are accessible to everybody everywhere (assuming access to some modern physical instruments and devices).
The so-called new SI will reduce all of the seven base units to seven fundamental constants of physics or basic properties of microscopic objects. Down on Earth, they started calling them “reference constants.”
Some people also call the new SI quantum SI because of its dependence on the Planck constant h and the elementary charge e. In addition to the importance of the Planck constant h in quantum mechanics, the following two quantum effects are connecting h and e: the Josephson effect and its associated Josephson constant KJ = 2 e / h, and the quantum Hall effect with the von Klitzing constant RK = h / e2. The quantum metrological triangle: connecting frequency and electric current through a singe electron tunneling device, connecting frequency and voltage through the Josephson effect, and connecting voltage and electric current through the quantum Hall effect will be a beautiful realization of electric quantities. (One day in the future, as Penin has pointed out, we will have to worry about second-order QED effects, but this will be many years from now.)
The BIPM already has a new logo for the future International System of Units:
New logo for the future International System of Units
Concretely, the proposal is:
1. The second will continue to be defined through cesium atom microwave radiation.
2. The meter will continue to be defined through an exactly defined speed of light.
3. The kilogram will be defined through an exactly defined value of the Planck constant.
4. The ampere will be defined through an exactly defined value of the elementary charge.
5. The kelvin will be defined through an exactly defined value of the Boltzmann constant.
6. The mole will be defined through an exact (counting) value.
7. The candela will be defined through an exact value of the candela steradian-to-watt ratio at a fixed frequency (already now the case).
I highly recommend a reading of the draft of the new SI brochure. Laplace and I have discussed it a lot here in heaven, and (modulo some small issues) we love it. Here is a quick word cloud summary of the new SI brochure:
Word cloud summary of new SI brochure
Before I forget, and before continuing the kilogram discussion, some comments on the other units.
The second
I still remember when we discussed introducing metric time in the 1790s: a 10-hour day, with 100 minutes per hour, and 100 seconds per minute, and we were so excited by this prospect. In hindsight, this wasn’t such a good idea. The habits of people are sometimes too hard to change. And I am so glad I could get Albert Einstein interested in the whole metrology over the past 50 years. We have had so many discussions about the meaning of time and that the second measures local time, and the difference between measurable local time and coordinate time. But this is a discussion for another day. The uncertainty of a second is today less than 10−16. Maybe one day in the future, cesium will be replaced by aluminum or other elements to achieve 100 to 1,000 times smaller uncertainties. But this does not alter the spirit of the new SI; it’s just a small technical change. (For a detailed history of the second, see this article.)
Clearly, today’s definition of second is much better than one that depends on the Earth. At a time when stock market prices are compared at the microsecond level, the change of the length of a day due to earthquakes, polar melting, continental drift, and other phenomena over a century is quite large:
Change in the length of a day over time
The mole
I have heard some chemists complain that their beloved unit, the mole, introduced into the SI only in 1971, will become trivialized. In the currently used SI, the mole relates to an actual chemical, carbon-12. In the new SI, it will be just a count of objects. A true chemical equivalent to a baker’s dozen, the chemist’s dozen. Based on the Avogadro constant, the mole is crucial in connecting the micro world with the macro world. A more down-to-Earth definition of the mole matters for such quantitative values—for example, pH values. The second is the SI base unit of time; the mole is the SI base unit of the physical quantity, or amount of substance:
Mole is the SI base unit of the physical quantity
But not everybody likes the term “amount of substance.” Even this year (2016), alternative names are being proposed, e.g. stoichiometric amount. Over the last decades, a variety of names have been proposed to replace “amount of substance.” Here are some examples:
Alternative names for "amount of substance"
But the SI system only defines the unit “mole.” The naming of the physical quantity that is measured in moles is up to the International Union of Pure and Applied Chemistry.
For recent discussions from this year, see the article by Leonard, “Why Is ‘Amount of Substance’ So Poorly Understood? The Mysterious Avogadro Constant Is the Culprit!”, and the article by Giunta, “What’s in a Name? Amount of Substance, Chemical Amount, and Stoichiometric Amount.”
Wouldn’t it be nice if we could have made a “perfect cube” (number) that would represent the Avogadro number? Such a representation would be easy to conceptualize. This was suggested a few years back, and at the time was compatible with the value of the Avogadro constant, and would have been a cube of edge length 84,446,888 items. I asked Srinivasa Ramanujan, while playing a heavenly round of cricket with him and Godfrey Harold Hardy, his longtime friend, what’s special about 84,446,888, but he hasn’t come up with anything deep yet. He said that 84,446,888=2^3*17*620933, and that 620,933 appears starting at position 1,031,622 in the decimal digits of 𝜋, but I can’t see any metrological relevance in this. With the latest value of the Avogadro constant, no third power of an integer number falls into the possible values, so no wonder there is nothing special.
Here is the latest CODATA (Committee on Data for Science and Technology) value from the NIST Reference on Constants, Units, and Uncertainty:
Latest CODATA value from NIST Reference on Constants, Units, and Uncertainty
The candidate number 84,446,885 cubed is too small, and adding a one gives too large a number:
Candidate number 84,446,885
Interestingly, if we would settle for a body-centered lattice, with one additional atom per unit cell, then we could still maintain a cube interpretation:
Maintaining a cube interpretation with a body-centered lattice
A face-centered lattice would not work, either:
Using a face-centered lattice
But a diamond (silicon) lattice would work:
Diamond (silicon) lattice
To summarize:
Lattice summary
Here is a little trivia:
Sometime amid the heights of the Cold War, the accepted value of the Avogadro constant suddenly changed in the third digit! This was quite a change, considering that there is currently a lingering controversy regarding the discrepancy in the sixth digit. Can you explain the sudden decrease in Avogadro constant during the Cold War?
Do you know the answer? If not, see here or here.
But I am diverting from my main thread of thoughts. As I am more interested in the mechanical units anyway, I will let my old friend Antoine Lavoisier judge the new mole definition, as he was the chemist on our team.
The kelvin
Josiah Willard Gibbs even convinced me that temperature should be defined mechanically. I am still trying to understand John von Neumann’s opinion on this subject, but because I never fully understand his evening lectures on type II and type III factors, I don’t have a firm opinion on the kelvin. Different temperatures correspond to inequivalent representations of the algebras. As I am currently still working my way through Ruetsche’s book, I haven’t made my mind up on how to best define the kelvin from an algebraic quantum field theory point of view. I had asked John for his opinion of a first-principle evaluation of h / k based on KMS states and Tomita–Takesaki theory, and even he wasn’t sure about it. He told me some things about thermal time and diamond temperature that I didn’t fully understand.
And then there is the possibility of deriving the value of the Boltzmann constant. Even 40 years after the Koppe–Huber paper, it is not clear if it is possible. It is a subject I am still pondering, and I am taking various options into account. As mentioned earlier, the meaning of temperature and how to define its units are not fully clear to me. There is no question that the new definition of the kelvin will be a big step forward, but I don’t know if it will be the end of the story.
The ampere
This is one of the most direct, intuitive, and beautiful definitions in the new SI: the current is just the number of electrons that flow per second. Defining the value of the ampere through the number of elementary charges moved around is just a stroke of genius. When it was first suggested, Robert Andrews Millikan up here was so happy he invited many of us to an afternoon gathering in his yard. In practice (and in theoretical calculations), we have to exercise a bit more care, as we mainly measure the electric current of electrons in crystalline objects, and electrons are no longer “bare” electrons, but quasiparticles. But we’ve known since 1959, thanks to Walter Kohn, that we shouldn’t worry too much about this, and expect the charge of the electron in a crystal to be the same as the charge of a bare electron. As an elementary charge is a pretty small charge, the issue of measuring fractional charges as currents is not a practical one for now. I personally feel that Robert’s contribution to determining the value of the physical constants in the beginning of the twentieth century are not pointed out enough (Robert Andrews really knew what he was doing).
The candela
No, you will not get me started on my opinion the candela. Does it deserve to be a base unit? The whole story of human-centered physiological units is a complicated one. Obviously they are enormously useful. We all see and hear every day, even every second. But what if the human race continues to develop (in Darwin’s sense)? How will it fit together with our “for all time” mantra? I have my thoughts on this, but laying them out here and now would sidetrack me from my main discussion topic for today.
Why seven base units?
I also want to mention that originally I was very concerned about the introduction of some of the additional units that are in use today. In endless discussions with my chess partner Carl Friedrich Gauss here in heaven, he had originally convinced me that we can reduce all measurements of electric quantities to measurements of mechanical properties, and I already was pretty fluent in his CGS system, that originally I did not like it at all. But as a human-created unit system, it should be as useful as possible, and if seven units do the job best, it should be seven. In principle one could even eliminate a mass unit and express a mass through time and length. In addition to just being impractical, I strongly believe this is conceptually not the right approach. I recently discussed this with Carl Friedrich. He said he had the idea of just using time and length in the late 1820s, but abandoned such an approach. While alive, Carl Friedrich never had the opportunity to discuss the notion of mass as a synthetic a priori with Immanual, over the last century the two (Carl Friedrich and Immanuel) agreed on mass as an a priori (at least in this universe).
Our motto for the original metric system was, “For all time, for all people.” The current SI already realizes “for all people,” and by grounding the new SI in the fundamental constants of physics, the first promise “for all time” will finally become true. You cannot imagine what this means to me. If at all, fundamental constants seem to change maximally with rates on the order of 10–18 per year. This is many orders of magnitude away from the currently realized precisions for most units.
Granted, some things will get a bit numerically more cumbersome in the new SI. If we take the current CODATA values as exact values, then, for instance, the von Klitzing constant e2/h will be a big fraction:
von Klitzing contant with current CODATA values and exact values as a big fraction
The integer part of the last result is, of course, 25,812Ω. Now, is this a periodic decimal fraction or a terminating fraction? The prime factorization of the denominator tells us that it is periodic:
Prime factorization of the denominator tells us that it is periodic
Progress is good, but as happens so often, it comes at a price. While the new constant-based definitions of the SI units are beautiful, they are a bit harder to understand, and physics and chemistry teachers will have to come up with some innovative ways to explain the new definitions to pupils. (For recent first attempts, see this paper and this paper.)
And in how many textbooks have I seen that the value of the magnetic constant (permeability of the vacuum) μ0 is 4 𝜋 10–7 N / A2? The magnetic and the electric constants will in the new SI become measured quantities with an error term. Concretely, from the current exact value:
Current exact value
With the Planck constant h exactly and the elementary charge e exactly, the value of μ0 would incur the uncertainty of the fine-structure constant α. Fortunately, the dimensionless fine-structure constant α is one of the best-known constants:
Dimensionless fine-structure constant alpha
But so what? Textbook publishers will not mind having a reason to print new editions of all their books. They will like it—a reason to sell more new books.
With μ0 a measured quantity in the future, I predict one will see many more uses of the current underdog of the fundamental constant, the impedance of the vacuum Z in the future:
Impedance of the vacuum Z
I applaud all physicists and metrologist for the hard work they’ve carried out in continuation of my committee’s work over the last 225 years, which culminated in the new, physical constant-based definitions of the units. So do my fellow original committee members. These definitions are beautiful and truly forever.
(I know it is a bit indiscreet to reveal this, but Joseph Louis Lagrange told me privately that he regrets a bit that we did not introduce base and derived units as such in the 1790s. Now with the Planck constant being too important for the new SI, he thought we should have had a named base unit for the action (the time integral over his Lagrangian). And then make mass a derived quantity. While this would be the high road of classical mechanics, he does understand that a base unit for the action would not have become popular with farmers and peasants as a daily unit needed for masses.)
I don’t have the time today to go into any detailed discussion of the quarterly garden fests that Percy Williams Bridgman holds. As my schedule allows, I try to participate in every single one of them. It is also so intellectually stimulating to listen to the general discussions about the pros and cons of alternative unit systems. As you can imagine, Julius Wallot, Jan de Boer, Edward Guggenheim, William Stroud, Giovanni Giorgi, Otto Hölder, Rudolf Fleischmann, Ulrich Stille, Hassler Whitney, and Chester Page are, not unexpectedly, most outspoken at these parties. The discussion about coherence and completeness of unit systems and what is a physical quantity go on and on. At the last event, the discussion of whether probability is or is not a physical quantity went on for six hours, with no decision at the end. I suggested inviting Richard von Mises and Hans Reichenbach the next time. They might have something to contribute. At the parties, Otto always complains that mathematicians do not care enough anymore about units and unit systems as they did in the past, and he is so happy to see at least theoretical physicists pick up the topic from time to time, like the recent vector-based differentiation of physical quantities or the recent paper on the general structure of unit systems. And when he saw in an article from last year’s Dagstuhl proceedings that modern type theory met units and physical dimensions, he was the most excited he had been in decades.
Interestingly, basically the same discussions came up three years ago (and since then regularly) in the monthly mountain walks that Claude Shannon organizes. Leo Szilard argues that the “bit” has to become a base unit of the SI in the future. In his opinion, information as a physical quantity has been grossly underrated.
Once again: the new SI will be just great! There are a few more details that I would like to see changed. The current status of the radian and the steradian, which SP 811 now defines as derived units, saying, “The radian and steradian are special names for the number one that may be used to convey information about the quantity concerned.” But I see with satisfaction that the experts are discussing this topic recently quite in detail.
To celebrate the upcoming new SI here in heaven, we held a crowd-based fundraiser to celebrate this event. We raised enough funds to actually hire the master himself, Michelangelo. He will be making a sculpture. Some early sketches shown to the committee (I am fortunate to have the honorary chairmanship) are intriguing. I am sure it will be an eternal piece rivaling the David. One day every human will have the chance to see it (may it be a long time until then, dependent on your current age and your smoking habits). In addition to the constants and the units on their own, he plans to also work Planck himself, Boltzmann, and Avogadro into the sculpture, as these are the only three constants named after a person. Max was immediately accessible to model, but we are still having issues getting permission for Boltzmann to leave hell for a while to be a model. (Millikan and Fletcher were, understandably, a bit disappointed.) Ironically, it was Paul Adrien Maurice Dirac who came up with a great idea on how to convince Lucifer to get Boltzmann a Sabbath-ical. Ironically—because Paul himself is not so keen on the new SI because of the time dependence of the constants themselves over billions of years. But anyway, Paul’s clever idea was to point out that three fundamental constants, the Planck constant (6.62… × 1034 J · s), the Avogradro constant (6.02… × 1023 / mol), and the gravitational constant (6.6… × 10–11 m3 / (kg · s)) all start with the digit 6. And forming the number of the beast, 666, through three fundamental constants really made an impression on Lucifer, and I expect him to approve Ludwig’s temporary leave.
As an ex-mariner with an affinity for the oceans, I also pointed out to Lucifer that the mean ocean depth is exactly 66% of his height (2,443 m, according to a detailed re-analysis of Dante’s Divine Comedy). He liked this cute fact so much that he owes me a favor.
Mean depth of the oceans
So far, Lucifer insists on having the combination G(me / (h k))1/2 on the sculpture. For obvious reasons:
Lucifer's favorite combination
We will see how this discussion turns out. As there is really nothing wrong with this combination, even if it is not physically meaningful, we might agree to his demands.
All of the new SI 2018 committee up here has also already agreed on the music, we will play Wojciech Kilar’s Sinfonia de motu, which uniquely represents the physical constants as a musical composition using only the notes c, g, e, h (b-flat in the English-speaking world), and a (where a represents the cesium atom). And we could convince Rainer Maria Rilke to write a poem for the event. Needless to say, Wojciech, who has now been with us for more than two years, agreed, and even offered to compose an exact version.
Down on Earth, the arrival of the constants-based units will surely also be celebrated in many ways and many places. I am looking forward especially to the documentary The State of the Unit, which will be about the history of the kilogram and its redefinition through the Planck constant.
The path to the redefinition of the kilogram
As I already touched on, the most central point of the new SI will be the new definition of the kilogram. After all, the kilogram is the one artifact still present in the current SI that should be eliminated. In addition to the kilogram itself, many more derived units depend on it, say, the volt: 1 volt = 1 kilogram meters2/(ampere second3). Redefining the kilogram will make many (at least the theoretically inclined) electricians happy. Electrician have been using their exact conventional values for 25 years.
Exact conventional values
The value resulting from the convential value for the von Klitzing constant and the Josephson constant is very near to the latest CODATA value of the Planck constant:
Value resulting from the convential value for the von Klitzing constant and the Josephson constant
A side note on the physical quantity that the kilogram represents: The kilogram is the SI base unit for the physical quantity mass. Mass is most relevant for mechanics. Through Newton’s second law, Newton's second law, mass is intimately related to force. Assume we have understood length and time (and so also acceleration). What is next in line, force or mass? William Francis Magie wrote in 1912:
It would be very improper to dogmatize, and I shall accordingly have to crave your pardon for a frequent expression of my own opinion, believing it less objectionable to be egotistic than to be dogmatic…. The first question which I shall consider is that raised by the advocates of the dynamical definition of force, as to the order in which the concepts of force and mass come in thought when one is constructing the science of mechanics, or in other words, whether force or mass is the primary concept…. He [Newton] further supplies the measurement of mass as a fundamental quantity which is needed to establish the dynamical measure of force…. I cannot find that Lagrange gives any definition of mass…. To get the measure of mass we must start with the intuitional knowledge of force, and use it in the experiments by which we first define and then measure mass…. Now owing to the permanency of masses of matter it is convenient to construct our system of units with a mass as one of the fundamental units.
And Henri Poincaré in his Science and Method says, “Knowing force, it is easy to define mass; this time the definition should be borrowed from dynamics; there is no way of doing otherwise, since the end to be attained is to give understanding of the distinction between mass and weight. Here again, the definition should be led up to by experiments.”
While I always had an intuitive feeling for the meaning of mass in mechanics, up until the middle of the twentieth century, I never was able to put it into a crystal-clear statement. Only over the last decades, with the help of Valentine Bargmann and Jean-Marie Souriau did I fully understand the role of mass in mechanics: mass is an element in the second cohomology group of the Lie algebra of the Galilei group.
Mass as a physical quantity manifests itself in different domains of physics. In classical mechanics it is related to dynamics, in general relativity to the curvature of space, and in quantum field theory mass occurs as one of the Casimir operators of the Poincaré group.
In our weekly “Philosophy of Physics” seminar, this year led by Immanuel himself, Hans Reichenbach, and Carl Friedrich von Weizsäcker (Pascual Jordan suggested this Dreimännerführung of the seminars), we discuss the nature of mass in five seminars. The topics for this year’s series are mass superselection rules in nonrelativistic and relativistic theories, the concept and uses of negative mass, mass-time uncertainty relations, non-Higgs mechanisms for mass generation, and mass scaling in biology and sports. I need at least three days of preparation for each seminar, as the recommended reading list is more than nine pages—and this year they emphasize the condensed matter appearance of these phenomena a lot! I am really looking forward to this year’s mass seminars; I am sure that I will learn a lot about the nature of mass. I hope Ehrenfest, Pauli, and Landau don’t constantly interrupt the speakers, as they did last year (the talk on mass in general relativity was particularly bad). In the last seminar of the series, I have to give my talk. In addition to metabolic scaling laws, my favorite example is the following:
Shaking frequency of wet animal
I also intend to speak about the recently found predator-prey power laws.
For sports, I already have a good example inspired by Texier et al.: the relation between the mass of a sports ball and its maximal speed. The following diagram lets me conjecture speedmax~ln(mass). In the downloadable notebook, mouse over to see the sport, the mass of the ball, and the top speeds:
Mass of sports ball and its maximal speed
For the negative mass seminar, we had some interesting homework: visualize the trajectories of a classical point particle with complex mass in a double-well potential. As I had seen some of Bender’s papers on complex energy trajectories, the trajectories I got for complex masses did not surprise me:
Trajectories for complex masses
End side note.
The complete new definition reads thus: The kilogram, kg, is the unit of mass; its magnitude is set by fixing the numerical value of the Planck constant to be equal to exactly 6.62606X*10–34 when it is expressed in the unit s–1 · m2 · kg, which is equal to J · s. Here X stands for some digits soon to be explicitly stated that will represent the latest experimental values.
And the kilogram cylinder can finally retire as the world’s most precious artifact. I expect soon after this event the international kilogram prototype will finally be displayed in the Louvre. As the Louvre had been declared “a place for bringing together monuments of all the sciences and arts” in May 1791 and opened in 1793, all of us on the committee agreed that one day, when the original kilogram was to be replaced with something else, it would end up in the Louvre. Ruling the kingdom of mass for more than a century, IPK deserves its eternal place as a true monument of the sciences. I will make a bet—in a few years the retired kilogram, under its three glass domes, will become one of the Louvre’s most popular objects. And the queue that physicists, chemists, mathematicians, engineers, and metrologists will form to see it will, in a few years, be longer than the queue for the Mona Lisa. I would also make a bet that the beautiful miniature kilogram replicas will within a few years become the best-selling item in the Louvre’s museum store:
Miniature kilogram replicas
At the same time, as a metrologist, maybe the international kilogram prototype should stay where it is for another 50 years, so that it can be measured against a post-2018 kilogram made from an exact value of the Planck constant. Then we would finally know for sure if the international kilogram prototype is/was really losing weight.
Let me quickly recapitulate the steps toward the new “electronic” kilogram.
Intuitively, one could have thought to define the kilogram through the Avogadro constant as a certain number of atoms of, say, 12C. But because of binding energies and surface effects in a pile of carbon (e.g. diamond, graphene) made up from n = round(1 kg / m (12C)) atoms to realize the mass of one kilogram, all the n carbon-12 atoms would have to be well separated. Otherwise we would have a mass defect (remember Albert’s famous E = m c2 formula), and the mass equivalent for one kilogram or compact carbon versus the same number of individual, well-separated atoms is on the order of 10–10. Using the carbon-carbon bond energry, here is an estimation of the mass difference:
Estimation of the mass difference using the carbon-carbon bond energy
A mass difference of this size can for a 1 kg weight can be detected without problems with a modern mass comparator.
To give a sense of scale, this would be equivalent to the (Einsteinian) relativistic mass conversion of the energy expenditure of fencing for most of a day:
Energy expenditure of fencing for most of a day
This does not mean one could not define a kilogram through the mass of an atom or a fraction of it. Given the mass of a carbon atom m (12C), the atomic mass constant u = m (12C) / 12 follows, and using u we can easily connect to the Planck constant:
Connecting to the Planck constant
I read with great interest the recent comparison of using different sets of constants for the kilogram definition. Of course, if the mass of a 12C atom would be the defined value, then the Planck constant would become a measured, meaning nonexact, value. For me, having an exact value for the Planck constant is aesthetically preferable.
I have been so excited over the last decade following the steps toward the redefinition of the kilogram. For more than 20 years now, there has been a light visible at the end of the tunnel that would eliminate the one kilogram from its throne.
And when I read 11 years ago the article by Ian Mills, Peter Mohr, Terry Quinn, Barry Taylor, and Edwin Williams entitled “Redefinition of the Kilogram: A Decision Whose Time Has Come” in Metrologia (my second-favorite, late-morning Tuesday monthly read, after the daily New Arrivals, a joint publication of Hells’ Press, the Heaven Publishing Group, Jannah Media, and Deva University Press), I knew that soon my dreams would come true. The moment I read the Appendix A.1 Definitions that fix the value of the Planck constant h, I knew that was the way to go. While the idea had been floating around for much longer, it now became a real program to be implemented within a decade (give or take a few years).
James Clerk Maxwell wrote in his 1873 A Treatise on Electricity and Magnetism:
In framing a universal system of units we may either deduce the unit of mass in this way from those of length and time already defined, and this we can do to a rough approximation in the present state of science; or, if we expect soon to be able to determine the mass of a single molecule of a standard substance, we may wait for this determination before fixing a universal standard of mass.
Until around 2005, James Clerk thought that mass should be defined through the mass of an atom, but he came around over the last decade and now favors the definition through Planck’s constant.
In a discussion with Albert Einstein and Max Planck (I believe this was in the early seventies) in a Vienna-style coffee house (Max loves the Sachertorte and was so happy when Franz and Eduard Sacher opened their now-famous HHS (“Heavenly Hotel Sacher”)), Albert suggested using his two famous equations, E = m c2 and E = h f, to solve for m to get m = h f / c2. So, if we define h as was done with c, then we know m because we can measure frequencies pretty well. (Compton was arguing that this is just his equation rewritten, and Niels Bohr was remarking that we cannot really trust E = m c2 because of its relatively weak experimental verification, but I think he was just mocking Einstein, retaliating for some of the Solvay Conference Gedankenexperiment discussions. And of course, Bohr could not resist bringing up Δm Δt ~ h / c2 as a reason why we cannot define the second and the kilogram independently, as one implies an error in the other for any finite mass measurement time. But Léon Rosenfeld convinced Bohr that this is really quite remote, as for a day measurement time this limits the mass measurement precision to about 10–52 kg for a kilogram mass m.)
An explicit frequency equivalent f = m c2 / h is not practical for a mass of a kilogram as it would mean f ~ 1.35 1050 Hz, which is far, far too large for any experiment, dwarfing even the Planck frequency by about seven orders of magnitude. But some recent experiments from Berkeley from the last few years will maybe allow the use of such techniques at the microscopic scale. For more than 25 years now, in every meeting of the HPS (Heavenly Physical Society), Louis de Broglie insists on these frequencies being real physical processes, not just convenient mathematical tools.
So we need to know the value of the Planck constant h. Still today, the kilogram is defined as the mass of the IPK. As a result, we can measure the value of h using the current definition of the kilogram. Once we know the value of h to a few times 10–8 (this is basically where we are right now), we will then define a concrete value of h (very near or at the measured value). From then on, the kilogram will become implicitly defined through the value of the Planck constant. At the transition, the two definitions overlap in their uncertainties, and no discontinuities arise for any derived quantities. The international prototype has lost over the last 100 years on the order of 50 μg weight, which is a relative change of 5 × 10–8, so a value for the Planck constant with an error less than 2 × 10–8 does guarantee that the mass of objects will not change in a noticeable manner.
Looking back over the last 116 years, the value of the Planck constant gained about seven digits in precision. A real success story! In his paper “Ueber das Gesetz der Energieverteilung im Normalspectrum,” Max Planck for the first time used the symbol h, and gave for the first time a numerical value for the Planck constant (in a paper published a few months earlier, Max used the symbol b instead of h):
Excerpts from "Ueber das Gesetz der Energieverteilung im Normalspectrum"
(I had asked Max why he choose the symbol h, and he said he can’t remember anymore. Anyway, he said it was a natural choice in conjunction with the symbol k for the Boltzmann constant. Sometimes one reads today that h was used to express the German word Hilfsgrösse (auxiliary helping quantity); Max said that this was possible, and that he really doesn’t remember.)
In 1919, Raymond Thayer Birge published the first detailed comparison of various measurements of the Planck constant:
Various measurements of the Planck constant
From Planck’s value 6.55 × 10–34 J · s to the 2016 value 6.626070073(94) × 10–34 J · s, amazing measurement progress has been made.
The next interactive Demonstration allows you to zoom in and see the progress in measuring h over the last century. Mouse over the Bell curves (indicating the uncertainties of the values) in the notebook to see the experiment (for detailed discussions of many of the experiments for determining h, see this paper):
History of measurement of the Planck constant h
There have been two major experiments carried out over the last few years that my original group eagerly followed from the heavens: the watt balance experiment (actually, there is more than one of them—one at NIST, two in Paris, one in Bern…) and the Avogadro project. As a person who built mechanical measurements when I was alive, I personally love the watt balance experiment. Building a mechanical device that through a clever trick by Bryan Kibble eliminates an unknown geometric quantity gets my applause. The recent do-it-yourself LEGO home version is especially fun. With an investment of a few hundred dollars, everybody can measure the Planck constant at home! The world has come a long way since my lifetime. You could perhaps even check your memory stick before and after you put a file on it and see if its mass has changed.
But my dear friend Lavoisier, not unexpectedly, always loved the Avogadro project that determines the value of the Avogadro constant to high precision. Having 99.995% pure silicon makes the heart of a chemist beat faster. I deeply admire the efforts (and results) in making nearly perfect spheres out of them. The product of the Avogadro constant with the Planck constant NA h is related to the Rydberg constant. Fortunately, as we saw above, the Rydberg constant is known to about 11 digits; this means that knowing NA h to a high precision allows us to find the value of our beloved Planck constant h to high precision. In my lifetime, we started to understand the nature of the chemical elements. We knew nothing about isotopes yet—if you had told me that there are more than 20 silicon isotopes, I would not even have understood the statement:
Silicon isotopes
I am deeply impressed how mankind today can even sort the individual atoms by their neutron count. The silicon spheres of the Avogadro project are 99.995 % silicon 28—much, much more than the natural fraction of this isotope:
Silicon spheres of the Avogadro project
While the highest-end beam balances and mass comparators achieve precisions of 10–11, they can only compare masses but not realize one. Once the Planck constant has a fixed value using the watt balance, a mass can be constructively realized.
I personally think the Planck constant is one of the most fascinating constants. It reigns in the micro world and is barely visible at macroscopic scales directly, yet every macroscopic object holds together just because of it.
A few years ago I was getting quite concerned that our dream of eternal unit definitions would never be realized. I could not get a good night’s sleep when the value for the Planck constant from the watt balance experiments and the Avogadro silicon sphere experiments were far apart. How relieved I was to see that over the last few years the discrepancies were resolved! And now the working mass is again in sync with the international prototype.
Before ending, let me say a few words about the Planck constant itself. The Planck constant is the archetypal quantity that one expects to appear in quantum-mechanical phenomena. And when the Planck constant goes to zero, we recover classical mechanics (in a singular limit). This is what I myself thought until recently. But since I go to the weekly afternoon lectures of Vladimir Arnold, which he started giving in the summer of 2010 after getting settled up here, I now have strong reservations against such simplistic views. In his lecture about high-dimensional geometry, he covered the symplectic camel; since then, I view the Heisenberg uncertainty relations more as a classical relic than a quantum property. And since Werner Heisenberg recently showed me the Brodsky–Hoyer paper on ħ expansions, I have a much more reserved view on the BZO cube (the Bronshtein–Zelmanov–Okun cGh physics cube). And let’s not forget recent attempts to express quantum mechanics without reference to Planck’s constant at all. While we understand a lot about the Planck constant, its obvious occurrences and uses (such as a “conversion factor” between frequency and energy of photons in a vacuum), I think its deepest secrets have not yet been discovered. We will need a long ride on a symplectic camel into the deserts of hypothetical multiverses to unlock it. And Paul Dirac thinks that the role of the Planck constant in classical mechanics is still not well enough understood.
For the longest time, Max himself thought that in phase space (classical or through a Wigner transform), the minimal volume would be on the order of his constant h. As one of the fathers of quantum mechanics, Max follows the conceptual developments still today, especially the decoherence program. How amazed was he when sub-h structures were discovered 15 years ago. Eugene Wigner told me that he had conjectured such fine structures since the late 1930s. Since then, he has loved to play around with plotting Wigner functions for all kind of hypergeometric potentials and quantum carpets. His favorite is still the Duffing oscillator’s Wigner function. A high-precision solution of the time-dependent Schrödinger equations followed by a fractional Fourier transform-based Wigner function construction can be done in a straightforward and fast way. Here is how a Gaussian initial wavepacket looks after three periods of the external force. The blue rectangle is an area with in the x p plane of area h:
How Gaussian initial wavepacket looks after three periods of the external force
Here are some zoomed-in (colored according to the sign of the Wigner function) images of the last Wigner function. Each square has an area of 4 h and shows a variety of sub-Planckian structures:
Zoomed-in images of the last Wigner function
For me, the forthcoming definition of the kilogram through the Planck constant is a great intellectual and technological achievement of mankind. It represents two centuries of hard work at metrological institutes, and cements some of the deepest physical truths found in the twentieth century into the foundations of our unit system. At once a whole slew of units, unit conversions, and fundamental constants will be known with greater precision. (Make sure you get a new CODATA sheet after the redefinition and have the pocket card with the new constant values with you always until you know all the numbers by heart!) This will open a path to new physics and new technologies. In case you make your own experiments determining the values of the constants, keep in mind that the deadline for the inclusion of your values is July 1, 2017.
The transition from the platinum-iridium kilogram, historically denoted platinum-iridium kilogram, to the kilogram based on the Planck constant h can be nicely visualized graphically as a 3D object that contains both characters. Rotating it shows a smooth transition of the projection shape from platinum-iridium kilogram to h representing over 200 years of progress in metrology and physics:
3D object of both the platinum-iridium kilogram and the Planck constant h
The interested reader can order a beautiful, shiny, 3D-printed version here. It will make a perfect gift for your significant other (or ask your significant other to get you one) for Christmas to be ready for the 2018 redefinition, and you can show public support for it as a pendent or as earrings. (Available in a variety of metals, platinum is, obviously, the most natural choice, and it is under $5k—but the $82.36 polished silver version looks pretty nice too.)
Here are some images of golden-looking versions of KToh3D (up here, gold, not platinum is the preferred metal color):
Golden-looking versions of KToh3D
I realize that not everybody is (or can be) as excited as I am about these developments. But I see forward to the year 2018 when, after about 225 years, the kilogram as a material artifact will retire and a fundamental constant will replace it. The new SI will base our most important measurement standards on twenty-first century technology.
If the reader has questions or comments, don’t hesitate to email me at jeancharlesdeborda@gmail.com; based on recent advances in the technological implications of EPR=ER, we now have a much faster and more direct connection to Earth.
À tous les temps, à tous les peuples!
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Leave a Comment
Marc E. Janety
Monsieur, Je vous remercie pour le “shoutout”. –Janety, Saint-Germain des Prés, Paris, France
Posted by Marc E. Janety May 20, 2016 at 8:25 am
Jean-Charles de Borda
Michael, it was great working with you. One of my best collaborations.
Posted by Jean-Charles de Borda May 20, 2016 at 9:06 am
Very nice post! I’m going to read one of the suggested books!
Posted by Lou May 21, 2016 at 9:00 am
Citizen Trallès
Not sure that I agree with Citizen Borda on absolutely everything, but I respect his views (comme toujours).
Posted by Citizen Trallès May 21, 2016 at 9:53 am
Fantastic article Michael!
It will take me at least a month just to absorb its many interconnecting ideas.
It is amazing how even the 21st century is still dominated by the ideas of nineteenth century French, German, Italian and English mathematicians and physicists. Northern Europe and Russia still love their scientists, unfortunately not so much here in America anymore.
Posted by Michael May 22, 2016 at 1:15 pm
David Carraher
Michael, The breadth and depth of this article boggle the mind! I love the links you included for additional reading and consultation. We at the Poincaré Institute for Mathematics Education have been working hard to make the study of quantities integral to K-12 mathematics. There many useful leads in your article. Thanks.
Posted by David Carraher May 23, 2016 at 2:25 pm
I started reading… kept on reading… and realized that this is a mini-novel. Added to Pocket! Can’t wait to finish it.
Posted by constantine June 2, 2016 at 1:14 pm
Rob Ryan
Posted by Rob Ryan June 25, 2016 at 3:27 pm
Posted by Raza July 27, 2016 at 3:24 am
Bruce Camber
What a mensch!
“What’s this?” was my first response,
then when you settle into the rhyme and scheme,
it makes you smile, laugh, and learn.
Sweet. Oh, so sweet.
A wonderful on the edge of knowledge!
Posted by Bruce Camber February 28, 2017 at 3:04 pm
Bruce Camber
That last sentence should read:
A wonderful walk on the edges of knowledge!
Posted by Bruce Camber February 28, 2017 at 3:06 pm
Leave a comment in reply to Bruce Camber |
c891ec0c594f24d8 | Normal mode
From Wikipedia, the free encyclopedia
(Redirected from Modes of vibration)
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A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at the fixed frequencies. These fixed frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies. A physical object, such as a building, bridge, or molecule, has a set of normal modes and their natural frequencies that depend on its structure, materials and boundary conditions. When relating to music, normal modes of vibrating instruments (strings, air pipes, drums, etc.) are called "harmonics" or "overtones".
A flash photo of a cup of black coffee vibrating in normal modes
General definitions[edit]
In physics and engineering, for a dynamical system according to wave theory, a mode is a standing wave state of excitation, in which all the components of the system will be affected sinusoidally under a specified fixed frequency.
Because no real system can perfectly fit under the standing wave framework, the mode concept is taken as a general characterization of specific states of oscillation, thus treating the dynamic system in a linear fashion, in which linear superposition of states can be performed.
As classical examples, there are:
• In a mechanical dynamical system, a vibrating rope is the most clear example of a mode, in which the rope is the medium, the stress on the rope is the excitation, and the displacement of the rope with respect to its static state is the modal variable.
• In an acoustic dynamical system, a single sound pitch is a mode, in which the air is the medium, the sound pressure in the air is the excitation, and the displacement of the air molecules is the modal variable.
• In a structural dynamical system, a high tall building oscillating under its most flexural axis is a mode, in which all the material of the building -under the proper numerical simplifications- is the medium, the seismic/wind/environmental solicitations are the excitations and the displacements are the modal variable.
• In an electrical dynamical system, a resonant cavity made of thin metal walls, enclosing a hollow space, for a particle accelerator is a pure standing wave system, and thus an example of a mode, in which the hollow space of the cavity is the medium, the RF source (a Klystron or another RF source) is the excitation and the electromagnetic field is the modal variable.
• The concept of normal modes also finds application in optics, quantum mechanics, and molecular dynamics.
Most dynamical system can be excited under several modes. Each mode is characterized by one or several frequencies, according to the modal variable field. For example, a vibrating rope in the 2D space is defined by a single-frequency (1D axial displacement), but a vibrating rope in the 3D space is defined by two frequencies (2D axial displacement).
For a given amplitude on the modal variable, each mode will store a specific amount of energy, because of the sinusoidal excitation.
From all the modes of a dynamical system, the normal or dominant mode of a system will be the mode storing the minimum amount of energy for a given amplitude of the modal variable. Or equivalently, for a given stored amount of energy, will be the mode imposing the maximum amplitude of the modal variable.
Mode numbers[edit]
In a system with two or more dimensions, such as the pictured disk, each dimension is given a mode number. Using polar coordinates, we have a radial coordinate and an angular coordinate. If one measured from the center outward along the radial coordinate one would encounter a full wave, so the mode number in the radial direction is 2. The other direction is trickier, because only half of the disk is considered due to the anti-symmetric (also called skew-symmetry) nature of a disk's vibration in the angular direction. Thus, measuring 180° along the angular direction you would encounter a half wave, so the mode number in the angular direction is 1. So the mode number of the system is 2–1 or 1–2, depending on which coordinate is considered the "first" and which is considered the "second" coordinate (so it is important to always indicate which mode number matches with each coordinate direction).
In linear systems each mode is entirely independent of all other modes. In general all modes have different frequencies (with lower modes having lower frequencies) and different mode shapes.
When expanded to a two dimensional system, these nodes become lines where the displacement is always zero. If you watch the animation above you will see two circles (one about halfway between the edge and center, and the other on the edge itself) and a straight line bisecting the disk, where the displacement is close to zero. In an idealized system these lines equal zero exactly, as shown to the right.
In mechanical systems[edit]
Coupled oscillators[edit]
Consider two equal bodies (not affected by gravity), each of mass m, attached to three springs, each with spring constant k. They are attached in the following manner, forming a system that is physically symmetric:
Coupled Harmonic Oscillator.svg
If one denotes acceleration (the second derivative of x(t) with respect to time) as , the equations of motion are:
Substituting these into the equations of motion gives us:
And in matrix representation:
If the matrix on the left is invertible, the unique solution is the trivial solution (A1A2) = (x1x2) = (0,0). The non trivial solutions are to be found for those values of ω whereby the matrix on the left is singular i.e. is not invertible. It follows that the determinant of the matrix must be equal to 0, so:
Solving for , we have two positive solutions:
The first normal mode is:
Which corresponds to both masses moving in the same direction at the same time. This mode is called antisymmetric.
The second normal mode is:
This corresponds to the masses moving in the opposite directions, while the center of mass remains stationary. This mode is called symmetric.
Standing waves[edit]
The general form of a standing wave is:
Elastic solids[edit]
According to quantum theory, the mean energy of a normal vibrational mode of a crystalline solid with characteristic frequency ν is:
By knowing the thermodynamic formula,
the entropy per normal mode is:
The free energy is:
which, for kT >> , tends to:
In quantum mechanics[edit]
In quantum mechanics, a state of a system is described by a wavefunction which solves the Schrödinger equation. The square of the absolute value of , i.e.
Usually, when involving some sort of potential, the wavefunction is decomposed into a superposition of energy eigenstates, each oscillating with frequency of . Thus, one may write
In seismology[edit]
See also[edit]
• Blevins, Robert D. (2001). Formulas for natural frequency and mode shape (Reprint ed.). Malabar, Florida: Krieger Pub. ISBN 978-1575241845.
• Tzou, H.S.; Bergman, L.A., eds. (2008). Dynamics and Control of Distributed Systems. Cambridge [England]: Cambridge University Press. ISBN 978-0521033749.
• Shearer, Peter M. (2009). Introduction to seismology (2nd ed.). Cambridge: Cambridge University Press. pp. 231–237. ISBN 9780521882101.
External links[edit] |
91b085093d0ad73e | A Big State-Space of Consciousness
Kenneth Shinozuka of Blank Horizons asks: Andrés, how long do you think it’ll take to fully map out the state space of consciousness? A thousand or a million years?
The state-space of consciousness is unimaginably large (and yet finite)
I think we will discover the core principles of a foundational theory of consciousness within a century or so. That is, we might find plausible solutions to Mike Johnsons’ 8 subproblems of consciousness and experimentally verify a specific formal theory of consciousness before 2100. That said, there is a very large distance between proving a certain formal theory of consciousness and having a good grasp of the state-space of consciousness.
Knowing Maxwell’s equations gives you a formal theory of electromagnetism. But even then, photons are hidden as an implication of the formalism; you need to do some work to find them in it. And that’s the tip of the iceberg; you would also find hidden in the formalism an array of exotic electromagnetic behavior that arise in unusual physical conditions such as those produced by metamaterials. The formalism is a first step to establish the fundamental constraints for what’s possible. What follows is filling in the gaps between the limits of physical possibility, which is a truly fantastical enterprise considering the range of possible permutations.Island_of_Stability_derived_from_Zagrebaev
A useful analogy here might be: even though we know all of the basic stable elements and many of their properties, we have only started mapping out the space of possible small molecules (e.g. there are ~10^60 bioactive drugs that have never been tested), and have yet to even begin the project in earnest of understanding what proteins can do. Or consider the number of options there are to make high-entropy alloys (alloys made with five or more metals). Or all the ways in which snowflakes of various materials can form, meaning that even when you are studying a single material it can form crystal structures of an incredibly varied nature. And then take into account the emergence of additional collective properties: physical systems can display a dazzling array of emergent exotic effects, from superconductivity and superradiance to Bose-Einstein condensates and fusion chain reactions. Exploring the state-space of material configurations and their emergent properties entails facing a combinatorial explosion of unexpected phenomena.
And this is the case in physics even though we know for a fact that there are only a bit over a hundred possible building blocks (i.e. the elements).
In the province of the mind, we do not yet have even that level of understanding. When it comes to the state-space of consciousness we do not have a corresponding credible “periodic table of qualia”. The range of possible experiences in normal everyday life is astronomical. Even so, the set of possible human sober experiences is a vanishing fraction of the set of possible DMT trips, which is itself a vanishing fraction of the set of possible DMT + LSD + ketamine + TMS + optogenetics + Generalized Wada Test + brain surgery experiences. Brace yourself for a state-space that grows supergeometrically with each variable you introduce.
If we are to truly grasp the state-space of consciousness, we should also take into account non-human animal qualia. And then further still, due to dual-aspect monism, we will need to go into things like understanding that high-entropy alloys themselves have qualia, and then Jupiter Brains, and Mike’s Fraggers, and Black Holes, and quantum fields in the inflation period, and so on. This entails a combinatorial explosion of the likes I don’t believe anyone is really grasping at the moment. We are talking about a monumental “monster” state-space far beyond the size of even the wildest dreams of full-time dreamers. So, I’d say -honestly- I think that mapping out the state-space of consciousness is going to take millions of years.
But isn’t the state-space of consciousness infinite, you ask?
Alas, no. There are two core limiting factors here – one is the speed of light (which entails the existence of gravitational collapse and hence limits to how much matter you can arrange in complex ways before a black hole arises) and the second one is quantum (de)coherence. If phenomenal binding requires fundamental physical properties such as quantum coherence, there will be a maximum limit to how much matter you can bind into a unitary “moment of experience“. Who knows what the limit is! But I doubt it’s the size of a galaxy – perhaps it is more like a Jupiter Brain, or maybe just the size of a large building. This greatly reduces the state-space of consciousness; after all, something finite, no matter how large, is infinitely smaller than something infinite!
But what if reality is continuous? Doesn’t that entail an infinite state-space?
I do not think that the discrete/continuous distinction meaningfully impacts the size of the state-space of consciousness. The reason is that at some point of degree of similarity between experiences you get “just noticeable differences” (JNDs). Even with the tiniest hint of true continuity in consciousness, the state-space would be infinite as a result. But the vast majority of those differences won’t matter: they can be swept under the rug to an extent because they can’t actually be “distinguished from the inside”. To make a good discrete approximation of the state-space, we would just need to divide the state-space into regions of equal area such that their diameter is a JND.15965332_1246551232103698_2088025318638395407_n
In summary, the state-space of consciousness is insanely large but not infinite. While I do think it is possible that the core underlying principles of consciousness (i.e. an empirically-adequate formalism) will be discovered this century or the next, I do not anticipate a substantive map of the state-space of consciousness to be available anytime soon. A truly comprehensive map would, I suspect, be only possible after millions of years of civilizational investment on the task.
Toy Story 4 – 20 Movie Review
by Frank Yang (post)
Toy Story was my favorite movie growing up. I had the entire collection. The fact that the toys could think without a brain made me explore dualism, monism, the existence of God, and nofap (see next slides). Toy Story 4 is weird as fuck for a Disney movie. Due to the psychedelics Renaissance and mass awakening, people want everything to be increasingly trippy.
Forky is probably the weirdest Disney character of all time. It’s like the producers and writers got together in a room and brainstormed, “emm how can we strip a character down to its bare minimum materially to embody pure Being and Nothingness, and all he wanted was to go back to the Source”? The toys are on their way to realizations with Woody and Buzz self-inquiring about the distinction between the voice in their heads and the voice from their voice box. Woody eventually dissolved the part of his ego attached to having an owner by the end, but is still asleep because he still believes he is a toy.
Toy Story 15 will eventually be about enlightenment.
Buzz will be the first one to wake up since he always had a hunch that he wasn’t a toy and is obsessed with infinity.
Buzz screams at Woody, ‘You are not a toy, but infinite Consciousness.’
Maybe by Toy Story 18 both the toys and their owners can break through the layer of illusion that separates them and finally rejoice and communicate with each other after realizing they are made up of the same pixels, floating inside the same bubble of Divine imagination with limitless possibilities.
In Toy Story 20, every object inside the screen – toys and kids, trees, shoes and houses all combine force and congeal their pixels into One, exists the screen and merge with the audience in an Absolute orgy where all dualities collapse.
We’re left with an empty screen; the good old Witness. McDonald manufactures blank screen keychains to go along with happy meals and all the kids thought they got woke.
But when my grandson brings one home I’ll smash the little screens with a hammer “the Observer is the last stand against freedom!” I yell.
And then he was enlightened.
#toystory代購 #jumpman #thefappening
Three Interesting Math Problems to Work on While on LSD
1. Let P be a simple polygon with n>3 sides. A simple polygon is a polygon that does not self-intersect, but it is not necessarily convex. Prove that no matter the shape of P, there is always a diagonal (a segment that connects two vertices of P without intersecting any of its sides) that divides P into two polygons, both of which have at least n/3 sides.
2. Let A and B be two points in the plane. Using only a compass and a straightedge, find the point C which is the exact middle point between A and B. Now do the same thing, but using only a compass.
3. There are 17 point-sized light-houses in the plane. Assume that each of these lighthouses can shine light in any direction with an angle of 2*pi/17. Prove that no matter the position of each lighthouse, it is always possible to choose the angles at which they shine their light such that every point in the plane is illuminated (point-sized lighthouses don’t cast shadows).
In Selective Enhancement of Specific Capacities Through Psychedelic Training, Willis Harman and James Fadiman outline the results of a study about the potential use of psychedelics for problem solving. In the study, scientists, engineers, mathematicians, and designers took either 100 micrograms of LSD or 200mg of mescaline and worked on a problem they were personally invested in and which they had not been able to solve for at least 3 months. According to Fadiman, 9 out of 10 participants came up with a solution to the problem that was validated by the participant’s professional colleagues.
The three problems above are not easy, but they are also not insanely difficult. If it means anything to you, their level of difficulty might be around that of a problem 1 or 4 of an IMO, with the advantage that you do not need any fancy math to solve them (high-school math is more than sufficient). I do not know if solving these problems is easier or harder on psychedelics, but I figure I would share them as possible Schelling points for “challenging math problems to think about while on psychedelics” to see if anyone reports benefits from such a setup. I personally like these problems, and I can assure you that they do have interesting and clever solutions. Assuming you are already planning on taking a psychedelic substance in the future: I would recommend trying to solve one of these problems for at least 1 hour while sober, and then setting aside at least 30 minutes (preferably 1 hour) while on a psychedelic and giving it your full attention. Please let me know if you either solve the problem or get an interesting insight from such an exercise. I am particularly curious to hear about *what aspects* of the psychedelic state seemed to be either beneficial or detrimental in solving these problems. Even if you do not solve the problem, you may be able to think about it in new ways and derive useful insights. Again, if you do so, let me know as well.
Realms as Interpretive Lenses
How people in different (Buddhist) realms interpret pain:
1) Heavenly Realm / God Realm: Pain is impermanent. It’s a trick of the mind. A method to help us wake up and realize who we truly are. [said while peacefully unaware of actual pain due to the formidable amounts of pleasure and distractions on hand]
2) Asura Realm / Titan Realm: Pain is a tool to succeed. It is a challenge to be overcome at a personal level, and a weapon to be used against one’s enemies. If I didn’t suffer intensely for the things that I achieved, would they mean anything? [said while experiencing intense cravings for social recognition and the need to feel superbly significant]
3) Animal Realm: Pain is the separation from my pleasures of the day to day. My morning coffee, interrupted by a call. My conversations with a friend, when someone’s bad luck is brought up. The annoying commercials in-between the chunks of TV I like. [said while snoozing the alarm for the 4th time in a row]
4) Hell Realm: Pain is reality in and of itself. Life is suffering. And if it isn’t at the moment, that’s just temporary good luck. Happiness is merely the absence of suffering; happiness is therefore as good as nonexistence. [said while waiting in the ER while experiencing a kidney stone]
5) Hungry Ghost Realm: Pain is realizing that only 10 out of the 15 people who RSVP’ed to my party showed up. It is the feeling of noticing that the Pringles are almost gone. The feeling that you get when you make out with someone and only get to 2nd base when you could have gotten to 3rd or 4th. [said while scrolling Reddit for the 3rd hour in a row].
6) Human Realm: Pain is a healthy signaling mechanism. When you look at it scientifically, it is just a negative reinforcement signal that propagates throughout your nervous system in order to prevent the chain of causes that led to the current state. It’s nothing to worry about, just as you shouldn’t worry about the weather or the shape of the solar system. [said while dispassionately reading a neuroscience textbook].
See also: Traps of the God Realm and The Penfield Mood Organ
The Resonance and Vibration of [Phenomenal] Objects
25th February, 2007
Michael: Why would it have a perfume, Silver Star?
It is up to you!
It depends what you put into that box.
Jane: Do objects emit music as well?
Jane: Yes, thank you.
Silver Star: Was there anything else?
Jane: No.
Gaining Root Access to Your World Simulation
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Happiness and Harmony: A Marriage Made in Heaven
But What if Our Scientific World Picture is Wrong?
Me: Thank you!
– Conversation with God, Burning Man 2017
Infinite Bliss!
A Non-Circular Solution to the Measurement Problem: If the Superposition Principle is the Bedrock of Quantum Mechanics Why Do We Experience Definite Outcomes?
Source: Quora question – “Scientifically speaking, how serious is the measurement problem concerning the validity of the various interpretations in quantum mechanics?
David Pearce responds [emphasis mine]:
It’s serious. Science should be empirically adequate. Quantum mechanics is the bedrock of science. The superposition principle is the bedrock of quantum mechanics. So why don’t we ever experience superpositions? Why do experiments have definite outcomes? “Schrödinger’s cat” isn’t just a thought-experiment. The experiment can be done today. If quantum mechanics is complete, then microscopic superpositions should rapidly be amplified via quantum entanglement into the macroscopic realm of everyday life.
Copenhagenists are explicit. The lesson of quantum mechanics is that we must abandon realism about the micro-world. But Schrödinger’s cat can’t be quarantined. The regress spirals without end. If quantum mechanics is complete, the lesson of Schrödinger’s cat is that if one abandons realism about a micro-world, then one must abandon realism about a macro-world too. The existence of an objective physical realm independent of one’s mind is certainly a useful calculational tool. Yet if all that matters is empirical adequacy, then why invoke such superfluous metaphysical baggage? The upshot of Copenhagen isn’t science, but solipsism.
There are realist alternatives to quantum solipsism. Some physicists propose that we modify the unitary dynamics to prevent macroscopic superpositions. Roger Penrose, for instance, believes that a non-linear correction to the unitary evolution should be introduced to prevent superpositions of macroscopically distinguishable gravitational fields. Experiments to (dis)confirm the Penrose-Hameroff Orch-OR conjecture should be feasible later this century. But if dynamical collapse theories are wrong, and if quantum mechanics is complete (as most physicists believe), then “cat states” should be ubiquitous. This doesn’t seem to be what we experience.
Everettians are realists, in a sense. Unitary-only QM says that there are quasi-classical branches of the universal wavefunction where you open an infernal chamber and see a live cat, other decohered branches where you see a dead cat; branches where you perceive the detection of a spin-up electron that has passed through a Stern–Gerlach device, other branches where you perceive the detector recording a spin-down electron; and so forth. I’ve long been haunted by a horrible suspicion that unitary-only QM is right, though Everettian QM boggles the mind (cfUniverseSplitter). Yet the heart of the measurement problem from the perspective of empirical science is that one doesn’t ever see superpositions of live-and-dead cats, or detect superpositions of spin-up-and-spin-down electrons, but only definite outcomes. So the conjecture that there are other, madly proliferating decohered branches of the universal wavefunction where different versions of you record different definite outcomes doesn’t solve the mystery of why anything anywhere ever seems definite to anyone at all. Therefore, the problem of definite outcomes in QM isn’t “just” a philosophical or interpretational issue, but an empirical challenge for even the most hard-nosed scientific positivist. “Science” that isn’t empirically adequate isn’t science: it’s metaphysics. Some deeply-buried background assumption(s) or presupposition(s) that working physicists are making must be mistaken. But which? To quote the 2016 International Workshop on Quantum Observers organized by the IJQF,
“…the measurement problem in quantum mechanics is essentially the determinate-experience problem. The problem is to explain how the linear quantum dynamics can be compatible with the existence of our definite experience. This means that in order to finally solve the measurement problem it is necessary to analyze the observer who is physically in a superposition of brain states with definite measurement records. Indeed, such quantum observers exist in all main realistic solutions to the measurement problem, including Bohm’s theory, Everett’s theory, and even the dynamical collapse theories. Then, what does it feel like to be a quantum observer?
Indeed. Here I’ll just state rather than argue my tentative analysis.
Monistic physicalism is true. Quantum mechanics is formally complete. There is no consciousness-induced collapse the wave function, no “hidden variables”, nor any other modification or supplementation of the unitary Schrödinger dynamics. The wavefunction evolves deterministically according to the Schrödinger equation as a linear superposition of different states. Yet what seems empirically self-evident, namely that measurements always find a physical system in a definite state, is false(!) The received wisdom, repeated in countless textbooks, that measurements always find a physical system in a definite state reflects an erroneous theory of perception, namely perceptual direct realism. As philosophers (e.g. the “two worlds” reading of Kant) and even poets (“The brain is wider than the sky…”) have long realised, the conceptual framework of perceptual direct realism is untenable. Only inferential realism about mind-independent reality is scientifically viable. Rather than assuming that superpositions are never experienced, suspend disbelief and consider the opposite possibility. Only superpositions are ever experienced. “Observations” are superpositions, exactly as unmodified and unsupplemented quantum mechanics says they should be: the wavefunction is a complete representation of the physical state of a system, including biological minds and the pseudo-classical world-simulations they run. Not merely “It is the theory that decides what can be observed” (Einstein); quantum theory decides the very nature of “observation” itself. If so, then the superposition principle underpins one’s subjective experience of definite, well-defined classical outcomes (“observations”), whether, say, a phenomenally-bound live cat, or the detection of a spin-up electron that has passed through a Stern–Gerlach device, or any other subjectively determinate outcome. If one isn’t dreaming, tripping or psychotic, then within one’s phenomenal world-simulation, the apparent collapse of a quantum state (into one of the eigenstates of the Hermitian operator associated with the relevant observable in accordance with a probability calculated as the squared absolute value of a complex probability amplitude) consists of fleeting uncollapsed neuronal superpositions within one’s CNS. To solve the measurement problem, the neuronal vehicle of observation and its subjective content must be distinguished. The universality of the superposition principle – not its unexplained breakdown upon “observation” – underpins one’s classical-seeming world-simulation. What naïvely seems to be the external world, i.e. one’s egocentric world-simulation, is what linear superpositions of different states feel like “from the inside”: the intrinsic nature of the physical. The otherwise insoluble binding problem in neuroscience and the problem of definite outcomes in QM share a solution.
Yes, for sure: this minimum requirement for a successful resolution of the mystery is satisfied (“If at first the idea is not absurd, then there is no hope for it”– Einstein, again). The raw power of environmentally-induced decoherence in a warm environment like the CNS makes the conjecture intuitively flaky. Assuming unitary-only QM, the effective theoretical lifetime of neuronal “cat states” in the CNS is less than femtoseconds. Neuronal superpositions of distributed feature-processors are intuitively just “noise”, not phenomenally-bound perceptual objects. At best, the idea that sub-femtosecond neuronal superpositions could underpin our experience of law-like classicality is implausible. Yet we’re not looking for plausible theories but testable theories. Every second of selection pressure in Zurek’s sense (cf. “Quantum Darwinism”) sculpting one’s neocortical world-simulation is more intense and unremitting than four billion years of evolution as conceived by Darwin. My best guess is that interferometry will disclose a perfect structural match. If the non-classical interference signature doesn’t yield a perfect structural match, then dualism is true.
Is the quantum-theoretic version of the intrinsic nature argument for non-materialist physicalism – more snappily, “Schrödinger’s neurons” – a potential solution to the measurement problem? Or a variant of the “word salad” interpretation of quantum mechanics?
Sadly, I can guess.
But if there were one experiment that I could do, one loophole I’d like to see closed via interferometry, then this would be it.
Psychedelic Turk: A Platform for People on Altered States of Consciousness
An interesting variable is how much external noise is optimal for peak processing. Some, like Kafka, insisted that “I need solitude for my writing; not ‘like a hermit’ – that wouldn’t be enough – but like a dead man.” Others, like von Neumann, insisted on noisy settings: von Neumann would usually work with the TV on in the background, and when his wife moved his office to a secluded room on the third floor, he reportedly stormed downstairs and demanded “What are you trying to do, keep me away from what’s going on?” Apparently, some brains can function with (and even require!) high amounts of sensory entropy, whereas others need essentially zero. One might look for different metastable thresholds and/or convergent cybernetic targets in this case.
– Mike Johnson, A future for neuroscience
My drunk or high Tweets are my best work.
– Joe Rogan, Vlog#18
Mechanical Turk is a service that makes outsourcing simple tasks to a large number of people extremely easy. The only constraint is that the tasks outsourced ought to be the sort of thing that can be explained and performed within a browser in less than 10 minutes, which in practice is not a strong constraint for most tasks you would outsource anyway. This service is in fact a remarkably effective way to accelerate the testing of digital prototypes at a reasonable price.
I think the core idea has incredible potential in the field of interest we explore in this blog. Namely, consciousness research and the creation of consciousness technologies. Mechanical Turk is already widely used in psychology, but its usefulness could be improved further. Here is an example: Imagine an extension to Mechanical Turk in which one could choose to have the tasks completed (or attempted) by people in non-ordinary states of consciousness.
Demographic Breakdown
With Mechanical Turk you can already ask for people who belong to specific demographic categories to do your task. For example, some academics are interested in the livelihoods of people within certain ages, NLP researchers might need native speakers of a particular language, and people who want to proof-read a text may request users who have completed an undergraduate degree. The demographic categories are helpful but also coarse. In practice they tend to be used as noisy proxies for more subtle attributes. If we could multiply the categories, which ones would give the highest bang for the buck? I suspect there is a lot of interesting information to be gained from adding categories like personality, cognitive organization, and emotional temperament. What else?
States of Consciousness as Points of View
One thing to consider is that the value of a service like Mechanical Turk comes in part from the range of “points of view” that the participants bring. After all, ensemble models that incorporate diverse types of modeling approaches and datasets usually dominate in real-world machine learning competitions (e.g. Kaggle). Analogously, for a number of applications, getting feedback from someone who thinks differently than everyone already consulted is much more valuable than consulting hundreds of people similar to those already queried. Human minds, insofar as they are prediction machines, can be used as diverse models. A wide range of points of view expands the perspectives used to draw inferences, and in many real-world conditions this will be beneficial for the accuracy of an aggregated prediction. So what would a radical approach to multiplying such “points of view” entail? Arguably a very efficient way of doing so would involve people who inhabit extraordinarily different states of consciousness outside the “typical everyday” mode of being.
Jokingly, I’d very much like to see the “wisdom of the crowds enhanced with psychedelic points of view” expressed in mainstream media. I can imagine an anchorwoman on CNN saying: “according to recent polls 30% of people agree that X, now let’s break this down by state of consciousness… let’s see what the people on acid have to say… ” I would personally be very curious to hear how “the people on acid” are thinking about certain issues relative to e.g. a breakdown of points of view by political affiliation. Leaving jokes aside, why would this be a good idea? Why would anyone actually build this?
I posit that a “Mechanical Turk for People on Psychedelics” would benefit the requesters, the workers, and outsiders. Let’s start with the top three benefits for requesters: better art and marketing, enhanced problem solving, and accelerating the science of consciousness. For workers, the top reason would be making work more interesting, stimulating, and enjoyable. And from the point of view of outsiders, we could anticipate some positive externalities such as improved foundational science, accelerated commercial technology development, and better prediction markets. Let’s dive in:
Benefits to Requesters
Art and Marketing
A reason why a service like this might succeed commercially comes from the importance of understanding one’s audience in art and marketing. For example, if one is developing a product targeted to people who have a hangover (e.g. “hangover remedies”), one’s best bet would be to see how people who actually are hungover resonate with the message. Asking people who are drunk, high on weed, on empathogenic states, on psychedelics, specific psychiatric medications, etc. could certainly find its use in marketing research for sports, comedy, music shows, etc.
Basically, when the product is consumed in the sort of events in which people frequently avoid being sober for the occasion, doing market research on the same people sober might produce misleading results. What percent of concert-goers are sober the entire night? Or people watching the World Cup final? Clearly, a Mechanical Turk service with diverse states of consciousness has the potential to improve marketing epistemology.
On the art side, people who might want to be the next Alex Grey or Android Jones would benefit from prototyping new visual styles on crowds of people who are on psychedelics (i.e. the main consumers of such artistic styles).
As an aside, I would like to point out that in my opinion, artists who create audio or images that are expected to be consumed by people in altered states of consciousness have some degree of responsibility in ensuring that they are not particularly upsetting to people in such states. Indeed, some relatively innocent sounds and images might cause a lot of anxiety or trigger negative states in people on psychedelics due to the way they are processed in such states. With a Mechanical Turk for psychedelics, artists could reduce the risk of upsetting festival/concert goers who partake in psychedelic perception by screening out offending stimuli.
Problem Solving
On a more exciting note, there are a number of indications that states of consciousness as alien as those induced by major psychedelics are at times computationally suited to solve information processing tasks in competitive ways. Here are two concrete examples: First, in the sixties there was some amount of research performed on psychedelics for problem solving. A notorious example would be the 1966 study conducted by Willis Harman & James Fadiman in which mescaline was used to aid scientists, engineers, and designers in solving concrete technical problems with very positive outcomes. And second, in How to Secretly Communicate with People on LSD we delved into ways that messages could be encoded in audio-visual stimuli in such a way that only people high on psychedelics could decode them. We called this type of information concealment Psychedelic Cryptography:
These examples are just proofs of concept that there probably are a multitude of tasks for which minds under various degrees of psychedelic alteration outperform those minds in sober states. In turn, it may end up being profitable to recruit people on such states to complete your tasks when they are genuinely better at them than the sober competition. How to know when to use which state of consciousness? The system could include an algorithm that samples people from various states of consciousness to identify the most promising states to solve your particular problem and then assign the bulk of the task to them.
All of this said, the application I find the most exciting is…
Accelerating the Science of Consciousness
The psychedelic renaissance is finally getting into the territory of performance enhancement in altered states. For example, there is an ongoing study that evaluates how microdosing impacts how one plays Go, and another one that uses a self-blinding protocol to assess how microdosing influences cognitive abilities and general wellbeing.
A whole lot of information about psychedelic states can be gained by doing browser experiments with people high on them. From sensory-focused studies such as visual psychophysics and auditory hedonics to experiments involving higher-order cognition and creativity, internet-based studies of people on altered states can shed a lot of light on how the mind works. I, for one, would love to estimate the base-rate of various wallpaper symmetry groups in psychedelic visuals (cf. Algorithmic Reduction of Psychedelic States), and to study the way psychedelic states influence the pleasantness of sound. There may be no need to spend hundreds of thousands of dollars in experiments that study those questions when the cost of asking people who are on psychedelics to do tasks can be amortized by having them participate in hundreds of studies on e.g. a single LSD session.
Quantifying Bliss (36)
17 wallpaper symmetry groups
This kind of research platform would also shed light on how experiences of mental illness compare with altered states of consciousness and allow us to place the effects of common psychiatric medications on a common “map of mental states”. Let me explain. While recreational materials tend to produce the largest changes to people’s conscious experience, it should go without saying that a whole lot of psychiatric medications have unusual effects on one’s state of consciousness. For example: Most people have a hard time pin-pointing the effect of beta blockers on their experience, but it is undeniable that such compounds influence brain activity and there are suggestions that they may have long-term mood effects. Many people do report specific changes to their experience related to beta blockers, and experienced psychonauts can often compare their effects to other drugs that they may use as benchmarks. By conducting psychophysical experiments on people who are taking various major psychoactives, one would get an objective benchmark for how the mind is altered along a wide range of dimensions by each of these substances. In turn, this generalized Mechanical Turk would enable us to pin-point where much more subtle drugs fall along on this space (cf. State-Space of Drug Effects).
In other words, this platform may be revolutionary when it comes to data collection and bench-marking for psychiatric drugs in general. That said, since these compounds are more often than not used daily for several months rather than briefly or as needed, it would be hard to see how the same individual performs a certain task while on and off the medicine. This could be addressed by implementing a system allowing requesters to ask users for follow up experiments if/when the user changes his or her drug regimen.
Benefit to Users
As claimed earlier on, we believe that this type of platform would make work more enjoyable, stimulating, and interesting for many users. Indeed, there does seem to be a general trend of people wanting to contribute to science and culture by sharing their experiences in non-ordinary states of consciousness. For instance, the wonderful artists at r/replications try to make accurate depiction of various unusual states of consciousness for free. There is even an initiative to document the subjective effects of various compounds by grounding trip reports on a subjective effects index. The point being that if people are willing to share their experience and time on psychedelic states of consciousness for free, chances are that they will not complain if they can also earn money with this unusual hobby.
LSD replication (source: r/replications)
We also know from many artists and scientists that normal everyday states of consciousness are not always the best for particular tasks. By expanding the range of states of consciousness with economic advantages, we would be allowing people to perform at their best. You may not be allowed to conduct your job while high at your workplace even if you perform it better that way. But with this kind of platform, you would have the freedom to choose the state of consciousness that optimizes your performance and be paid in kind.
Possible Downsides
It is worth mentioning that there would be challenges and negative aspects too. In general, we can probably all agree that it would suck to have to endure advertisement targeted to your particular state of consciousness. If there is a way to prevent this from happening I would love to hear it. Unfortunately, I assume that marketing will sooner or later catch on to this modus operandi, and that a Mechanical Turk for people on altered states would be used for advertisement before anything else. Making better targeted ads, it turns out, is a commercially viable way of bootstrapping all sorts of novel systems. But better advertisement indeed puts us at higher risk of being taken over by pure replicators in the broader scope, so it is worth being cautious with this application.
In the worst case scenario, we discover that very negative states of consciousness dominate other states in the arena of computational efficiency. In this scenario, the abilities useful to survive in the mental economy of the future happen to be those that employ suffering in one way or another. In that case, the evolutionary incentive gradients would lead to terrible places. For example, future minds might end up employing massive amounts of suffering to “run our servers”, so to speak. Plus, these minds would have no choice because if they don’t then they would be taken over by other minds that do, i.e. this is a race to the bottom. Scenarios like this have been considered before (1, 2, 3), and we should not ignore their warning signs.
Of course this can only happen if there are indeed computational benefits to using consciousness for information processing tasks to begin with. At Qualia Computing we generally assume that the unity of consciousness confers unique computational benefits. Hence, I would expect any outright computational use of states of consciousness is likely to involve a lot of phenomenal binding. Hence, at the evolutionary limit, conscious super-computers would probably be super-sentient. That said, the optimal hedonic tone of the minds with the highest computational efficiency is less certain. This complex matter will be dealt with elsewhere.
Concluding Discussion
Reverse Engineering Systems
A lot of people would probably agree that a video of Elon Musk high on THC may have substantially higher value than many videos of him sober. A lot of this value comes from the information gained about him by having a completely new point of view (or projection) of his mind. Reverse-engineering systems involves doing things to them to change the way they operate in order to try to reconstruct how they are put together. The same is true for the mind and the computational benefits of consciousness more broadly.
The Cost of a State of Consciousness
Another important consideration would be cost assignment for different states of consciousness. I imagine that the going rates for participants on various states would highly depend on the kind of application and profitability of these states. The price would reach a stable point that balances the usability of a state of consciousness for various tasks (demand) and its overall supply.
For problem solving in some specialized applications, for example, I could imagine “mathematician on DMT” to be a high-end sort of state of consciousness priced very highly. For example, foundational consciousness research and phenomenological studies might find such participants to be extremely valuable, as they might be helpful analyzing novel mathematical ideas and using their mathematical expertise to describe the structure of such experiences (cf. Hyperbolic Geometry of DMT Experiences).
Unfortunately, if the demand for high-end rational psychonauts never truly picks up, one might expect that people who could become professional rational psychonauts will instead work for Google or Facebook or some other high-paying company. More so, due to Lemon Markets people who do insist on hiring rational psychonauts will most likely be disappointed. Sasha Shulgin and his successors will probably only participate in such markets if the rewards are high enough to justify using their precious time on novel alien states of consciousness to do your experiment rather than theirs.
In the ideal case this type of platform might function as a spring-board to generate a critical mass of active rational psychonauts who could do each other’s experiments and replicate the results of underground researchers.
Quality Metrics
Accurately matching the task with the state of consciousness would be critical. For example, you might not necessarily want someone who is high on a large dose of acid to take a look at your tax returns*. Perhaps for mundane tasks one would want people who are on states of optimal arousal (e.g. modafinil). As mentioned earlier, a system that identifies the most promising states of consciousness for your task would be a key feature of the platform.
If we draw inspiration from the original service, we could try to make an analogous system to “Mechanical Turk Masters“. Here the service charges a higher price for requesting people who have been vetted as workers who produce high quality output. To be a Master one needs to have a high task-approval rating and have completed an absurd number of them. Perhaps top score boards and public requester prices for best work would go a long way in keeping the quality of psychedelic workers at a high level.
In practice, given the population base of people who would use this service, I would predict that to a large extent the most successful tasks in terms of engagement from the user-base will be those that have nerd-sniping qualities.** That is, make tasks that are especially fun to complete on psychedelics (and other altered states) and you would most likely get a lot of high quality work. In turn, this platform would generate the best outcomes when the tasks submitted are both fun and useful (hence benefiting both workers and requesters alike).
Keeping Consciousness Useful
Finally, we think that this kind of platform would have a lot of long-term positive externalities. In particular, making a wider range of states of consciousness economically useful goes in the general direction of keeping consciousness relevant in the future. In the absence of selection pressures that make consciousness economically useful (and hence useful to stay alive and reproduce), we can anticipate a possible drift from consciousness being somewhat in control (for now) to a point where only pure replicators matter.
Bonus content
If you are concerned with social power in a post-apocalyptic landscape, it is important that you figure out a way to induce psychedelic experiences in such a way that they cannot easily be used as weapons. E.g. it would be key to only have physiologically safe (e.g. not MDMA) and low-potency (e.g. not LSD) materials in a Mad Max scenario. For the love of God, please avoid stockpiling compounds that are both potent and physiologically dangerous (e.g. NBOMes) in your nuclear bunker! Perhaps high-potency materials could still work out if they are blended in hard-to-separate ways with fillers, but why risk it? I assume that becoming a cult leader would not be very hard if one were the only person who can procure reliable mystical experiences for people living in most post-apocalyptic scenarios. For best results make sure that the cause of the post-apocalyptic state of the world is a mystery to its inhabitants, such as in the documentary Gurren Lagann, and the historical monographs written by Philip K. Dick.
*With notable exceptions. For example, some regular cannabis users do seem to concentrate better while on manageable amounts of THC, and if the best tax attorney in your vicinity willing to do your taxes is in this predicament, I’d suggest you don’t worry too much about her highness.
**If I were a philosopher of science I would try to contribute a theory for scientific development based on nerd-sniping. Basically, how science develops is by the dynamic way in which scientists at all points are following the nerd-sniping gradient. Scientists are typically people who have their curiosity lever all the way to the top. It’s not so much that they choose their topics strategically or at random. It’s not so much a decision as it is a compulsion. Hence, the sociological implementation of science involves a collective gradient ascent towards whatever is nerd-sniping given the current knowledge. In turn, the generated knowledge from the intense focus on some area modifies what is known and changes the nerd-sniping landscape, and science moves on to other topics.
The Qualia Explosion
Extract from “Humans and Intelligent Machines: Co-Evolution, Fusion or Replacement?” (talk) by David Pearce
Supersentience: Turing plus Shulgin?
Compared to the natural sciences (cf. the Standard Model in physics) or computing (cf. the Universal Turing Machine), the “science” of consciousness is pre-Galilean, perhaps even pre-Socratic. State-enforced censorship of the range of subjective properties of matter and energy in the guise of a prohibition on psychoactive experimentation is a powerful barrier to knowledge. The legal taboo on the empirical method in consciousness studies prevents experimental investigation of even the crude dimensions of the Hard Problem, let alone locating a solution-space where answers to our ignorance might conceivably be found.
Singularity theorists are undaunted by our ignorance of this fundamental feature of the natural world. Instead, the Singularitarians offer a narrative of runaway machine intelligence in which consciousness plays a supporting role ranging from the minimal and incidental to the completely non-existent. However, highlighting the Singularity movement’s background assumptions about the nature of mind and intelligence, not least the insignificance of the binding problem to AGI, reveals why FUSION and REPLACEMENT scenarios are unlikely – though a measure of “cyborgification” of sentient biological robots augmented with ultrasmart software seems plausible and perhaps inevitable.
If full-spectrum superintelligence does indeed entail navigation and mastery of the manifold state-spaces of consciousness, and ultimately a seamless integration of this knowledge with the structural understanding of the world yielded by the formal sciences, then where does this elusive synthesis leave the prospects of posthuman superintelligence? Will the global proscription of radically altered states last indefinitely?
Social prophecy is always a minefield. However, there is one solution to the indisputable psychological health risks posed to human minds by empirical research into the outlandish state-spaces of consciousness unlocked by ingesting the tryptaminesphenylethylaminesisoquinolines and other pharmacological tools of sentience investigation. This solution is to make “bad trips” physiologically impossible – whether for individual investigators or, in theory, for human society as a whole. Critics of mood-enrichment technologies sometimes contend that a world animated by information-sensitive gradients of bliss would be an intellectually stagnant society: crudely, a Brave New World. On the contrary, biotech-driven mastery of our reward circuitry promises a knowledge explosion in virtue of allowing a social, scientific and legal revolution: safe, full-spectrum biological superintelligence. For genetic recalibration of hedonic set-points – as distinct from creating uniform bliss – potentially leaves cognitive function and critical insight both sharp and intact; and offers a launchpad for consciousness research in mind-spaces alien to the drug-naive imagination. A future biology of invincible well-being would not merely immeasurably improve our subjective quality of life: empirically, pleasure is the engine of value-creation. In addition to enriching all our lives, radical mood-enrichment would permit safe, systematic and responsible scientific exploration of previously inaccessible state-spaces of consciousness. If we were blessed with a biology of invincible well-being, exotic state-spaces would all be saturated with a rich hedonic tone.
Until this hypothetical world-defining transition, pursuit of the rigorous first-person methodology and rational drug-design strategy pioneered by Alexander Shulgin in PiHKAL and TiHKAL remains confined to the scientific counterculture. Investigation is risky, mostly unlawful, and unsystematic. In mainstream society, academia and peer-reviewed scholarly journals alike, ordinary waking consciousness is assumed to define the gold standard in which knowledge-claims are expressed and appraised. Yet to borrow a homely-sounding quote from Einstein, “What does the fish know of the sea in which it swims?” Just as a dreamer can gain only limited insight into the nature of dreaming consciousness from within a dream, likewise the nature of “ordinary waking consciousness” can only be glimpsed from within its confines. In order to scientifically understand the realm of the subjective, we’ll need to gain access to all its manifestations, not just the impoverished subset of states of consciousness that tended to promote the inclusive fitness of human genes on the African savannah.
Why the Proportionality Thesis Implies an Organic Singularity
So if the preconditions for full-spectrum superintelligence, i.e. access to superhuman state-spaces of sentience, remain unlawful, where does this roadblock leave the prospects of runaway self-improvement to superintelligence? Could recursive genetic self-editing of our source code repair the gap? Or will traditional human personal genomes be policed by a dystopian Gene Enforcement Agency in a manner analogous to the coercive policing of traditional human minds by the Drug Enforcement Agency?
Even in an ideal regulatory regime, the process of genetic and/or pharmacological self-enhancement is intuitively too slow for a biological Intelligence Explosion to be a live option, especially when set against the exponential increase in digital computer processing power and inorganic AI touted by Singularitarians. Prophets of imminent human demise in the face of machine intelligence argue that there can’t be a Moore’s law for organic robots. Even the Flynn Effect, the three-points-per-decade increase in IQ scores recorded during the 20th century, is comparatively puny; and in any case, this narrowly-defined intelligence gain may now have halted in well-nourished Western populations.
However, writing off all scenarios of recursive human self-enhancement would be premature. Presumably, the smarter our nonbiological AI, the more readily AI-assisted humans will be able recursively to improve our own minds with user-friendly wetware-editing tools – not just editing our raw genetic source code, but also the multiple layers of transcription and feedback mechanisms woven into biological minds. Presumably, our ever-smarter minds will be able to devise progressively more sophisticated, and also progressively more user-friendly, wetware-editing tools. These wetware-editing tools can accelerate our own recursive self-improvement – and manage potential threats from nonfriendly AGI that might harm rather than help us, assuming that our earlier strictures against the possibility of digital software-based unitary minds were mistaken. MIRI rightly call attention to how small enhancements can yield immense cognitive dividends: the relatively short genetic distance between humans and chimpanzees suggests how relatively small enhancements can exert momentous effects on a mind’s general intelligence, thereby implying that AGIs might likewise become disproportionately powerful through a small number of tweaks and improvements. In the post-genomic era, presumably exactly the same holds true for AI-assisted humans and transhumans editing their own minds. What David Chalmers calls the proportionality thesis, i.e. increases in intelligence lead to proportionate increases in the capacity to design intelligent systems, will be vindicated as recursively self-improving organic robots modify their own source code and bootstrap our way to full-spectrum superintelligence: in essence, an organic Singularity. And in contrast to classical digital zombies, superficially small molecular differences in biological minds can result in profoundly different state-spaces of sentience. Compare the ostensibly trivial difference in gene expression profiles of neurons mediating phenomenal sight and phenomenal sound – and the radically different visual and auditory worlds they yield.
Compared to FUSION or REPLACEMENT scenarios, the AI-human CO-EVOLUTION conjecture is apt to sound tame. The likelihood our posthuman successors will also be our biological descendants suggests at most a radical conservativism. In reality, a post-Singularity future where today’s classical digital zombies were superseded merely by faster, more versatile classical digital zombies would be infinitely duller than a future of full-spectrum supersentience. For all insentient information processors are exactly the same inasmuch as the living dead are not subjects of experience. They’ll never even know what it’s like to be “all dark inside” – or the computational power of phenomenal object-binding that yields illumination. By contrast, posthuman superintelligence will not just be quantitatively greater but also qualitatively alien to archaic Darwinian minds. Cybernetically enhanced and genetically rewritten biological minds can abolish suffering throughout the living world and banish experience below “hedonic zero” in our forward light-cone, an ethical watershed without precedent. Post-Darwinian life can enjoy gradients of lifelong blissful supersentience with the intensity of a supernova compared to a glow-worm. A zombie, on the other hand, is just a zombie – even if it squawks like Einstein. Posthuman organic minds will dwell in state-spaces of experience for which archaic humans and classical digital computers alike have no language, no concepts, and no words to describe our ignorance. Most radically, hyperintelligent organic minds will explore state-spaces of consciousness that do not currently play any information-signalling role in living organisms, and are impenetrable to investigation by digital zombies. In short, biological intelligence is on the brink of a recursively self-amplifying Qualia Explosion – a phenomenon of which digital zombies are invincibly ignorant, and invincibly ignorant of their own ignorance. Humans too of course are mostly ignorant of what we’re lacking: the nature, scope and intensity of such posthuman superqualia are beyond the bounds of archaic human experience. Even so, enrichment of our reward pathways can ensure that full-spectrum biological superintelligence will be sublime.
Image Credit: MohammadReza DomiriGanji
Everything in a Nutshell
Image credit: Joseph Matthias Young
Why I think the Foundational Research Institute should rethink its approach
by Mike Johnson
The following is my considered evaluation of the Foundational Research Institute, circa July 2017. I discuss its goal, where I foresee things going wrong with how it defines suffering, and what it could do to avoid these problems.
TL;DR version: functionalism (“consciousness is the sum-total of the functional properties of our brains”) sounds a lot better than it actually turns out to be in practice. In particular, functionalism makes it impossible to define ethics & suffering in a way that can mediate disagreements.
I. What is the Foundational Research Institute?
The Foundational Research Institute (FRI) is a Berlin-based group that “conducts research on how to best reduce the suffering of sentient beings in the near and far future.” Executive Director Max Daniel introduced them at EA Global Boston as “the only EA organization which at an organizational level has the mission of focusing on reducing s-risk.” S-risks are, according to Daniel, “risks where an adverse outcome would bring about suffering on an astronomical scale, vastly exceeding all suffering that has existed on Earth so far.”
Essentially, FRI wants to become the research arm of suffering-focused ethics, and help prevent artificial general intelligence (AGI) failure-modes which might produce suffering on a cosmic scale.
What I like about FRI:
While I have serious qualms about FRI’s research framework, I think the people behind FRI deserve a lot of credit- they seem to be serious people, working hard to build something good. In particular, I want to give them a shoutout for three things:
• First, FRI takes suffering seriously, and I think that’s important. When times are good, we tend to forget how tongue-chewingly horrific suffering can be. S-risks seem particularly horrifying.
• Second, FRI isn’t afraid of being weird. FRI has been working on s-risk research for a few years now, and if people are starting to come around to the idea that s-risks are worth thinking about, much of the credit goes to FRI.
• Third, I have great personal respect for Brian Tomasik, one of FRI’s co-founders. I’ve found him highly thoughtful, generous in debates, and unfailingly principled. In particular, he’s always willing to bite the bullet and work ideas out to their logical end, even if it involves repugnant conclusions.
What is FRI’s research framework?
FRI believes in analytic functionalism, or what David Chalmers calls “Type-A materialism”. Essentially, what this means is there’s no ’theoretical essence’ to consciousness; rather, consciousness is the sum-total of the functional properties of our brains. Since ‘functional properties’ are rather vague, this means consciousness itself is rather vague, in the same way words like “life,” “justice,” and “virtue” are messy and vague.
Brian suggests that this vagueness means there’s an inherently subjective, perhaps arbitrary element to how we define consciousness:
Analytic functionalism looks for functional processes in the brain that roughly capture what we mean by words like “awareness”, “happy”, etc., in a similar way as a biologist may look for precise properties of replicators that roughly capture what we mean by “life”. Just as there can be room for fuzziness about where exactly to draw the boundaries around “life”, different analytic functionalists may have different opinions about where to define the boundaries of “consciousness” and other mental states. This is why consciousness is “up to us to define”. There’s no hard problem of consciousness for the same reason there’s no hard problem of life: consciousness is just a high-level word that we use to refer to lots of detailed processes, and it doesn’t mean anything in addition to those processes.
Finally, Brian argues that the phenomenology of consciousness is identical with the phenomenology of computation:
I know that I’m conscious. I also know, from neuroscience combined with Occam’s razor, that my consciousness consists only of material operations in my brain — probably mostly patterns of neuronal firing that help process inputs, compute intermediate ideas, and produce behavioral outputs. Thus, I can see that consciousness is just the first-person view of certain kinds of computations — as Eliezer Yudkowsky puts it, “How An Algorithm Feels From Inside“. Consciousness is not something separate from or epiphenomenal to these computations. It is these computations, just from their own perspective of trying to think about themselves.
In other words, consciousness is what minds compute. Consciousness is the collection of input operations, intermediate processing, and output behaviors that an entity performs.
And if consciousness is all these things, so too is suffering. Which means suffering is computational, yet also inherently fuzzy, and at least a bit arbitrary; a leaky high-level reification impossible to speak about accurately, since there’s no formal, objective “ground truth”.
II. Why do I worry about FRI’s research framework?
In short, I think FRI has a worthy goal and good people, but its metaphysics actively prevent making progress toward that goal. The following describes why I think that, drawing heavily on Brian’s writings (of FRI’s researchers, Brian seems the most focused on metaphysics):
Note: FRI is not the only EA organization which holds functionalist views on consciousness; much of the following critique would also apply to e.g. MIRI, FHI, and OpenPhil. I focus on FRI because (1) Brian’s writings on consciousness & functionalism have been hugely influential in the community, and are clear enough *to* criticize; (2) the fact that FRI is particularly clear about what it cares about- suffering- allows a particularly clear critique about what problems it will run into with functionalism; (3) I believe FRI is at the forefront of an important cause area which has not crystallized yet, and I think it’s critically important to get these objections bouncing around this subcommunity.
Objection 1: Motte-and-bailey
Brian: “Consciousness is not a thing which exists ‘out there’ or even a separate property of matter; it’s a definitional category into which we classify minds. ‘Is this digital mind really conscious?’ is analogous to ‘Is a rock that people use to eat on really a table?’ [However,] That consciousness is a cluster in thingspace rather than a concrete property of the world does not make reducing suffering less important.”
The FRI model seems to imply that suffering is ineffable enough such that we can’t have an objective definition, yet sufficiently effable that we can coherently talk and care about it. This attempt to have it both ways seems contradictory, or at least in deep tension.
Indeed, I’d argue that the degree to which you can care about something is proportional to the degree to which you can define it objectively. E.g., If I say that “gnireffus” is literally the most terrible thing in the cosmos, that we should spread gnireffus-focused ethics, and that minimizing g-risks (far-future scenarios which involve large amounts of gnireffus) is a moral imperative, but also that what is and what and isn’t gnireffus is rather subjective with no privileged definition, and it’s impossible to objectively tell if a physical system exhibits gnireffus, you might raise any number of objections. This is not an exact metaphor for FRI’s position, but I worry that FRI’s work leans on the intuition that suffering is real and we can speak coherently about it, to a degree greater than its metaphysics formally allow.
Max Daniel (personal communication) suggests that we’re comfortable with a degree of ineffability in other contexts; “Brian claims that the concept of suffering shares the allegedly problematic properties with the concept of a table. But it seems a stretch to say that the alleged tension is problematic when talking about tables. So why would it be problematic when talking about suffering?” However, if we take the anti-realist view that suffering is ‘merely’ a node in the network of language, we have to live with the consequences of this: that ‘suffering’ will lose meaning as we take it away from the network in which it’s embedded (Wittgenstein). But FRI wants to do exactly this, to speak about suffering in the context of AGIs, simulated brains, even video game characters.
We can be anti-realists about suffering (suffering-is-a-node-in-the-network-of-language), or we can argue that we can talk coherently about suffering in novel contexts (AGIs, mind crime, aliens, and so on), but it seems inherently troublesome to claim we can do both at the same time.
Objection 2: Intuition duels
Two people can agree on FRI’s position that there is no objective fact of the matter about what suffering is (no privileged definition), but this also means they have no way of coming to any consensus on the object-level question of whether something can suffer. This isn’t just an academic point: Brian has written extensively about how he believes non-human animals can and do suffer extensively, whereas Yudkowsky (who holds computationalist views, like Brian) has written about how he’s confident that animals are not conscious and cannot suffer, due to their lack of higher-order reasoning.
And if functionalism is having trouble adjudicating the easy cases of suffering–whether monkeys can suffer, or whether dogs can— it doesn’t have a sliver of a chance at dealing with the upcoming hard cases of suffering: whether a given AGI is suffering, or engaging in mind crime; whether a whole-brain emulation (WBE) or synthetic organism or emergent intelligence that doesn’t have the capacity to tell us how it feels (or that we don’t have the capacity to understand) is suffering; if any aliens that we meet in the future can suffer; whether changing the internal architecture of our qualia reports means we’re also changing our qualia; and so on.
In short, FRI’s theory of consciousness isn’t actually a theory of consciousness at all, since it doesn’t do the thing we need a theory of consciousness to do: adjudicate disagreements in a principled way. Instead, it gives up any claim on the sorts of objective facts which could in principle adjudicate disagreements.
This is a source of friction in EA today, but it’s mitigated by the sense that
(1) The EA pie is growing, so it’s better to ignore disagreements than pick fights;
(2) Disagreements over the definition of suffering don’t really matter yet, since we haven’t gotten into the business of making morally-relevant synthetic beings (that we know of) that might be unable to vocalize their suffering.
If the perception of one or both of these conditions change, the lack of some disagreement-adjudicating theory of suffering will matter quite a lot.
Objection 3: Convergence requires common truth
Mike: “[W]hat makes one definition of consciousness better than another? How should we evaluate them?”
Brian: “Consilience among our feelings of empathy, principles of non-discrimination, understandings of cognitive science, etc. It’s similar to the question of what makes one definition of justice or virtue better than another.”
Brian is hoping that affective neuroscience will slowly converge to accurate views on suffering as more and better data about sentience and pain accumulates. But convergence to truth implies something (objective) driving the convergence- in this way, Brian’s framework still seems to require an objective truth of the matter, even though he disclaims most of the benefits of assuming this.
Objection 4: Assuming that consciousness is a reification produces more confusion, not less
Brian: “Consciousness is not a reified thing; it’s not a physical property of the universe that just exists intrinsically. Rather, instances of consciousness are algorithms that are implemented in specific steps. … Consciousness involves specific things that brains do.”
Brian argues that we treat conscious/phenomenology as more ‘real’ than it is. Traditionally, whenever we’ve discovered something is a leaky reification and shouldn’t be treated as ‘too real’, we’ve been able to break it down into more coherent constituent pieces we can treat as real. Life, for instance, wasn’t due to élan vital but a bundle of self-organizing properties & dynamics which generally co-occur. But carrying out this “de-reification” process on consciousness– enumerating its coherent constituent pieces– has proven difficult, especially if we want to preserve some way to speak cogently about suffering.
Speaking for myself, the more I stared into the depths of functionalism, the less certain everything about moral value became– and arguably, I see the same trajectory in Brian’s work and Luke Muehlhauser’s report. Their model uncertainty has seemingly become larger as they’ve looked into techniques for how to “de-reify” consciousness while preserving some flavor of moral value, not smaller. Brian and Luke seem to interpret this as evidence that moral value is intractably complicated, but this is also consistent with consciousness not being a reification, and instead being a real thing. Trying to “de-reify” something that’s not a reification will produce deep confusion, just as surely trying to treat a reification as ‘more real’ than it actually is will.
Edsger W. Dijkstra famously noted that “The purpose of abstraction is not to be vague, but to create a new semantic level in which one can be absolutely precise.” And so if our ways of talking about moral value fail to ‘carve reality at the joints’- then by all means let’s build better ones, rather than giving up on precision.
Objection 5: The Hard Problem of Consciousness is a red herring
Brian spends a lot of time discussing Chalmers’ “Hard Problem of Consciousness”, i.e. the question of why we’re subjectively conscious, and seems to base at least part of his conclusion on not finding this question compelling— he suggests “There’s no hard problem of consciousness for the same reason there’s no hard problem of life: consciousness is just a high-level word that we use to refer to lots of detailed processes, and it doesn’t mean anything in addition to those processes.” I.e., no ‘why’ is necessary; when we take consciousness and subtract out the details of the brain, we’re left with an empty set.
But I think the “Hard Problem” isn’t helpful as a contrastive centerpiece, since it’s unclear what the problem is, and whether it’s analytic or empirical, a statement about cognition or about physics. At the Qualia Research Institute (QRI), we don’t talk much about the Hard Problem; instead, we talk about Qualia Formalism, or the idea that any phenomenological state can be crisply and precisely represented by some mathematical object. I suspect this would be a better foil for Brian’s work than the Hard Problem.
Objection 6: Mapping to reality
Brian argues that consciousness should be defined at the functional/computational level: given a Turing machine, or neural network, the right ‘code’ will produce consciousness. But the problem is that this doesn’t lead to a theory which can ‘compile’ to physics. Consider the following:
Imagine you have a bag of popcorn. Now shake it. There will exist a certain ad-hoc interpretation of bag-of-popcorn-as-computational-system where you just simulated someone getting tortured, and other interpretations that don’t imply that. Did you torture anyone? If you’re a computationalist, no clear answer exists- you both did, and did not, torture someone. This sounds like a ridiculous edge-case that would never come up in real life, but in reality it comes up all the time, since there is no principled way to *objectively derive* what computation(s) any physical system is performing.
I don’t think this is an outlandish view of functionalism; Brian suggests much the same in How to Interpret a Physical System as a Mind“Physicalist views that directly map from physics to moral value are relatively simple to understand. Functionalism is more complex, because it maps from physics to computations to moral value. Moreover, while physics is real and objective, computations are fictional and ‘observer-relative’ (to use John Searle’s terminology). There’s no objective meaning to ‘the computation that this physical system is implementing’ (unless you’re referring to the specific equations of physics that the system is playing out).”
Gordon McCabe (McCabe 2004) provides a more formal argument to this effect— that precisely mapping between physical processes and (Turing-level) computational processes is inherently impossible— in the context of simulations. First, McCabe notes that:
[T]here is a one-[to-]many correspondence between the logical states [of a computer] and the exact electronic states of computer memory. Although there are bijective mappings between numbers and the logical states of computer memory, there are no bijective mappings between numbers and the exact electronic states of memory.
This lack of an exact bijective mapping means that subjective interpretation necessarily creeps in, and so a computational simulation of a physical system can’t be ‘about’ that system in any rigorous way:
In a computer simulation, the values of the physical quantities possessed by the simulated system are represented by the combined states of multiple bits in computer memory. However, the combined states of multiple bits in computer memory only represent numbers because they are deemed to do so under a numeric interpretation. There are many different interpretations of the combined states of multiple bits in computer memory. If the numbers represented by a digital computer are interpretation-dependent, they cannot be objective physical properties. Hence, there can be no objective relationship between the changing pattern of multiple bit-states in computer memory, and the changing pattern of quantity-values of a simulated physical system.
McCabe concludes that, metaphysically speaking,
A digital computer simulation of a physical system cannot exist as, (does not possess the properties and relationships of), anything else other than a physical process occurring upon the components of a computer. In the contemporary case of an electronic digital computer, a simulation cannot exist as anything else other than an electronic physical process occurring upon the components and circuitry of a computer.
Where does this leave ethics? In Flavors of Computation Are Flavors of Consciousness, Brian notes that “In some sense all I’ve proposed here is to think of different flavors of computation as being various flavors of consciousness. But this still leaves the question: Which flavors of computation matter most? Clearly whatever computations happen when a person is in pain are vastly more important than what’s happening in a brain on a lazy afternoon. How can we capture that difference?”
But if Brian grants the former point- that “There’s no objective meaning to ‘the computation that this physical system is implementing’”– then this latter task of figuring out “which flavors of computation matter most” is provably impossible. There will always be multiple computational (and thus ethical) interpretations of a physical system, with no way to figure out what’s “really” happening. No way to figure out if something is suffering or not. No consilience; not now, not ever.
Note: despite apparently granting the point above, Brian also remarks that:
I should add a note on terminology: All computations occur within physics, so any computation is a physical process. Conversely, any physical process proceeds from input conditions to output conditions in a regular manner and so is a computation. Hence, the set of computations equals the set of physical processes, and where I say “computations” in this piece, one could just as well substitute “physical processes” instead.
This seems to be (1) incorrect, for the reasons I give above, or (2) taking substantial poetic license with these terms, or (3) referring to hypercomputation (which might be able to salvage the metaphor, but would invalidate many of FRI’s conclusions dealing with the computability of suffering on conventional hardware).
This objection may seem esoteric or pedantic, but I think it’s important, and that it ripples through FRI’s theoretical framework with disastrous effects.
Objection 7: FRI doesn’t fully bite the bullet on computationalism
Brian suggests that “flavors of computation are flavors of consciousness” and that some computations ‘code’ for suffering. But if we do in fact bite the bullet on this metaphor and place suffering within the realm of computational theory, we need to think in “near mode” and accept all the paradoxes that brings. Scott Aaronson, a noted expert on quantum computing, raises the following objections to functionalism:
I’m guessing that many people in this room side with Dennett, and (not coincidentally, I’d say) also with Everett. I certainly have sympathies in that direction too. In fact, I spent seven or eight years of my life as a Dennett/Everett hardcore believer. But, while I don’t want to talk anyone out of the Dennett/Everett view, I’d like to take you on a tour of what I see as some of the extremely interesting questions that that view leaves unanswered. I’m not talking about “deep questions of meaning,” but about something much more straightforward: what exactly does a computational process have to do to qualify as “conscious”?
There’s this old chestnut, what if each person on earth simulated one neuron of your brain, by passing pieces of paper around. It took them several years just to simulate a single second of your thought processes. Would that bring your subjectivity into being? Would you accept it as a replacement for your current body? If so, then what if your brain were simulated, not neuron-by-neuron, but by a gigantic lookup table? That is, what if there were a huge database, much larger than the observable universe (but let’s not worry about that), that hardwired what your brain’s response was to every sequence of stimuli that your sense-organs could possibly receive. Would that bring about your consciousness? Let’s keep pushing: if it would, would it make a difference if anyone actually consulted the lookup table? Why can’t it bring about your consciousness just by sitting there doing nothing?
To these standard thought experiments, we can add more. Let’s suppose that, purely for error-correction purposes, the computer that’s simulating your brain runs the code three times, and takes the majority vote of the outcomes. Would that bring three “copies” of your consciousness into being? Does it make a difference if the three copies are widely separated in space or time—say, on different planets, or in different centuries? Is it possible that the massive redundancy taking place in your brain right now is bringing multiple copies of you into being?
Maybe my favorite thought experiment along these lines was invented by my former student Andy Drucker. In the past five years, there’s been a revolution in theoretical cryptography, around something called Fully Homomorphic Encryption (FHE), which was first discovered by Craig Gentry. What FHE lets you do is to perform arbitrary computations on encrypted data, without ever decrypting the data at any point. So, to someone with the decryption key, you could be proving theorems, simulating planetary motions, etc. But to someone without the key, it looks for all the world like you’re just shuffling random strings and producing other random strings as output.
You can probably see where this is going. What if we homomorphically encrypted a simulation of your brain? And what if we hid the only copy of the decryption key, let’s say in another galaxy? Would this computation—which looks to anyone in our galaxy like a reshuffling of gobbledygook—be silently producing your consciousness?
When we consider the possibility of a conscious quantum computer, in some sense we inherit all the previous puzzles about conscious classical computers, but then also add a few new ones. So, let’s say I run a quantum subroutine that simulates your brain, by applying some unitary transformation U. But then, of course, I want to “uncompute” to get rid of garbage (and thereby enable interference between different branches), so I apply U-1. Question: when I apply U-1, does your simulated brain experience the same thoughts and feelings a second time? Is the second experience “the same as” the first, or does it differ somehow, by virtue of being reversed in time? Or, since U-1U is just a convoluted implementation of the identity function, are there no experiences at all here?
Here’s a better one: many of you have heard of the Vaidman bomb. This is a famous thought experiment in quantum mechanics where there’s a package, and we’d like to “query” it to find out whether it contains a bomb—but if we query it and there is a bomb, it will explode, killing everyone in the room. What’s the solution? Well, suppose we could go into a superposition of querying the bomb and not querying it, with only ε amplitude on querying the bomb, and √(1-ε2) amplitude on not querying it. And suppose we repeat this over and over—each time, moving ε amplitude onto the “query the bomb” state if there’s no bomb there, but moving ε2 probability onto the “query the bomb” state if there is a bomb (since the explosion decoheres the superposition). Then after 1/ε repetitions, we’ll have order 1 probability of being in the “query the bomb” state if there’s no bomb. By contrast, if there is a bomb, then the total probability we’ve ever entered that state is (1/ε)×ε2 = ε. So, either way, we learn whether there’s a bomb, and the probability that we set the bomb off can be made arbitrarily small. (Incidentally, this is extremely closely related to how Grover’s algorithm works.)
OK, now how about the Vaidman brain? We’ve got a quantum subroutine simulating your brain, and we want to ask it a yes-or-no question. We do so by querying that subroutine with ε amplitude 1/ε times, in such a way that if your answer is “yes,” then we’ve only ever activated the subroutine with total probability ε. Yet you still manage to communicate your “yes” answer to the outside world. So, should we say that you were conscious only in the ε fraction of the wavefunction where the simulation happened, or that the entire system was conscious? (The answer could matter a lot for anthropic purposes.)
To sum up: Brian’s notion that consciousness is the same as computation raises more issues than it solves; in particular, the possibility that if suffering is computable, it may also be uncomputable/reversible, would suggest s-risks aren’t as serious as FRI treats them.
Objection 8: Dangerous combination
Three themes which seem to permeate FRI’s research are:
(1) Suffering is the thing that is bad.
(2) It’s critically important to eliminate badness from the universe.
(3) Suffering is impossible to define objectively, and so we each must define what suffering means for ourselves.
Taken individually, each of these seems reasonable. Pick two, and you’re still okay. Pick all three, though, and you get A Fully General Justification For Anything, based on what is ultimately a subjective/aesthetic call.
Much can be said in FRI’s defense here, and it’s unfair to single them out as risky: in my experience they’ve always brought a very thoughtful, measured, cooperative approach to the table. I would just note that ideas are powerful, and I think theme (3) is especially pernicious if incorrect.
III. QRI’s alternative
Analytic functionalism is essentially a negative hypothesis about consciousness: it’s the argument that there’s no order to be found, no rigor to be had. It obscures this with talk of “function”, which is a red herring it not only doesn’t define, but admits is undefinable. It doesn’t make any positive assertion. Functionalism is skepticism- nothing more, nothing less.
But is it right?
Ultimately, I think these a priori arguments are much like people in the middle ages arguing whether one could ever formalize a Proper System of Alchemy. Such arguments may in many cases hold water, but it’s often difficult to tell good arguments apart from arguments where we’re just cleverly fooling ourselves. In retrospect, the best way to *prove* systematized alchemy was possible was to just go out and *do* it, and invent Chemistry. That’s how I see what we’re doing at QRI with Qualia Formalism: we’re assuming it’s possible to build stuff, and we’re working on building the object-level stuff.
What we’ve built with QRI’s framework
Note: this is a brief, surface-level tour of our research; it will probably be confusing for readers who haven’t dug into our stuff before. Consider this a down-payment on a more substantial introduction.
My most notable work is Principia Qualia, in which I lay out my meta-framework for consciousness (a flavor of dual-aspect monism, with a focus on Qualia Formalism) and put forth the Symmetry Theory of Valence (STV). Essentially, the STV is an argument that much of the apparent complexity of emotional valence is evolutionarily contingent, and if we consider a mathematical object isomorphic to a phenomenological experience, the mathematical property which corresponds to how pleasant it is to be that experience is the object’s symmetry. This implies a bunch of testable predictions and reinterpretations of things like what ‘pleasure centers’ do (Section XI; Section XII). Building on this, I offer the Symmetry Theory of Homeostatic Regulation, which suggests understanding the structure of qualia will translate into knowledge about the structure of human intelligence, and I briefly touch on the idea of Neuroacoustics.
Likewise, my colleague Andrés Gómez Emilsson has written about the likely mathematics of phenomenology, including The Hyperbolic Geometry of DMT Experiences, Tyranny of the Intentional Object, and Algorithmic Reduction of Psychedelic States. If I had to suggest one thing to read in all of these links, though, it would be the transcript of his recent talk on Quantifying Bliss, which lays out the world’s first method to objectively measure valence from first principles (via fMRI) using Selen Atasoy’s Connectome Harmonics framework, the Symmetry Theory of Valence, and Andrés’s CDNS model of experience.
These are risky predictions and we don’t yet know if they’re right, but we’re confident that if there is some elegant structure intrinsic to consciousness, as there is in many other parts of the natural world, these are the right kind of risks to take.
I mention all this because I think analytic functionalism- which is to say radical skepticism/eliminativism, the metaphysics of last resort- only looks as good as it does because nobody’s been building out any alternatives.
IV. Closing thoughts
FRI is pursuing a certain research agenda, and QRI is pursuing another, and there’s lots of value in independent explorations of the nature of suffering. I’m glad FRI exists, everybody I’ve interacted with at FRI has been great, I’m happy they’re focusing on s-risks, and I look forward to seeing what they produce in the future.
On the other hand, I worry that nobody’s pushing back on FRI’s metaphysics, which seem to unavoidably lead to the intractable problems I describe above. FRI seems to believe these problems are part of the territory, unavoidable messes that we just have to make philosophical peace with. But I think that functionalism is a bad map, that the metaphysical messes it leads to are much worse than most people realize (fatal to FRI’s mission), and there are other options that avoid these problems (which, to be fair, is not to say they have no problems).
Ultimately, FRI doesn’t owe me a defense of their position. But if they’re open to suggestions on what it would take to convince a skeptic like me that their brand of functionalism is viable, or at least rescuable, I’d offer the following:
Re: Objection 1 (motte-and-bailey), I suggest FRI should be as clear and complete as possible in their basic definition of suffering. In which particular ways is it ineffable/fuzzy, and in which particular ways is it precise? What can we definitely say about suffering, and what can we definitely never determine? Preregistering ontological commitments and methodological possibilities would help guard against FRI’s definition of suffering changing based on context.
Re: Objection 2 (intuition duels), FRI may want to internally “war game” various future scenarios involving AGI, WBE, etc, with one side arguing that a given synthetic (or even extraterrestrial) organism is suffering, and the other side arguing that it isn’t. I’d expect this would help diagnose what sorts of disagreements future theories of suffering will need to adjudicate, and perhaps illuminate implicit ethical intuitions. Sharing the results of these simulated disagreements would also be helpful in making FRI’s reasoning less opaque to outsiders, although making everything transparent could lead to certain strategic disadvantages.
Re: Objection 3 (convergence requires common truth), I’d like FRI to explore exactly what might drive consilience/convergence in theories of suffering, and what precisely makes one theory of suffering better than another, and ideally to evaluate a range of example theories of suffering under these criteria.
Re: Objection 4 (assuming that consciousness is a reification produces more confusion, not less), I would love to see a historical treatment of reification: lists of reifications which were later dissolved (e.g., élan vital), vs scattered phenomena that were later unified (e.g., electromagnetism). What patterns do the former have, vs the latter, and why might consciousness fit one of these buckets better than the other?
Re: Objection 5 (the Hard Problem of Consciousness is a red herring), I’d like to see a more detailed treatment of what kinds of problem people have interpreted the Hard Problem as, and also more analysis on the prospects of Qualia Formalism (which I think is the maximally-empirical, maximally-charitable interpretation of the Hard Problem). It would be helpful for us, in particular, if FRI preregistered their expectations about QRI’s predictions, and their view of the relative evidence strength of each of our predictions.
Re: Objection 6 (mapping to reality), this is perhaps the heart of most of our disagreement. From Brian’s quotes, he seems split on this issue; I’d like clarification about whether he believes we can ever precisely/objectively map specific computations to specific physical systems, and vice-versa. And if so— how? If not, this seems to propagate through FRI’s ethical framework in a disastrous way, since anyone can argue that any physical system does, or does not, ‘code’ for massive suffering, and there’s no principled way to derive any ‘ground truth’ or even pick between interpretations in a principled way (e.g. my popcorn example). If this isn’t the case— why not?
Brian has suggested that “certain high-level interpretations of physical systems are more ‘natural’ and useful than others” (personal communication); I agree, and would encourage FRI to explore systematizing this.
It would be non-trivial to port FRI’s theories and computational intuitions to the framework of “hypercomputation”– i.e., the understanding that there’s a formal hierarchy of computational systems, and that Turing machines are only one level of many– but it may have benefits too. Namely, it might be the only way they could avoid Objection 6 (which I think is a fatal objection) while still allowing them to speak about computation & consciousness in the same breath. I think FRI should look at this and see if it makes sense to them.
Re: Objection 7 (FRI doesn’t fully bite the bullet on computationalism), I’d like to see responses to Aaronson’s aforementioned thought experiments.
Re: Objection 8 (dangerous combination), I’d like to see a clarification about why my interpretation is unreasonable (as it very well may be!).
In conclusion- I think FRI has a critically important goal- reduction of suffering & s-risk. However, I also think FRI has painted itself into a corner by explicitly disallowing a clear, disagreement-mediating definition for what these things are. I look forward to further work in this field.
Mike Johnson
Qualia Research Institute
Acknowledgements: thanks to Andrés Gómez Emilsson, Brian Tomasik, and Max Daniel for reviewing earlier drafts of this.
My sources for FRI’s views on consciousness:
Flavors of Computation are Flavors of Consciousness:
Is There a Hard Problem of Consciousness?
Consciousness Is a Process, Not a Moment
How to Interpret a Physical System as a Mind
Dissolving Confusion about Consciousness
Debate between Brian & Mike on consciousness:
Max Daniel’s EA Global Boston 2017 talk on s-risks:
Multipolar debate between Eliezer Yudkowsky and various rationalists about animal suffering:
The Internet Encyclopedia of Philosophy on functionalism:
Gordon McCabe on why computation doesn’t map to physics:
Toby Ord on hypercomputation, and how it differs from Turing’s work:
Luke Muehlhauser’s OpenPhil-funded report on consciousness and moral patienthood:
Scott Aaronson’s thought experiments on computationalism:
Selen Atasoy on Connectome Harmonics, a new way to understand brain activity:
My work on formalizing phenomenology:
My meta-framework for consciousness, including the Symmetry Theory of Valence:
My hypothesis of homeostatic regulation, which touches on why we seek out pleasure:
My exploration & parametrization of the ‘neuroacoustics’ metaphor suggested by Atasoy’s work:
My colleague Andrés’s work on formalizing phenomenology:
A model of DMT-trip-as-hyperbolic-experience:
June 2017 talk at Consciousness Hacking, describing a theory and experiment to predict people’s valence from fMRI data:
A parametrization of various psychedelic states as operators in qualia space:
A brief post on valence and the fundamental attribution error:
A summary of some of Selen Atasoy’s current work on Connectome Harmonics: |
20fc4cee8978eee3 | Journal of Nonlocality and Remote Mental Interactions, Vol. I Nr. 1, January 2002
TGD inspired theory of consciousness
Matti Pitkänen
Postal address:
Department of Physical Sciences, High Energy Physics Division, PL 64, FIN-00014, University of Helsinki, Finland.
Home address:
Kadermonkatu 16, 10900, Hanko, Finland
TGD inspired theory of consciousness relies on two identifications. The identification of quantum jump between quantum histories as a moment of consciousness reduces quantum measurement theory to fundamental physics and solves long list of paradoxes of modern physics. The identification of self as a subsystem able to remain unentangled in subsequent quantum jumps provides a quantum theory of observer and one can identify self also as a fundamental statistical ensemble. The generalization of spacetime concept so that it allows both real and p-adic regions unable to entangle mutually is a crucial prequisite for the notion of self and p-adic spacetime regions define cognitive representations. In this article these notions are analyzed in more detail. Quantum jump between quantum histories decomposes into three parts corresponding to the unitary U process of Penrose followed by the TGD counterpart of state function reduction in turn followed by a cascade of self measurements giving rise to the TGD counterpart of state preparation. The dynamics of self measurement is determined by the so called Negentropy Maximization Principle (NMP).
Selves form a hierarchy and self experiences its subselves as mental images and is in turn a mental image of a higher level self. The experience of self is a statistical average over quantum jumps occurred after the last 'wake-up' and the theory of qualia can be formulated in terms of statistical physics. Self experiences the subselves of subself as a statistical average so that self hierarchy means also an abstraction hierarchy. Self has as its geometric correlate so called mindlike spacetime sheets of finite duration with respect to geometric time (as opposed to subjective time determined by the sequence of quantum jumps) and one can understand how psychological time and its arrow emerge. The new view about time has rather dramatic implications: the civilizations of the geometric past and future exist subjectively now so that one can speak about four-dimensional society. The paradigm of four-dimensional brain in turn provides a completely new view about long term memories. Crucial element behind all these developments is the classical non-determinism of the fundamental variational principle determining the dynamics of the spacetime surfaces.
Table of contents
1. Introduction
2.Quantum jump as a moment of consciousness
2.1 The anatomy of quantum jump and connection with quantum measurement theory
2.2 Negentropy Maximization Principle
2.3 The new view about time
3. Quantum self
3.1 Self as a subsystem able to remain unentangled
3.2 How the contents of consciousness of self are determined
3.3 Selves self-organize
3.4 Self hierarchy
3.5 Self as a statistical ensemble and qualia
3.6 Self as a moral agent
About notation
I have not been able to avoid totally the use of greek letters and mathematical symbols in the text. I have chosen to represent them in latex code since it is probably familiar to many readers. Thus greek letters are denoted by symbols like \Psi, \alpha, \Delta, \tau. ^n signifies upper index n (say in symbol M^4 for Minkowski space or in n:th power x^n of x). Lower index n is expressed as _n (say x_n or CP_2). Square root of x is expressed as \sqrt{x}. Sum of elements x_n is expressed as SUM_n x_n. x propto y reads x is proportional to y. X times Y denotes Cartesian product of spaces X and Y and x times y denotes the ordinary product of numbers x and y. x \pm y denotes for x plusminus y. x\simeq y can be read as x=about y. and x\sim y can be read as x =roughly about y. \infty denotes infinity.
1. Introduction
T(opological)G(eometro)D(ynamics) inspired theory of consciousness can be regarded also as a generalization of the quantum measurement theory. The connection comes from the identification of quantum jump as a moment of consciousness and the replacement of the external observer with the notion of 'self' defined as a subsystem able to remain unentangled during subsequent quantum jumps. This generalization of quantum measurement theory opens the black boxes of of state function reduction and preparation by combining them in the notion of quantum jump between quantum histories.
The basic new elements as compared to the standard physics based theories of consciousness are the new view about time and quantum state allowing to resolve the basic paradoxes of modern physics, the notion of manysheeted spacetime; the non-determinism of the fundamental variational principle determining the dynamics of the spacetime surfaces; and p-adic numbers.
1. Quantum states as quantum histories
General coordinate invariance forces to replace quantum state as time=constant snapshot with entire quantum history with can be regarded as a generalition for the solution of Schrödinger equation describing entire universe. Classical histories correspond to spacetime surfaces.
2. Nondeterminism of quantum jump is outside the realm of spacetime and state space
Since quantum jumps occur between quantum histories, the non-determinism of quantum jump is outside the spacetime and the space of quantum states. This solves the basic paradox of the quantum measurement theory. Time evolution by quantum jumps, subjective time development, corresponds to hopping in the space of solutions of the field equations.
3. Two times, two causalities
This view forces to differentiate between subjective time and geometric time. Geometric time is the fourth coordinate for spacetime surfaces whereas subjective time corresponds to a sequence of quantum jumps identified as moments of consciousness. The complete space-time democracy has most profound implications concerning the interpretation of the theory.
The generalization of the spacetime concept involving in an essential manner also the classical non-determinism of the basic variational principle defining spacetime surface X^4(X^3) associated with a given 3-surface X^3, allows to understand how the correspondence between geometric and subjective time emerges. The point is that mindlike spacetime sheets with finite geometric time duration and well defined temporal center of mass coordinate become possible. These mindlike spacetime sheets serve as geometric correlates for conscious selves and one can understand the emergence of the psychological time and its arrow.
4. p-Adic physics as physics of cognition
p-Adic numbers (completions of rationals) is also something essentially new. The very definition of the concept of self as a system able to remain unentangled during subsequent quantum jumps requires p-adic numbers. The reason is that the un-entangled state of two subsystems is unstable unless they correspond to different number fields in which case entanglement is not possible at all. In purely real context the only self would be the entire Universe: subselves inside real self are p-adic islands in the sea of real numbers.
The inherent non-determinism of the p-adic field equations is identified as non-determinism of imagination which is an essential element of cognition. p-Adic spacetime regions represent the 'mind stuff', geometric correlate for cognition, they are however not conscious. The transformations of intentions to actions occur in quantum jumps in which p-adic spacetime region is replaced with a real one whereas sensory input transforms to thought in the reverse transition. This mechanism should apply not only to the ordinary volitional acts but also to various forms of psychokinesis. p-Adic spacetime regions are obviously the TGD counterpart for the mind stuff of Descartes and dualism relates material world and cognitive representations which both are Zombies.
5. Connection with statistical physics and self-organization
The notion of self as a system able to remain unentangled and able to perform quantum jumps in this state implies also a deep connection with statistical physics. Self corresponds to the sequence of quantum jumps for subsystem and the final states of these quantum jumps define what might be called a fundamental statistical ensemble. The contents of consciousness of self are determined as statistical averages over experiences associated with individual quantum jumps. Self has subselves and expreriences them as mental images. Self experiences also subselves of subself as a statistical ensemble providing kind of abstraction. Selves participate to each quantum jump and thus self-organize. This leads to a quantum level model of self-organization and Darwinian selection can be seen as due to the dissipation accompanying self-organizing systems.
6. The notion of manysheeted spacetime and biosystems as macroscopic quantum systems
Concerning the concrete applications of the theory at the level of biosystems and brain, the notion of manysheeted spacetime is of crucial importance since it makes possible to understand how biosystems manage to be macroscopic quantum systems. Also classical color force and Z^0 force play a key role in the new physics associated with the living matter. The implications of the theory are rather far-reaching and strongly encourage to give up the cherished belief about brain as a seat of consciousness. In TGD universe our selves involve in an essential manner electromagnetic field structures (topological field quanta) having size measured using Earth size as a unit. Our physical bodies can be seen as kind of sensory and motor organs of these electromagnetic selves. In particular, physical death can be seen only as a death of a mental image about the physical body.
2. Quantum jump as a moment of consciousness
The notions of quantum jump and self are the basic concepts of TGD inspired theory of consciousness. Quantum jump is the microtemporal building block of conscious experience and relates to the theory of consciousness much like physics at Planck length scale relates to the macroscopic physics. Self corresponds to the macrotemporal aspects of conscious experience and statistical ensemble aspect is crucial in the theory of consciousness.
2.1 The anatomy of quantum jump and connection with quantum measurement theory
Quantum jump was originally seen as something totally irreducible. Gradually the rich substructure of quantum jump has revealed itself.
First of all, quantum jump decomposes into informational time development
\Psi_i--> U\Psi_i
followed by the TGD counterpart of state function reduction realized as a localization in zero modes which correspond to non-quantum fluctuating degrees of freedom of configuration space of 3-surfaces (see the first part of [TGD] and of [cbook]):
U\Psi_i--> \Psi_f^0 .
The assumption that the localization occurs in the zero modes of the configuration space poses a very important consistency condition on U. U must effectively correspond to a flow in zero modes such that there is one-one correlation between the quantum numbers \alpha in quantum fluctuating degrees of freedom in some state basis and the values z of the zero modes in state U\Psi_i:
\alpha<---> z (\alpha) .
This together with the fact that zero modes are effectively classical variables, implies that the localization in zero modes can be identified as the TGD counterpart for the state function reduction. The state function reduction is followed by a cascade of self measurements in quantum fluctuating degrees of freedom (the values of the zero modes do not change during this stage)
\Psi_f^0--> .... --> \Psi_f ,
whose dynamics is governed by the Negentropy Maximization Principle (NMP, see the chapter "NMP" of [cbookI]). At least formally, this process is analogous to analysis at level of cognition leads to a completely unentangled state (apart from entanglement present in bound states) identifiable as a prepared state. It must be emphasized that self measurement is microtemporal aspect of consciousness and does not directly relate to our conscious experience.
A good metaphor for quantum jump is as Djinn leaving the bottle (informational time development), fulfilling the wish (quantum jump involving choice) and returning to, possibly new, bottle (localization in zero modes and subsequent state preparation process). One could formally regard each quantum jump as a TGD counterpart of a quantum computation lasting infinitely long time t--> \infty followed by a state preparation of the initial state of the next quantum computation.
2.2 Negentropy Maximization Principle
The dynamics of self measurements is governed by Negentropy Maximization Principle (NMP, see the chapter "NMP" of [cbookI]), which specifies which subsystems of self are subject to quantum measurement in a given self measurement. NMP can be regarded as a basic law for the dynamics of quantum jumps and states that the information content of the conscious experience is maximized. In p-adic context NMP dictates the dynamics of cognition.
a) NMP applies to each self with fixed values of zero modes separately and is therefore in a well-defined sense a local principle. Every self in \Psi_{f}^0 participates in self measurement sequence \Psi_{f}^0--> ... \Psi_f, which means that some subsystem of the self is quantum measured.
b) A quantum jump for a given irreducible self X corresponds to a measurement of the density matrix for some subsystem Y of X. In this measurement subsystem Y goes to an eigenstate of the density matrix and Y becomes unentangled. Same happens to the complement of Y inside X. The amount of entanglement is measured by entanglement entropy S and S vanishes for the final state of the quantum jump. Thus S can be regarded as negentropy gain having interpretation as some kind of conscious information, or rather, reduction of dis-information. The conscious experience must be assigned with self X. One cannot associate it with the measured subsystem or its complement inside self since they are in a completely symmetric position since diagonalized density matrices are identical. Hence there is no manner to tell which is the measured system and which the measuring subsystem and one must define self measurement as creating an unentangled subsystem-complement pair inside a self and identify the self as the conscious measurer. In state function reduction zero modes could be regarded as representing dynamical degrees of freedom of measurer.
c) NMP states that the entanglement entropy reduction associated with the conscious experience of an irreducible self X is maximal. Interpreting entanglement negentropy gain as a conscious information, on can say that we live in (or create) the best possible world. Only the quantum jumps giving rise to maximum information content of conscious experience occur (it must be noticed however that one can assign several types of information measures with conscious experience). This requirement fixes the quantum measured subsystem Y of given self uniquely unless there are several subsystems giving rise to same maximum negentropy gain: in this case any of the quantum jumps occurs with same probability.
2.2 The new view about time
The understanding of the relationship between subjective and geometric time leads to the notion of psychological time involving in an an essential manner the new view about spacetime, in particular the idea about mindlike spacetime sheet (defined as spacetime sheet having finite time-duration) as a geometric correlate of self (see the chapter "Time and Consciousness" of [cbookI]). One can understand psychological time as a temporal center of mass coordinate for the cognitive spacetime sheet. The arrow of psychological time can be understood as resulting from a drift towards the geometric future induced by the the geometry of the future lightcone. The simplest guess is that the average increment of the geometric time per quantum jump is given by
\Delta t =k \tau ,
where k is a numerical constant and \tau some fundamental time scale, most naturally of the order of CP_2 time \tau_{CP_2} about 10^4 Planck times. This means 2^{127}\simeq 10^{38} quantum jumps per .1 seconds so that psychological time is effectively continuous. It must be emphasized, that this identification is based on a dimensional and aesthetic argument and one must keep mind open for the possibility that the duration of the psychological cronon is dynamical and depends on the geometrical size and other properties of self.
The notion of psychological time forces to view the entire manysheeted spacetime surface as a living system so that the standard notion of linear time is illusory and reflects the restricted information content of our conscious experience rather than fundamental 4-dimensional reality. The paradigm of 4-dimensional brain provides completely new understanding of the long term memory. No memory storage of information about the geometric past to the geometric now is needed and one avoids the basic difficulties of neural net models (new memories tend to destroy the old ones). There are two kinds of memories, geometric and subjective. Subjective memory is about real events and its duration is that of subself responsible for the mental image. Geometric memory provides a narrative which changes when geometric past changes in quantum jumps: geometric memory of childhood is about the childhood subjectively now, not the real childhood. There are also two kinds of causalities: the causality of passive events and the causality of active deeds. Various causal anomalies [Deeeke,BR1,BR2} to be discussed in more detail in forthcoming article can be understood from the fact that also the geometric past changes in each quantum jump. A mind-boggling possibility is that this effect could occur in a time scale of a year [Peoch] and be testable (see the chapters "Time and Consciousness" and "Quantum Model for Cognition" of [cbookI]).
3. Quantum self
The notion of self was originally forced by the paradoxes resulting in the attempt to understand consciousness in terms of quantum jumps alone. The concept of self has developed gradually during years and the recent view is probably not be yet the final one. The connection with statistical physics and self-organization theory however encourage to think that basic ideas are sound.
3.1 Self as a subsystem able to remain unentangled
A natural identification of self is as a sub-Universe behaving autonomously. Thus sub-systems able to remain un-entangled are natural candidates for selves. In purely real context the generation of an even slightest entanglement kills self and the subsystems (other than entire Universe) able to remain unentangled under the action of U are extremely rare. As already described, the identification of the geometric correlates of selves as regions of spacetime appearing in the decomposition of spacetime into regions belonging to various number fields solves this problem: entanglement does not simply occur between different number fields. One could critisize this assumption: rational numbers are common to both reals and p-adics and if entanglement coefficients are rational, entanglement between different number fields might be possible. It is however difficult to understand how the notion of the Hilbert space inner product could make sense unless the whole quantum theory reduces to the field of rationals.
The basic prediction is the existence of infinite hierarchy of selves and this has rather dramatic consequences. At the top of the infinite hierarchy is entire Universe, which might be called God. This structure cannot entangled with any larger structure of same kind so that this self can be said to live eternal life. God abstracts all experiences in the infinite hierarchy of subselves to single experience. If infinite primes are allowed, as required by simple physical arguments, God corresponds to infinite p-adic prime characterizing entire universe and since this prime grows, also God evolves.
3.2 How the contents of consciousness of self are determined
In the following basic aspects about how the contents of consciousness of self are determined are discussed.
1. Summation hypothesis, binding, and statistical averaging of experiences
Subsystem X possessing self behaves essentially as a separate sub-Universe with respect to NMP. Also the subselves of X_i of X have their own experiences. The question is: how the experience of X and experiences of X_i are related? The following basic hypothesis provides a possible answer to this question.
a) X experiences the subselves X_i as separate mental images superposed to the pure self experience of X: this is natural since subselves are unentangled and hence behave like separate sub-Universes. These subselves are bound in the sense that self experiences them simultaneously.
b) The experiences of self X about the experiences of its subselves X_i are abstractions. Subself X_i experiences its subselves X_{ij} as separate mental images. X however experiences them as a single mental image representing what it is to be a subself of X_i, that is the average < X_{ij}> of the mental images X_{ij}. Thus the mental images of sub-sub-...selves of X are smoothed out to an average mental image and become effectively unconcious to X. Averaging hypothesis generalizes quantum statistical determinism to the level of subjective experience and is analogous to the hypothesis about averaging related to temporal binding.
When self has no subselves, the experience of self reduces to pure awareness without any mental images. In case of real selves these mental images are p-adic and thus represent thoughts: thus the empty mind in a state of Oneness means getting rid of thoughts. An interesting question is what kind of experience self decomposing to several subselves, each in state of whole-body consciousness, has: there is no averaging involved so that the mental images of self could be identical with the experiences of subselves.
Temporal binding with averaging implies that the experiences of the individual selves are reliable and abstraction brings in the possibility of quantum statistical determinism at the level of ensembles. The inability to perceive the flickering of light when the frequency of the flickering is larger than about 16-18 Hz is consistent with the hypothesis that sensory subselves (mental images) have a duration of order .1 seconds and that temporal averaging indeed occurs. Our self can have duration much longer than .1 seconds. For instance, the duration of the ordinary wake-up period could determine the duration of our self. The duration could be even longer: sleep could actually involve awareness and the lack of the sensory memories from sleep period could create the illusion about sleep as an unconscious state. The subjecto-temporal sequence of subselves of a finite duration is experienced as a sequence of separate mental images: this makes possible to remember the digits of a phone number despite the presence of the temporal averaging.
Summation hypothesis and temporal binding with averaging imply a hierarchy of conscious experiences with increasingly richer but abstracted contents. Also we are mental images of some higher level self.
I ended up with p-adic physics originally as a successful model of elementary particle masses (see the fourth part of "p-Adic TGD" of [padTGD]). The only possible interpretation for this success is that this model is a model of a cognitive model so that p-adic physics and cognition are present already at elementary particle level. This also explains the selection of p-adic primes corresponding to p-adic length scale hypothesis as a a result of fight for survival at elementary particle level. Without this selection electron would have practically continuous mass spectrum.
2. Binding of the experiencers by entanglement
The binding of experiencers is also possible and this process gives rise to what is usually understood with binding in neuroscience context.
a) The simplest assumption is that the binding of selves by quantum entanglement means that they lose their consciousness. In the case of subselves entanglement means binding of separate mental images to single mental image. This process naturally corresponds to the formation of wholes from their parts at the level of conscious experiences. The formation of a mental image (subself) representing word from the mental images representing letters is example of this process. The information about various areas of brain (there are about separate visual areas) could bind by entanglement mechanism. Also the fusion of the left and right visual fields to a single visual field could occur via the entanglement of the corresponding subselves. Right--left entanglement might occur already at the neuronal level.
b) Quantum entanglement could make possible communication between selves belonging to different levels of the self hierarchy: for instance, part of brain representing subself could entangle with a higher level self and mediate communications to those parts of brain which are awake (the semitrance mechanism discussed in the last part of [cbookII]).
c) It seems that also subselves of separate selves could entangle. This could make possible shared experiences. Telepathy could be based on this mechanism. Communications might involve entanglement between subselves: classically communication would involve generation of spacetime sheet containing ME serving as a join along boundaries bond connecting the regions representing the subselves of sender and receiver. In the final state this ME would disappear but leave subself which has received the message (during communication stage subself would become unconscious).
d) The non-determinism of Kähler action makes possible timelike entanglement. Long term memories could be seen as shared experiences in which the self at geometric now shares the experience of self in the geometric past. Laser mirrors defined by parallel MEs accompanying magnetic flux tubes could be the realization of this mechanism and present at all levels of the self hierarchy, even at DNA level.
The synchronous neuronal firing with the amazing precision of order millisecond [Engel] could be the neural correlate of entanglement between different areas of brain where subselves representing mental images could be located. Z^0 MEs giving rise to ZEG could provide the needed synchronizer at frequency of about one kHz, which corresponds to the duration of the bit of the memetic codeword. In p-adic state they would define cognitive representations and be passive whereas in the real state they would become active and synchronize neuronal firing (the coupling to Z^0 fields is strongest in cellular length scale). p-Adic Z^0 MEs would mimic the neuronal activity and transform to real MEs resonantly when the oscillation frequency is about kHz. Thus synchrony would be generated in phase transition like manner with a neuronal oscillation at kHz frequency serving as a seed. This vision is described in more detail in the chapter "Spectroscopy of consciousness" of [cbookII].
3.3 Selves self-organize
Subjective time development by quantum jumps implies quantum self-organization which can be regarded as a sequence of quantum jumps between quantum histories (see the chapter "Quantum Theory of Self-Organization" of [cbookI]). This evolution corresponds to a sequence of macroscopic spacetime surfaces associated with the final state quantum histories. Quantum jumps imply dissipation at fundamental level. Dissipation serves as a Darwinian selector of self-organization patterns, which can represent both genes and memes. In particular, one can understand how habits, skills and behavioural patterns are gradually learned. Protein folding occurring to very few final state patterns suggests itself as resulting from self-organization process (proteins would be thus conscious selves).
3.4 Self hierarchy
The notion of self hierarchy, starting from elementary particle level and having entire Universe at the top, is a highly nontrivial prediction of TGD inspired theory of consciousness. Self hierarchy is very much analogous to the hierarchy of subprograms of a computer program and defines a hierarchy of increasingly abstract experiences. Self hierarchy allows to understand computational aspects of brain functioning although connectionistic picture realized as quantum association network seems to work at various levels of the hierarchy (see the chapter "Quantum Model for Intelligent Systems" of [cbookI]). Topological field quanta of em fields (MEs and magnetic flux tube structures) are an part of the self hierarchy and this encourages to give up the view that consciousness is a purely brain centered phenomenon (wavelength of 10 Hz EEG wave has size scale of Earth). Self hierarchy is also crucial for the model of the sensory qualia.
3.5 Self as a statistical ensemble and qualia
The notion of self means possible fundamental identification of two kinds of ensembles: the subjecto-temporal ensemble defined by the quantum jumps occurred after the last 'wake-up' and spatial ensembles defined by the subselves of self defining mental images of self as statistical averages over experiences of subselves of subselves. This leads to the hypothesis that qualia correspond to average increments of quantum numbers and zero modes in quantum jumps. The sharpness of a given quale is determined by the entropy of the distribution for the quantum number increments of given type. At the statistical level qualia correspond to average rates of the change of quantum numbers and zero modes. The rates of change for entropy type variables associated with subselves are assumed to define emotional qualia. This picture is consistent with the assignment of qualia to quantum phase transitions.
The sequence of quantum jumps defining self defines also a sequence of maximally unentangled quantum states resulting in the state preparation process governed by NMP. This set of states, which grows in size quantum jump by quantum jump, defines in a natural manner a statistical ensemble identifiable as the fundamental realization of the otherwise fictive notion of statistical ensemble fundamental in the formulation of statistical physics. There are actually two statistical ensembles: the first one being associated with the final states of quantum jump and the second one being associated with the values of zero modes resulting in quantum jump. As far as conscious experience is involved, it however seems that it is the increments of quantum numbers and zero modes which are the relevant statistical variables.
This observation anchors the theory of conscious experience to statistical physics (see the chapter "General Theory of Qualia" of [cbookII]). For instance, the increments of zero modes resp. quantum numbers are responsible for geometric resp. non-geometric qualia. More precisely, the gradients with respect to subjective time for the zero modes and for the net quantum numbers associated with selves correspond to qualia. One can classify non-geometric qualia to entropy gradients associated with various increments (emotions in accordance with the fact that peptides are both informational molecules and molecules of emotion); kinestetic qualia (sense of pressure and force and, more generally, gradient of any conserved (with respect to geometric time) quantity associated with self with respect to subjective time); and generalized chemical qualia (rates for the changes of numbers of particles with various quantum numbers). Various entropies associated with self and subselves in turn characterize the sharpness of the mental images, and one can relate concepts like attentiveness, alertness and the level of arousal to these variables.
It must be however emphasized that quantum number increments alone need not determine entirely the contents of conscious experience. There is an infinite number of possible quantum jump sequences between two states \Psi_i and \Psi_f. This is also the case for diagonal quantum jumps \Psi_i-->.....\Psi_i. The idea that diagonal quantum jumps, and more generally, quantum jump sequences leading from \Psi_i back to \Psi_i, could give all possible conscious information about given quantum history \Psi_i is attractive. The requirement that diagonal quantum jumps give information about \Psi_i suggests that quantum jumps give also other conscious information than the information coded to the quantum number and zero mode increments. For instance, the average over the cascade of self measurements might have interpretation as a counterpart of conscious analysis.
3.6 Self as a moral agent
One could argue that the randomness of the quantum jump means that moral choices are impossible. Volition can however be associated with the quantum jumps in which p-adic spacetime sheet representing intention is transformed to real spacetime sheet representing real action.
p-Adic evolution defines the fundamental value of the quantum ethics. The selections which tend to increase the value of the p-adic prime represent good deeds since they mean evolution. The values of this ethics are not in the physical world but in the quantum jumps defining the subjective reality. The p-adic prime associated with entire universe is literally infinite (for the theory of infinite primes, see the chapter "Infinite primes and consciousness" of [cbookII] was originally motivated by consciousness theory). Infinite primes have however decomposition into finite primes in a well-defined sense and the increase of the infinite prime in a statistical sense implies the increase of finite composite primes and the appearence of new spacetime regions characterized by finite primes. A physical correlate for the increase of finite p-adic prime is the gradual growth of say cell or biological organism whereas the creation of new organism is a correlate for generation of a spacetime region characterized by p-adic prime.
Selves can make plans since they have 4-dimensional geometric memory (conscious experience contains information about a four-dimensional spacetime region, rather than only time=constant snapshot, and gives rise to a "prophecy", a prediction for the future and past, which would be reliable if the world were completely classical). Intentions, plans and anticipations are represented by p-adic spacetime regions simulating real regions. Selves can make decisions and select between various classical macroscopic time developments. Selves are able to remember their choices since they have subjective memories about the previous quantum jumps. Thus selves are genuine moral agents if they can experience directly that increase of p is good and decrease of p is bad.
I am grateful to Lian Sidorov for a considerable help and encouragement during the preparation of the manuscript as well as for very stimulating discussions.
4. Bibliography
[BR1] D. J. Bierman and D. I. Radin (1997), Anomalous Anticipatory Response on Randomized Future Conditions, Perceptual and Motor Skills, 84, pp. 689-690.
[BR2] D.J Bierman and D. I. Radin (1998), Anomalous unconscious emotional responses: Evidence for a reversal of the arrow of time .
[Deeke] L. Deeke, B. Götzinger and H. H. Kornhuber (1976), Voluntary finger movements in man: cerebral potentials and theory, Biol. Cybernetics, 23, 99.
[Engel] A. K. Engel et al(2000) Temporal Binding, Binocular Rivalry, and Consciousness
[Peoch] R. Peoch (1995), Network (the journal of Medical Network edited by Peter Fenwick), vol. 62. For a popular article about animal-robot interactions see .
[TGD] M. Pitkänen (1990) Topological Geometrodynamics Internal Report HU-TFT-IR-90-4 (Helsinki University). The online version of the book is at
[padTGD] M. Pitkänen (1995) Topological Geometrodynamics and p-Adic Numbers. Internal Report HU-TFT-IR-95-5 (Helsinki University). The online version of the book is at
[cbookI] M. Pitkänen (1998) TGD inspired theory of consciousness with applications to biosystems
[cbookII] M. Pitkänen (2001), Genes, Memes, Qualia, and Semitrance . |
eeaf233a5f0f45cd | You are currently browsing the tag archive for the ‘limiting absorption principle’ tag.
Igor Rodnianski and I have just uploaded to the arXiv our paper “Effective limiting absorption principles, and applications“, submitted to Communications in Mathematical Physics. In this paper we derive limiting absorption principles (of type discussed in this recent post) for a general class of Schrödinger operators {H = -\Delta + V} on a wide class of manifolds, namely the asymptotically conic manifolds. The precise definition of such manifolds is somewhat technical, but they include as a special case the asymptotically flat manifolds, which in turn include as a further special case the smooth compact perturbations of Euclidean space {{\bf R}^n} (i.e. the smooth Riemannian manifolds that are identical to {{\bf R}^n} outside of a compact set). The potential {V} is assumed to be a short range potential, which roughly speaking means that it decays faster than {1/|x|} as {x \rightarrow \infty}; for several of the applications (particularly at very low energies) we need to in fact assume that {V} is a strongly short range potential, which roughly speaking means that it decays faster than {1/|x|^2}.
To begin with, we make no hypotheses about the topology or geodesic geometry of the manifold {M}; in particular, we allow {M} to be trapping in the sense that it contains geodesic flows that do not escape to infinity, but instead remain trapped in a bounded subset of {M}. We also allow the potential {V} to be signed, which in particular allows bound states (eigenfunctions of negative energy) to be created. For standard technical reasons we restrict attention to dimensions three and higher: {d \geq 3}.
It is well known that such Schrödinger operators {H} are essentially self-adjoint, and their spectrum consists of purely absolutely continuous spectrum on {(0,+\infty)}, together with possibly some eigenvalues at zero and negative energy (and at zero energy and in dimensions three and four, there are also the possibility of resonances which, while not strictly eigenvalues, have a somewhat analogous effect on the dynamics of the Laplacian and related objects, such as resolvents). In particular, the resolvents {R(\lambda \pm i\epsilon) := (H - \lambda \mp i\epsilon)^{-1}} make sense as bounded operators on {L^2(M)} for any {\lambda \in {\bf R}} and {\epsilon > 0}. As discussed in the previous blog post, it is of interest to obtain bounds for the behaviour of these resolvents, as this can then be used via some functional calculus manipulations to obtain control on many other operators and PDE relating to the Schrödinger operator {H}, such as the Helmholtz equation, the time-dependent Schrödinger equation, and the wave equation. In particular, it is of interest to obtain limiting absorption estimates such as
\displaystyle \| R(\lambda \pm i\epsilon) f \|_{H^{0,-1/2-\sigma}(M)} \leq C(M,V,\lambda,\sigma) \| f \|_{H^{0,1/2+\sigma}(M)} \ \ \ \ \ (1)
for {\lambda \in {\bf R}} (and particularly in the positive energy regime {\lambda>0}), where {\sigma,\epsilon > 0} and {f} is an arbitrary test function. The constant {C(M,V,\lambda,\sigma)} needs to be independent of {\epsilon} for such estimates to be truly useful, but it is also of interest to determine the extent to which these constants depend on {M}, {V}, and {\lambda}. The dependence on {\sigma} is relatively uninteresting and henceforth we will suppress it. In particular, our paper focused to a large extent on quantitative methods that could give effective bounds on {C(M,V,\lambda)} in terms of quantities such as the magnitude {A} of the potential {V} in a suitable norm.
It turns out to be convenient to distinguish between three regimes:
• The high-energy regime {\lambda \gg 1};
• The medium-energy regime {\lambda \sim 1}; and
• The low-energy regime {0 < \lambda \ll 1}.
Our methods actually apply more or less uniformly to all three regimes, but the nature of the conclusions is quite different in each of the three regimes.
The high-energy regime {\lambda \gg 1} was essentially worked out by Burq, although we give an independent treatment of Burq’s results here. In this regime it turns out that we have an unconditional estimate of the form (1) with a constant of the shape
\displaystyle C(M,V,\lambda) = C(M,A) e^{C(M,A) \sqrt{\lambda}}
where {C(M,A)} is a constant that depends only on {M} and on a parameter {A} that controls the size of the potential {V}. This constant, while exponentially growing, is still finite, which among other things is enough to rule out the possibility that {H} contains eigenfunctions (i.e. point spectrum) embedded in the high-energy portion of the spectrum. As is well known, if {M} contains a certain type of trapped geodesic (in particular those arising from positively curved portions of the manifold, such as the equator of a sphere), then it is possible to construct pseudomodes {f} that show that this sort of exponential growth is necessary. On the other hand, if we make the non-trapping hypothesis that all geodesics in {M} escape to infinity, then we can obtain a much stronger high-energy limiting absorption estimate, namely
\displaystyle C(M,V,\lambda,\sigma) = C(M,A) \lambda^{-1/2}.
The exponent {1/2} here is closely related to the standard fact that on non-trapping manifolds, there is a local smoothing effect for the time-dependent Schrödinger equation that gains half a derivative of regularity (cf. previous blog post). In the high-energy regime, the dynamics are well-approximated by semi-classical methods, and in particular one can use tools such as the positive commutator method and pseudo-differential calculus to obtain the desired estimates. In case of trapping one also needs the standard technique of Carleman inequalities to control the compact (and possibly trapping) core of the manifold, and in particular needing the delicate two-weight Carleman inequalities of Burq.
In the medium and low energy regimes one needs to work harder. In the medium energy regime {\lambda \sim 1}, we were able to obtain a uniform bound
\displaystyle C(M,V,\lambda) \leq C(M,A)
for all asymptotically conic manifolds (trapping or not) and all short-range potentials. To establish this bound, we have to supplement the existing tools of the positive commutator method and Carleman inequalities with an additional ODE-type analysis of various energies of the solution {u = R(\lambda \pm i\epsilon) f} to a Helmholtz equation on large spheres, as will be discussed in more detail below the fold.
The methods also extend to the low-energy regime {0 < \lambda \ll 1}. Here, the bounds become somewhat interesting, with a subtle distinction between effective estimates that are uniform over all potentials {V} which are bounded in a suitable sense by a parameter {A} (e.g. obeying {|V(x)| \leq A \langle x \rangle^{-2-2\sigma}} for all {x}), and ineffective estimates that exploit qualitative properties of {V} (such as the absence of eigenfunctions or resonances at zero) and are thus not uniform over {V}. On the effective side, and for potentials that are strongly short range (at least at local scales {|x| = O(\lambda^{-1/2})}; one can tolerate merely short-range behaviour at more global scales, but this is a technicality that we will not discuss further here) we were able to obtain a polynomial bound of the form
\displaystyle C(M,V,\lambda) \leq C(M,A) \lambda^{-C(M,A)}
that blew up at a large polynomial rate at the origin. Furthermore, by carefully designing a sequence of potentials {V} that induce near-eigenfunctions that resemble two different Bessel functions of the radial variable glued together, we are able to show that this type of polynomial bound is sharp in the following sense: given any constant {C > 0}, there exists a sequence {V_n} of potentials on Euclidean space {{\bf R}^d} uniformly bounded by {A}, and a sequence {\lambda_n} of energies going to zero, such that
\displaystyle C({\bf R}^d,V_n,\lambda_n) \geq \lambda_n^{-C}.
This shows that if one wants bounds that are uniform in the potential {V}, then arbitrary polynomial blowup is necessary.
Interestingly, though, if we fix the potential {V}, and then ask for bounds that are not necessarily uniform in {V}, then one can do better, as was already observed in a classic paper of Jensen and Kato concerning power series expansions of the resolvent near the origin. In particular, if we make the spectral assumption that {V} has no eigenfunctions or resonances at zero, then an argument (based on (a variant of) the Fredholm alternative, which as discussed in this recent blog post gives ineffective bounds) gives a bound of the form
\displaystyle C(M,V,\lambda) \leq C(M,V) \lambda^{-1/2}
in the low-energy regime (but note carefully here that the constant {C(M,V)} on the right-hand side depends on the potential {V} itself, and not merely on the parameter {A} that upper bounds it). Even if there are eigenvalues or resonances, it turns out that one can still obtain a similar bound but with an exponent of {\lambda^{-3/2}} instead of {\lambda^{-1/2}}. This limited blowup at infinity is in sharp contrast to the arbitrarily large polynomial blowup rate that can occur if one demands uniform bounds. (This particular subtlety between uniform and non-uniform estimates confused us, by the way, for several weeks; for a long time we thought that we had somehow found a contradiction between our results and the results of Jensen and Kato.)
As applications of our limiting absorption estimates, we give local smoothing and dispersive estimates for solutions (as well as the closely related RAGE type theorems) to the time-dependent Schrödinger and wave equations, and also reprove standard facts about the spectrum of Schrödinger operators in this setting.
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Perhaps the most fundamental differential operator on Euclidean space {{\bf R}^d} is the Laplacian
\displaystyle \Delta := \sum_{j=1}^d \frac{\partial^2}{\partial x_j^2}.
The Laplacian is a linear translation-invariant operator, and as such is necessarily diagonalised by the Fourier transform
\displaystyle \hat f(\xi) := \int_{{\bf R}^d} f(x) e^{-2\pi i x \cdot \xi}\ dx.
Indeed, we have
\displaystyle \widehat{\Delta f}(\xi) = - 4 \pi^2 |\xi|^2 \hat f(\xi)
for any suitably nice function {f} (e.g. in the Schwartz class; alternatively, one can work in very rough classes, such as the space of tempered distributions, provided of course that one is willing to interpret all operators in a distributional or weak sense).
Because of this explicit diagonalisation, it is a straightforward manner to define spectral multipliers {m(-\Delta)} of the Laplacian for any (measurable, polynomial growth) function {m: [0,+\infty) \rightarrow {\bf C}}, by the formula
\displaystyle \widehat{m(-\Delta) f}(\xi) := m( 4\pi^2 |\xi|^2 ) \hat f(\xi).
(The presence of the minus sign in front of the Laplacian has some minor technical advantages, as it makes {-\Delta} positive semi-definite. One can also define spectral multipliers more abstractly from general functional calculus, after establishing that the Laplacian is essentially self-adjoint.) Many of these multipliers are of importance in PDE and analysis, such as the fractional derivative operators {(-\Delta)^{s/2}}, the heat propagators {e^{t\Delta}}, the (free) Schrödinger propagators {e^{it\Delta}}, the wave propagators {e^{\pm i t \sqrt{-\Delta}}} (or {\cos(t \sqrt{-\Delta})} and {\frac{\sin(t\sqrt{-\Delta})}{\sqrt{-\Delta}}}, depending on one’s conventions), the spectral projections {1_I(\sqrt{-\Delta})}, the Bochner-Riesz summation operators {(1 + \frac{\Delta}{4\pi^2 R^2})_+^\delta}, or the resolvents {R(z) := (-\Delta-z)^{-1}}.
Each of these families of multipliers are related to the others, by means of various integral transforms (and also, in some cases, by analytic continuation). For instance:
1. Using the Laplace transform, one can express (sufficiently smooth) multipliers in terms of heat operators. For instance, using the identity
\displaystyle \lambda^{s/2} = \frac{1}{\Gamma(-s/2)} \int_0^\infty t^{-1-s/2} e^{-t\lambda}\ dt
(using analytic continuation if necessary to make the right-hand side well-defined), with {\Gamma} being the Gamma function, we can write the fractional derivative operators in terms of heat kernels:
\displaystyle (-\Delta)^{s/2} = \frac{1}{\Gamma(-s/2)} \int_0^\infty t^{-1-s/2} e^{t\Delta}\ dt. \ \ \ \ \ (1)
2. Using analytic continuation, one can connect heat operators {e^{t\Delta}} to Schrödinger operators {e^{it\Delta}}, a process also known as Wick rotation. Analytic continuation is a notoriously unstable process, and so it is difficult to use analytic continuation to obtain any quantitative estimates on (say) Schrödinger operators from their heat counterparts; however, this procedure can be useful for propagating identities from one family to another. For instance, one can derive the fundamental solution for the Schrödinger equation from the fundamental solution for the heat equation by this method.
3. Using the Fourier inversion formula, one can write general multipliers as integral combinations of Schrödinger or wave propagators; for instance, if {z} lies in the upper half plane {{\bf H} := \{ z \in {\bf C}: \hbox{Im} z > 0 \}}, one has
\displaystyle \frac{1}{x-z} = i\int_0^\infty e^{-itx} e^{itz}\ dt
for any real number {x}, and thus we can write resolvents in terms of Schrödinger propagators:
\displaystyle R(z) = i\int_0^\infty e^{it\Delta} e^{itz}\ dt. \ \ \ \ \ (2)
In a similar vein, if {k \in {\bf H}}, then
\displaystyle \frac{1}{x^2-k^2} = \frac{i}{k} \int_0^\infty \cos(tx) e^{ikt}\ dt
for any {x>0}, so one can also write resolvents in terms of wave propagators:
\displaystyle R(k^2) = \frac{i}{k} \int_0^\infty \cos(t\sqrt{-\Delta}) e^{ikt}\ dt. \ \ \ \ \ (3)
4. Using the Cauchy integral formula, one can express (sufficiently holomorphic) multipliers in terms of resolvents (or limits of resolvents). For instance, if {t > 0}, then from the Cauchy integral formula (and Jordan’s lemma) one has
\displaystyle e^{itx} = \frac{1}{2\pi i} \lim_{\epsilon \rightarrow 0^+} \int_{\bf R} \frac{e^{ity}}{y-x+i\epsilon}\ dy
for any {x \in {\bf R}}, and so one can (formally, at least) write Schrödinger propagators in terms of resolvents:
\displaystyle e^{-it\Delta} = - \frac{1}{2\pi i} \lim_{\epsilon \rightarrow 0^+} \int_{\bf R} e^{ity} R(y+i\epsilon)\ dy. \ \ \ \ \ (4)
5. The imaginary part of {\frac{1}{\pi} \frac{1}{x-(y+i\epsilon)}} is the Poisson kernel {\frac{\epsilon}{\pi} \frac{1}{(y-x)^2+\epsilon^2}}, which is an approximation to the identity. As a consequence, for any reasonable function {m(x)}, one has (formally, at least)
\displaystyle m(x) = \lim_{\epsilon \rightarrow 0^+} \frac{1}{\pi} \int_{\bf R} (\hbox{Im} \frac{1}{x-(y+i\epsilon)}) m(y)\ dy
which leads (again formally) to the ability to express arbitrary multipliers in terms of imaginary (or skew-adjoint) parts of resolvents:
\displaystyle m(-\Delta) = \lim_{\epsilon \rightarrow 0^+} \frac{1}{\pi} \int_{\bf R} (\hbox{Im} R(y+i\epsilon)) m(y)\ dy. \ \ \ \ \ (5)
Among other things, this type of formula (with {-\Delta} replaced by a more general self-adjoint operator) is used in the resolvent-based approach to the spectral theorem (by using the limiting imaginary part of resolvents to build spectral measure). Note that one can also express {\hbox{Im} R(y+i\epsilon)} as {\frac{1}{2i} (R(y+i\epsilon) - R(y-i\epsilon))}.
Remark 1 The ability of heat operators, Schrödinger propagators, wave propagators, or resolvents to generate other spectral multipliers can be viewed as a sort of manifestation of the Stone-Weierstrass theorem (though with the caveat that the spectrum of the Laplacian is non-compact and so the Stone-Weierstrass theorem does not directly apply). Indeed, observe the *-algebra type properties
\displaystyle e^{s\Delta} e^{t\Delta} = e^{(s+t)\Delta}; \quad (e^{s\Delta})^* = e^{s\Delta}
\displaystyle e^{is\Delta} e^{it\Delta} = e^{i(s+t)\Delta}; \quad (e^{is\Delta})^* = e^{-is\Delta}
\displaystyle e^{is\sqrt{-\Delta}} e^{it\sqrt{-\Delta}} = e^{i(s+t)\sqrt{-\Delta}}; \quad (e^{is\sqrt{-\Delta}})^* = e^{-is\sqrt{-\Delta}}
\displaystyle R(z) R(w) = \frac{R(w)-R(z)}{z-w}; \quad R(z)^* = R(\overline{z}).
Because of these relationships, it is possible (in principle, at least), to leverage one’s understanding one family of spectral multipliers to gain control on another family of multipliers. For instance, the fact that the heat operators {e^{t\Delta}} have non-negative kernel (a fact which can be seen from the maximum principle, or from the Brownian motion interpretation of the heat kernels) implies (by (1)) that the fractional integral operators {(-\Delta)^{-s/2}} for {s>0} also have non-negative kernel. Or, the fact that the wave equation enjoys finite speed of propagation (and hence that the wave propagators {\cos(t\sqrt{-\Delta})} have distributional convolution kernel localised to the ball of radius {|t|} centred at the origin), can be used (by (3)) to show that the resolvents {R(k^2)} have a convolution kernel that is essentially localised to the ball of radius {O( 1 / |\hbox{Im}(k)| )} around the origin.
In this post, I would like to continue this theme by using the resolvents {R(z) = (-\Delta-z)^{-1}} to control other spectral multipliers. These resolvents are well-defined whenever {z} lies outside of the spectrum {[0,+\infty)} of the operator {-\Delta}. In the model three-dimensional case {d=3}, they can be defined explicitly by the formula
\displaystyle R(k^2) f(x) = \int_{{\bf R}^3} \frac{e^{ik|x-y|}}{4\pi |x-y|} f(y)\ dy
whenever {k} lives in the upper half-plane {\{ k \in {\bf C}: \hbox{Im}(k) > 0 \}}, ensuring the absolute convergence of the integral for test functions {f}. (In general dimension, explicit formulas are still available, but involve Bessel functions. But asymptotically at least, and ignoring higher order terms, one simply replaces {\frac{e^{ik|x-y|}}{4\pi |x-y|}} by {\frac{e^{ik|x-y|}}{c_d |x-y|^{d-2}}} for some explicit constant {c_d}.) It is an instructive exercise to verify that this resolvent indeed inverts the operator {-\Delta-k^2}, either by using Fourier analysis or by Green’s theorem.
Henceforth we restrict attention to three dimensions {d=3} for simplicity. One consequence of the above explicit formula is that for positive real {\lambda > 0}, the resolvents {R(\lambda+i\epsilon)} and {R(\lambda-i\epsilon)} tend to different limits as {\epsilon \rightarrow 0}, reflecting the jump discontinuity in the resolvent function at the spectrum; as one can guess from formulae such as (4) or (5), such limits are of interest for understanding many other spectral multipliers. Indeed, for any test function {f}, we see that
\displaystyle \lim_{\epsilon \rightarrow 0^+} R(\lambda+i\epsilon) f(x) = \int_{{\bf R}^3} \frac{e^{i\sqrt{\lambda}|x-y|}}{4\pi |x-y|} f(y)\ dy
\displaystyle \lim_{\epsilon \rightarrow 0^+} R(\lambda-i\epsilon) f(x) = \int_{{\bf R}^3} \frac{e^{-i\sqrt{\lambda}|x-y|}}{4\pi |x-y|} f(y)\ dy.
Both of these functions
\displaystyle u_\pm(x) := \int_{{\bf R}^3} \frac{e^{\pm i\sqrt{\lambda}|x-y|}}{4\pi |x-y|} f(y)\ dy
solve the Helmholtz equation
\displaystyle (-\Delta-\lambda) u_\pm = f, \ \ \ \ \ (6)
but have different asymptotics at infinity. Indeed, if {\int_{{\bf R}^3} f(y)\ dy = A}, then we have the asymptotic
\displaystyle u_\pm(x) = \frac{A e^{\pm i \sqrt{\lambda}|x|}}{4\pi|x|} + O( \frac{1}{|x|^2}) \ \ \ \ \ (7)
as {|x| \rightarrow \infty}, leading also to the Sommerfeld radiation condition
\displaystyle u_\pm(x) = O(\frac{1}{|x|}); \quad (\partial_r \mp i\sqrt{\lambda}) u_\pm(x) = O( \frac{1}{|x|^2}) \ \ \ \ \ (8)
where {\partial_r := \frac{x}{|x|} \cdot \nabla_x} is the outgoing radial derivative. Indeed, one can show using an integration by parts argument that {u_\pm} is the unique solution of the Helmholtz equation (6) obeying (8) (see below). {u_+} is known as the outward radiating solution of the Helmholtz equation (6), and {u_-} is known as the inward radiating solution. Indeed, if one views the function {u_\pm(t,x) := e^{-i\lambda t} u_\pm(x)} as a solution to the inhomogeneous Schrödinger equation
\displaystyle (i\partial_t + \Delta) u_\pm = - e^{-i\lambda t} f
and using the de Broglie law that a solution to such an equation with wave number {k \in {\bf R}^3} (i.e. resembling {A e^{i k \cdot x}} for some amplitide {A}) should propagate at (group) velocity {2k}, we see (heuristically, at least) that the outward radiating solution will indeed propagate radially away from the origin at speed {2\sqrt{\lambda}}, while inward radiating solution propagates inward at the same speed.
There is a useful quantitative version of the convergence
\displaystyle R(\lambda \pm i\epsilon) f \rightarrow u_\pm, \ \ \ \ \ (9)
known as the limiting absorption principle:
Theorem 1 (Limiting absorption principle) Let {f} be a test function on {{\bf R}^3}, let {\lambda > 0}, and let {\sigma > 0}. Then one has
\displaystyle \| R(\lambda \pm i\epsilon) f \|_{H^{0,-1/2-\sigma}({\bf R}^3)} \leq C_\sigma \lambda^{-1/2} \|f\|_{H^{0,1/2+\sigma}({\bf R}^3)}
for all {\epsilon > 0}, where {C_\sigma > 0} depends only on {\sigma}, and {H^{0,s}({\bf R}^3)} is the weighted norm
\displaystyle \|f\|_{H^{0,s}({\bf R}^3)} := \| \langle x \rangle^s f \|_{L^2_x({\bf R}^3)}
and {\langle x \rangle := (1+|x|^2)^{1/2}}.
This principle allows one to extend the convergence (9) from test functions {f} to all functions in the weighted space {H^{0,1/2+\sigma}} by a density argument (though the radiation condition (8) has to be adapted suitably for this scale of spaces when doing so). The weighted space {H^{0,-1/2-\sigma}} on the left-hand side is optimal, as can be seen from the asymptotic (7); a duality argument similarly shows that the weighted space {H^{0,1/2+\sigma}} on the right-hand side is also optimal.
We prove this theorem below the fold. As observed long ago by Kato (and also reproduced below), this estimate is equivalent (via a Fourier transform in the spectral variable {\lambda}) to a useful estimate for the free Schrödinger equation known as the local smoothing estimate, which in particular implies the well-known RAGE theorem for that equation; it also has similar consequences for the free wave equation. As we shall see, it also encodes some spectral information about the Laplacian; for instance, it can be used to show that the Laplacian has no eigenvalues, resonances, or singular continuous spectrum. These spectral facts are already obvious from the Fourier transform representation of the Laplacian, but the point is that the limiting absorption principle also applies to more general operators for which the explicit diagonalisation afforded by the Fourier transform is not available. (Igor Rodnianski and I are working on a paper regarding this topic, of which I hope to say more about soon.)
In order to illustrate the main ideas and suppress technical details, I will be a little loose with some of the rigorous details of the arguments, and in particular will be manipulating limits and integrals at a somewhat formal level.
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cfea71b73ba440c0 | Agronomy and Horticulture Department
Date of this Version
Published in PHYSICAL REVIEW A 79, 023403 (2009). Copyright ©2009 The American Physical Society. Used by permission.
Three alternative forms of harmonic spectra, based on the dipole moment, dipole velocity, and dipole acceleration, are compared by a numerical solution of the Schrödinger equation for a hydrogen atom interacting with a linearly polarized laser pulse, whose electric field is given by E(t)= E0f(t)cos(ω0t + η) with Gaussian carrier envelope f(t) = exp(−t22). The carrier frequency ω0 is fixed to correspond to a wavelength of 800 nm. Spectra for a selection of pulses, for which the intensity I0=cε0E20, duration T∞ δ, and carrier-envelope phase η are systematically varied, show that, depending on η, all three forms are in good agreement for “weak” pulses with I0 < Ib, the over-barrier ionization threshold, but that marked differences among the three appear as the pulse becomes shorter and stronger (I0 >Ib). Except for scalings by powers of the harmonic frequency, the three forms differ from one another only by “limit contributions” proportional to the expectation values of the dipole moment ‹z(tf)› or dipole velocity ‹z(tf)› at the end (tf) of the pulse. For long, weak pulses the limit contributions are negligible, whereas for short, strong ones they are not. In the short, strong limit, where ‹z(tf)› ≠ 0 and therefore ‹z(t)› may increase without bound (i.e., the atom may ionize), depending on η, an “infinite-time” spectrum based on the acceleration form provides a convenient computational pathway to the corresponding infinite-time dipole-velocity spectrum, which is related directly to the experimentally measured “harmonic photon number spectrum” (HPNS). For short, intense pulses the HPNS is quite sensitive to η and exhibits not only the usual odd harmonics but also even ones. The analysis also reveals that most of the harmonic photons are emitted during the passage of the pulse. Because of the divergence of ‹z(t)› the dipolemoment form does not provide a numerically reliable route to the harmonic spectrum for very short (fewcycle), very intense laser pulses. |
6f0b21c9911e27bd | Readings and Lecture Notes
Lecture notes (with blanks) are provided for each lecture. Students are expected to follow along during the lecture in order to fill in the blanks in the notes.
Readings are from the required textbook:
Amazon logo Atkins, Peter, and Loretta Jones. Chemical Principles: The Quest for Insight. 4th ed. New York, NY: W.H. Freeman and Company, 2007. ISBN: 9781429209656.
The reading assignment listed for the first session is a review of information you are expected to know before you begin the class. This information is not discussed during lecture.
In addition, no lecture notes were provided for the first session. The handout associated with that lecture is an overview of the class format and expectations.
L1 The importance of chemical principles Section A.1
Sections B.3-B.4
Sections C-H
Sections L-M
L2 Discovery of electron and nucleus, need for quantum mechanics Sections A.2-A.3
Sections B.1-B.2
Section 1.1
L3 Wave-particle duality of light Sections 1.2 and 1.4 (PDF)
L4 Wave-particle duality of matter, Schrödinger equation Sections 1.5-1.6 (PDF)
L5 Hydrogen atom energy levels Sections 1.3, 1.7 up to equation 9b, and 1.8 (PDF)
L6 Hydrogen atom wavefunctions (orbitals) Section 1.9 (PDF - 1.2 MB)
L7 p-orbitals Sections 1.10-1.11 (PDF)
L8 Multelectron atoms and electron configurations Sections 1.12-1.13 (PDF)
L9 Periodic trends Sections 1.14-1.18, and 1.20 (PDF - 1.6 MB)
L10 Periodic trends continued; Covalent bonds Sections 2.5-2.6, and 2.14-2.16 (PDF - 1.6 MB)
L11 Lewis structures Sections 2.7-2.8 (PDF)
L12 Exceptions to Lewis structure rules; Ionic bonds Sections 2.3 and 2.9-2.12 (PDF - 1.1 MB)
L13 Polar covalent bonds; VSEPR theory Sections 3.1-3.2 (PDF - 5.1 MB)
L14 Molecular orbital theory Sections 3.8-3.11 (PDF)
L15 Valence bond theory and hybridization Sections 3.4-3.7 (PDF - 1.0 MB)
L16 Determining hybridization in complex molecules; Termochemistry and bond energies/bond enthalpies Sections 6.13, 6.15-6.18, and 6.20 (PDF)
L17 Entropy and disorder Sections 7.1-7.2, 7.8, 7.12-7.13, and 7.15 (PDF)
L18 Free energy and control of spontaneity Section 7.16 (PDF)
L19 Chemical equilibrium Sections 9.0-9.9 (PDF)
L20 Le Chatelier's principle and applications to blood-oxygen levels Sections 9.10-9.13 (PDF)
L21 Acid-base equilibrium: Is MIT water safe to drink? Chapter 10 (PDF)
L22 Chemical and biological buffers Chapters 10 and 11 (PDF)
L23 Acid-base titrations Chapter 11 (PDF)
L24 Balancing oxidation/reduction equations Section K
Chapter 12
L25 Electrochemical cells Chapter 12 (PDF)
L26 Chemical and biological oxidation/reduction reactions Chapter 12 (PDF)
L27 Transition metals and the treatment of lead poisoning pp. 669-681 (PDF)
L28 Crystal field theory pp. 681-683 (PDF - 1.4 MB)
L29 Metals in biology pp. 631-637 (PDF - 1.2 MB)
L30 Magnetism and spectrochemical theory Chapter 16 (PDF)
L31 Rate laws Sections 13.1-13.5 (PDF)
L32 Nuclear chemistry and elementary reactions pp. 498-501 and 660-664 (PDF)
L33 Reaction mechanism pp. 549-552 (PDF)
L34 Temperature and kinetics Sections 13.11-13.13 (PDF)
L35 Enzyme catalysis Sections 13.14-13.15 (PDF)
L36 Biochemistry (PDF) |
7bc4b3e985c0b65b | Monday, March 20, 2006
A universe of Qualia
In my previous posting I applied Tegmark's idea that every mathematical model is a universe, to humans. This leads to the conclusion that we can think of our minds as universes in their own right. If we think of the universe we live in, we usually think of the objects we see around us, their properties and how they behave.
In case of our mind considered as a universe, the laws of physics are contained in an exact description of the way our neurons in our brain interact with each other. This description is, of course, enormously complicated. Alternatively, we could think of the neurons in our brain as simulating ''emergent laws of physics'' that describe the qualia we experience.
Just like one can do organic chemistry without solving the Schrödinger equation for complex organic molecules, we can talk about how we feel, what we see etc. without referring to what exactly our neurons are doing in our brains. We can thus think of the qualia as ''events'' in our personal universe. These are described by ''effective laws of physics'', analogously to the imprecise laws of, say, organic chemistry or biology.
Since we experience the qualia and not the fundamental processes that give rise to the qualia (this follows from the Simulation Argument: If the brain were simulated on some computer, it would have the same consciousness), we should consider the qualia as fundamental objects of our personal universe. The universe on the level of the qualia is where the mind really resides. It is here that the notions of pain, anger, happiness, colors etc. exist.
Blogger QUASAR9 said...
Hi Count seeing the universe with the eyes is relative, our eyes can and do deceive us. The same with thoughts we can like Quixote be fighting windmills.
But physical 'reality' ie a brick wall, no matter whether we have 20/20 vision, whether we are partially sighted, whether we are totally blind, or whether our minds are troubled or otherwise distracted, if we walk into a brick wall we shall know we have walked into one.
You'll be surprised how many people walk into lampposts, even among those with 20/20 vision, no not because they weren't looking in that direction (in front of them) but because they didn't see it (didn't even see it coming).
Not because of the 'blind' spot, but because their focus, or thoughts were on something other than wahat was in front of them.
Incidentally, have you ever pulled up at a roundabout, there is a car in front, you look (left) in EU (right) in uk, no traffic on roundabout, so you start to move forward, only to slam the brakes on when you realise the vehicle in front has not moved. You (brain) just assumed that because you could see it was ok to go the chap in front would see it too, and respond at the same speed as you.
Of course some people travel thru life seldom encountering a red light, or gettin caught in traffic, whilst others go from one red light to the next.
And some people drive accelerating braking accelerating braking in urban traffic, whilst taxi drivers have developed the skill of going with the 'flow' and often arriving at the destination in less time, with less stress and less wear on their selves and vehicles.
But I digress, what I meant is that if there is something solid there, there are no X-men that can walk through it, the wall is there whether you can see or whether you are blind. Ram raiders used to and do get over the problem of walls or reinforced glass by using 4x4's with bull bars. lol! Laters ... Q
Wed Jun 14, 06:23:00 AM PDT
Blogger Faust said...
Hi Count,
I have just opened up my own blog; you may find it interesting.
Name: 'Space - Time - Matter'
p.s.: Are you a physics (grad) student? I want to get primarily physics and math students to my site.
Fri Oct 06, 07:03:00 AM PDT
Blogger Count Iblis said...
Quasar9, I agree with your analyses. An interesting question is why is there a physical world and why can't be like the X-men you mention? I'll elaborate on that in a next posting.
Mon Oct 09, 06:56:00 PM PDT
Blogger Count Iblis said...
Hi Faust, I'll visit your Blog. I've a Ph.D. in physics. On this blog I only explore metaphysical ideas that are not (yet) publishable :)
Mon Oct 09, 06:57:00 PM PDT
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Quantum Gates Simulator Based on DSP TI6711
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Quantum theory has found a new field of application in the information and computation fields during recent years. We developed a Quantum Gate Simulator based on the Digital Signal Processor (DSP) DSP TI6711 using the Hamiltonian in the time- dependent Schrödinger equation. The Hamiltonian describes the Quantum System by manipulating a Quantum Bit (QuBit) using unitary matrices. Gates simulated are conditional NOT operation, Controlled-NOT Gate, Multi-bit Controlled-NOT Gate or Toffoli gate, Rotation Gate or Hadamard transform and twiddle gate, all useful in quantum computation due to their inherently reversible characteristic. With the simulation process, we have obtained approximately 95% fidelity action of the gate on an arbitrary two and three QuBit input state. We have determined an average error probability bounded above by 0.07 ± 0.01.
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1a751e1c114f00b4 | Eigenvalues and eigenvectors
From Wikipedia, the free encyclopedia
Jump to: navigation, search
"Characteristic root" redirects here. For other uses, see Characteristic root (disambiguation).
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that does not change its direction when that linear transformation is applied to it. More formally, if T is a linear transformation from a vector space V over a field F into itself and v is a vector in V that is not the zero vector, then v is an eigenvector of T if T(v) is a scalar multiple of v. This condition can be written as the equation
where λ is a scalar in the field F, known as the eigenvalue, characteristic value, or characteristic root associated with the eigenvector v.
If the vector space V is finite-dimensional, then the linear transformation T can be represented as a square matrix A, and the vector v by a column vector, rendering the above mapping as a matrix multiplication on the left hand side and a scaling of the column vector on the right hand side in the equation
There is a correspondence between n by n square matrices and linear transformations from an n-dimensional vector space to itself. For this reason, it is equivalent to define eigenvalues and eigenvectors using either the language of matrices or the language of linear transformations.[1][2]
Geometrically an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction that is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed.[3]
Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix eigen- is adopted from the German word eigen for "proper", "inherent"; "own", "individual", "special"; "specific", "peculiar", or "characteristic".[4] Originally utilized to study principal axes of the rotational motion of rigid bodies, eigenvalues and eigenvectors have a wide range of applications, for example in stability analysis, vibration analysis, atomic orbitals, facial recognition, and matrix diagonalization.
In essence, an eigenvector v of a linear transformation T is a non-zero vector that, when T is applied to it, does not change direction. Applying T to the eigenvector only scales the eigenvector by the scalar value λ, called an eigenvalue. This condition can be written as the equation
referred to as the eigenvalue equation or eigenequation. In general, λ may be any scalar. For example, λ may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or complex.
In this shear mapping the red arrow changes direction but the blue arrow does not. The blue arrow is an eigenvector of this shear mapping because it doesn't change direction, and since its length is unchanged, its eigenvalue is 1.
The Mona Lisa example pictured at right provides a simple illustration. Each point on the painting can be represented as a vector pointing from the center of the painting to that point. The linear transformation in this example is called a shear mapping. Points in the top half are moved to the right and points in the bottom half are moved to the left proportional to how far they are from the horizontal axis that goes through the middle of the painting. The vectors pointing to each point in the original image are therefore tilted right or left and made longer or shorter by the transformation. Notice that points along the horizontal axis do not move at all when this transformation is applied. Therefore, any vector that points directly to the right or left with no vertical component is an eigenvector of this transformation because the mapping does not change its direction. Moreover, these eigenvectors all have an eigenvalue equal to one because the mapping does not change their length, either.
Linear transformations can take many different forms, mapping vectors in a variety of vector spaces, so the eigenvectors can also take many forms. For example, the linear transformation could be a differential operator like , in which case the eigenvectors are functions called eigenfunctions that are scaled by that differential operator, such as
Alternatively, the linear transformation could take the form of an n by n matrix, in which case the eigenvectors are n by 1 matrices that are also referred to as eigenvectors. If the linear transformation is expressed in the form of an n by n matrix A, then the eigenvalue equation above for a linear transformation can be rewritten as the matrix multiplication
where the eigenvector v is an n by 1 matrix. For a matrix, eigenvalues and eigenvectors can be used to decompose the matrix, for example by diagonalizing it.
Eigenvalues and eigenvectors give rise to many closely related mathematical concepts, and the prefix eigen- is applied liberally when naming them:
• The set of all eigenvectors of a linear transformation, each paired with its corresponding eigenvalue, is called the eigensystem of that transformation.[5][6]
• The set of all eigenvectors of T corresponding to the same eigenvalue, together with the zero vector, is called an eigenspace or characteristic space of T.[7][8]
• If the set of eigenvectors of T form a basis of the domain of T, then this basis is called an eigenbasis.
Eigenvalues are often introduced in the context of linear algebra or matrix theory. Historically, however, they arose in the study of quadratic forms and differential equations.
In the 18th century Euler studied the rotational motion of a rigid body and discovered the importance of the principal axes.[9] Lagrange realized that the principal axes are the eigenvectors of the inertia matrix.[10] In the early 19th century, Cauchy saw how their work could be used to classify the quadric surfaces, and generalized it to arbitrary dimensions.[11] Cauchy also coined the term racine caractéristique (characteristic root) for what is now called eigenvalue; his term survives in characteristic equation.[12][13]
Fourier used the work of Laplace and Lagrange to solve the heat equation by separation of variables in his famous 1822 book Théorie analytique de la chaleur.[14] Sturm developed Fourier's ideas further and brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that real symmetric matrices have real eigenvalues.[11] This was extended by Hermite in 1855 to what are now called Hermitian matrices.[12] Around the same time, Brioschi proved that the eigenvalues of orthogonal matrices lie on the unit circle,[11] and Clebsch found the corresponding result for skew-symmetric matrices.[12] Finally, Weierstrass clarified an important aspect in the stability theory started by Laplace by realizing that defective matrices can cause instability.[11]
In the meantime, Liouville studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called Sturm–Liouville theory.[15] Schwarz studied the first eigenvalue of Laplace's equation on general domains towards the end of the 19th century, while Poincaré studied Poisson's equation a few years later.[16]
At the start of the 20th century, Hilbert studied the eigenvalues of integral operators by viewing the operators as infinite matrices.[17] He was the first to use the German word eigen, which means "own", to denote eigenvalues and eigenvectors in 1904,[18] though he may have been following a related usage by Helmholtz. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is standard today.[19]
The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when Von Mises published the power method. One of the most popular methods today, the QR algorithm, was proposed independently by John G.F. Francis[20] and Vera Kublanovskaya[21] in 1961.[22]
Eigenvalues and eigenvectors of matrices[edit]
Eigenvalues and eigenvectors are often introduced to students in the context of linear algebra courses focused on matrices.[23][24] Furthermore, linear transformations can be represented using matrices,[1][2] which is especially common in numerical and computational applications.[25]
Matrix A acts by stretching the vector x, not changing its direction, so x is an eigenvector of A.
Consider n-dimensional vectors that are formed as a list of n scalars, such as the three-dimensional vectors
These vectors are said to be scalar multiples of each other, or parallel or collinear, if there is a scalar λ such that
In this case λ = −1/20.
Now consider the linear transformation of n-dimensional vectors defined by an n by n matrix A,
where, for each row,
If it occurs that v and w are scalar multiples, that is if
then v is an eigenvector of the linear transformation A and the scale factor λ is the eigenvalue corresponding to that eigenvector. Equation (1) is the eigenvalue equation for the matrix A.
Equation (1) can be stated equivalently as
where I is the n by n identity matrix.
Eigenvalues and the characteristic polynomial[edit]
Equation (2) has a non-zero solution v if and only if the determinant of the matrix (A − λI) is zero. Therefore, the eigenvalues of A are values of λ that satisfy the equation
Using Leibniz' rule for the determinant, the left hand side of Equation (3) is a polynomial function of the variable λ and the degree of this polynomial is n, the order of the matrix A. Its coefficients depend on the entries of A, except that its term of degree n is always (−1)nλn. This polynomial is called the characteristic polynomial of A. Equation (3) is called the characteristic equation or the secular equation of A.
The fundamental theorem of algebra implies that the characteristic polynomial of an n by n matrix A, being a polynomial of degree n, can be factored into the product of n linear terms,
where each λi may be real but in general is a complex number. The numbers λ1, λ2, ... λn, which may not all have distinct values, are roots of the polynomial and are the eigenvalues of A.
As a brief example, which is described in more detail in the examples section later, consider the matrix
Taking the determinant of (M − λI), the characteristic polynomial of M is
Setting the characteristic polynomial equal to zero, it has roots at λ = 1 and λ = 3, which are the two eigenvalues of M. The eigenvectors corresponding to each eigenvalue can be found by solving for the components of v in the equation Mv = λv. In this example, the eigenvectors are any non-zero scalar multiples of
If the entries of the matrix A are all real numbers, then the coefficients of the characteristic polynomial will also be real numbers, but the eigenvalues may still have non-zero imaginary parts. The entries of the corresponding eigenvectors therefore may also have non-zero imaginary parts. Similarly, the eigenvalues may be irrational numbers even if all the entries of A are rational numbers or even if they are all integers. However, if the entries of A are all algebraic numbers, which include the rationals, the eigenvalues are complex algebraic numbers.
The non-real roots of a real polynomial with real coefficients can be grouped into pairs of complex conjugates, namely with the two members of each pair having imaginary parts that differ only in sign and the same real part. If the degree is odd, then by the intermediate value theorem at least one of the roots is real. Therefore, any real matrix with odd order has at least one real eigenvalue, whereas a real matrix with even order may not have any real eigenvalues. The eigenvectors associated with these complex eigenvalues are also complex and also appear in complex conjugate pairs.
Algebraic multiplicity[edit]
Let λi be an eigenvalue of an n by n matrix A. The algebraic multiplicity μAi) of the eigenvalue is its multiplicity as a root of the characteristic polynomial, that is, the largest integer k such that (λ − λi)k divides evenly that polynomial.[8][26][27]
Suppose a matrix A has dimension n and dn distinct eigenvalues. Whereas Equation (4) factors the characteristic polynomial of A into the product of n linear terms with some terms potentially repeating, the characteristic polynomial can instead be written as the product d terms each corresponding to a distinct eigenvalue and raised to the power of the algebraic multiplicity,
If d = n then the right hand side is the product of n linear terms and this is the same as Equation (4). The size of each eigenvalue's algebraic multiplicity is related to the dimension n as
If μAi) = 1, then λi is said to be a simple eigenvalue.[27] If μAi) equals the geometric multiplicity of λi, γAi), defined in the next section, then λi is said to be a semisimple eigenvalue.
Eigenspaces, geometric multiplicity, and the eigenbasis for matrices[edit]
Given a particular eigenvalue λ of the n by n matrix A, define the set E to be all vectors v that satisfy Equation (2),
On one hand, this set is precisely the kernel or nullspace of the matrix (A − λI). On the other hand, by definition, any non-zero vector that satisfies this condition is an eigenvector of A associated with λ. So, the set E is the union of the zero vector with the set of all eigenvectors of A associated with λ, and E equals the nullspace of (A − λI). E is called the eigenspace or characteristic space of A associated with λ.[7][8] In general λ is a complex number and the eigenvectors are complex n by 1 matrices. A property of the nullspace is that it is a linear subspace, so E is a linear subspace of ℂn.
Because the eigenspace E is a linear subspace, it is closed under addition. That is, if two vectors u and v belong to the set E, written (u,v) ∈ E, then (u + v) ∈ E or equivalently A(u + v) = λ(u + v). This can be checked using the distributive property of matrix multiplication. Similarly, because E is a linear subspace, it is closed under scalar multiplication. That is, if vE and α is a complex number, v) ∈ E or equivalently Av) = λ(αv). This can be checked by noting that multiplication of complex matrices by complex numbers is commutative. As long as u + v and αv are not zero, they are also eigenvectors of A associated with λ.
The dimension of the eigenspace E associated with λ, or equivalently the maximum number of linearly independent eigenvectors associated with λ, is referred to as the eigenvalue's geometric multiplicity γA(λ). Because E is also the nullspace of (A − λI), the geometric multiplicity of λ is the dimension of the nullspace of (A − λI), also called the nullity of (A − λI), which relates to the dimension and rank of (A - λI) as
Because of the definition of eigenvalues and eigenvectors, an eigenvalue's geometric multiplicity must be at least one, that is, each eigenvalue has at least one associated eigenvector. Furthermore, an eigenvalue's geometric multiplicity cannot exceed its algebraic multiplicity. Additionally, recall that an eigenvalue's algebraic multiplicity cannot exceed n.
The condition that γA(λ) ≤ μA(λ) can be proven by considering a particular eigenvalue ξ of A and diagonalizing the first γA(ξ) columns of A with respect to ξ's eigenvectors, described in a later section. The resulting similar matrix B is block upper triangular, with its top left block being the diagonal matrix ξIγA(ξ). As a result, the characteristic polynomial of B will have a factor of (ξ - λ)γA(ξ). The other factors of the characteristic polynomial of B are not known, so the algebraic multiplicity of ξ as an eigenvalue of B is no less than the geometric multiplicity of ξ as an eigenvalue of A. The last element of the proof is the property that similar matrices have the same characteristic polynomial.
Suppose A has dn distinct eigenvalues λ1, λ2, ..., λd, where the geometric multiplicity of λi is γAi). The total geometric multiplicity of A,
is the dimension of the union of all the eigenspaces of A's eigenvalues, or equivalently the maximum number of linearly independent eigenvectors of A. If γA = n, then
• The union of the eigenspaces of all of A's eigenvalues is the entire vector space ℂn
• A basis of ℂn can be formed from n linearly independent eigenvectors of A; such a basis is called an eigenbasis
• Any vector in ℂn can be written as a linear combination of eigenvectors of A
Additional properties of eigenvalues[edit]
Let A be an arbitrary n by n matrix of complex numbers with eigenvalues λ1, λ2, ..., λn. Each eigenvalue appears μAi) times in this list, where μAi) is the eigenvalue's algebraic multiplicity. The following are properties of this matrix and its eigenvalues:
• The trace of A, defined as the sum of its diagonal elements, is also the sum of all eigenvalues,
• The determinant of A is the product of all its eigenvalues,
• The eigenvalues of the kth power of A, i.e. the eigenvalues of Ak, for any positive integer k, are λ1k, λ2k, ..., λnk.
• The matrix A is invertible if and only if every eigenvalue is nonzero.
• If A is invertible, then the eigenvalues of A−1 are 1/λ1, 1/λ2, ..., 1/λn and each eigenvalue's geometric multiplicity coincides. Moreover, since the characteristic polynomial of the inverse is the reciprocal polynomial of the original, the eigenvalues share the same algebraic multiplicity.
• If A is equal to its conjugate transpose A*, or equivalently if A is Hermitian, then every eigenvalue is real. The same is true of any symmetric real matrix.
• If A is not only Hermitian but also positive-definite, positive-semidefinite, negative-definite, or negative-semidefinite, then every eigenvalue is positive, non-negative, negative, or non-positive, respectively.
• If A is unitary, every eigenvalue has absolute value |λi| = 1.
Left and right eigenvectors[edit]
Many disciplines traditionally represent vectors as matrices with a single column rather than as matrices with a single row. For that reason, the word "eigenvector" in the context of matrices almost always refers to a right eigenvector, namely a column vector that right multiples the n by n matrix A in the defining equation, Equation (1),
The eigenvalue and eigenvector problem can also be defined for row vectors that left multiply matrix A. In this formulation, the defining equation is
where κ is a scalar and u is a 1 by n matrix. Any row vector u satisfying this equation is called a left eigenvector of A and κ is its associated eigenvalue. Taking the conjugate transpose of this equation,
Comparing this equation to Equation (1), the left eigenvectors of A are the conjugate transpose of the right eigenvectors of A*. The eigenvalues of the left eigenvectors are the solution of the characteristic polynomial |A* − κ*I|=0. Because the identity matrix is Hermitian and |M*| = |M|* for a square matrix M, the eigenvalues of the left eigenvectors of A are the complex conjugates of the eigenvalues of the right eigenvectors of A. Recall that if A is a real matrix, all of its complex eigenvalues appear in complex conjugate pairs. Therefore, the eigenvalues of the left and right eigenvectors of a real matrix are the same. Similarly, if A is a real matrix, all of its complex eigenvectors also appear in complex conjugate pairs. Therefore, the left eigenvectors simplify to the transpose of the right eigenvectors of AT if A is real.
Diagonalization and the eigendecomposition[edit]
Suppose the eigenvectors of A form a basis, or equivalently A has n linearly independent eigenvectors v1, v2, ..., vn with associated eigenvalues λ1, λ2, ..., λn. The eigenvalues need not be distinct. Define a square matrix Q whose columns are the n linearly independent eigenvectors of A,
Since each column of Q is an eigenvector of A, right multiplying A by Q scales each column of Q by its associated eigenvalue,
With this in mind, define a diagonal matrix Λ where each diagonal element Λii is the eigenvalue associated with the ith column of Q. Then
Because the columns of Q are linearly independent, Q is invertible. Right multiplying both sides of the equation by Q−1,
or by instead left multiplying both sides by Q−1,
A can therefore be decomposed into a matrix composed of its eigenvectors, a diagonal matrix with its eigenvalues along the diagonal, and the inverse of the matrix of eigenvectors. This is called the eigendecomposition and it is a similarity transformation. Such a matrix A is said to be similar to the diagonal matrix Λ or diagonalizable. The matrix Q is the change of basis matrix of the similarity transformation. Essentially, the matrices A and Λ represent the same linear transformation expressed in two different bases. The eigenvectors are used as the basis when representing the linear transformation as Λ.
Conversely, suppose a matrix A is diagonalizable. Let P be a non-singular square matrix such that P−1AP is some diagonal matrix D. Left multiplying both by P, AP = PD. Each column of P must therefore be an eigenvector of A whose eigenvalue is the corresponding diagonal element of D. Since the columns of P must be linearly independent for P to be invertible, there exist n linearly independent eigenvectors of A. It then follows that the eigenvectors of A form a basis if and only if A is diagonalizable.
A matrix that is not diagonalizable is said to be defective. For defective matrices, the notion of eigenvectors generalizes to generalized eigenvectors and the diagonal matrix of eigenvalues generalizes to the Jordan normal form. Over an algebraically closed field, any matrix A has a Jordan normal form and therefore admits a basis of generalized eigenvectors and a decomposition into generalized eigenspaces.
Variational characterization[edit]
Main article: Min-max theorem
In the Hermitian case, eigenvalues can be given a variational characterization. The largest eigenvalue of is the maximum value of the quadratic form . A value of that realizes that maximum, is an eigenvector.
Matrix examples[edit]
Two-dimensional matrix example[edit]
The transformation matrix A = preserves the direction of vectors parallel to vλ=1 = [1 −1]T (in purple) and vλ=3 = [1 1]T (in blue). The vectors in red are not parallel to either eigenvector, so, their directions are changed by the transformation. See also: An extended version, showing all four quadrants.
Consider the matrix
The figure on the right shows the effect of this transformation on point coordinates in the plane. The eigenvectors v of this transformation satisfy Equation (1), and the values of λ for which the determinant of the matrix (A − λI) equals zero are the eigenvalues.
Taking the determinant to find characteristic polynomial of A,
For λ = 1, Equation (2) becomes,
Any non-zero vector with v1 = −v2 solves this equation. Therefore,
is an eigenvector of A corresponding to λ = 1, as is any scalar multiple of this vector.
For λ = 3, Equation (2) becomes
Any non-zero vector with v1 = v2 solves this equation. Therefore,
is an eigenvector of A corresponding to λ = 3, as is any scalar multiple of this vector.
Thus, the vectors vλ=1 and vλ=3 are eigenvectors of A associated with the eigenvalues λ = 1 and λ = 3, respectively.
Three-dimensional matrix example[edit]
Consider the matrix
The characteristic polynomial of A is
The roots of the characteristic polynomial are 2, 1, and 11, which are the only three eigenvalues of A. These eigenvalues correspond to the eigenvectors and , or any non-zero multiple thereof.
Three-dimensional matrix example with complex eigenvalues[edit]
Consider the cyclic permutation matrix
This matrix shifts the coordinates of the vector up by one position and moves the first coordinate to the bottom. Its characteristic polynomial is 1 − λ3, whose roots are
where i = is the imaginary unit.
For the real eigenvalue λ1 = 1, any vector with three equal non-zero entries is an eigenvector. For example,
For the complex conjugate pair of imaginary eigenvalues, note that
Therefore, the other two eigenvectors of A are complex and are and with eigenvalues λ2 and λ3, respectively. Note that the two complex eigenvectors also appear in a complex conjugate pair,
Diagonal matrix example[edit]
Matrices with entries only along the main diagonal are called diagonal matrices. The eigenvalues of a diagonal matrix are the diagonal elements themselves. Consider the matrix
The characteristic polynomial of A is
which has the roots λ1 = 1, λ2 = 2, and λ3 = 3. These roots are the diagonal elements as well as the eigenvalues of A.
Each diagonal element corresponds to an eigenvector whose only non-zero component is in the same row as that diagonal element. In the example, the eigenvalues correspond to the eigenvectors,
respectively, as well as scalar multiples of these vectors.
Triangular matrix example[edit]
A matrix whose elements above the main diagonal are all zero is called a lower triangular matrix, while a matrix whose elements below the main diagonal are all zero is called an upper triangular matrix. As with diagonal matrices, the eigenvalues of triangular matrices are the elements of the main diagonal.
Consider the lower triangular matrix,
The characteristic polynomial of A is
These eigenvalues correspond to the eigenvectors,
respectively, as well as scalar multiples of these vectors.
Matrix with repeated eigenvalues example[edit]
As in the previous example, the lower triangular matrix
has a characteristic polynomial that is the product of its diagonal elements,
The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The algebraic multiplicity of each eigenvalue is 2; in other words they are both double roots. The sum of the algebraic multiplicities of each distinct eigenvalue is μA = 4 = n, the order of the characteristic polynomial and the dimension of A.
On the other hand, the geometric multiplicity of the eigenvalue 2 is only 1, because its eigenspace is spanned by just one vector [0 1 -1 1]T and is therefore 1-dimensional. Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by just one vector [0 0 0 1]T. The total geometric multiplicity γA is 2, which is the smallest it could be for a matrix with two distinct eigenvalues. Geometric multiplicities are defined in a later section.
Eigenvalues and eigenfunctions of differential operators[edit]
Main article: Eigenfunction
The definitions of eigenvalue and eigenvectors of a linear transformation T remains valid even if the underlying vector space is an infinite-dimensional Hilbert or Banach space. A widely used class of linear transformations acting on infinite-dimensional spaces are the differential operators on function spaces. Let D be a linear differential operator on the space C of infinitely differentiable real functions of a real argument t. The eigenvalue equation for D is the differential equation
The functions that satisfy this equation are eigenvectors of D and are commonly called eigenfunctions.
Derivative operator example[edit]
Consider the derivative operator with eigenvalue equation
This differential equation can be solved by multiplying both sides by dt/f(t) and integrating. Its solution, the exponential function
is the eigenfunction of the derivative operator. Note that in this case the eigenfunction is itself a function of its associated eigenvalue. In particular, note that for λ = 0 the eigenfunction f(t) is a constant.
The main eigenfunction article gives other examples.
General definition[edit]
The concept of eigenvalues and eigenvectors extends naturally to arbitrary linear transformations on arbitrary vector spaces. Let V be any vector space over some field K of scalars, and let T be a linear transformation mapping V into V,
We say that a non-zero vector vV is an eigenvector of T if and only if there exists a scalar λ ∈ K such that
This equation is called the eigenvalue equation for T, and the scalar λ is the eigenvalue of T corresponding to the eigenvector v. Note that T(v) is the result of applying the transformation T to the vector v, while λv is the product of the scalar λ with v.[33]
Eigenspaces, geometric multiplicity, and the eigenbasis[edit]
Given an eigenvalue λ, consider the set
which is the union of the zero vector with the set of all eigenvectors associated with λ. E is called the eigenspace or characteristic space of T associated with λ.
By definition of a linear transformation,
for (x,y) ∈ V and α ∈ K. Therefore, if u and v are eigenvectors of T associated with eigenvalue λ, namely (u,v) ∈ E, then
So, both u + v and αv are either zero or eigenvectors of T associated with λ, namely (u+vv) ∈ E, and E is closed under addition and scalar multiplication. The eigenspace E associated with λ is therefore a linear subspace of V.[8][34][35] If that subspace has dimension 1, it is sometimes called an eigenline.[36]
The geometric multiplicity γT(λ) of an eigenvalue λ is the dimension of the eigenspace associated with λ, i.e., the maximum number of linearly independent eigenvectors associated with that eigenvalue.[8][27] By the definition of eigenvalues and eigenvectors, γT(λ) ≥ 1 because every eigenvalue has at least one eigenvector.
The eigenspaces of T always form a direct sum. As a consequence, eigenvectors of different eigenvalues are always linearly independent. Therefore, the sum of the dimensions of the eigenspaces cannot exceed the dimension n of the vector space on which T operates, and there cannot be more than n distinct eigenvalues.[37]
Any subspace spanned by eigenvectors of T is an invariant subspace of T, and the restriction of T to such a subspace is diagonalizable. Moreover, if the entire vector space V can be spanned by the eigenvectors of T, or equivalently if the direct sum of the eigenspaces associated with all the eigenvalues of T is the entire vector space V, then a basis of V called an eigenbasis can be formed from linearly independent eigenvectors of T. When T admits an eigenbasis, T is diagonalizable.
Zero vector as an eigenvector[edit]
While the definition of an eigenvector used in this article excludes the zero vector, it is possible to define eigenvalues and eigenvectors such that the zero vector is an eigenvector.[38]
Consider again the eigenvalue equation, Equation (5). Define an eigenvalue to be any scalar λ ∈ K such that there exists a non-zero vector vV satisfying Equation (5). It is important that this version of the definition of an eigenvalue specify that the vector be non-zero, otherwise by this definition the zero vector would allow any scalar in K to be an eigenvalue. Define an eigenvector v associated with the eigenvalue λ to be any vector that, given λ, satisfies Equation (5). Given the eigenvalue, the zero vector is among the vectors that satisfy Equation (5), so the zero vector is included among the eigenvectors by this alternate definition.
Spectral theory[edit]
Main article: Spectral theory
If λ is an eigenvalue of T, then the operator (T − λI) is not one-to-one, and therefore its inverse (T − λI)−1 does not exist. The converse is true for finite-dimensional vector spaces, but not for infinite-dimensional vector spaces. In general, the operator (T − λI) may not have an inverse even if λ is not an eigenvalue.
For this reason, in functional analysis eigenvalues can be generalized to the spectrum of a linear operator T as the set of all scalars λ for which the operator (T − λI) has no bounded inverse. The spectrum of an operator always contains all its eigenvalues but is not limited to them.
Associative algebras and representation theory[edit]
One can generalize the algebraic object that is acting on the vector space, replacing a single operator acting on a vector space with an algebra representation – an associative algebra acting on a module. The study of such actions is the field of representation theory.
The representation-theoretical concept of weight is an analog of eigenvalues, while weight vectors and weight spaces are the analogs of eigenvectors and eigenspaces, respectively.
Dynamic equations[edit]
The simplest difference equations have the form
The solution of this equation for x in terms of t is found by using its characteristic equation
which can be found by stacking into matrix form a set of equations consisting of the above difference equation and the k–1 equations giving a k-dimensional system of the first order in the stacked variable vector in terms of its once-lagged value, and taking the characteristic equation of this system's matrix. This equation gives k characteristic roots for use in the solution equation
A similar procedure is used for solving a differential equation of the form
Main article: Eigenvalue algorithm
The eigenvalues of a matrix can be determined by finding the roots of the characteristic polynomial. Explicit algebraic formulas for the roots of a polynomial exist only if the degree is 4 or less. According to the Abel–Ruffini theorem there is no general, explicit and exact algebraic formula for the roots of a polynomial with degree 5 or more.
It turns out that any polynomial with degree is the characteristic polynomial of some companion matrix of order . Therefore, for matrices of order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula, and must therefore be computed by approximate numerical methods.
In theory, the coefficients of the characteristic polynomial can be computed exactly, since they are sums of products of matrix elements; and there are algorithms that can find all the roots of a polynomial of arbitrary degree to any required accuracy.[39] However, this approach is not viable in practice because the coefficients would be contaminated by unavoidable round-off errors, and the roots of a polynomial can be an extremely sensitive function of the coefficients (as exemplified by Wilkinson's polynomial).[39]
Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the advent of the QR algorithm in 1961. [39] Combining the Householder transformation with the LU decomposition results in an algorithm with better convergence than the QR algorithm.[citation needed] For large Hermitian sparse matrices, the Lanczos algorithm is one example of an efficient iterative method to compute eigenvalues and eigenvectors, among several other possibilities.[39]
Once the (exact) value of an eigenvalue is known, the corresponding eigenvectors can be found by finding non-zero solutions of the eigenvalue equation, that becomes a system of linear equations with known coefficients. For example, once it is known that 6 is an eigenvalue of the matrix
we can find its eigenvectors by solving the equation , that is
This matrix equation is equivalent to two linear equations
that is
Both equations reduce to the single linear equation . Therefore, any vector of the form , for any non-zero real number , is an eigenvector of with eigenvalue .
The matrix above has another eigenvalue . A similar calculation shows that the corresponding eigenvectors are the non-zero solutions of , that is, any vector of the form , for any non-zero real number .
Some numeric methods that compute the eigenvalues of a matrix also determine a set of corresponding eigenvectors as a by-product of the computation.
Eigenvalues of geometric transformations[edit]
The following table presents some example transformations in the plane along with their 2×2 matrices, eigenvalues, and eigenvectors.
scaling unequal scaling rotation horizontal shear hyperbolic rotation
illustration Equal scaling (homothety) Vertical shrink ('"`UNIQ--postMath-00000069-QINU`"') and horizontal stretch ('"`UNIQ--postMath-0000006A-QINU`"') of a unit square. Rotation by 50 degrees
Horizontal shear mapping
algebraic multipl.
geometric multipl.
eigenvectors All non-zero vectors
Note that the characteristic equation for a rotation is a quadratic equation with discriminant , which is a negative number whenever is not an integer multiple of 180°. Therefore, except for these special cases, the two eigenvalues are complex numbers, ; and all eigenvectors have non-real entries. Indeed, except for those special cases, a rotation changes the direction of every nonzero vector in the plane.
A linear transformation that takes a square to a rectangle of the same area (a squeeze mapping) has reciprocal eigenvalues.
Schrödinger equation[edit]
The wavefunctions associated with the bound states of an electron in a hydrogen atom can be seen as the eigenvectors of the hydrogen atom Hamiltonian as well as of the angular momentum operator. They are associated with eigenvalues interpreted as their energies (increasing downward: ) and angular momentum (increasing across: s, p, d, ...). The illustration shows the square of the absolute value of the wavefunctions. Brighter areas correspond to higher probability density for a position measurement. The center of each figure is the atomic nucleus, a proton.
An example of an eigenvalue equation where the transformation is represented in terms of a differential operator is the time-independent Schrödinger equation in quantum mechanics:
where , the Hamiltonian, is a second-order differential operator and , the wavefunction, is one of its eigenfunctions corresponding to the eigenvalue , interpreted as its energy.
However, in the case where one is interested only in the bound state solutions of the Schrödinger equation, one looks for within the space of square integrable functions. Since this space is a Hilbert space with a well-defined scalar product, one can introduce a basis set in which and can be represented as a one-dimensional array (i.e., a vector) and a matrix respectively. This allows one to represent the Schrödinger equation in a matrix form.
The bra–ket notation is often used in this context. A vector, which represents a state of the system, in the Hilbert space of square integrable functions is represented by . In this notation, the Schrödinger equation is:
where is an eigenstate of and represents the eigenvalue. is an observable self adjoint operator, the infinite-dimensional analog of Hermitian matrices. As in the matrix case, in the equation above is understood to be the vector obtained by application of the transformation to .
Molecular orbitals[edit]
In quantum mechanics, and in particular in atomic and molecular physics, within the Hartree–Fock theory, the atomic and molecular orbitals can be defined by the eigenvectors of the Fock operator. The corresponding eigenvalues are interpreted as ionization potentials via Koopmans' theorem. In this case, the term eigenvector is used in a somewhat more general meaning, since the Fock operator is explicitly dependent on the orbitals and their eigenvalues. Thus, if one wants to underline this aspect, one speaks of nonlinear eigenvalue problems. Such equations are usually solved by an iteration procedure, called in this case self-consistent field method. In quantum chemistry, one often represents the Hartree–Fock equation in a non-orthogonal basis set. This particular representation is a generalized eigenvalue problem called Roothaan equations.
Geology and glaciology[edit]
In geology, especially in the study of glacial till, eigenvectors and eigenvalues are used as a method by which a mass of information of a clast fabric's constituents' orientation and dip can be summarized in a 3-D space by six numbers. In the field, a geologist may collect such data for hundreds or thousands of clasts in a soil sample, which can only be compared graphically such as in a Tri-Plot (Sneed and Folk) diagram,[40][41] or as a Stereonet on a Wulff Net.[42]
The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space. The three eigenvectors are ordered by their eigenvalues ;[43] then is the primary orientation/dip of clast, is the secondary and is the tertiary, in terms of strength. The clast orientation is defined as the direction of the eigenvector, on a compass rose of 360°. Dip is measured as the eigenvalue, the modulus of the tensor: this is valued from 0° (no dip) to 90° (vertical). The relative values of , , and are dictated by the nature of the sediment's fabric. If , the fabric is said to be isotropic. If , the fabric is said to be planar. If , the fabric is said to be linear.[44]
Principal component analysis[edit]
PCA of the multivariate Gaussian distribution centered at with a standard deviation of 3 in roughly the direction and of 1 in the orthogonal direction. The vectors shown are unit eigenvectors of the (symmetric, positive-semidefinite) covariance matrix scaled by the square root of the corresponding eigenvalue. (Just as in the one-dimensional case, the square root is taken because the standard deviation is more readily visualized than the variance.
The eigendecomposition of a symmetric positive semidefinite (PSD) matrix yields an orthogonal basis of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix is used in multivariate analysis, where the sample covariance matrices are PSD. This orthogonal decomposition is called principal components analysis (PCA) in statistics. PCA studies linear relations among variables. PCA is performed on the covariance matrix or the correlation matrix (in which each variable is scaled to have its sample variance equal to one). For the covariance or correlation matrix, the eigenvectors correspond to principal components and the eigenvalues to the variance explained by the principal components. Principal component analysis of the correlation matrix provides an orthonormal eigen-basis for the space of the observed data: In this basis, the largest eigenvalues correspond to the principal components that are associated with most of the covariability among a number of observed data.
Principal component analysis is used to study large data sets, such as those encountered in bioinformatics, data mining, chemical research, psychology, and in marketing. PCA is popular especially in psychology, in the field of psychometrics. In Q methodology, the eigenvalues of the correlation matrix determine the Q-methodologist's judgment of practical significance (which differs from the statistical significance of hypothesis testing; cf. criteria for determining the number of factors). More generally, principal component analysis can be used as a method of factor analysis in structural equation modeling.
Vibration analysis[edit]
Mode Shape of a Tuning Fork at Eigenfrequency 440.09 Hz
Main article: Vibration
Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many degrees of freedom. The eigenvalues are the natural frequencies (or eigenfrequencies) of vibration, and the eigenvectors are the shapes of these vibrational modes. In particular, undamped vibration is governed by
that is, acceleration is proportional to position (i.e., we expect to be sinusoidal in time).
In dimensions, becomes a mass matrix and a stiffness matrix. Admissible solutions are then a linear combination of solutions to the generalized eigenvalue problem
where is the eigenvalue and is the (imaginary) angular frequency. Note that the principal vibration modes are different from the principal compliance modes, which are the eigenvectors of alone. Furthermore, damped vibration, governed by
leads to a so-called quadratic eigenvalue problem,
This can be reduced to a generalized eigenvalue problem by clever use of algebra at the cost of solving a larger system.
The orthogonality properties of the eigenvectors allows decoupling of the differential equations so that the system can be represented as linear summation of the eigenvectors. The eigenvalue problem of complex structures is often solved using finite element analysis, but neatly generalize the solution to scalar-valued vibration problems.
Eigenfaces as examples of eigenvectors
Main article: Eigenface
In image processing, processed images of faces can be seen as vectors whose components are the brightnesses of each pixel.[45] The dimension of this vector space is the number of pixels. The eigenvectors of the covariance matrix associated with a large set of normalized pictures of faces are called eigenfaces; this is an example of principal components analysis. They are very useful for expressing any face image as a linear combination of some of them. In the facial recognition branch of biometrics, eigenfaces provide a means of applying data compression to faces for identification purposes. Research related to eigen vision systems determining hand gestures has also been made.
Similar to this concept, eigenvoices represent the general direction of variability in human pronunciations of a particular utterance, such as a word in a language. Based on a linear combination of such eigenvoices, a new voice pronunciation of the word can be constructed. These concepts have been found useful in automatic speech recognition systems for speaker adaptation.
Tensor of moment of inertia[edit]
In mechanics, the eigenvectors of the moment of inertia tensor define the principal axes of a rigid body. The tensor of moment of inertia is a key quantity required to determine the rotation of a rigid body around its center of mass.
Stress tensor[edit]
In solid mechanics, the stress tensor is symmetric and so can be decomposed into a diagonal tensor with the eigenvalues on the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation, the stress tensor has no shear components; the components it does have are the principal components.
In spectral graph theory, an eigenvalue of a graph is defined as an eigenvalue of the graph's adjacency matrix , or (increasingly) of the graph's Laplacian matrix due to its Discrete Laplace operator, which is either (sometimes called the combinatorial Laplacian) or (sometimes called the normalized Laplacian), where is a diagonal matrix with equal to the degree of vertex , and in , the th diagonal entry is . The th principal eigenvector of a graph is defined as either the eigenvector corresponding to the th largest or th smallest eigenvalue of the Laplacian. The first principal eigenvector of the graph is also referred to merely as the principal eigenvector.
The principal eigenvector is used to measure the centrality of its vertices. An example is Google's PageRank algorithm. The principal eigenvector of a modified adjacency matrix of the World Wide Web graph gives the page ranks as its components. This vector corresponds to the stationary distribution of the Markov chain represented by the row-normalized adjacency matrix; however, the adjacency matrix must first be modified to ensure a stationary distribution exists. The second smallest eigenvector can be used to partition the graph into clusters, via spectral clustering. Other methods are also available for clustering.
Basic reproduction number[edit]
The basic reproduction number () is a fundamental number in the study of how infectious diseases spread. If one infectious person is put into a population of completely susceptible people, then is the average number of people that one typical infectious person will infect. The generation time of an infection is the time, , from one person becoming infected to the next person becoming infected. In a heterogeneous population, the next generation matrix defines how many people in the population will become infected after time has passed. is then the largest eigenvalue of the next generation matrix.[46][47]
See also[edit]
1. ^ a b Herstein (1964, pp. 228,229)
2. ^ a b Nering (1970, p. 38)
3. ^ Burden & Faires (1993, p. 401)
4. ^ Betteridge (1965)
5. ^ Press (2007, p. 536)
6. ^ Wolfram Research, Inc. (2010) Eigenvector. Accessed on 2016-04-01.
7. ^ a b Anton (1987, pp. 305,307)
8. ^ a b c d e Nering (1970, p. 107)
9. ^ Note:
• In 1751, Leonhard Euler proved that any body has a principal axis of rotation: Leonhard Euler (presented: October 1751 ; published: 1760) "Du mouvement d'un corps solide quelconque lorsqu'il tourne autour d'un axe mobile" (On the movement of any solid body while it rotates around a moving axis), Histoire de l'Académie royale des sciences et des belles lettres de Berlin, pp.176-227. On p. 212, Euler proves that any body contains a principal axis of rotation: "Théorem. 44. De quelque figure que soit le corps, on y peut toujours assigner un tel axe, qui passe par son centre de gravité, autour duquel le corps peut tourner librement & d'un mouvement uniforme." (Theorem. 44. Whatever be the shape of the body, one can always assign to it such an axis, which passes through its center of gravity, around which it can rotate freely and with a uniform motion.)
• In 1755, Johann Andreas Segner proved that any body has three principal axes of rotation: Johann Andreas Segner, Specimen theoriae turbinum [Essay on the theory of tops (i.e., rotating bodies)] ( Halle ("Halae"), (Germany) : Gebauer, 1755). On p. XXVIIII (i.e., 29), Segner derives a third-degree equation in t, which proves that a body has three principal axes of rotation. He then states (on the same page): "Non autem repugnat tres esse eiusmodi positiones plani HM, quia in aequatione cubica radices tres esse possunt, et tres tangentis t valores." (However, it is not inconsistent [that there] be three such positions of the plane HM, because in cubic equations, [there] can be three roots, and three values of the tangent t.)
• The relevant passage of Segner's work was discussed briefly by Arthur Cayley. See: A. Cayley (1862) "Report on the progress of the solution of certain special problems of dynamics," Report of the Thirty-second meeting of the British Association for the Advancement of Science; held at Cambridge in October 1862, 32 : 184-252 ; see especially pages 225-226.
10. ^ See Hawkins 1975, §2
11. ^ a b c d See Hawkins 1975, §3
12. ^ a b c See Kline 1972, pp. 807–808
13. ^ Augustin Cauchy (1839) "Mémoire sur l'intégration des équations linéaires" (Memoir on the integration of linear equations), Comptes rendus, 8 : 827-830, 845-865, 889-907, 931-937. From p. 827: "On sait d'ailleurs qu'en suivant la méthode de Lagrange, on obtient pour valeur générale de la variable prinicipale une fonction dans laquelle entrent avec la variable principale les racines d'une certaine équation que j'appellerai l'équation caractéristique, le degré de cette équation étant précisément l'order de l'équation différentielle qu'il s'agit d'intégrer." (On knows, moreover, that by following Lagrange's method, one obtains for the general value of the principal variable a function in which there appear, together with the principal variable, the roots of a certain equation that I will call the "characteristic equation", the degree of this equation being precisely the order of the differential equation that must be integrated.)
14. ^ See Kline 1972, p. 673
15. ^ See Kline 1972, pp. 715–716
16. ^ See Kline 1972, pp. 706–707
17. ^ See Kline 1972, p. 1063
18. ^ See:
• David Hilbert (1904) "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. (Erste Mitteilung)" (Fundamentals of a general theory of linear integral equations. (First report)), Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (News of the Philosophical Society at Göttingen, mathematical-physical section), pp. 49-91. From page 51: "Insbesondere in dieser ersten Mitteilung gelange ich zu Formeln, die die Entwickelung einer willkürlichen Funktion nach gewissen ausgezeichneten Funktionen, die ich Eigenfunktionen nenne, liefern: … (In particular, in this first report I arrive at formulas that provide the [series] development of an arbitrary function in terms of some distinctive functions, which I call eigenfunctions: … ) Later on the same page: "Dieser Erfolg ist wesentlich durch den Umstand bedingt, daß ich nicht, wie es bisher geschah, in erster Linie auf den Beweis für die Existenz der Eigenwerte ausgehe, … " (This success is mainly attributable to the fact that I do not, as it has happened until now, first of all aim at a proof of the existence of eigenvalues, … )
• For the origin and evolution of the terms eigenvalue, characteristic value, etc., see: Earliest Known Uses of Some of the Words of Mathematics (E)
19. ^ See Aldrich 2006
20. ^ Francis, J. G. F. (1961), "The QR Transformation, I (part 1)", The Computer Journal, 4 (3): 265–271, doi:10.1093/comjnl/4.3.265 and Francis, J. G. F. (1962), "The QR Transformation, II (part 2)", The Computer Journal, 4 (4): 332–345, doi:10.1093/comjnl/4.4.332
21. ^ Kublanovskaya, Vera N. (1961), "On some algorithms for the solution of the complete eigenvalue problem", USSR Computational Mathematics and Mathematical Physics, 3: 637–657 . Also published in: "О некоторых алгорифмах для решения полной проблемы собственных значений" [On certain algorithms for the solution of the complete eigenvalue problem], Журнал вычислительной математики и математической физики (Journal of Computational Mathematics and Mathematical Physics), 1 (4): 555–570, 1961
22. ^ See Golub & van Loan 1996, §7.3; Meyer 2000, §7.3
23. ^ Cornell University Department of Mathematics (2016) Lower-Level Courses for Freshmen and Sophomores. Accessed on 2016-03-27.
24. ^ University of Michigan Mathematics (2016) Math Course Catalogue. Accessed on 2016-03-27.
25. ^ Press (2007, pp. 38)
26. ^ Fraleigh (1976, p. 358)
27. ^ a b c Golub & Van Loan (1996, p. 316)
28. ^ a b Beauregard & Fraleigh (1973, p. 307)
29. ^ Herstein (1964, p. 272)
30. ^ Nering (1970, pp. 115–116)
31. ^ Herstein (1964, p. 290)
32. ^ Nering (1970, p. 116)
33. ^ See Korn & Korn 2000, Section 14.3.5a; Friedberg, Insel & Spence 1989, p. 217
34. ^ Shilov 1977, p. 109
35. ^ Lemma for the eigenspace
36. ^ Schaum's Easy Outline of Linear Algebra, p. 111
37. ^ For a proof of this lemma, see Roman 2008, Theorem 8.2 on p. 186; Shilov 1977, p. 109; Hefferon 2001, p. 364; Beezer 2006, Theorem EDELI on p. 469; and Lemma for linear independence of eigenvectors
38. ^ Axler, Sheldon, "Ch. 5", Linear Algebra Done Right (2nd ed.), p. 77
39. ^ a b c d Trefethen, Lloyd N.; Bau, David (1997), Numerical Linear Algebra, SIAM
40. ^ Graham, D.; Midgley, N. (2000), "Graphical representation of particle shape using triangular diagrams: an Excel spreadsheet method", Earth Surface Processes and Landforms, 25 (13): 1473–1477, Bibcode:2000ESPL...25.1473G, doi:10.1002/1096-9837(200012)25:13<1473::AID-ESP158>3.0.CO;2-C
41. ^ Sneed, E. D.; Folk, R. L. (1958), "Pebbles in the lower Colorado River, Texas, a study of particle morphogenesis", Journal of Geology, 66 (2): 114–150, Bibcode:1958JG.....66..114S, doi:10.1086/626490
42. ^ Knox-Robinson, C.; Gardoll, Stephen J. (1998), "GIS-stereoplot: an interactive stereonet plotting module for ArcView 3.0 geographic information system", Computers & Geosciences, 24 (3): 243, Bibcode:1998CG.....24..243K, doi:10.1016/S0098-3004(97)00122-2
43. ^ Stereo32 software
44. ^ Benn, D.; Evans, D. (2004), A Practical Guide to the study of Glacial Sediments, London: Arnold, pp. 103–107
45. ^ Xirouhakis, A.; Votsis, G.; Delopoulus, A. (2004), Estimation of 3D motion and structure of human faces (PDF), National Technical University of Athens
46. ^ Diekmann O, Heesterbeek JA, Metz JA (1990), "On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations", Journal of Mathematical Biology, 28 (4): 365–382, doi:10.1007/BF00178324, PMID 2117040
47. ^ Odo Diekmann; J. A. P. Heesterbeek (2000), Mathematical epidemiology of infectious diseases, Wiley series in mathematical and computational biology, West Sussex, England: John Wiley & Sons
External links[edit]
"Eigenvalue (of a matrix)". PlanetMath.
Demonstration applets[edit] |
061e934a4549dc3e | Pedia View . com
Open Source Encyclopedia
A glass tube containing a glowing green electron beam
Experiments with a Crookes tube first demonstrated the particle nature of electrons. In this illustration, the profile of the Maltese-cross-shaped target is projected against the tube face at right by a beam of electrons.[1]
Composition Elementary particle[2]
Statistics Fermionic
Generation First
Interactions Gravity, Electromagnetic, Weak
Symbol e, β
Antiparticle Positron (also called antielectron)
Theorized Richard Laming (1838–1851),[3]
G. Johnstone Stoney (1874) and others.[4][5]
Discovered J. J. Thomson (1897)[6]
Mass 9.10938291(40)×10−31 kg[7]
5.4857990946(22)×10−4 u[7]
[1,822.8884845(14)]−1 u[note 1]
0.510998928(11) MeV/c2[7]
Electric charge −1 e[note 2]
−1.602176565(35)×10−19 C[7]
−4.80320451(10)×10−10 esu
Magnetic moment −1.00115965218076(27) μB[7]
Spin 12
The electron (symbol: e) is a subatomic particle with a negative elementary electric charge.[8]
Electrons, which belong to the first generation of the lepton particle family,[10] participate in gravitational, electromagnetic and weak interactions.[11] Like all matter, they have quantum mechanical properties of both particles and waves, so they can collide with other particles and can be diffracted like light. However, this duality is best demonstrated in experiments with electrons, due to their tiny mass. Since an electron is a fermion, no two electrons can occupy the same quantum state, in accordance with the Pauli exclusion principle.[10]
The concept of an indivisible quantity of electric charge was theorized to explain the chemical properties of atoms, beginning in 1838 by British natural philosopher Richard Laming;[4] the name electron was introduced for this charge in 1894 by Irish physicist George Johnstone Stoney. The electron was identified as a particle in 1897 by J. J. Thomson and his team of British physicists.[6][12][13]
In many physical phenomena, such as electricity, magnetism, and thermal conductivity, electrons play an essential role. An electron in motion relative to an observer generates a magnetic field, and will be deflected by external magnetic fields. When an electron is accelerated, it can absorb or radiate energy in the form of photons. Electrons, together with atomic nuclei made of protons and neutrons, make up atoms. However, electrons contribute less than 0.06% to an atom's total mass. The attractive Coulomb force between an electron and a proton causes electrons to be bound into atoms. The exchange or sharing of the electrons between two or more atoms is the main cause of chemical bonding.[14]
Electrons may be created through beta decay of radioactive isotopes and in high-energy collisions, for instance when cosmic rays enter the atmosphere. Electrons may be destroyed through annihilation with positrons, and may be absorbed during nucleosynthesis in stars. Laboratory instruments are capable of containing and observing individual electrons as well as electron plasma, whereas dedicated telescopes can detect electron plasma in outer space. Electrons have many applications, including in electronics, welding, cathode ray tubes, electron microscopes, radiation therapy, lasers, gaseous ionization detectors and particle accelerators.
The ancient Greeks noticed that amber attracted small objects when rubbed with fur. Along with lightning, this phenomenon is one of humanity's earliest recorded experiences with electricity.[15] In his 1600 treatise De Magnete, the English scientist William Gilbert coined the New Latin term electricus, to refer to this property of attracting small objects after being rubbed.[16] Both electric and electricity are derived from the Latin ēlectrum (also the root of the alloy of the same name), which came from the Greek word for amber, ήλεκτρον (ēlektron).
In the early 1700s, Francis Hauksbee and French chemist C. F. du Fay independently discovered what they believed to be two kinds of frictional electricity; one generated from rubbing glass, the other from rubbing resin. From this, Du Fay theorized that electricity consists of two electrical fluids, "vitreous" and "resinous", that are separated by friction and that neutralize each other when combined.[17] A decade later Benjamin Franklin proposed that electricity was not from different types of electrical fluid, but the same electrical fluid under different pressures. He gave them the modern charge nomenclature of positive and negative respectively.[18] Franklin thought of the charge carrier as being positive, but he did not correctly identify which situation was a surplus of the charge carrier, and which situation was a deficit.[19]
Between 1838 and 1851, British natural philosopher Richard Laming developed the idea that an atom is composed of a core of matter surrounded by subatomic particles that had unit electric charges.[3] Beginning in 1846, German physicist William Weber theorized that electricity was composed of positively and negatively charged fluids, and their interaction was governed by the inverse square law. After studying the phenomenon of electrolysis in 1874, Irish physicist George Johnstone Stoney suggested that there existed a "single definite quantity of electricity", the charge of a monovalent ion. He was able to estimate the value of this elementary charge e by means of Faraday's laws of electrolysis.[20] However, Stoney believed these charges were permanently attached to atoms and could not be removed. In 1881, German physicist Hermann von Helmholtz argued that both positive and negative charges were divided into elementary parts, each of which "behaves like atoms of electricity".[4]
In 1891 Stoney coined the term electron to describe these elementary charges, writing later in 1894: "... an estimate was made of the actual amount of this most remarkable fundamental unit of electricity, for which I have since ventured to suggest the name electron".[21] The word electron is a combination of the words electr(ic) and (i)on.[22] The suffix -on which is now used to designate other subatomic particles, such as a proton or neutron, is in turn derived from electron.[23][24]
A round glass vacuum tube with a glowing circular beam inside
A beam of electrons deflected in a circle by a magnetic field[25]
The German physicist Johann Wilhelm Hittorf undertook the study of electrical conductivity in rarefied gases. In 1869, he discovered a glow emitted from the cathode that increased in size with decrease in gas pressure. In 1876, the German physicist Eugen Goldstein showed that the rays from this glow cast a shadow, and he dubbed the rays cathode rays.[26] During the 1870s, the English chemist and physicist Sir William Crookes developed the first cathode ray tube to have a high vacuum inside.[27] He then showed that the luminescence rays appearing within the tube carried energy and moved from the cathode to the anode. Furthermore, by applying a magnetic field, he was able to deflect the rays, thereby demonstrating that the beam behaved as though it were negatively charged.[28][29] In 1879, he proposed that these properties could be explained by what he termed 'radiant matter'. He suggested that this was a fourth state of matter, consisting of negatively charged molecules that were being projected with high velocity from the cathode.[30]
The German-born British physicist Arthur Schuster expanded upon Crookes' experiments by placing metal plates parallel to the cathode rays and applying an electric potential between the plates. The field deflected the rays toward the positively charged plate, providing further evidence that the rays carried negative charge. By measuring the amount of deflection for a given level of current, in 1890 Schuster was able to estimate the charge-to-mass ratio of the ray components. However, this produced a value that was more than a thousand times greater than what was expected, so little credence was given to his calculations at the time.[28][31]
In 1892 Hendrik Antoon Lorentz suggested that the mass of these particles (electrons) could be a consequence of their electric charge.[32]
In 1896, the British physicist J. J. Thomson, with his colleagues John S. Townsend and H. A. Wilson,[12] performed experiments indicating that cathode rays really were unique particles, rather than waves, atoms or molecules as was believed earlier.[6] Thomson made good estimates of both the charge e and the mass m, finding that cathode ray particles, which he called "corpuscles," had perhaps one thousandth of the mass of the least massive ion known: hydrogen.[6][13] He showed that their charge to mass ratio, e/m, was independent of cathode material. He further showed that the negatively charged particles produced by radioactive materials, by heated materials and by illuminated materials were universal.[6][33] The name electron was again proposed for these particles by the Irish physicist George F. Fitzgerald, and the name has since gained universal acceptance.[28]
Robert Millikan
While studying naturally fluorescing minerals in 1896, the French physicist Henri Becquerel discovered that they emitted radiation without any exposure to an external energy source. These radioactive materials became the subject of much interest by scientists, including the New Zealand physicist Ernest Rutherford who discovered they emitted particles. He designated these particles alpha and beta, on the basis of their ability to penetrate matter.[34] In 1900, Becquerel showed that the beta rays emitted by radium could be deflected by an electric field, and that their mass-to-charge ratio was the same as for cathode rays.[35] This evidence strengthened the view that electrons existed as components of atoms.[36][37]
The electron's charge was more carefully measured by the American physicists Robert Millikan and Harvey Fletcher in their oil-drop experiment of 1909, the results of which were published in 1911. This experiment used an electric field to prevent a charged droplet of oil from falling as a result of gravity. This device could measure the electric charge from as few as 1–150 ions with an error margin of less than 0.3%. Comparable experiments had been done earlier by Thomson's team,[6] using clouds of charged water droplets generated by electrolysis,[12] and in 1911 by Abram Ioffe, who independently obtained the same result as Millikan using charged microparticles of metals, then published his results in 1913.[38] However, oil drops were more stable than water drops because of their slower evaporation rate, and thus more suited to precise experimentation over longer periods of time.[39]
Around the beginning of the twentieth century, it was found that under certain conditions a fast-moving charged particle caused a condensation of supersaturated water vapor along its path. In 1911, Charles Wilson used this principle to devise his cloud chamber, allowing the tracks of charged particles, such as fast-moving electrons, to be photographed.[40]
Atomic theory
Three concentric circles about a nucleus, with an electron moving from the second to the first circle and releasing a photon
The Bohr model of the atom, showing states of electron with energy quantized by the number n. An electron dropping to a lower orbit emits a photon equal to the energy difference between the orbits.
By 1914, experiments by physicists Ernest Rutherford, Henry Moseley, James Franck and Gustav Hertz had largely established the structure of an atom as a dense nucleus of positive charge surrounded by lower-mass electrons.[41] In 1913, Danish physicist Niels Bohr postulated that electrons resided in quantized energy states, with the energy determined by the angular momentum of the electron's orbits about the nucleus. The electrons could move between these states, or orbits, by the emission or absorption of photons at specific frequencies. By means of these quantized orbits, he accurately explained the spectral lines of the hydrogen atom.[42] However, Bohr's model failed to account for the relative intensities of the spectral lines and it was unsuccessful in explaining the spectra of more complex atoms.[41]
Chemical bonds between atoms were explained by Gilbert Newton Lewis, who in 1916 proposed that a covalent bond between two atoms is maintained by a pair of electrons shared between them.[43] Later, in 1927, Walter Heitler and Fritz London gave the full explanation of the electron-pair formation and chemical bonding in terms of quantum mechanics.[44] In 1919, the American chemist Irving Langmuir elaborated on the Lewis' static model of the atom and suggested that all electrons were distributed in successive "concentric (nearly) spherical shells, all of equal thickness".[45] The shells were, in turn, divided by him in a number of cells each containing one pair of electrons. With this model Langmuir was able to qualitatively explain the chemical properties of all elements in the periodic table,[44] which were known to largely repeat themselves according to the periodic law.[46]
In 1924, Austrian physicist Wolfgang Pauli observed that the shell-like structure of the atom could be explained by a set of four parameters that defined every quantum energy state, as long as each state was inhabited by no more than a single electron. (This prohibition against more than one electron occupying the same quantum energy state became known as the Pauli exclusion principle.)[47] The physical mechanism to explain the fourth parameter, which had two distinct possible values, was provided by the Dutch physicists Samuel Goudsmit and George Uhlenbeck. In 1925, Goudsmit and Uhlenbeck suggested that an electron, in addition to the angular momentum of its orbit, possesses an intrinsic angular momentum and magnetic dipole moment.[41][48] The intrinsic angular momentum became known as spin, and explained the previously mysterious splitting of spectral lines observed with a high-resolution spectrograph; this phenomenon is known as fine structure splitting.[49]
Quantum mechanics
In his 1924 dissertation Recherches sur la théorie des quanta (Research on Quantum Theory), French physicist Louis de Broglie hypothesized that all matter possesses a De Broglie wave similar to light.[50] That is, under the appropriate conditions, electrons and other matter would show properties of either particles or waves. The corpuscular properties of a particle are demonstrated when it is shown to have a localized position in space along its trajectory at any given moment.[51] Wave-like nature is observed, for example, when a beam of light is passed through parallel slits and creates interference patterns. In 1927, the interference effect was found in a beam of electrons by English physicist George Paget Thomson with a thin metal film and by American physicists Clinton Davisson and Lester Germer using a crystal of nickel.[52]
A symmetrical blue cloud that decreases in intensity from the center outward
In quantum mechanics, the behavior of an electron in an atom is described by an orbital, which is a probability distribution rather than an orbit. In the figure, the shading indicates the relative probability to "find" the electron, having the energy corresponding to the given quantum numbers, at that point.
De Broglie's prediction of a wave nature for electrons led Erwin Schrödinger to postulate a wave equation for electrons moving under the influence of the nucleus in the atom. In 1926, this equation, the Schrödinger equation, successfully described how electron waves propagated.[53] Rather than yielding a solution that determined the location of an electron over time, this wave equation also could be used to predict the probability of finding an electron near a position, especially a position near where the electron was bound in space, for which the electron wave equations did not change in time. This approach led to a second formulation of quantum mechanics (the first being by Heisenberg in 1925), and solutions of Schrödinger's equation, like Heisenberg's, provided derivations of the energy states of an electron in a hydrogen atom that were equivalent to those that had been derived first by Bohr in 1913, and that were known to reproduce the hydrogen spectrum.[54] Once spin and the interaction between multiple electrons were considered, quantum mechanics later allowed the configuration of electrons in atoms with higher atomic numbers than hydrogen to be successfully predicted.[55]
In 1928, building on Wolfgang Pauli's work, Paul Dirac produced a model of the electron – the Dirac equation, consistent with relativity theory, by applying relativistic and symmetry considerations to the hamiltonian formulation of the quantum mechanics of the electro-magnetic field.[56] In order to resolve some problems within his relativistic equation, in 1930 Dirac developed a model of the vacuum as an infinite sea of particles having negative energy, which was dubbed the Dirac sea. This led him to predict the existence of a positron, the antimatter counterpart of the electron.[57] This particle was discovered in 1932 by Carl D. Anderson, who proposed calling standard electrons negatrons, and using electron as a generic term to describe both the positively and negatively charged variants.
In 1947 Willis Lamb, working in collaboration with graduate student Robert Rutherford, found that certain quantum states of hydrogen atom, which should have the same energy, were shifted in relation to each other, the difference being the Lamb shift. About the same time, Polykarp Kusch, working with Henry M. Foley, discovered the magnetic moment of the electron is slightly larger than predicted by Dirac's theory. This small difference was later called anomalous magnetic dipole moment of the electron. This difference was later explained by the theory of quantum electrodynamics, developed by Sin-Itiro Tomonaga, Julian Schwinger and Richard P. Feynman in the late 1940s.[58]
Particle accelerators
With the development of the particle accelerator during the first half of the twentieth century, physicists began to delve deeper into the properties of subatomic particles.[59] The first successful attempt to accelerate electrons using electromagnetic induction was made in 1942 by Donald Kerst. His initial betatron reached energies of 2.3 MeV, while subsequent betatrons achieved 300 MeV. In 1947, synchrotron radiation was discovered with a 70 MeV electron synchrotron at General Electric. This radiation was caused by the acceleration of electrons, moving near the speed of light, through a magnetic field.[60]
With a beam energy of 1.5 GeV, the first high-energy particle collider was ADONE, which began operations in 1968.[61] This device accelerated electrons and positrons in opposite directions, effectively doubling the energy of their collision when compared to striking a static target with an electron.[62] The Large Electron-Positron Collider (LEP) at CERN, which was operational from 1989 to 2000, achieved collision energies of 209 GeV and made important measurements for the Standard Model of particle physics.[63][64]
Confinement of individual electrons
Individual electrons can now be easily confined in ultra small (L=20 nm, W=20 nm) CMOS transistors operated at cryogenic temperature over a range of −269 °C (4 K) to about −258 °C (15 K).[65] The electron wavefunction spreads in a semiconductor lattice and negligibly interacts with the valence band electrons, so it can be treated in the single particle formalism, by replacing its mass with the effective mass tensor.
A table with four rows and four columns, with each cell containing a particle identifier
Standard Model of elementary particles. The electron is at lower left.
In the Standard Model of particle physics, electrons belong to the group of subatomic particles called leptons, which are believed to be fundamental or elementary particles. Electrons have the lowest mass of any charged lepton (or electrically charged particle of any type) and belong to the first-generation of fundamental particles.[66] The second and third generation contain charged leptons, the muon and the tau, which are identical to the electron in charge, spin and interactions, but are more massive. Leptons differ from the other basic constituent of matter, the quarks, by their lack of strong interaction. All members of the lepton group are fermions, because they all have half-odd integer spin; the electron has spin 12.[67]
Fundamental properties
The invariant mass of an electron is approximately 9.109×10−31 kilograms,[68] or 5.489×10−4 atomic mass units. On the basis of Einstein's principle of mass–energy equivalence, this mass corresponds to a rest energy of 0.511 MeV. The ratio between the mass of a proton and that of an electron is about 1836.[9][69] Astronomical measurements show that the proton-to-electron mass ratio has held the same value for at least half the age of the universe, as is predicted by the Standard Model.[70]
Electrons have an electric charge of −1.602×10−19 coulomb,[68] which is used as a standard unit of charge for subatomic particles. Within the limits of experimental accuracy, the electron charge is identical to the charge of a proton, but with the opposite sign.[71] As the symbol e is used for the elementary charge, the electron is commonly symbolized by e, where the minus sign indicates the negative charge. The positron is symbolized by e+ because it has the same properties as the electron but with a positive rather than negative charge.[67][68]
The electron has an intrinsic angular momentum or spin of 12.[68] This property is usually stated by referring to the electron as a spin-12 particle.[67] For such particles the spin magnitude is 32 ħ.[note 3] while the result of the measurement of a projection of the spin on any axis can only be ±ħ2. In addition to spin, the electron has an intrinsic magnetic moment along its spin axis.[68] It is approximately equal to one Bohr magneton,[72][note 4] which is a physical constant equal to 9.27400915(23)×10−24 joules per tesla.[68] The orientation of the spin with respect to the momentum of the electron defines the property of elementary particles known as helicity.[73]
The electron has no known substructure.[2][74] Hence, it is defined or assumed to be a point particle with a point charge and no spatial extent.[10] Observation of a single electron in a Penning trap shows the upper limit of the particle's radius is 10−22 meters.[75] There is a physical constant called the "classical electron radius", with the much larger value of 2.8179×10−15 m. However, the terminology comes from a simplistic calculation that ignores the effects of quantum mechanics; in reality, the so-called classical electron radius has little to do with the true fundamental structure of the electron.[76][note 5]
There are elementary particles that spontaneously decay into less massive particles. An example is the muon, which decays into an electron, a neutrino and an antineutrino, with a mean lifetime of 2.2×10−6 seconds. However, the electron is thought to be stable on theoretical grounds: the electron is the least massive particle with non-zero electric charge, so its decay would violate charge conservation.[77] The experimental lower bound for the electron's mean lifetime is 4.6×1026 years, at a 90% confidence level.[78][79]
Quantum properties
As with all particles, electrons can act as waves. This is called the wave–particle duality and can be demonstrated using the double-slit experiment. The wave-like nature of the electron allows it to pass through two parallel slits simultaneously, rather than just one slit as would be the case for a classical particle. In quantum mechanics, the wave-like property of one particle can be described mathematically as a complex-valued function, the wave function, commonly denoted by the Greek letter psi (ψ). When the absolute value of this function is squared, it gives the probability that a particle will be observed near a location—a probability density.[80]
A three dimensional projection of a two dimensional plot. There are symmetric hills along one axis and symmetric valleys along the other, roughly giving a saddle-shape
Example of an antisymmetric wave function for a quantum state of two identical fermions in a 1-dimensional box. If the particles swap position, the wave function inverts its sign.
Electrons are identical particles because they cannot be distinguished from each other by their intrinsic physical properties. In quantum mechanics, this means that a pair of interacting electrons must be able to swap positions without an observable change to the state of the system. The wave function of fermions, including electrons, is antisymmetric, meaning that it changes sign when two electrons are swapped; that is, ψ(r1, r2) = −ψ(r2, r1), where the variables r1 and r2 correspond to the first and second electrons, respectively. Since the absolute value is not changed by a sign swap, this corresponds to equal probabilities. Bosons, such as the photon, have symmetric wave functions instead.[80]
In the case of antisymmetry, solutions of the wave equation for interacting electrons result in a zero probability that each pair will occupy the same location or state. This is responsible for the Pauli exclusion principle, which precludes any two electrons from occupying the same quantum state. This principle explains many of the properties of electrons. For example, it causes groups of bound electrons to occupy different orbitals in an atom, rather than all overlapping each other in the same orbit.[80]
Virtual particles
Physicists believe that empty space may be continually creating pairs of virtual particles, such as a positron and electron, which rapidly annihilate each other shortly thereafter.[81] The combination of the energy variation needed to create these particles, and the time during which they exist, fall under the threshold of detectability expressed by the Heisenberg uncertainty relation, ΔE · Δt ≥ ħ. In effect, the energy needed to create these virtual particles, ΔE, can be "borrowed" from the vacuum for a period of time, Δt, so that their product is no more than the reduced Planck constant, ħ6.6×10−16 eV·s. Thus, for a virtual electron, Δt is at most 1.3×10−21 s.[82]
A sphere with a minus sign at lower left symbolizes the electron, while pairs of spheres with plus and minus signs show the virtual particles
While an electron–positron virtual pair is in existence, the coulomb force from the ambient electric field surrounding an electron causes a created positron to be attracted to the original electron, while a created electron experiences a repulsion. This causes what is called vacuum polarization. In effect, the vacuum behaves like a medium having a dielectric permittivity more than unity. Thus the effective charge of an electron is actually smaller than its true value, and the charge decreases with increasing distance from the electron.[83][84] This polarization was confirmed experimentally in 1997 using the Japanese TRISTAN particle accelerator.[85] Virtual particles cause a comparable shielding effect for the mass of the electron.[86]
The interaction with virtual particles also explains the small (about 0.1%) deviation of the intrinsic magnetic moment of the electron from the Bohr magneton (the anomalous magnetic moment).[72][87] The extraordinarily precise agreement of this predicted difference with the experimentally determined value is viewed as one of the great achievements of quantum electrodynamics.[88]
In classical physics, the angular momentum and magnetic moment of an object depend upon its physical dimensions. Hence, the concept of a dimensionless electron possessing these properties might seem inconsistent. The apparent paradox can be explained by the formation of virtual photons in the electric field generated by the electron. These photons cause the electron to shift about in a jittery fashion (known as zitterbewegung),[89] which results in a net circular motion with precession. This motion produces both the spin and the magnetic moment of the electron.[10][90] In atoms, this creation of virtual photons explains the Lamb shift observed in spectral lines.[83]
An electron generates an electric field that exerts an attractive force on a particle with a positive charge, such as the proton, and a repulsive force on a particle with a negative charge. The strength of this force is determined by Coulomb's inverse square law.[91] When an electron is in motion, it generates a magnetic field.[92] The Ampère-Maxwell law relates the magnetic field to the mass motion of electrons (the current) with respect to an observer. It is this property of induction which supplies the magnetic field that drives an electric motor.[93] The electromagnetic field of an arbitrary moving charged particle is expressed by the Liénard–Wiechert potentials, which are valid even when the particle's speed is close to that of light (relativistic).
A graph with arcs showing the motion of charged particles
A particle with charge q (at left) is moving with velocity v through a magnetic field B that is oriented toward the viewer. For an electron, q is negative so it follows a curved trajectory toward the top.
When an electron is moving through a magnetic field, it is subject to the Lorentz force that exerts an influence in a direction perpendicular to the plane defined by the magnetic field and the electron velocity. This centripetal force causes the electron to follow a helical trajectory through the field at a radius called the gyroradius. The acceleration from this curving motion induces the electron to radiate energy in the form of synchrotron radiation.[94][95][note 6] The energy emission in turn causes a recoil of the electron, known as the Abraham–Lorentz–Dirac Force, which creates a friction that slows the electron. This force is caused by a back-reaction of the electron's own field upon itself.[96]
In quantum electrodynamics the electromagnetic interaction between particles is mediated by photons. An isolated electron that is not undergoing acceleration is unable to emit or absorb a real photon; doing so would violate conservation of energy and momentum. Instead, virtual photons can transfer momentum between two charged particles. It is this exchange of virtual photons that, for example, generates the Coulomb force.[97] Energy emission can occur when a moving electron is deflected by a charged particle, such as a proton. The acceleration of the electron results in the emission of Bremsstrahlung radiation.[98]
A curve shows the motion of the electron, a red dot shows the nucleus, and a wiggly line the emitted photon
Here, Bremsstrahlung is produced by an electron e deflected by the electric field of an atomic nucleus. The energy change E2 − E1 determines the frequency f of the emitted photon.
An inelastic collision between a photon (light) and a solitary (free) electron is called Compton scattering. This collision results in a transfer of momentum and energy between the particles, which modifies the wavelength of the photon by an amount called the Compton shift.[note 7] The maximum magnitude of this wavelength shift is h/mec, which is known as the Compton wavelength.[99] For an electron, it has a value of 2.43×10−12 m.[68] When the wavelength of the light is long (for instance, the wavelength of the visible light is 0.4–0.7 μm) the wavelength shift becomes negligible. Such interaction between the light and free electrons is called Thomson scattering or Linear Thomson scattering.[100]
The relative strength of the electromagnetic interaction between two charged particles, such as an electron and a proton, is given by the fine-structure constant. This value is a dimensionless quantity formed by the ratio of two energies: the electrostatic energy of attraction (or repulsion) at a separation of one Compton wavelength, and the rest energy of the charge. It is given by α ≈ 7.297353×10−3, which is approximately equal to 1137.[68]
When electrons and positrons collide, they annihilate each other, giving rise to two or more gamma ray photons. If the electron and positron have negligible momentum, a positronium atom can form before annihilation results in two or three gamma ray photons totalling 1.022 MeV.[101][102] On the other hand, high-energy photons may transform into an electron and a positron by a process called pair production, but only in the presence of a nearby charged particle, such as a nucleus.[103][104]
In the theory of electroweak interaction, the left-handed component of electron's wavefunction forms a weak isospin doublet with the electron neutrino. This means that during weak interactions, electron neutrinos behave like electrons. Either member of this doublet can undergo a charged current interaction by emitting or absorbing a W and be converted into the other member. Charge is conserved during this reaction because the W boson also carries a charge, canceling out any net change during the transmutation. Charged current interactions are responsible for the phenomenon of beta decay in a radioactive atom. Both the electron and electron neutrino can undergo a neutral current interaction via a Z0 exchange, and this is responsible for neutrino-electron elastic scattering.[105]
Atoms and molecules
A table of five rows and five columns, with each cell portraying a color-coded probability density
Probability densities for the first few hydrogen atom orbitals, seen in cross-section. The energy level of a bound electron determines the orbital it occupies, and the color reflects the probability to find the electron at a given position.
An electron can be bound to the nucleus of an atom by the attractive Coulomb force. A system of one or more electrons bound to a nucleus is called an atom. If the number of electrons is different from the nucleus' electrical charge, such an atom is called an ion. The wave-like behavior of a bound electron is described by a function called an atomic orbital. Each orbital has its own set of quantum numbers such as energy, angular momentum and projection of angular momentum, and only a discrete set of these orbitals exist around the nucleus. According to the Pauli exclusion principle each orbital can be occupied by up to two electrons, which must differ in their spin quantum number.
Electrons can transfer between different orbitals by the emission or absorption of photons with an energy that matches the difference in potential.[106] Other methods of orbital transfer include collisions with particles, such as electrons, and the Auger effect.[107] In order to escape the atom, the energy of the electron must be increased above its binding energy to the atom. This occurs, for example, with the photoelectric effect, where an incident photon exceeding the atom's ionization energy is absorbed by the electron.[108]
The orbital angular momentum of electrons is quantized. Because the electron is charged, it produces an orbital magnetic moment that is proportional to the angular momentum. The net magnetic moment of an atom is equal to the vector sum of orbital and spin magnetic moments of all electrons and the nucleus. The magnetic moment of the nucleus is negligible compared with that of the electrons. The magnetic moments of the electrons that occupy the same orbital (so called, paired electrons) cancel each other out.[109]
The chemical bond between atoms occurs as a result of electromagnetic interactions, as described by the laws of quantum mechanics.[110] The strongest bonds are formed by the sharing or transfer of electrons between atoms, allowing the formation of molecules.[14] Within a molecule, electrons move under the influence of several nuclei, and occupy molecular orbitals; much as they can occupy atomic orbitals in isolated atoms.[111] A fundamental factor in these molecular structures is the existence of electron pairs. These are electrons with opposed spins, allowing them to occupy the same molecular orbital without violating the Pauli exclusion principle (much like in atoms). Different molecular orbitals have different spatial distribution of the electron density. For instance, in bonded pairs (i.e. in the pairs that actually bind atoms together) electrons can be found with the maximal probability in a relatively small volume between the nuclei. On the contrary, in non-bonded pairs electrons are distributed in a large volume around nuclei.[112]
Four bolts of lightning strike the ground
A lightning discharge consists primarily of a flow of electrons.[113] The electric potential needed for lightning may be generated by a triboelectric effect.[114][115]
If a body has more or fewer electrons than are required to balance the positive charge of the nuclei, then that object has a net electric charge. When there is an excess of electrons, the object is said to be negatively charged. When there are fewer electrons than the number of protons in nuclei, the object is said to be positively charged. When the number of electrons and the number of protons are equal, their charges cancel each other and the object is said to be electrically neutral. A macroscopic body can develop an electric charge through rubbing, by the triboelectric effect.[116]
Independent electrons moving in vacuum are termed free electrons. Electrons in metals also behave as if they were free. In reality the particles that are commonly termed electrons in metals and other solids are quasi-electrons—quasiparticles, which have the same electrical charge, spin and magnetic moment as real electrons but may have a different mass.[117] When free electrons—both in vacuum and metals—move, they produce a net flow of charge called an electric current, which generates a magnetic field. Likewise a current can be created by a changing magnetic field. These interactions are described mathematically by Maxwell's equations.[118]
At a given temperature, each material has an electrical conductivity that determines the value of electric current when an electric potential is applied. Examples of good conductors include metals such as copper and gold, whereas glass and Teflon are poor conductors. In any dielectric material, the electrons remain bound to their respective atoms and the material behaves as an insulator. Most semiconductors have a variable level of conductivity that lies between the extremes of conduction and insulation.[119] On the other hand, metals have an electronic band structure containing partially filled electronic bands. The presence of such bands allows electrons in metals to behave as if they were free or delocalized electrons. These electrons are not associated with specific atoms, so when an electric field is applied, they are free to move like a gas (called Fermi gas)[120] through the material much like free electrons.
Because of collisions between electrons and atoms, the drift velocity of electrons in a conductor is on the order of millimeters per second. However, the speed at which a change of current at one point in the material causes changes in currents in other parts of the material, the velocity of propagation, is typically about 75% of light speed.[121] This occurs because electrical signals propagate as a wave, with the velocity dependent on the dielectric constant of the material.[122]
Metals make relatively good conductors of heat, primarily because the delocalized electrons are free to transport thermal energy between atoms. However, unlike electrical conductivity, the thermal conductivity of a metal is nearly independent of temperature. This is expressed mathematically by the Wiedemann–Franz law,[120] which states that the ratio of thermal conductivity to the electrical conductivity is proportional to the temperature. The thermal disorder in the metallic lattice increases the electrical resistivity of the material, producing a temperature dependence for electrical current.[123]
When cooled below a point called the critical temperature, materials can undergo a phase transition in which they lose all resistivity to electrical current, in a process known as superconductivity. In BCS theory, this behavior is modeled by pairs of electrons entering a quantum state known as a Bose–Einstein condensate. These Cooper pairs have their motion coupled to nearby matter via lattice vibrations called phonons, thereby avoiding the collisions with atoms that normally create electrical resistance.[124] (Cooper pairs have a radius of roughly 100 nm, so they can overlap each other.)[125] However, the mechanism by which higher temperature superconductors operate remains uncertain.
Electrons inside conducting solids, which are quasi-particles themselves, when tightly confined at temperatures close to absolute zero, behave as though they had split into two other quasiparticles: spinons and holons.[126][127] The former carries spin and magnetic moment, while the latter electrical charge.
Motion and energy
According to Einstein's theory of special relativity, as an electron's speed approaches the speed of light, from an observer's point of view its relativistic mass increases, thereby making it more and more difficult to accelerate it from within the observer's frame of reference. The speed of an electron can approach, but never reach, the speed of light in a vacuum, c. However, when relativistic electrons—that is, electrons moving at a speed close to c—are injected into a dielectric medium such as water, where the local speed of light is significantly less than c, the electrons temporarily travel faster than light in the medium. As they interact with the medium, they generate a faint light called Cherenkov radiation.[128]
The plot starts at zero and curves sharply upward toward the right
Lorentz factor as a function of velocity. It starts at value 1 and goes to infinity as v approaches c.
The effects of special relativity are based on a quantity known as the Lorentz factor, defined as \scriptstyle\gamma=1/ \sqrt{ 1-{v^2}/{c^2} } where v is the speed of the particle. The kinetic energy Ke of an electron moving with velocity v is:
\displaystyle K_{\mathrm{e}} = (\gamma - 1)m_{\mathrm{e}} c^2,
where me is the mass of electron. For example, the Stanford linear accelerator can accelerate an electron to roughly 51 GeV.[129] Since an electron behaves as a wave, at a given velocity it has a characteristic de Broglie wavelength. This is given by λe = h/p where h is the Planck constant and p is the momentum.[50] For the 51 GeV electron above, the wavelength is about 2.4×10−17 m, small enough to explore structures well below the size of an atomic nucleus.[130]
A photon strikes the nucleus from the left, with the resulting electron and positron moving off to the right
Pair production caused by the collision of a photon with an atomic nucleus
The Big Bang theory is the most widely accepted scientific theory to explain the early stages in the evolution of the Universe.[131] For the first millisecond of the Big Bang, the temperatures were over 10 billion Kelvin and photons had mean energies over a million electronvolts. These photons were sufficiently energetic that they could react with each other to form pairs of electrons and positrons. Likewise, positron-electron pairs annihilated each other and emitted energetic photons:
γ + γe+ + e
An equilibrium between electrons, positrons and photons was maintained during this phase of the evolution of the Universe. After 15 seconds had passed, however, the temperature of the universe dropped below the threshold where electron-positron formation could occur. Most of the surviving electrons and positrons annihilated each other, releasing gamma radiation that briefly reheated the universe.[132]
For reasons that remain uncertain, during the process of leptogenesis there was an excess in the number of electrons over positrons.[133] Hence, about one electron in every billion survived the annihilation process. This excess matched the excess of protons over anti-protons, in a condition known as baryon asymmetry, resulting in a net charge of zero for the universe.[134][135] The surviving protons and neutrons began to participate in reactions with each other—in the process known as nucleosynthesis, forming isotopes of hydrogen and helium, with trace amounts of lithium. This process peaked after about five minutes.[136] Any leftover neutrons underwent negative beta decay with a half-life of about a thousand seconds, releasing a proton and electron in the process,
np + e + ν
For about the next 300,000400,000 years, the excess electrons remained too energetic to bind with atomic nuclei.[137] What followed is a period known as recombination, when neutral atoms were formed and the expanding universe became transparent to radiation.[138]
Roughly one million years after the big bang, the first generation of stars began to form.[138] Within a star, stellar nucleosynthesis results in the production of positrons from the fusion of atomic nuclei. These antimatter particles immediately annihilate with electrons, releasing gamma rays. The net result is a steady reduction in the number of electrons, and a matching increase in the number of neutrons. However, the process of stellar evolution can result in the synthesis of radioactive isotopes. Selected isotopes can subsequently undergo negative beta decay, emitting an electron and antineutrino from the nucleus.[139] An example is the cobalt-60 (60Co) isotope, which decays to form nickel-60 (60Ni).[140]
A branching tree representing the particle production
An extended air shower generated by an energetic cosmic ray striking the Earth's atmosphere
At the end of its lifetime, a star with more than about 20 solar masses can undergo gravitational collapse to form a black hole.[141] According to classical physics, these massive stellar objects exert a gravitational attraction that is strong enough to prevent anything, even electromagnetic radiation, from escaping past the Schwarzschild radius. However, it is believed that quantum mechanical effects may allow Hawking radiation to be emitted at this distance. Electrons (and positrons) are thought to be created at the event horizon of these stellar remnants.
When pairs of virtual particles (such as an electron and positron) are created in the vicinity of the event horizon, the random spatial distribution of these particles may permit one of them to appear on the exterior; this process is called quantum tunnelling. The gravitational potential of the black hole can then supply the energy that transforms this virtual particle into a real particle, allowing it to radiate away into space.[142] In exchange, the other member of the pair is given negative energy, which results in a net loss of mass-energy by the black hole. The rate of Hawking radiation increases with decreasing mass, eventually causing the black hole to evaporate away until, finally, it explodes.[143]
Cosmic rays are particles traveling through space with high energies. Energy events as high as 3.0×1020 eV have been recorded.[144] When these particles collide with nucleons in the Earth's atmosphere, a shower of particles is generated, including pions.[145] More than half of the cosmic radiation observed from the Earth's surface consists of muons. The particle called a muon is a lepton which is produced in the upper atmosphere by the decay of a pion.
πμ + ν
A muon, in turn, can decay to form an electron or positron.[146]
μe + ν
+ ν
A swirling green glow in the night sky above snow-covered ground
Aurorae are mostly caused by energetic electrons precipitating into the atmosphere.[147]
Remote observation of electrons requires detection of their radiated energy. For example, in high-energy environments such as the corona of a star, free electrons form a plasma that radiates energy due to Bremsstrahlung radiation. Electron gas can undergo plasma oscillation, which is waves caused by synchronized variations in electron density, and these produce energy emissions that can be detected by using radio telescopes.[148]
The frequency of a photon is proportional to its energy. As a bound electron transitions between different energy levels of an atom, it will absorb or emit photons at characteristic frequencies. For instance, when atoms are irradiated by a source with a broad spectrum, distinct absorption lines will appear in the spectrum of transmitted radiation. Each element or molecule displays a characteristic set of spectral lines, such as the hydrogen spectral series. Spectroscopic measurements of the strength and width of these lines allow the composition and physical properties of a substance to be determined.[149][150]
In laboratory conditions, the interactions of individual electrons can be observed by means of particle detectors, which allow measurement of specific properties such as energy, spin and charge.[108] The development of the Paul trap and Penning trap allows charged particles to be contained within a small region for long durations. This enables precise measurements of the particle properties. For example, in one instance a Penning trap was used to contain a single electron for a period of 10 months.[151] The magnetic moment of the electron was measured to a precision of eleven digits, which, in 1980, was a greater accuracy than for any other physical constant.[152]
The first video images of an electron's energy distribution were captured by a team at Lund University in Sweden, February 2008. The scientists used extremely short flashes of light, called attosecond pulses, which allowed an electron's motion to be observed for the first time.[153][154]
The distribution of the electrons in solid materials can be visualized by angle-resolved photoemission spectroscopy (ARPES). This technique employs the photoelectric effect to measure the reciprocal space—a mathematical representation of periodic structures that is used to infer the original structure. ARPES can be used to determine the direction, speed and scattering of electrons within the material.[155]
Plasma applications
Particle beams
A violet beam from above produces a blue glow about a Space shuttle model
During a NASA wind tunnel test, a model of the Space Shuttle is targeted by a beam of electrons, simulating the effect of ionizing gases during re-entry.[156]
Electron beams are used in welding,[157] which allows energy densities up to 107 W·cm−2 across a narrow focus diameter of 0.1–1.3 mm and usually does not require a filler material. This welding technique must be performed in a vacuum, so that the electron beam does not interact with the gas prior to reaching the target, and it can be used to join conductive materials that would otherwise be considered unsuitable for welding.[158][159]
Electron beam lithography (EBL) is a method of etching semiconductors at resolutions smaller than a micron.[160] This technique is limited by high costs, slow performance, the need to operate the beam in the vacuum and the tendency of the electrons to scatter in solids. The last problem limits the resolution to about 10 nm. For this reason, EBL is primarily used for the production of small numbers of specialized integrated circuits.[161]
Electron beam processing is used to irradiate materials in order to change their physical properties or sterilize medical and food products.[162] In radiation therapy, electron beams are generated by linear accelerators for treatment of superficial tumors. Because an electron beam only penetrates to a limited depth before being absorbed, typically up to 5 cm for electron energies in the range 5–20 MeV, electron therapy is useful for treating skin lesions such as basal-cell carcinomas. An electron beam can be used to supplement the treatment of areas that have been irradiated by X-rays.[163][164]
Particle accelerators use electric fields to propel electrons and their antiparticles to high energies. As these particles pass through magnetic fields, they emit synchrotron radiation. The intensity of this radiation is spin dependent, which causes polarization of the electron beam—a process known as the Sokolov–Ternov effect.[note 8] The polarized electron beams can be useful for various experiments. Synchrotron radiation can also be used for cooling the electron beams, which reduces the momentum spread of the particles. Once the particles have accelerated to the required energies, separate electron and positron beams are brought into collision. The resulting energy emissions are observed with particle detectors and are studied in particle physics.[165]
Low-energy electron diffraction (LEED) is a method of bombarding a crystalline material with a collimated beam of electrons, then observing the resulting diffraction patterns to determine the structure of the material. The required energy of the electrons is typically in the range 20–200 eV.[166] The reflection high-energy electron diffraction (RHEED) technique uses the reflection of a beam of electrons fired at various low angles to characterize the surface of crystalline materials. The beam energy is typically in the range 8–20 keV and the angle of incidence is 1–4°.[167][168]
The electron microscope directs a focused beam of electrons at a specimen. As the beam interacts with the material, some electrons change their properties, such as movement direction, angle, relative phase and energy. By recording these changes in the electron beam, microscopists can produce atomically resolved image of the material.[169] In blue light, conventional optical microscopes have a diffraction-limited resolution of about 200 nm.[170] By comparison, electron microscopes are limited by the de Broglie wavelength of the electron. This wavelength, for example, is equal to 0.0037 nm for electrons accelerated across a 100,000-volt potential.[171] The Transmission Electron Aberration-Corrected Microscope is capable of sub-0.05 nm resolution, which is more than enough to resolve individual atoms.[172] This capability makes the electron microscope a useful laboratory instrument for high resolution imaging. However, electron microscopes are expensive instruments that are costly to maintain.
There are two main types of electron microscopes: transmission and scanning. Transmission electron microscopes function in a manner similar to overhead projector, with a beam of electrons passing through a slice of material then being projected by lenses on a photographic slide or a charge-coupled device. In scanning electron microscopes, the image is produced by rastering a finely focused electron beam, as in a TV set, across the studied sample. The magnifications range from 100× to 1,000,000× or higher for both microscope types. The scanning tunneling microscope uses quantum tunneling of electrons from a sharp metal tip into the studied material and can produce atomically resolved images of its surface.[173][174][175]
Other applications
In the free-electron laser (FEL), a relativistic electron beam is passed through a pair of undulators containing arrays of dipole magnets, whose fields are oriented in alternating directions. The electrons emit synchrotron radiation, which, in turn, coherently interacts with the same electrons. This leads to the strong amplification of the radiation field at the resonance frequency. FEL can emit a coherent high-brilliance electromagnetic radiation with a wide range of frequencies, from microwaves to soft X-rays. These devices can be used in the future for manufacturing, communication and various medical applications, such as soft tissue surgery.[176]
Electrons are at the heart of cathode ray tubes, which have been used extensively as display devices in laboratory instruments, computer monitors and television sets.[177] In a photomultiplier tube, every photon striking the photocathode initiates an avalanche of electrons that produces a detectable current pulse.[178] Vacuum tubes use the flow of electrons to manipulate electrical signals, and they played a critical role in the development of electronics technology. However, they have been largely supplanted by solid-state devices such as the transistor.[179]
See also
1. ^ The fractional version's denominator is the inverse of the decimal value (along with its relative standard uncertainty of 4.2×10−13 u).
2. ^ The electron's charge is the negative of elementary charge, which has a positive value for the proton.
3. ^ This magnitude is obtained from the spin quantum number as
S & = \sqrt{s(s + 1)} \cdot \frac{h}{2\pi} \\
& = \frac{\sqrt{3}}{2} \hbar \\
for quantum number s = 12.
See: Gupta, M.C. (2001). Atomic and Molecular Spectroscopy. New Age Publishers. p. 81. ISBN 81-224-1300-5 [Amazon-US | Amazon-UK].
4. ^ Bohr magneton:
5. ^ The classical electron radius is derived as follows. Assume that the electron's charge is spread uniformly throughout a spherical volume. Since one part of the sphere would repel the other parts, the sphere contains electrostatic potential energy. This energy is assumed to equal the electron's rest energy, defined by special relativity (E = mc2).
From electrostatics theory, the potential energy of a sphere with radius r and charge e is given by:
E_{\mathrm p} = \frac{e^2}{8\pi \varepsilon_0 r},
where ε0 is the vacuum permittivity. For an electron with rest mass m0, the rest energy is equal to:
\textstyle E_{\mathrm p} = m_0 c^2,
where c is the speed of light in a vacuum. Setting them equal and solving for r gives the classical electron radius.
See: Haken, H.; Wolf, H.C.; Brewer, W.D. (2005). The Physics of Atoms and Quanta: Introduction to Experiments and Theory. Springer. p. 70. ISBN 3-540-67274-5 [Amazon-US | Amazon-UK].
6. ^ Radiation from non-relativistic electrons is sometimes termed cyclotron radiation.
7. ^ The change in wavelength, Δλ, depends on the angle of the recoil, θ, as follows,
\textstyle \Delta \lambda = \frac{h}{m_{\mathrm{e}}c} (1 - \cos \theta),
where c is the speed of light in a vacuum and me is the electron mass. See Zombeck (2007: 393, 396).
8. ^ The polarization of an electron beam means that the spins of all electrons point into one direction. In other words, the projections of the spins of all electrons onto their momentum vector have the same sign.
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10851fd047d19c63 | Volume 91 Issue 36 | Web Exclusive
Issue Date: September 9, 2013
Cover Stories: Nine For Ninety
Trying To Explain A Bond
Department: Science & Technology
C&EN asked chemists who spend most of their time working with and thinking about chemical bonding to answer this question: What do you think a chemical bond is in reality? The chemists were prompted to consider how they might describe a chemical bond to a nonchemist, to a nonscientist, or to a family member. They were also asked to consider how a chemical bond impacts our daily lives. Add your thoughts on the chemical bond in the comments below.
Anthony J. Arduengo III, University of Alabama, studies the chemistry of compounds with abnormal valency and bonding arrangements and discovered the first isolable N-heterocyclic carbene.
Humankind’s ability to understand and transform matter through chemistry is as much the reason for the success of our species as the opposable thumb and language. Whether this chemistry was mastering combustion (fire) to release chemically stored energy (breaking and making bonds) to provide heat and light at will, or whether one chooses a suitable material (wood or stone, for example) to construct durable housing, tools, and devices to serve the interest of the human community, it’s all chemistry. When it comes to describing a chemical bond, I worry about being constrained by semantics, so I tend to have a rather broad view. I’ve often thought about applying a definition that is energy related—that is, a chemical bond must have a dissociation energy greater that X kcal/mol. But I’m not sure what constructive purpose such a definition would serve, and I can see how it might constrain creative thought. You certainly don’t want to ever get me started talking about valency or aromaticity. I’d as soon discuss the number of angels on a pinhead. Strangely, I’m much more comfortable discussing unusual valency, which seems to imply that I have some preconceived notion of what valence is—I don’t, but I’m happy to accept the usual historical definitions and claim all exceptions as “unusual.” But when talking to a layman, I have no problem in stating that a chemical bond is what holds materials (chemicals) together and gives them their properties.
Polly L. Arnold, University of Edinburgh, focuses on the synthesis of unusual metal complexes of rare earths, actinides, and early transition metals as catalysts.
A bond is the sharing of electrons between two or more atoms, and it doesn’t matter which atoms they belonged to originally or how many electrons are involved. To my mother, who struggles with the difference between atoms and molecules, I refer to bonds as the “glue” between atoms. The variety of glues on the market allows for some simple analogies to be made, ranging from Post-it note-strength hydrogen bonds to the water-soluble glues that hold ionic salts together and superglued polypropylene skeletons.
If we completely understood everything about chemical bonding, we’d do a lot less exploring and have long-list orders for drugs and chelators that we had to make. I’d probably become quite depressed.
Gregory H. Robinson, University of Georgia, is a specialist in the structure and bonding of organometallic compounds, in particular compounds containing multiple bonds between heavier main-group elements.
Chemists have seemingly always been grappling with the definition of a chemical bond. I simply believe that it is a cooperative truce among two atoms. Most of the time the truce concerns a pair of electrons, but on occasion the truce can involve two or three or more pairs of electrons. It’s almost as if the atoms are holding the electrons captive until something better comes along.
Martyn Poliakoff, University of Nottingham, is a pioneer in the field of green chemistry, in particular for chemical applications of supercritical fluids, and a cocreator of the Periodic Table of Videos.
My feeling is that most chemists don’t define bonds. They just know what bonds are when they see them. But many chemists anguish over whether particular atoms in their latest crystal structure are really bonded. The biggest surprise in my research career has been the invention of STM, AFM, and other scanning techniques. We can now actually “see” molecules and their bonds, and they look just as fuzzy as we imagined. The recent Science paper using AFM to show individual molecules before and after they react is something that I was brought up to think was quite impossible. Do the images teach us much about bonding? No, but chemistry is an immensely visual subject and, if we are honest, most of our images are theoretical constructs. So it’s quite reassuring that molecules do really appear to resemble our pictures.
James L. Marshall and Virginia (Jenny) Marshall, University of North Texas. The Marshalls’ special collection of the elements features samples of minerals collected at the sites from which all the natural elements were originally discovered.
A real impediment to the correct understanding of chemical bonding in the early 1800s was the commonplace belief that like atoms could not combine with one another. This was the logical conclusion ensuing from electrolysis experiments of metals, historically credited to Humphry Davy (1778–1829) but in fact pioneered by Jöns Jakob Berzelius (1779–1848). Berzelius noticed that in a battery the “alkalis and earths” were drawn toward the negative pole, and that oxygen, acids, and oxidized substances migrated to the positive pole. He was very impressed by the electrical forces needed to rip apart these reactive metals, and he championed the hypothesis of “electrochemical dualism,” wherein the forces holding atoms together were positive-like (the metals) and negative-like (the nonmetals). Berzelius was thus the first to conceptualize ionic bonding, and he assumed this bonding occurred in all compounds. The idea that elemental hydrogen and chlorine were H2 and Cl2 never occurred to him, and his ideas held sway through the first half of the 19th century.
One of the first to understand “like”-bonding was Jean Baptiste André Dumas (1800–84). In his mid-20s, Dumas was asked to explore the reason why the burning candles in the Tuileries Palace were emitting obnoxious odors. Dumas is best known today for his method of molecular weight determination, currently used in undergraduate chemistry labs. He found that these candles had been bleached by chlorine and that the irritating stench was hydrogen chloride. His curiosity piqued, he followed up with research that allowed him to conclude that either H (positive-like) or Cl (negative-like) could combine with carbon in a similar way, without losing the general physical properties of the compound. That is, the compound still acted like an organic substance. By 1828 he introduced the terms “molécule chimique” (atoms) and “molécule physique” (true molecules). Thus, he formulated the idea that there must be an additional type of bonding, which today we recognize as covalent bonding.
Richard Eisenberg, University of Rochester, studies inorganic and organometallic compounds applied to photochemistry and solar energy conversion and is past editor-in-chief of ACS’s journal Inorganic Chemistry.
In teaching bonding of homonuclear diatomic molecules to first-year students in chemistry, I am always fascinated by the electronic structure of molecular oxygen. It possesses unpaired electrons when a Lewis structure suggests otherwise, and because of those unpaired electrons it can be trapped at low temperature between the poles of a magnet as a blue liquid. This is extraordinary. The paramagnetism makes O2 relatively unreactive. In fact, there is an excited state of O2 that lies close in energy to the ground state, and it can be generated relatively easily. This singlet form of O2 is exceedingly reactive. I tell students that if O2 really existed as a singlet molecule rather than as a triplet with two unpaired electrons, life as we know it would never exist because we would all be turned into inorganic oxide solids. It is the triplet state of O2 that allows life to survive on this planet, and it is molecular orbital theory that allows us to understand why.
Debbie C. Crans, Colorado State University, studies the chemistry and biochemistry of vanadium and other transition metals, fueled by their applications in medicine and their mechanisms of toxicity.
Chemical bonding is the association between atoms facilitated by electrons, and as such produces inherent properties manifested in chemical structure, stability, and reactivity. Structure: When electrons fill up the space between atoms they create a material, which is characterized as having a bond between the two or three components in question. Electrons and bonds are therefore responsible for the shape and volume of molecules. Stability: Bonding can be envisioned as the glue that keeps the different parts of the molecules together. When bond distances are near the ideal, the molecule is stable. Reactivity: Reactivity is a direct consequence of the nature of elements, molecular shape, and bonding. Weak bonds are readily and rapidly broken and can be formed as a result of, for example, reactive forms of elements, undesirable shapes, and long bonds.
Bonding is defined differently in the life sciences and even within each field of chemistry. At a general level, a stick-and-marbles model set can be used to envision molecules and their properties. However, that description does not properly describe the wave nature of electrons and their probabilistic location. Organic chemists use the simple hybridization explanations to describe and understand bonding to tetrahedral carbon. Physical chemists use mathematical and statistical equations to explain the electronic properties of materials. Each description and approach to bonding has strengths and limitations. Organic chemists are simplifying systems and as a result can work with complex molecules. Inorganic chemists concern themselves with a large range of different elements and embrace the relativistic aspects, but as a consequence of the diversity lack the well-developed framework to make predictions that organic chemists have. Physical chemists embrace the mathematical and electronic details but ignore facts such as shape and 3-D occupancy of space, and as a result address mainly electronic and statistical properties. Yet, any discovery exploring these parameters is critical for the future progress of chemistry.
Leo Manzer, an expert in catalysis, is a retired DuPont chief scientist (DuPont Fellow) and is currently head of the chemistry consulting firm Catalytic Insights.
When I think of developments that have impacted society and involved chemical bonding, I think of the development of catalysts in the petroleum industry to convert crude oil to transportation fuels such as gasoline, diesel, and jet fuel. This has largely involved breaking and making C–C and C–H bonds. Without the ability to selectively crack or break the chemical bonds in the viscous oil that comes out of the ground to meet the stringent needs of engine producers, we might still be traveling on coal-fired trains and riding horses or bicycles. In addition, during this bond-breaking process, contaminants such as nitrogen and sulfur are reduced to extremely low levels so that the combustion products don’t contaminate the environment as they did in the early days. Still, new challenges are being tackled by chemists as they learn how to catalytically convert renewable feedstocks such as sugars and wood chips into renewable fuels and chemicals. This involves new technologies for the conversion of C–O and C–OH bonds to C–C bonds.
I also think of the amazing use of chlorofluorocarbons (CFCs) to provide refrigeration to prevent food spoilage; air-conditioning to cool office buildings, homes, and cars; highly insulating foams for energy efficiency; and solvents for circuit board cleaning that helped the computer industry grow. Early on, CFCs were manufactured by carefully and safely creating chlorine and fluorine bonds to carbon. After more than 50 years of production, it was recognized that these CFCs were depleting the ozone layer. Industrial scientists quickly learned how to make new fluorocarbons without chlorine. The ability of chemists to identify, prepare, and selectively eliminate C–Cl bonds on a large scale allowed society to continue operating with little disruption.
These technologies were truly major advances in catalysis and bond manipulation.
Alexander I. Boldyrev, Utah State University, studies theoretical and computational chemistry of new compounds, and is coorganizer of the International Conference on Chemical Bonding.
Chemical bonding is at the heart of our chemical language, and it is extremely important for teaching. When I introduce myself to a nonchemical audience I frequently hear a comment that “I hated chemistry. I really did not understand it.” Our science indeed is very complicated, and a big part of the problem is how we teach chemical bonding. We teach the “golden chemical bonding model” based on Lewis structures. That is a simple concept, and it’s easy to teach. Then we have aromaticity, which is a quite fuzzy concept. People are still arguing about how to recognize aromaticity. Then, we teach valence bond theory, where bonds start to jump from one part of a molecule to another. That confuses freshman students, especially those who were not previously exposed to chemistry. Finally, we teach molecular orbital theory. Now, chemical bonds completely disappear from the picture and we have orbitals instead. I am not surprised that many students, even if they passed general chemistry classes, are still confused about chemical bonding. I personally believe we need to develop a comprehensive chemical bonding theory that will be able to describe most of chemistry. That will help to teach our beautiful science. It will help build financial support of our science and our standing in society.
Akira Sekiguchi, University of Tsukuba, is a specialist in organosilicon chemistry and multiple bonding in main-group elements.
To date, there are many classes of chemical compounds that do not conform to the standard definitions of covalent and ionic bonds. These are the so-called nonclassical compounds. Among them are odd electron bonds such as radical species, hypervalent bonds in molecules with an expanded octet, electron-deficient bonds such as three-center two-electron bonds commonly found in boranes, singlet biradicaloid bonds in the highly strained cluster hydrocarbons propellanes, trans-bent multiple bonds between the heavier main-group elements, a covalent form of the ionic bond in pyramidal shaped hydrocarbons, and more. However, even given the number of these nonclassical compounds and nontrivial bonding situations in them, I still favor the general definition of the chemical bond as the attractive interaction of electrons provided by the participating atoms. This is just like a handshake between two atoms: each atom stretches its “arm” (electron) toward the other, and when these two “arms” (electrons) meet, then the “handshake” (chemical bond) takes place.
Kendall N. Houk, University of California, Los Angeles, solves problems in organic and bioorganic chemistry using theoretical and computational methods.
A chemical bond is what holds atoms together in molecules. Bonds arise from the electrostatic forces between positively charged atomic nuclei and negatively charged electrons (the positions of which in space are determined by quantum mechanics). Pretty simple, except for the parenthetical phrase!
Cathleen M. Crudden, Queen’s University, Kingston, Ontario, focuses on chiral catalysis in organic synthesis and served as 2012 president of the Canadian Society for Chemistry.
The accurate depiction of bonding arrangements in molecules is critical to chemists and chemistry. Take valency, which is something that chemists feel we understand. Carbon likes four bonds, nitrogen three, oxygen two. Simple. Yet, for those of us who teach second-year organic chemistry and see errant pentavalent carbons on a regular basis, molecules that don’t fit our typical idea of valency can be challenging! But there’s no doubt that some such odd molecules do exist. When atoms don’t follow the normal rules, we like to think we understand the consequences. For example, carbon with only two bonds is not a happy creature. These so-called carbenes tend to dimerize, cyclopropanate, insert into C–H bonds, or generally react with just about anything at hand. Until recently, the idea of creating a class of carbenes that can be stored, crystallized, or even distilled would have been largely unthinkable. However, placing two heteroatoms such as nitrogen on the carbene carbon and then confining these two heteroatoms in a ring made the unthinkable a reality. N-heterocyclic carbenes and their derivatives are not only more stable than typical carbenes, they are exceptionally valuable ligands for transition metals, serve as organocatalysts, and have increasing applications in materials chemistry. Thus it seems that taming divalent carbon has been not only a fun endeavor but a useful one as well.
But if asked to explain bonding to a nonscientist, I would probably equate chemical bonds to molecular glue. Carbon-hydrogen bonds would be like superglue, very strong and difficult to break with ordinary methods. The significant amount of energy stored in carbon-hydrogen bonds is one of the things that makes hydrocarbons (perhaps unfortunately) such spectacular fuels because this energy is released when they are burned. Bonds that are significantly weaker, such as hydrogen bonds, are also important. Although individually weak, when a multitude of them act in concert, the effect is dramatic. Consider the fact that water, with only three atoms and a molecular weight of 18, boils at 100 °C. Compare this with methane that has no hydrogen bonding ability, which boils at –164 °C, not so far from liquid nitrogen at –195 °C. It is hydrogen bonding that is responsible for this dramatic change in properties, so tea and coffee drinkers should thank the humble hydrogen bond for the fact that water has the perfect boiling point!
Arnold L. Rheingold, University of California, San Diego, is one of the world’s most prolific crystallographers.
A chemical bond forms when two or more atoms in close proximity achieve a lower overall energy either by creating new orbitals encompassing multiple nuclei or by the transfer of one or more electrons from one atom to another. If asked to state this in a way that my grandmother might understand it, I’d say: “You and grandpa have been together for more than 50 years. Over that time the two of you have functioned better as a team than you would have separately, and during that time you have shared just about everything. The two of you are clearly happier together than you would have been apart. On the other hand, I have known many couples that are just as happy by clearly defining separate roles for themselves by dividing responsibilities. In the world of chemistry, you and grandpa formed a share-all covalent bond, while other couples, for whom functions are kept separate, form an ionic bond.”
Marcetta Y. Darensbourg, Texas A&M University, carries out the synthesis of transition-metal catalysts as mimics for natural hydrogenase enzymes for producing hydrogen.
As only he can, the great professor Harry Gray of Caltech sometimes describes how inorganic chemists account for inexplicable structure and bonding mysteries: “If the Jahn-Teller effect doesn’t work, we go for π-backbonding!” While the former [geometrical distortion of a molecule] is rarely appropriate in my research, the π-backbonding idea definitely applies.
The beautiful simplicity of dative bonding in classical coordination chemistry had to make way for electron delocalization in the huge class of metal carbonyls developed by German chemists in the 1930s and onward. How does it work? Compounds with metals in impressively low oxidation states, nickel(0), cobalt(–1), and iron(–2) in Ni(CO)4, Co(CO)4– and, Fe(CO)42–, follow a pattern of 18-electron counts about the metals. The CO ligands donate two electrons each from a lone pair “forward” to the metal to form a σ-bond, and combined with the d-electrons on the metal, it usually equals 18 electrons. Additionally there are empty antibonding π orbitals on CO that “back”-accept those d-electrons from the electron-rich metals, resulting in decreasing the CO bond order and stabilizing the metal-carbon bond. This push-pull effect is also seen in any metal-ligand system where empty π antibonding orbitals of (most usually) carbon-based ligands, such as olefins or cyclic aromatics, are an energetic match of the filled metal’s d-orbital set.
Synthetic organic chemists and the chemical industry have capitalized on the metal’s way of binding and activating CO and olefins in this manner. But what’s amazing for my research, an organoiron unit, Fe(CN)2(CO), as in the piano-stool organometallic complex, (C5H5)Fe(CO)(CN)2−, is completely analogous to one found in biology. The active sites of hydrogenase enzymes, natural biocatalysts that facilitate production or use of hydrogen, contain iron-carbonyl bonds replete with this π backbonding. As a synthetic inorganic chemist, I can use my infrared spectrometer to contrast my synthetic analogs with biological moieties. Harry is correct––it works!
Douglass W. Stephan, University of Toronto, conducts research in inorganic main-group and organometallic chemistry and is the discoverer of frustrated Lewis pair reactive molecules.
Chemical bonds are the glue that bind atoms together into the ensembles that are molecules. Understanding the nature of bonds, how they influence the properties of molecules, and developing methods to judiciously reconfigure bonds among atoms are the goals of chemistry. These studies can lead to dramatic new technologies that give us such things as pharmaceuticals, flat-screen TVs, and plastics. However, the study of chemical bonds provides insights well beyond the pragmatic. Everything we make, everything we do, and everything we are requires the breaking and making of chemical bonds to supply the energy and yield the fruits of our labors. While chemistry is often focused on real-world goals, in a broader philosophical sense the study of chemical bonds is the study of all that is our world.
Robin D. Rogers, University of Alabama, Tuscaloosa, is director of the Center for Green Manufacturing and editor of the ACS journal Crystal Growth & Design.
Being somewhat of a contrarian, my thoughts on chemical bonding tend toward developments that would more accurately be said to challenge what a bond is, rather than to strictly define it. The development of supramolecular chemistry and the rising importance of coordination chemistry, coupled with hydrogen bonding and halogen bonding, have led to a new way of thinking about how to make molecules and design materials with defined macroscopic properties that goes beyond the classic covalent bond we all learned. This has led to rather heated arguments on exactly what constitutes a chemical bond. Is a bond to be defined by sharing of electrons? If so, how many electrons, and to what degree of sharing? How does this work when virtually every interaction between atoms or molecules involves some sharing of electron density in a continuum of states between absolutely no sharing to absolutely equal sharing? Perhaps it is time to refrain from restrictive nomenclature that limits our imagination by constraining our expectations. Combining all of our advancements in experimental and computational science, let’s try following the electron density and trying to relate properties not just with “ionic” or “covalent” bonds, but with the actual degree of electron density shared.
Anastassia N. Alexandrova, University of California, Los Angeles, conducts theoretical and computational chemistry of proteins, enzymes, catalytic surfaces, and clusters and is coorganizer of the International Conference on Chemical Bonding.
In defining a chemical bond, I tend to be conservative and refer to Pauling’s book “The Nature of the Chemical Bond.” There is a chemical bond between two atoms or groups of atoms in the case that the forces acting between them are such as to lead to the formation of an aggregate with sufficient stability to make it convenient for the chemist to consider it as an independent molecular species. This definition is both inclusive of various types of bonding (if you are a connoisseur) and simple for a nonscientist to grasp. The foundation of every chemical or physical phenomenon is electronic structure. But do we always have to run an electronic structure calculation or take a spectrum to gain a chemical insight? I hope not. For chemists, electronic structure traditionally and fruitfully translates into the qualitative language of chemical bonding. This language, though rooted in the solution of the Schrödinger equation, is simple enough to enable intuition and fast thinking about chemical systems. Whether you think of efficient heterogeneous catalysts, organic solar cells, or fluorescent labeling of biomolecules, consider bonding—it will help to guide and accelerate your progress. The theory of chemical bonding is undergoing rapid development, and it should be appreciated as a simple qualitative tool for the design of structures and properties of materials and molecules across the discipline.
Sason Shaik, Hebrew University of Jerusalem, carries out theoretical and computational chemistry with a focus on reactions of metalloenzymes and new chemical bonding concepts.
Given that every sticky interaction is called a chemical bond today, it is impossible to give an effective and productive description of “the chemical bond.” If, however, we focus on the most common bond that holds molecules together, this would be the electron-pair bond, which lowers the energy of the bonded fragments by 20 to 141 kcal/mol per bond. Accepting that the electron-pair bond leads to formation of molecules, it can be explained to nonscientists and to grandmothers, using the simple imagery of a game of Lego. Take two H atoms. Each one has an electron. If the two electrons have opposite spin around their axes, then when the H’s approach one another they click to make a bond. This click bond obeys magic numbers, which allow you to construct and conceptualize an infinite number of molecules. This is how I teach chemistry to humanities and social sciences students in my course at Hebrew University.
Considering how the chemical bond impacts our daily life and our existence, imagine if water molecules did not obey the rules of bonding and were linear instead of bent. Water would then most likely be a gas. Where would we then be? Take graphite and diamond. They have different bonding patterns, and as a result, graphite is cheap and ugly, while diamond is beautifully scintillating, has a high value, and evokes emotions such as falling in love. Think about the retinal molecule in our eyes. Because of its bonding in cis and trans structures, we can “see.” The eternal homochirality of living matter is a result of bonding that makes proteins and sugars persistently chiral. Think about DNA and RNA: Our genetic code is a chemical code of architecture and dynamics of weak bonds (hydrogen bonds). Matter without the chemical bond as we know it would be an atomic soup!
Gabriel Merino, Center for Research & Advanced Studies of the National Polytechnic Institute, in Mérida, Mexico, conducts computational chemistry to understand and predict new molecules.
The chemical bond is a fuzzy concept, explained by limited models, that has historically been full of intense debate and controversy. It is really complicated to understand that chemical bonding is a concept, and a chemical bond is not a real object. In this regard, the teaching of chemical bonding promotes many misconceptions when this point is not clear. If, from the beginning, we teach that chemistry is based on models, perhaps it will be simpler to understand why sometimes two models provide conflicting views for the same chemical problem. In my opinion, one way to understand chemical bonding is forcing bonding to extreme situations. Molecules under pressure, delocalized systems, nonclassical molecules, transition states, and many other challenging systems are strong motivations to many chemists all around the globe to continue our quest in better understanding this difficult, challenging, controversial, but fascinating cornerstone concept of chemistry: the chemical bond.
Roald Hoffmann, Cornell University, was a 1981 Nobel Laureate in Chemistry for his theories concerning the course of chemical reactions.
I think that any rigorous definition of a chemical bond is bound to be impoverishing, leaving one with the comfortable feeling, “yes (no), I have (do not have) a bond,” but little else. And yet the concept of a chemical bond, so essential to chemistry and with a venerable history, has life, generating controversy and incredible interest. My advice is this: Push the concept to its limits. Be aware of the different experimental and theoretical measures out there. Accept that at the limits a bond will be a bond by some criteria, maybe not others. Respect chemical tradition, relax, and instead of wringing your hands about how terrible it is that this concept cannot be unambiguously defined, have fun with the fuzzy richness of the idea.
Josef Michl, University of Colorado, conducts physical organic chemistry studies of organic and organometallic compounds, including molecular rotors and molecular circuits.
What I told my grandmother is that atoms are sticky. Once two of them stick together in a line, or three in a triangle, they require an effort to pull apart and are said to be connected through a bond. This effort can be small when a bond is weak and the atoms far apart (van der Waals attraction), or a little bigger (hydrogen bonds), or larger still (coordination), or huge when the atoms are really close (covalent, ionic, and multicenter). We indicate a bond between two atoms with a line or several lines, and can display the many bonds present in straight and branched chains, rings, and cages. It is harder to draw three-center bonds (sometimes we use dashed lines).
Why are atoms sticky? They contain heavy positive nuclei and light negative electrons flying about, unable to leave due to electrostatic attraction. When two or three atoms join, their electrons are attracted to each nucleus and repelled by each other. They modify their motion, lowering the total energy even though the nuclei repel. Pushing the atoms closer than the optimal distance would modify the electron motion differently, increase nuclear repulsion, and augment the energy again.
By now, my grandmother’s curiosity waned and I did not get a chance to tell her about quantum mechanics and about what the world would look like if atoms were not sticky.
Dean J. Tantillo, University of California, Davis, is a theoretical organic chemist studying the mechanisms of cascade polycyclization reactions and the design of new catalysts.
To me a bond is simply the attraction between atoms. This attraction may be covalent or ionic, or be labeled with a more specific name like sigma bond, pi bond, delta bond, conjugation, hyperconjugation, percaudal interaction, salt bridge, hydrogen bond, halogen bond, dative bond, charge shift bond, and on and on and on. I tend to think of bonding, rather than bonds. How much bonding? How large is the favorable interaction energy? What are the origins of the attraction? These concepts I digest and describe in terms chemists are generally comfortable thinking about—charges attracting, orbitals with appropriate numbers of electrons overlapping, and so forth. In short, it’s all about continua, in terms of strength, length, and relative contributions of different sources of attraction.
Joel S. Miller, University of Utah, studies multicenter carbon bonding as well as organic-based magnetic materials.
I think a chemical bond is a stabilizing, or attractive, interaction between atoms that significantly alters the properties of the atoms, leading to an independent or new species with new properties. This definition does not state how stabilizing or significant “significantly” has to be. One could limit the definition to something that can be put into a bottle, but justifiably others would argue against that limitation. In a more general way, a chemical bond is a construct, frequently a pictogram, used by chemists to understand the stronger attractive interactions among atoms that enable the organization and understanding of the structure, properties, reactivities, and interrelations for the growing myriad of substances. The concept of chemical bonding impacts our daily lives because it is a language enabling chemists to communicate among each other and facilitate the design and synthesis of improved substances that have benefited mankind. A chemical bond enables us to glean the order and complexity of how the basic building blocks, atoms, interact to form all substances. This language has enabled the broad enterprise of chemistry to rapidly develop and flourish into the central science.
Chemical & Engineering News
ISSN 0009-2347
Copyright © American Chemical Society
Leila (Fri May 30 08:30:33 EDT 2014)
This was a totally brilliant idea. I gained a whole lot from the incredibly different-seeming (to me) explanations from the different scientists. Thanks.
S.Senthilkumar (Tue Jul 22 12:50:53 EDT 2014)
Yes... this is what I want to know about chemical bonds from different souls. Excellent job.. Especially K.N.Houk thought was short and perfect.
Mark McAdon (Thu Jan 15 17:12:10 EST 2015)
Among friends, I would say that chemical bonds are theoretical constructs that are very useful in teaching and understanding about materials, chemicals, and chemical reactions. However, chemical bonds are sort of like Santa Claus - - they "exist" but they are not real.
For the chemists, I would revert to Linus Pauling's book, The Nature of the Chemical Bond, page 6. Linus Pauling is our Saint Nicholas - - the patron saint of chemists.
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f0feacc1021069b2 | atomic theory
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atomic theory, ancient philosophical speculation that all things can be accounted for by innumerable combinations of hard, small, indivisible particles (called atoms) of various sizes but of the same basic material; or the modern scientific theory of matter according to which the chemical elements that combine to form the great variety of substances consist themselves of aggregations of similar subunits (atoms) possessing nuclear and electron substructure characteristic of each element. The ancient atomic theory was proposed in the 5th century bc by the Greek philosophers Leucippus and Democritus and was revived in the 1st century bc by the Roman philosopher and poet Lucretius. The modern atomic theory, which has undergone continuous refinement, began to flourish at the beginning of the 19th century with the work of the English chemist John Dalton. The experiments of the British physicist Ernest Rutherford in the early 20th century on the scattering of alpha particles from a thin gold foil established the Rutherford atomic model of an atom as consisting of a central, positively charged nucleus containing nearly all the mass and surrounded by a cloud of negatively charged planetlike electrons.
With the advent of quantum mechanics and the Schrödinger equation in the 1920s, atomic theory became a precise mathematical science. Austrian physicist Erwin Schrödinger devised a partial differential equation for the quantum dynamics of atomic electrons, including the electrostatic repulsion of all the negatively charged electrons from each other and their attraction to the positively charged nucleus. The equation can be solved exactly for an atom containing only a single electron (hydrogen), and very close approximations can be found for atoms containing two or three electrons (helium and lithium). To the extent that the Schrödinger equation can be solved for more-complex cases, atomic theory is capable of predicting from first principles the properties of all atoms and their interactions. The recent availability of high-speed supercomputers to solve the Schrödinger equation has made possible accurate calculations of properties for atoms and molecules with ever larger numbers of electrons. Precise agreement with experiment is obtained if small corrections due to the effects of the theory of special relativity and quantum electrodynamics are also included.
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e9f12da03bc9bbff | bra intrusts
Bra-ket notation
Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets (chevrons) and vertical bars. It can also be used to denote abstract vectors and linear functionals in pure mathematics. It is so called because the inner product (or dot product) of two states is denoted by a bracket, langlephi|psirangle, consisting of a left part, langlephi|, called the bra, and a right part, |psirangle, called the ket. The notation was invented by Paul Dirac, and is also known as Dirac notation.
Bras and kets
Most common use: Quantum mechanics
In quantum mechanics, the state of a physical system is identified with a ray in a complex separable Hilbert space, mathcal{H}, or, equivalently, by a point in the projective Hilbert space of the system. Each vector in the ray is called a "ket" and written as |psirangle, which would be read as "ket psi ". (The ψ can be replaced by any symbols, letters, numbers, or even words—whatever serves as a convenient label for the ket.) The ket can be viewed as a column vector and (given a basis for the Hilbert space) written out in components,
|psirangle = (c_0, c_1, c_2, ...)^T,
when the considered Hilbert space is finite-dimensional. In infinite-dimensional spaces there are infinitely many components and the ket may be written in complex function notation, by prepending it with a bra (see below). For example,
langle x|psirangle = psi(x) = c e^{- ikx}.
Every ket |psirangle has a dual bra, written as langlepsi|. For example, the bra corresponding to the ket |psirangle above would be the row vector
langlepsi| = (c_0^*, c_1^*, c_2^*, ...).
This is a continuous linear functional from mathcal H to the complex numbers mathbb{C}, defined by:
langlepsi| : mathcal H to mathbb{C}: langle psi | left(|rhorangle right) = operatorname{IP}left(|psirangle ;,; |rhorangle right) for all kets |rhorangle
where operatorname{IP}(cdot , cdot ) denotes the inner product defined on the Hilbert space. Here an advantage of the bra-ket notation becomes clear: when we drop the parentheses (as is common with linear functionals) and meld the bars together we get langlepsi|rhorangle, which is common notation for an inner product in a Hilbert space. This combination of a bra with a ket to form a complex number is called a bra-ket or bracket.
The bra is simply the conjugate transpose (also called the Hermitian conjugate) of the ket and vice versa. The notation is justified by the Riesz representation theorem, which states that a Hilbert space and its dual space are isometrically conjugate isomorphic. Thus, each bra corresponds to exactly one ket, and vice versa. More precisely, if J: mathcal H rightarrow mathcal H^* is the Riesz isomorphism between mathcal H and its dual space, then forall phi in mathcal H: ; langlephi| = J(|phirangle).
Note that this only applies to states that are actually vectors in the Hilbert space. Non-normalizable states, such as those whose wavefunctions are Dirac delta functions or infinite plane waves, do not technically belong to the Hilbert space. So if such a state is written as a ket, it will not have a corresponding bra according to the above definition. This problem can be dealt with in either of two ways. First, since all physical quantum states are normalizable, one can carefully avoid non-normalizable states. Alternatively, the underlying theory can be modified and generalized to accommodate such states, as in the Gelfand-Naimark-Segal construction or rigged Hilbert spaces. In fact, physicists routinely use bra-ket notation for non-normalizable states, taking the second approach either implicitly or explicitly.
In quantum mechanics the expression langlephi|psirangle (mathematically: the coefficient for the projection of psi! onto phi!) is typically interpreted as the probability amplitude for the state psi! to collapse into the state phi.!
More general uses
Bra-ket notation can be used even if the vector space is not a Hilbert space. In any Banach space B, the vectors may be notated by kets and the continuous linear functionals by bras. Over any vector space without topology, we may also notate the vectors by kets and the linear functionals by bras. In these more general contexts, the bracket does not have the meaning of an inner product, because the Riesz representation theorem does not apply.
Linear operators
If A : HH is a linear operator, we can apply A to the ket |psirangle to obtain the ket (A|psirangle). Linear operators are ubiquitous in the theory of quantum mechanics. For example, observable physical quantities are represented by self-adjoint operators, such as energy or momentum, whereas transformative processes are represented by unitary linear operators such as rotation or the progression of time.
Operators can also be viewed as acting on bras from the right hand side. Composing the bra langlephi| with the operator A results in the bra bigg(langlephi|Abigg), defined as a linear functional on H by the rule
bigg(langlephi|Abigg) ; |psirangle = langlephi| ; bigg(A|psiranglebigg).
This expression is commonly written as
If the same state vector appears on both bra and ket side, this expression gives the expectation value, or mean or average value, of the observable represented by operator A for the physical system in the state |psirangle, written as
A convenient way to define linear operators on H is given by the outer product: if langlephi| is a bra and |psirangle is a ket, the outer product
|phirang lang psi|
denotes the rank-one operator that maps the ket |rhorangle to the ket |phiranglelanglepsi|rhorangle (where langlepsi|rhorangle is a scalar multiplying the vector |phirangle). One of the uses of the outer product is to construct projection operators. Given a ket |psirangle of norm 1, the orthogonal projection onto the subspace spanned by |psirangle is
Just as kets and bras can be transformed into each other (making |psirangle into langlepsi|) the element from the dual space corresponding with A|psirangle is langle psi | A^dagger where A denotes the Hermitian conjugate of the operator A.
It is usually taken as a postulate or axiom of quantum mechanics, that any operator corresponding to an observable quantity (shortly called observable) is self-adjoint, that is, it satisfies A = A. Then the identity
langle psi | A | psi rangle^star = langle psi |A^dagger |psi rangle = langle psi | A | psi rangle
holds (for the first equality, use the scalar product's conjugate symmetry and the conversion rule from the preceding paragraph). This implies that expectation values of observables are real.
Bra-ket notation was designed to facilitate the formal manipulation of linear-algebraic expressions. Some of the properties that allow this manipulation are listed herein. In what follows, c1 and c2 denote arbitrary complex numbers, c* denotes the complex conjugate of c, A and B denote arbitrary linear operators, and these properties are to hold for any choice of bras and kets.
• Since bras are linear functionals,
langlephi| ; bigg(c_1|psi_1rangle + c_2|psi_2rangle bigg) = c_1langlephi|psi_1rangle + c_2langlephi|psi_2rangle.
• By the definition of addition and scalar multiplication of linear functionals in the dual space,
bigg(c_1 langlephi_1| + c_2 langlephi_2|bigg) ; |psirangle = c_1 langlephi_1|psirangle + c_2langlephi_2|psirangle.
Given any expression involving complex numbers, bras, kets, inner products, outer products, and/or linear operators (but not addition), written in bra-ket notation, the parenthetical groupings do not matter (i.e., the associative property holds). For example:
lang psi| (A |phirang) = (lang psi|A)|phirang
(A|psirang)lang phi| = A(|psirang lang phi|)
and so forth. The expressions can thus be written, unambiguously, with no parentheses whatsoever. Note that the associative property does not hold for expressions that include non-linear operators, such as the antilinear time reversal operator in physics.
Hermitian conjugation
Bra-ket notation makes it particularly easy to compute the Hermitian conjugate (also called dagger, and denoted †) of expressions. The formal rules are:
• The Hermitian conjugate of a bra is the corresponding ket, and vice-versa.
• The Hermitian conjugate of a complex number is its complex conjugate.
• The Hermitian conjugate of the Hermitian conjugate of anything (linear operators, bras, kets, numbers) is itself—i.e.,
• Given any combination of complex numbers, bras, kets, inner products, outer products, and/or linear operators, written in bra-ket notation, its Hermitian conjugate can be computed by reversing the order of the components, and taking the Hermitian conjugate of each.
These rules are sufficient to formally write the Hermitian conjugate of any such expression; some examples are as follows:
• Kets:
left(c_1|psi_1rangle + c_2|psi_2rangleright)^dagger = c_1^* langlepsi_1| + c_2^* langlepsi_2|.
• Inner products:
lang phi | psi rang^* = lang psi|phirang
• Matrix elements:
lang phi| A | psi rang^* = lang psi | A^dagger |phi rang
lang phi| A^dagger B^dagger | psi rang^* = lang psi | BA |phi rang
• Outer products:
left((c_1|phi_1ranglang psi_1|) + (c_2|phi_2ranglangpsi_2|)right)^dagger = (c_1^* |psi_1ranglang phi_1|) + (c_2^*|psi_2ranglangphi_2|)
Composite bras and kets
Two Hilbert spaces V and W may form a third space V otimes W by a tensor product. In quantum mechanics, this is used for describing composite systems. If a system is composed of two subsystems described in V and W respectively, then the Hilbert space of the entire system is the tensor product of the two spaces. (The exception to this is if the subsystems are actually identical particles. In that case, the situation is a little more complicated.)
If |psirangle is a ket in V and |phirangle is a ket in W, the direct product of the two kets is a ket in V otimes W. This is written variously as
|psirangle|phirangle or |psirangle otimes |phirangle or |psi phirangle or |psi ,phirangle.
Representations in terms of bras and kets
In quantum mechanics, it is often convenient to work with the projections of state vectors onto a particular basis, rather than the vectors themselves. The reason is that the former are simply complex numbers, and can be formulated in terms of partial differential equations (see, for example, the derivation of the position-basis Schrödinger equation). This process is very similar to the use of coordinate vectors in linear algebra.
For instance, the Hilbert space of a zero-spin point particle is spanned by a position basis lbrace|mathbf{x}ranglerbrace, where the label x extends over the set of position vectors. Starting from any ket |psirangle in this Hilbert space, we can define a complex scalar function of x, known as a wavefunction:
psi(mathbf{x}) stackrel{text{def}}{=} lang mathbf{x}|psirang.
It is then customary to define linear operators acting on wavefunctions in terms of linear operators acting on kets, by
A psi(mathbf{x}) stackrel{text{def}}{=} lang mathbf{x}|A|psirang.
For instance, the momentum operator p has the following form:
mathbf{p} psi(mathbf{x}) stackrel{text{def}}{=} lang mathbf{x} |mathbf{p}|psirang = - i hbar nabla psi(x).
One occasionally encounters an expression like
- i hbar nabla |psirang.
This is something of an abuse of notation, though a fairly common one. The differential operator must be understood to be an abstract operator, acting on kets, that has the effect of differentiating wavefunctions once the expression is projected into the position basis:
- i hbar nabla langmathbf{x}|psirang.
For further details, see rigged Hilbert space.
The unit operator
Consider a complete orthonormal system (basis), { e_i | i in mathbb{N} }, for a Hilbert space H, with respect to the norm from an inner product langlecdot,cdotrangle. From basic functional analysis we know that any ket |psirangle can be written as
|psirangle = sum_{i in mathbb{N}} langle e_i | psi rangle | e_i rangle,
with langlecdot|cdotrangle the inner product on the Hilbert space. From the commutativity of kets with (complex) scalars now follows that
sum_{i in mathbb{N}} | e_i rangle langle e_i | = hat{1}
must be the unit operator, which sends each vector to itself. This can be inserted in any expression without affecting its value, for example
langle v | w rangle = langle v | sum_{i in mathbb{N}} | e_i rangle langle e_i | w rangle = langle v | sum_{i in mathbb{N}} | e_i rangle langle e_i | sum_{j in mathbb{N}} | e_j rangle langle e_j | w rangle = langle v | e_i rangle langle e_i | e_j rangle langle e_j | w rangle
where in the last identity Einstein summation convention has been used.
In quantum mechanics it often occurs that little or no information about the inner product langlepsi|phirangle of two arbitrary (state) kets is present, while it is possible to say something about the expansion coefficients langlepsi|e_irangle = langle e_i|psirangle^* and langle e_i|phirangle of those vectors with respect to a chosen (orthonormalized) basis. In this case it is particularly useful to insert the unit operator into the bracket one time or more.
Notation used by mathematicians
The object physicists are considering when using the "bra-ket" notation is a Hilbert space (a complete inner product space).
Let mathcal{H} be a Hilbert space and hinmathcal{H} . What physicists would denote as |hrangle is the vector itself. That is
(|hrangle)in mathcal{H} .
Let mathcal{H}^* be the dual space of mathcal{H} . This is the space of linear functionals on mathcal{H}. The isomorphism Phi:mathcal{H}tomathcal{H}^* is defined by Phi(h) = phi_h where for all ginmathcal{H} we have
phi_h(g) = mbox{IP}(h,g) = (h,g) = langle h,g rangle = langle h|g rangle ,
mbox{IP}(cdot,cdot), (cdot,cdot),langle cdot,cdot rangle, langle cdot | cdot rangle
are just different notations for expressing an inner product between two elements in a Hilbert space (or for the first three, in any inner product space). Notational confusion arises when identifying phi_h and g with langle h | and |g rangle respectively. This is because of literal symbolic substitutions. Let phi_h = H = langle h| and g=G=|grangle . This gives
phi_h(g) = H(g) = H(G)=langle h|(G) = langle h|(
One ignores the parentheses and removes the double bars. Some properties of this notation are convenient since we are dealing with linear operators and composition acts like a ring multiplication.
References and notes
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3ec5a1068e0380c7 | Friday, November 30, 2007 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Teppo Mattsson: Dark energy as a mirage
Is cosmological constant zero, after all? (PDF)
If this guy were right, the required changes would be tiny. Dark energy would go away, 90% of the mass would be dark matter, 10% would be baryonic, and the age of the Universe would jump to 14.8 billion years.
He says that as you observe distant objects, the Universe is getting emptier, due to a hierarchical clumping of matter, and in these emptier regions, the Hubble constant increases, mimicking the accelerating effects of the cosmological constant.
2007 Atlantic hurricane season: below forecasts
The 2007 Atlantic hurricane season that officially ends today (and it is no longer possible for an additional named storm to occur in time) was stronger than the 2006 Atlantic hurricane season but much weaker than what we saw two years ago.
Current hurricane info (bookmark)
In 2005, 2006, 2007, the total accumulated cyclone energy (ACE) was 248, 78.5, 67.5. You see a clear decreasing trend here: ACE even indicates that 2007 was even weaker than 2006. The median ACE index and the mean ACE index for the 1951-2005 period are 89.5 and 102.3, respectively. It means that according to ACE, both 2006 and 2007 were below the average.
Figure 1: H. Dean, the angriest hurricane of the season. ACE: over 33. Dean was the first male category 5 hurricane after four previous female ones - Emily, Katrina, Rita, Wilma - in 2005. But the surprise is not so overwhelming because Dean is really a self-described metrosexual. ;-)
The total damage was about USD 130 billion, 0.5 billion, 4 billion in 2005, 2006, 2007.
Klaus: Africa, don't rely on foreign aid
It has been ten years since the so-called Second Sarajevo Assassination, a forced resignation of the Czech prime minister Václav Klaus during his visit to Sarajevo that was justified by a financial scandal, one that was later demonstrated to be bogus.
The times are different now. President Klaus who was just nominated for the next term (the only candidate so far, despite many people who are desperately seeking Antiklaus) delivered his
speech in Lagos, Nigeria.
Klaus advised them to rely on their comparative advantages rather than foreign aid that is never really free, whose magnitude and importance is always overstated, and whose structure is always determined by the interest of donors. East Germany was used as a bad example of aid - a whole GDP of the Czech Republic was pumped to East Germany every year and they didn't make more progress than the Czech Republic that was getting no aid. Aid often makes actual useful developments impossible.
Also, the third world should determine its own optimal environmental, social, safety, labor, hygienic, and other standards, rather than to listen to someone else. Moreover, the best thing that the first world could do for Africa is to open its markets. You should see what Nigerian newspapers and their commenters write about the speech. Klaus's speech is accepted kind of enthusiastically and they seem to understand the main points and their power.
Hep-th papers on Friday
Below you find descriptions of the 17 today's papers on hep-th.
Bert Schroer dedicates 50+ pages to what he calls "significant conceptual differences" between quantum mechanics and quantum field theory. Needless to say, quantum field theory is a standard example of a quantum mechanical theory and the difference between quantum field theory and other quantum mechanical theories is purely dynamical, not conceptual. What defines a quantum theory are the postulates of quantum mechanics (Hilbert space, observables given by linear operators, evolution given by a unitary operator, probabilities given by expectation values of projection operators) that hold everywhere, including quantum field theory, plus a choice of dynamics on the Hilbert space (e.g. a Hamiltonian) that depends on a theory. Quantum field theory is thus just another example. Also, all the features of the uncertainty principle and localization that hold in non-relativistic quantum mechanics of particles may be derived from quantum field theory in the appropriate limit(s). The paper is a nonsensical stream of philosophical misinterpretations, misconceptions borrowed from the "real" algebraic quantum field theory, and buzzwords.
Alikram Aliev shows that the "g=2" gyromagnetic ratio for rotating charged black holes is surprisingly universal in general relativity, regardless of the asymptotic geometry, its curvature etc.: the value remarkably coincides with the value for the electron calculated from the Dirac equation. (Non-relativistic gyroscopes have "g=1".) It becomes "g=4" when two angular momenta coincide.
Arthur Sergyeyev and Pavel Krtouš study the Klein-Gordon equation on a multi-dimensional Kerr-NUT-dS or -AdS background. They find a complete set of many commuting angular-momentum-like operators and prove that they commute. This is done purely in the first-quantized setup because the second-quantized Hilbert space of course can't have a finite complete set of commuting observables. Moreover, it can't really be interpreted in the quantum fashion because the Klein-Gordon equation (or any other relativistic equation) can't really be used as a first-quantized physical Schrödinger equation. So to summarize, it is purely a work in general relativity & classical field theory, the word "operator" should be interpreted as nothing else than a mathematical (differential) operator and it is somewhat confusing why the paper is on hep-th.
Noboru Nakanishi seems to be unfamiliar with conventional renormalization and is troubled by the quadratic divergences of the Standard Model. So he rediscovers the Pauli-Villars regularization and interprets the wrong sign as a consequence of wrong statistics of these new complex fields, rather than a negative sign of the kinetic term. The author doesn't seem to be at home with quantum field theory, as highlighted e.g. by the fact that he or she doesn't use the term "Standard Model". He only cites three (not too relevant) papers besides his own and Pauli & Villars are not among them.
James Hartle, Stephen Hawking, and Thomas Hertog offer a possible solution to a problem of the Hartle-Hawking no-boundary proposal: that it predicts a very short inflation. They show a gauge-invariant, serious, "non-anthropic" calculation whose result is to add an additional factor of exp(3N) where N is the number of e-foldings to the probability of various classical solutions: similar factors may have appeared as results of anthropic hand-waving (or ingenious anthropic prophecies, if you wish). In the relevant physical context of a stringy-like landscape, a lot of inflation, starting near a de Sitter geometry at the saddle point, then follows. Surely one of the most interesting papers today.
Ee Chang-Young, Hoil Kim, and Hiroaki Nakajima construct a matrix representation of a super Heisenberg group that occurred in a stringy two-dimensional N=(2,2) deformed superspace describing D-branes on background Ramond-Ramond fields. Just like a background B-field forces bosonic coordinates to be non-commuting, a background RR-field makes the supercoordinates non-anti-commuting even though the math and limiting procedures are somewhat less clear.
Suresh Nampuri, Prasanta K.Tripathy, Sandip P. Trivedi ask, along the lines of "Dualities versus singularities", whether T-dualities - in their case those of type IIA on K3 times a two-torus - are enough to refractionalize a black hole with large D0-D4 or D0-D6 charges and bring those charges to Cardy's limit. The answer is "Yes" for non-supersymmetric black holes and "No" for generic supersymmetric black holes. The "Yes" answer might imply that the entropy of all extremal but non-supersymmetric black holes may be calculated.
Arzumanyan and 4 more Armenian authors compute the radiation from a charge that moves along a helix. Given the fact that the typical energy they consider is 10 MeV and the topic is more relevant for condensed matter physics (dielectric materials are needed) or something else, I don't think that the otherwise interesting paper should have appeared in a high-energy archive.
Cristina Zambon attempts to incorporate the so-called jump-defect, known from the sine-Gordon model, to the affine Toda field theories which are a complementary integrable description of similar physical systems.
B.M. Zupnik studies harmonic superspaces for three-dimensional theories. Harmonic superspace is a superspace that, in addition to anticommuting coordinates, contains additional bosonic coordinates spanning quotients of groups. His particular interest is in a non-Abelian Chern-Simons theory whose manifest supersymmetry from the superspace is N=5 but is extended to N=6.
Juraj Boháčik and Peter Prešnajder study the zero-spatial-dimensional anharmonic oscillator with a quartic interaction term using non-perturbative methods due to Gelfand and Yaglom. They offer a comprehensible proof of an equation that specifies corrections for such an oscillator. Again, it is interesting but not directly relevant for high-energy physicists.
A.T. Avelar et al. study topologically unusual soliton solutions to models with a single real scalar field. Their potential is a combination of (mostly fractional) powers of the field and the topologies include lumps with flat plateux at the top and lumps on top of another lump. The fact that they mention that the results may have applications to non-linear science highlights that this should probably not be a hep-th paper.
Kwan Sik Jeong studies supersymmetry breaking in KKLT-like models. It is being assumed that the source of the breaking is in a hidden, sequestered sector. The author argues that the impact of this breaking on the visible sector can be summarized in an F-term expectation value that is universal. Ratios of vevs and logarithms of ratios of various mass scales are the only thing that appear in the ultimate key formula.
Albion Lawrence, Tobias Sander, Michael B. Schulz, Brian Wecht look at type IIB string theory on a Calabi-Yau three-fold. Their aim is to find the spectrum of auxiliary fields which is not exactly a physically unique, objective, physical question but a particular natural answer may be useful. Indeed, it is useful and they argue that the expectation values of these auxiliary fields lead to deformed CFTs that add either the H-field (a field strength for the NS-NS B-field) or an SU(3) x SU(3) structure (different tangent bundles for left-movers and right-movers). Once these things are nonzero, generic vacua are non-geometric globally (although probably geometric locally, because of their starting point), a worldsheet argument suggests. Mirror symmetry is argued to hold beyond the (2,2) worldsheet supersymmetry and worldsheet instantons are presented as more important animals when their fluxes are turned on. One of the most interesting papers.
Niklas Beisert, Denis Erkal present the spin chains arising in the AdS/CFT correspondence as very special spin chains with non-nearest-neighbor interactions that nevertheless preserve the integrability of a simpler spin chain with nearest-neighbor interactions only. They can't prove the full integrability for the interesting cases that occur in string theory but they can do so for a seemingly similar gl(n) spin chain model with longer-range interactions. The proof is technically based on checking the Serre relations for a Yangian generator. A very interesting paper.
Borun D. Chowdhury and Samir D. Mathur study the fuzzball model of black holes. Now they look at radiation by these monsters. They derive the classical radiation emitted by these classical solutions (not suppressed by hbar) by combining the Hawking radiation into (very many) unstable modes of their individual geometries. I kind of feel that this was guaranteed to work because of the standard limiting relationships between classical and quantum systems but don't worry. They argue that this means that the information is manifestly preserved in the supergravity degrees of freedom. Well, I don't have any problem with the statement that these fuzzball geometries preserve the information or may behave as ordinary horizon-free solutions. They are ordinary, after all. What is missing for me is a proof that these fuzzball geometries conspire to behave like ordinary black holes in contexts where I want to believe that the black hole description is correct, e.g. after a collapse of a star. Also, I don't see any proof that all the relevant degrees of freedom that store the information about a black hole are geometric in character.
Chris Hull and R.A. Reid-Edwards discuss similar structures as Albion Lawrence et al. above, namely non-geometric compactifications. If the monodromy in such a background is taken from the T-duality group, such a background may be made similar to a geometric one by adding the T-dual coordinates besides the normal coordinates at each point. This has been discussed many times, even on this blog, and Hitchin was the most well-known guy who has advocated this viewpoint. Hull and Reid-Edwards think that one can also construct backgrounds that are non-geometric even locally, by thinking about the double as a Drinfeld double. I don't see this statement justified in the paper.
Thursday, November 29, 2007 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Al Gore & Pat Robertson
Tim Slagle, a political satirist, offers not only some comparisons of Gore and Robertson but also a funny albeit unflattering outsider's viewpoint on scientists in general.
Hat tip: Antagoniste
Bonus: You should see John Brignell's list of 600+ evil things caused by global warming, alphabetically sorted and including links to the mainstream media where the individual catastrophes are described. ;-)
The list works like a dictionary. Invent your favorite problem, for example salmonella, and find the word. Click it and you will see a proof that salmonella or anything else is caused by global warming. :-)
Jim Simons & string theory
Bloomberg & The International Herald Tribune
writes about some daily activities of Jim Simons. He divides his time between string theory, autism, and math education. Cumrun Vafa who is, much like Simons himself, a wizard is an important channel into string theory. And by the way, Simons also doubled his assets during the last year.
Paul Davies: Taking science on faith
Update: a list of wrong assumptions about science was added at the end of this essay. The article written on 11/25 was moved to the top as the most discussed recent text.
As far as I can say, The New York Times remain by far the best source of science news and opinions among the English-speaking newspapers.
Paul Davies' op-ed
meditates about the controversial question concerning the difference between science and religion. I agree with most things he writes. He starts with the idealized picture that many people believe to be true - namely a picture in which science and religion are sharply separated. Skepticism belongs to science while blind belief belongs to religion.
He instantly adds his main thesis that this picture is an oversimplification because science has its belief system, too. The first thing that a typical scientist - and especially a theoretical physicist - believes is that the questions he is trying to answer have coherent, understandable, and universally valid explanation.
Microsoft.NET service pack 1: update failure
If you failed to install an update for your Microsoft Windows yesterday, getting the error "0x80070002" or "80070002", you should know the following facts.
Windows Vista mostly runs the 3.0 version of the .NET framework but the version 1.1 usually also exists because many older applications use it. Even the 1.1 version is fully compatible with Windows Vista. The service pack 1 for this .NET framework 1.1 has been around since 2004. The only new development is that Microsoft has confirmed that the patch is Vista-compatible and published it via its update systems.
If you try to follow some automatic hints, e.g. to erase the Windows Update temporary files and download the patch from the scratch, you will fail again. So avoid this step. Instead, download and install the 10.2 MB service pack 1 manually:
Microsoft.NET framework 1.1 Service Pack 1: direct download
Disable your antivirus software and choose the language of your operating system. Neither of these two steps is probably necessary but I did both. After you are finished, click Yes to restart your computer. As soon as your service pack 1 is installed, Windows Update will be able to figure it out and it will describe the installation of this patch as "success" after you try to reinstall the available updates. It will offer you appropriate small patches later, for example KB 929729, an additional update of .NET 1.1 SP1 itself, will occur within a day.
Wednesday, November 28, 2007 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Enrico Fermi: an anniversary
Enrico Fermi (9/29/1901 in Rome, Italy - 11/28/1954 because of stomach cancer) was most likely the second most important Italian physicist after Galileo Galilei (check this list).
When he was 17 and he was entering the college in Pisa, he wrote an essay about a Fourier-series analysis of solutions to the partial differential equation describing... waves on a string. The examiner interviewed Fermi and determined that the essay would have been good enough for a PhD in Pisa.
The young 25-year-old professor did his most important purely theoretical work in fundamental physics in 1926 when he wrote the paper about the Fermi-Dirac statistics. The rest of his research life was dedicated to radioactivity. In 1938, right after the war started, he was wittily given a Nobel prize for induced radioactivity. ;-) But he was just getting started.
Did everyone accept Fermi's statements from the very beginning? You may guess what the answer is with these great minds. When he submitted his famous paper on beta decay to Nature, the editor rejected it because "it contained speculations which were too remote from reality". This is what the lagging, inferior minds say about cutting-edge research in theoretical physics in most cases, even today. The paper using the new term "neutrino" that Fermi invented (but Pauli got the idea in 1931) was therefore published in German and Italian before it appeared in English.
As Wikipedia argues, he never forgot this experience of being ahead of his time. His protégés were therefore told "to never be first; to try to be second". James DNA Watson was preaching pretty much the same thing to his protégés.
Nature eventually published his report on beta decay in January 1939. Once he left Italy for Columbia University at the very same time, in order to save his Jewish wife, he reproduced some other people's fission experiments. Fermi moved to Chicago where he built the first nuclear pile, a primitive nuclear reactor that went critical in December 1942 in a "squash court"; Russians translated the location as "pumpkin field". :-) Every step was carefully and brilliantly planned.
This wisdom and experiments were also useful during the Manhattan project in which Fermi, who became a U.S. citizen in 1944, assisted. When you summarize his work on nuclear technology, you will see that Fermi was the most practically talented man among the great 20th century theoretical physicists and the impact of his work makes the comments about his work being detached from reality doubly ludicrous.
He had to think, write, and do a lot to realize all his achievements associated with nuclear physics. But if we measure the impact on public's perception of scientifically loaded questions per word, his most influential results are three words from the 1950s: "Where are they?" The Fermi "paradox" shows that it is rather unlikely for the Universe to be filled with too many very advanced civilizations.
I have talked to a student of Fermi. She admired him and she was certainly not the only one.
Tuesday, November 27, 2007 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Most Germans are Celts
The Google News are now available in Czech:
Czech Google News
I am convinced that Google News are better than any individual source of news. And there are many special news in the Czech edition, too.
For example, we learn that Patrick Moore, a co-father of Greenpeace, met with President Klaus to support nuclear energy, industrial consumption of wood, genetically modified crops, chemical compounds protecting people against fires, industrial production of salmons, and our fight against the global warming religion.
This is how a true environmentalist should look like. The present generation should treat Patrick Moore as their role model. Incidentally, Patrick Moore was immensely influenced - and led to the green movement - by a Czech scientist and politician named Mr Vladimír Krajina (1905-1993) who emigrated to Canada in 1948. If you care, the word "Krajina" means "landscape". ;-)
Scafetta & West: Climate phenomenology
The Sun is a major player
Ada Lovelace died 155 years ago
Augusta Ada King née Byron, Countess of Lovelace, died of medicinal bloodletting associated with uterine cancer on November 27th, 1852, at the age of 37. She is considered to be the first programmer in the world.
Her father, the poet Lord Byron, called her "the princess of parallelograms", Charles Babbage called her "the enchantress of numbers", and she was one of the most mathematically gifted women of the history. She wrote the first computer program for the "Analytical Engine", a mechanical computer designed by Charles Babbage that was never built. The program was computing the coefficients of the expansion of the closed string vacuum in string field theory in the Schnabl gauge, the so-called Bernoulli numbers.
Some science historians argue that the program was written by Babbage himself and Ada hasn't really mastered some basic maths but the story in which her contributions were original sounds more sexy. Moreover, I find many champions of this viewpoint untrustworthy. On the other hand, it is also plausible that people like Ada Lovelace contributed to Babbage's design of the computer itself.
Alexei Zamolodchikov died
... see Asymptotia.
Monday, November 26, 2007 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Andrew Revkin asks James Hansen about holocaust
On his blog, Andrew Revkin from the New York Times discussed Hansen's description of coal-burning power plants as extermination camps.
Figure 1: Is global warming the new holocaust?
He reviews the very same formulations as we did. At the end of his text, Revkin reveals five questions that he has sent to Hansen:
1. Do you care whether holocaust survivors are offended?
2. Have you received complaints/support from any?
3. Is such a metaphor necessary to change the people?
4. Is it true that such analogies polarize and paralyze any discussion?
5. Who is the actual victim of your holocaust?
It might be interesting - and probably shocking - to see Hansen's answers to Revkin's questions. I would expect something like that:
Lucie Vondráčková: Strach
I feel that Slovak music remains overrepresented on this blog so let me offer a purely Czech song this time.
Ms Lucie Vondráčková (1980) is the niece of Mrs Helena Vondráčková, a top-tier Czechoslovak and Czech pop-music musician. Lucie's song "Strach" (Fear) remains at the top of many hitparades for much of this year (again).
This post with English lyrics
Fantastic journey: scales in Powerpoint
Has anyone seen the English version of this presentation?
Fantastic journey
The presentation includes typical pictures of various length scales from clusters of galaxies down to the size of quarks, including a Czech description.
Saturday, November 24, 2007 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Hansen: power plants = extermination camps
Source of correspondence (PDF)
In his recent testimony to the Iowa Utilities Board, Rev. James Hansen argued that the construction of a new coal-based power plant is equivalent to the holocaust. The trains that bring coal to the new power plant are nothing else than the death trains that were moving the Jews to extermination camps:
Kraig Naasz, the president and CEO of the U.S. National Mining Association, suggested that this comparison was both repellant and preposterous. It trivializes the suffering of millions of people while it irrationally evaluates the actual reasons behind various climate phenomena: one additional power plant in the U.S. surely can be no "tipping point" especially when China builds a new plant every week.
Figure 1: Kraig Naasz is the second man from the right. How many chestnut trees (and power plants) has Mr Hansen planted? ;-)
Naasz recommended Hansen to apologize to the hard-working men and women in the coal mining and railroad industries. What did Mr Hansen do? You may guess.
Sheldon Glashow: five minutes of video
You may watch four one-minute clips with Sheldon Glashow so that this co-father of the electroweak theory doesn't feel discriminated against: ;-)
1. The unification of the large and the small
2. The origin of the Universe
3. The four forces of Nature
4. Early work on unified theory
Alternative place to find the videos: Honeywellscience. On December 5th, 24:00 EST, right after Glashow's 75th birthday, there will be 90-minute-long videopodcast at
In the four videos above, you will probably not learn too much but there is a story in the last part. Julian Schwinger told Glashow, one of ten annoying students who wanted projects, to go and construct a unified electroweak theory. At the end, Glashow needed a help from his high-school buddy.
powered by ODEO Audio 1: Glashow describes the LHC as a possible end of particle physics. ;-) But he hopes for the best possible outcome: absolute confusion.
Amazon Kindle: an electronic book
People have been talking about electronic replacements for books for quite some time. But Amazon may be the first company to offer a viable realization of the idea: Amazon Kindle. It can't read PDF files and your right to read the e-books probably expire in a few years - and consumers give it 2.5 stars only - but a zeroth approximation for USD 400 is here. Click the picture to get to
Telegraph: Cosmologists are killing the Universe
Steve Heston has posted a comment telling us that
The Telegraph
argues that "mankind is shortening the universe's life" by observing the cosmological constant, bringing us closer to the doom. As soon as I saw his message, I would have accepted a 10:1 bet that the author had to be Roger Highfield because it is very unlikely that there exist two or more breathtakingly unreasonable people of this magnitude in the Telegraph. Of course, I would have won the bet.
Roger Highfield has already "informed" his readers that Einstein may have started the rot (the word "rot" means modern theoretical physics) and that Garrett Lisi has found a theory of everything.
The names of the two sensation-thirsty alternative "physicists" who wrote this complete absurdity into their preprint are
Lawrence Krauss, James Dent.
These "scientists" offer far too many basic misunderstandings of cosmology and quantum mechanics to discuss all of them. For example, without a glimpse of a rational reason and in contradiction with all existing theories and scenarios, anthropic or otherwise, they proclaim the typical lifetime of our Universe to be comparable to the Hubble time.
Leave string theory alone
Before you watch the video below, you should know Chris Crocker's famous, 13-million-visit defense of Britney Spears against the journalistic hyenes. Well, doesn't string theory need its own Chris Crocker? Yes, it does. Here it is:
But scooter, you should have been a bit more passionate! ;-) But at least, I am happy that you will now deal with all those Woits and all this crap. What a relief.
Radiation is not too deadly
1. Hiroshima in 1945
3. Chernobyl 1986
Geoengineering: a discussion
I am kind of interested what various readers think about geoengineering. Some of the questions are:
1. If you had the tools to achieve anything you want, what would be your optimal temperature and the concentration of basic compounds such as CO2? Is it higher or lower than the present values? Why do you think it is optimal?
2. How would you estimate the economic value of such an improvement? Describe your method and the result.
3. Would you find it OK to significantly change the composition of the ocean, e.g. by adding a lot of iron?
4. Would you find it OK to significantly change the composition of the atmosphere by sulphur oxides or particulate matter?
5. In the context of cloud-seeding, if you could pre-program when it is cloudy, what would be your schedule? You don't have to think about cooling the planet only - but about your general comfort.
6. If these global changes could be made much more rapidly than the changes that occur nowadays, or if someone installed some mirrors in space that could increase or decrease the amount of solar radiation, how would you regulate what is going on? What role would you assign to democracy or markets?
Joel Shapiro on the birth of string theory
Joel Shapiro, a very interesting early worker on string theory whom I know well from Rutgers (also as an excellent teacher, by the way), remembers the early days of string theory and it's a lot of fun.
At that time, the four interactions were really separated, both physiologically as well as sociologically. Shapiro was interested in "unified field theory" but his advisor never told him to study general relativity. ;-)
Instead, Shapiro started as a hadron phenomenologist. He improved the old Veneziano amplitude a bit and started to draw diagrams. Sy Pasternack insisted that "pomeronchukon" should be used instead of "pomeron" in his Physical Review. Pasternack should have cared about fixing his own silly name instead of screwing others. ;-)
Shapiro says that he still feels a certain kind of grumpiness :-) about a paper he wrote that was more cumbersome than another, more recent paper that became more famous - but truth to be told, I've never read either of these two original papers.
Thursday, November 22, 2007 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Arthur Eddington died 63 years ago
Arthur Stanley Eddington died on November 22nd, 1944. He was the most famous astrophysicist of the early 20th century and an interesting and sensible character who became a crackpot in the 1930s.
The Eddington limit, i.e. the maximum luminosity that can be obtained by accretion, is named after him.
Eddington and general relativity
Eddington was once asked by a journalist whether it was true that only three people understood general relativity. Of course, Eddington realized that such a statement was just another manifestation of journalistic stupidity but he answered with his famous question: "And who is the third?" :-)
Eddington, one of the most important early promoters of general relativity, made Einstein famous in 1919. His expedition confirmed Einstein's prediction that the light bending has to be twice as strong as Newton would have expected if he had assumed that light was a stream of massive particles moving at the light speed and influenced by Newton's gravitational force.
It seems rather likely today that Eddington's confirmation was a case of scientific misconduct. His accuracy couldn't have been good enough to make a decision. Nevertheless, the media were already powerful back in 1919. The London Times announced a "revolution in physics" on their title page. Masses started to adopt general relativity. Fortunately, general relativity was correct. But Einstein's remarkable intuition about classical physics was a more solid sociological explanation why it was correct than the journalists' belief in Eddington's statements.
Eddington's observation was overhyped
I find this phase transition irrational. Every good theoretical physicist who was interested in gravity and special relativity had to know back in 1915-1916 that general relativity was almost certainly the right theory of gravity because it was the only plausible theory of gravity at its level of complexity that agreed with special relativity as well as the equivalence principle. Moreover, it correctly postdicted the precession of Mercury's perihelion.
France: horses may replace buses and trucks
Reuters informs that French towns will be replacing vehicles, starting with school buses and refuse trucks, by horses. Seventy towns are ready to realize the plans of the Regional Horse Promotion Commission and thirty more will follow in a year.
In his 2003 speech at Caltech, Aliens Cause Global Warming, Michael Crichton was thinking what kind of a catastrophic problem the environmentalists in 1900 would predict for the year 2000. His best answer was exponentially increasing horse manure from their natural vehicles of choice.
As many science-fiction authors, Michael Crichton was right but he was 107 years ahead of his time. In reality, the tons of horse manure per squared meter are now predicted for 2107 rather than 2000. And Paris, not New York, will be the first city to experience this development.
The obsolete 20th century vehicles such as buses, trucks, and perhaps cars (for example Renaults that were never too good anyway) will be replaced by modern vehicles that emit not only CO2 but also CH4. The smoothly, exponentially increasing volume of excrements on the streets is referred to as "sustainable development" by Reuters. ;-)
Horses are beautiful animals but five million horses in Paris could be too much of a good thing. I have no doubts that there are places where horses would be more pleasant friends to do certain tasks. On the other hand, doubling or tripling the time that children need to get to school or from school could cause certain problems.
Wednesday, November 21, 2007 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Möbius transformations
This video is not quite a theory of everything but - as its creators forgot to tell you - it is about the tree level approximation of a theory of everything. The Möbius transformations are conformal i.e. angle-preserving, one-to-one maps from a sphere onto a sphere (or from a complex plane onto itself, or from anything of the same topology such as the Stanford bunny) which is why they are essential in perturbative string theory to bring a sphere diagram into a standard form.
Incidentally, the sphere is conformally equivalent to the plane because of Riemann's stereographic projection.
Tuesday, November 20, 2007 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Sidney Coleman: 1937-2007
Very sadly, Sidney Coleman died on Sunday morning. In sleep. Apparently peacefully. Arthur Jaffe's e-mail is here and there is an obituary in Chicago Tribune and Harvard Gazette.
In 2005, Arthur Jaffe and Barbara Drauschke organized Sidneyfest (official WWW) to celebrate Sidney's life achievements, his sense of humor, and his friendship.
Sidney was a physicist's physicist. He has been an excellent teacher, simplifier, expositor. His lectures at Harvard on quantum field theory were legendary and his notes were used by many of his successors including your humble correspondent for years. See the TeX edition of these notes and an earlier proposed Sidney Coleman Open Source Project. Many physicists have his Aspects of Symmetry with selected Erice lectures on their bookshelves.
With Abdus Salam, 1991 (taken from a today's article about crackpot Lisi)
But of course he has found some precious results, too. The Coleman-Mandula theorem showed that many superficially plausible ways to unify physics in the 1960s were guaranteed to be on the wrong track. Some people haven't understood the power of the theorem to day.
Related: ScienceNews celebrates Juan Maldacena: a decent article
The Coleman theorem proves that one cannot spontaneously break continuous symmetries in two dimensions or below. The Coleman-Weinberg potential pioneered loop corrections to the vacuum energy that can also induce new phase transitions. Coleman also found that the Thirring model was equivalent to the sine-Gordon model. He is the main physicist behind our understanding of the fate of false, unstable vacua. Add tadpoles (i.e. spermions), thin-wall Q-balls, many other papers about symmetry breaking, confinement, black holes, wormholes, parameterizing Lorentz symmetry breaking with Glashow... it's a lot of stuff.
Matory's disgraceful demagogy
The Harvard Crimson rightfully criticizes Lorand Matory's attempts to discredit friends of Israel as "enemies of free speech".
I find comments about free speech from a person such as Mr. Matory absolutely incredible. In 2005, Mr. Matory was the author of a truly disgusting "lack of confidence vote" against President Summers that was primarily justified by Lawrence Summers's polite and cautious remarks about the basic relationships between gender and innate aptitudes.
This guy, Mr. Matory, clearly thinks that free speech is something so intolerable that even the brilliant president of the world's most famous university must be eliminated when he dares to speak - or even ask - about things that every sane person knows anyway.
I have had huge personal problems with the voodoo expert myself. In 2005, after I criticized his resolution, he has harrassed all officials in the hierarchy above me behind the scenes and forced them to create problems for me. If you realize that all Harvard officials with a possible exception of Summers were (and are) cowards, you may guess what the result was.
Today, Mr. Matory, a fanatical anti-Semitic bigot, a freedom-hater, and the closest thing to Adolf Hitler that Harvard can offer, wants the FAS to "reaffirm its commitment to free speech and tolerance of minority views" which really means to "transform Harvard into loud headquarters of the world's anti-Israel movement".
Thankfully, the FAS has at least tabled the resolution. But until mechanisms will work not to allow crap like Mr. Matory to penetrate into influential places, similar ideological contamination of the Academia and the society and immoral ploys are guaranteed to continue.
And that's the memo.
Craig Loehle: Medieval Warm Period is back
Figure 1: Temperatures in the last two millenia.
See also: Loehle vs Schmidt, Loehle vs tree ring reconstructions
Monday, November 19, 2007 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Amazon: Zune beats iPod
I was almost certain that the moment would eventually arrive. It's here.
The brown Zune 30 GB is #1 bestseller in electronics at, ahead of all iPods. It is not so shocking because this particular color only costs USD 134, much less than the price of much weaker iPods. I am simply amazed how they can produce it for this ridiculously low amount of money.
Other colors besides brown are more expensive. And the 2nd generation Zune (above) costs USD 250, with 80 GB of disk space.
BBC HARDtalk with Václav Klaus
Part I
Part II
Home page of the program
Real Video
You can guess whether the BBC journalist and the Czech president agreed about every word or not. ;-)
I find the approach of the journalist somewhat incredible. It's the same kind of guys who like to say that George Bush is unprecedentedly stupid. But he finds it sensible to take the opinions of the same George Bush and other similar people and accuse Prof Václav Klaus of "plain arrogance" just because he doesn't agree with those fashionable talking points by all these lesser minds. It is apparently not "plain arrogance" to treat a European president in this way.
Their behavior is just an amazing combination of stupidity, intimidation, and hypocrisy.
After the second part of the video, Klaus answers that he has grandchildren. They won't know about the global warming debate in 30 years because it will be forgotten. But if they find something about it, they will just say that their granddad was right.
Finally, Stephen Sackur asks a few questions about the European constitution and the radar.
Type IIA vacua claimed to be cosmologically excluded
Take Barton Zwiebach's textbook on string theory and read his story about the type IIA braneworld scenarios with intersecting D6-branes. They surely look beautiful as a possible source of the Standard Model but are they correct?
Hertzberg, Kachru, Taylor, Tegmark
claim that they can rule out all types of type IIA models that have been constructed in literature, by their violation of cosmological requirements.
The slow-roll parameter epsilon that should be small is shown to be greater than 27/13 whenever the potential V is positive. You may view this inequality as a quantitative example of our generalized "weak gravity" principle. Slow-roll inflation and de Sitter vacua therefore become impossible. Their theorem makes some assumptions - such as the absence of NS5-branes. They sketch a possible class of models with extra features that could circumvent their no-go theorem. Well, with the right ingredients to obtain mirrors of the type IIB vacua claimed to be alive and well, it should be possible. ;-)
Sociologically speaking, you may want to know that these authors have been saying that type IIB vacua - the canonical KKLT-like and KKLLMT-like landscape - are the phenomenologically more acceptable ones for quite some time.
Sunday, November 18, 2007 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Niels Bohr died 45 years ago
Niels Bohr died on November 18th, 1962. Like most great 20th century theoretical physicists, this Danish physicist was a Jew.
While Bohr received his Nobel prize in 1922 for his old model of the Hydrogen atom - with ad hoc discretized Kepler orbits for electrons - that was superseded a few years after he was awarded the prize, it might be true that Bohr's role as a spiritual leader of the quantum community was more important.
Bohr is the father of the complementarity principle in quantum mechanics - something that only differs from the uncertainty principle by a different handwaving. And he became the ultimate lawmaker who codified the Copenhagen interpretation of quantum mechanics. As a boss, it was both his responsibility as well as pleasure to debate leading misguided contrarians, particularly Albert Einstein. He had to invent a lot of wise quotes, too.
Werner Heisenberg was his pupil. Their intellectual relationships were very intense and were only destroyed once the two great minds appeared in two different competing teams that were trying to create a nuclear weapon. Bohr was on the good side that eventually became the winning one, too.
Saturday, November 17, 2007 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Velvet Revolution: 18th anniversary
Video 1: Music and lyrics by Brontosaurs, a Czech folk and tramping band: one of the informal anthems of the pro-democratic demonstrations in 1989. The clip includes not-quite-English subtitles. ;-)
I was just watching a documentary about the Velvet Revolution in 1989 and there were many well-known things in it as well as some new ones.
The dissidents were remembering how hopeless the situation seemed even at the end of 1989 when communism in East Germany started to collapse. However, Czechs didn't seem to care. There was no visible momentum. There seemed to be no plausible mechanism that would lead to the end of the totalitarian system. I have had the same feeling at that time.
Kentucky, climate, and intimidation
If you want to see how intimidation by the climate change movement looks like, look at this story. There are hundreds of such stories every day but we have to pick an example. A Kentucky legislative committee had a hearing dedicated to climate change on Wednesday:
The Courier Journal
Jim Booch (DEM) was a co-chair of the committee and one of the guests, Lord Monckton, said a few very decent things - about Al Gore and others - that were however not great enough for the champions of the climate change catastrophe.
Jim Booch himself said that he had supported "their vice-president" nevertheless. He is also a "tree lover although not necessarily tree hugger" and wins most of the political battles for Democrats. Nevertheless, by having been a co-chair of an ideologically imperfect hearing, he has become a heretic anyway.
Two days later,
Lexington Herald Leader
ran a story asking whether Jim Booch should resign. To help the case, the journalist mentioned that USD 11,750 of donations during Booch's career could be linked to coal. Now, that's a lot of money, indeed - about 0.01% of what Al Gore has earned with his fraudulent theater and derived activities. Now, 0.01% doesn't seem as too much but when we talk about a heretic, the rules of mathematics change...
The talk about a resignation is of course absurd but you know that the power of the media is very large. Tens of thousands of people have surely begun to debate a fabricated question of his possible resignation. And when something is debated, it is gradually becoming a "reality" anyway.
The champions of the fight against climate change have high ideological standards, indeed. People must be 100% clean. A famous German political party in the 1930s allowed their new leader to be 1/4-Jewish. I am not sure whether this degree of tolerance could be found in the contemporary global warming movement. Jim Booch is untolerable for some people because he has been seen a few meters from Lord Monckton. Imagine what would happen if he were a cousin or a grandson of Lord Monckton! ;-)
15 minutes of fame: a definition
Andy Warhol, a Slovak American pop artist, coined the idiom "15 minutes of fame". No doubt, the modern media are creating a lot of short-lived hysterias, madness, misinformation, and cheap fads with no lasting value. But some of the quantitatively inclined readers don't know how this term is defined. Is there a graph that explains the concept? Well, there is one:
Click the graph to get to the source at Google Trends. The two peaks in the third quarter of 2006 correspond to the publication of "Not Even Wrong" and "The Trouble With Physics", two silly books attacking science. Besides the search queries "Woit" and "Smolin", I have included another query that is not subject to the 15-minute-of-fame effect as a reference.
Friday, November 16, 2007 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Symmetry and the Monster
A mathematics talk in RealVideo: use RealPlayer and open the following URL (write as one line):
The speaker, Mark Ronan, covers some history about the solutions to quintic equations, Galois's life, etc. Since 22:00 or so, he talks about the classification of finite groups. One minute later, the Monster enters the scene.
Hat tip: Prof John McKay, thanks!
A theory of everything triples traffic
A. Garrett Lisi doesn't know why bosons can't be added with fermions and why spinors can't be added with scalars and vectors, among many other things. But he has made it. ;-)
The term "a theory of everything" in the title together with a bunch of unusually stupid journalists is enough to do so.
A female surfer was picked instead of AGL to enforce affirmative action.
The Telegraph, The Ottawa Citizen, and other outlets have made this crackpot so famous that even humble blogs such as this one have seen their traffic triple yesterday as a consequence. On Thursday, TRF had over 9,000 unique visitors and on Friday over 14,000 unique visitors. Everyone is thrilled with the "E8 theory of everything". Well, people are rather excitable. ;-)
Good for them. Bad for the wisdom of the society.
Thursday, November 15, 2007 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Blauer Planet in grünen Fesseln
Was ist bedroht: Klima oder Freiheit?
And a climate conference
Buy at
Czech President Václav Klaus (whose book "Blue not Green Planet" was just published in German) and Petr Mach, the boss of the libertarian CEP think tank, have organized a pretty cool climate conference in Prague, showing seven scholars who can be counted as climate skeptics. There were 500 people watching and it was aired live on TV which is why it wasn't a good idea for me to be semi-gravel-voiced ;-) but otherwise my talk went just fine. I don't plan to see the conference again through TV archives. :-)
My PPT/DOC contribution in Czech: science basics of the climate debate
The ČT24 channel has attempted to preserve the people's sanity and ability to think independently against absurd proclamations about the "climate consensus" that some people are trying to import into the Czech Republic.
With all my respect to all of us, the Czech and other Central European speakers, the last two speeches by Julian Morris and Michael Walker (in English) were simply a different league. They're professional speakers, after all. I have learned some new things during the day, including conversations at a fun dinner in Sherwood, and other people have learned things, too.
Wednesday, November 14, 2007 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Uganda: Global warming makes girls hot
A short time ago, we saw that global warming has stopped circumcisions in Kenya. Consequently, young men are not ready to marry anyone which is why the girls marry older guys.
In neighboring Uganda, global warming causes a related problem, namely early marriages. It is also caused by global warming but the detailed mechanisms are different. It occurs because rich men are ready to marry young females while the girls' families need some money that were stolen from them by global warming. This is a conclusion of a scientific report funded by the United Nations that has identified "famine marriages", i.e. a new method for families to earn money and food by selling their daughters.
Global warming also increases the school dropout rates: click the cartoon above to zoom it in. It exposes people to sexually transmitted infections because when it is warm, people tend to undress their condoms, despite intense education by dancing Indian condoms. Other consequences of global warming are bush-burning (because they want to punish the bush that is clearly responsible for global warming and for the hot girls and because it should improve pastures), hunting of birds and animals, and diets.
Is there a consensus among skeptics?
Richard Black at BBC asks the question whether the climate skeptics have unified opinions about detailed questions about the climate.
I think that the correct answer is obviously "No". But unlike Richard Black, I don't think that it is a disadvantage or counter-argument of any kind. The differences between the skeptics pretty much reflect the amount of uncertainty about individual questions. If Richard Black or someone else believes that those 100% unified committees of communist parties or the unified body of believers in Al Qaeda make the opinions of these groups more likely, I beg to differ.
Not only skeptics have different opinions about detailed questions but individual skeptics are uncertain about individual questions themselves.
Skeptics have a unified opinion about the question that defines their skepticism. More concretely, they believe that climate change is not an urgent crisis that requires dramatic changes of the way how we live and how we use the fossil fuels.
But there are differences about pretty much every question simply because these questions are not settled and free people normally reach different conclusions when they analyze incomplete observations and incomplete theories. I know that this is an inconvenient truth for those who would like the opinions of the whole population to be unified but it is a truth nevertheless.
Also, there is a subtle problem with the questions that are usually not accurately enough formulated so that different people may mean different things by various words. Let us look at particular examples.
Is there global warming?
Well, I think that most skeptics will tell you that it is likely that the average temperature on Earth has increased during the last 100 years. But surely not all of them and frankly speaking, I have a full understanding for those who doubt it. The surface measurements don't seem too reliable because they are plagued by the urban heat island effects, human errors, and other things. It is conceivable that once a couple of these errors will be corrected, the warming that we like the quote today will go away or will at least be substantially reduced.
Some skeptics will tell you that the global temperature is not a terribly well-defined notion. Others will argue that the global character of the warming doesn't seem to be statistically significant. There have been many places that got cooler and the global average may be warmer simply because of statistical fluctuations: it is never guaranteed that the area of regions that get warmer must be equal to the are that gets cooler.
Many skeptics will protest that the choice of the 100-year timescale is an example of fine-tuning, cherry-picking, and cheating, and they are right. At different timescales, one cay see either warming or cooling. By emphasizing the importance of the 100-year timescale, we are already putting the "man-made" answer into the game as an assumption.
OK, I personally think that it is most likely that the warming trend in the last 100 years was close to 0.6°C per century.
Is this warming unprecedented?
Most skeptics, including myself, will say "No". They will tell you about dozens of types of climate changes in the past. Is it true that this 20th century warming is twice faster than the average warming or cooling during an average century in the last 1 million years? It might be. Was it the fastest centennial warming in the last 1 million of years? Probably not.
I think that it is obvious that people are guessing here. Climate has clearly been changing as we know from thousands of sources, experiments, and observations. But equally obviously, we don't have direct measurements of the decadal variations of temperatures 17,680 years ago, among other examples.
How the hell could there be any unity of opinions about it if we clearly have no data about it and no quantitative reliable theories either? Any group of people that is unified about these questions - such as decadal variations 17,680 years ago - that can be neither measured nor reliably calculated is simply a religious group. I think that the previous sentence is another statement that virtually all skeptics - and all sane people - will agree with.
Does the Sun's activity measurably influence the Earth's climate?
Surely, almost all skeptics, including myself will answer "Yes". Some influence is both explained physically as well as deduced from statistical analyses of the temperature records. The influence of galactic cosmic rays on the clouds and the climate is much more scientifically established than the role of the greenhouse effect. Again, most skeptics will agree. Look at the correlations in the papers by Svensmark and Friis-Christensen: they are simply impressively accurate.
The question about the cosmic and solar influence is just a quantitative one, just like the greenhouse effect. When combined with complex phenomena and feedbacks in the atmosphere, both solar activity and cosmic rays as well as the greenhouse effect have some effect on the climate.
The legitimate question is how large it is in both cases. Anyone who tries to make this question dogmatic and binary - that the answer is "Only one is correct and you shall never believe other Gods" is a religious bigot. Such bigots may exist on both sides but I happen to know many more greenhouse bigots than solar bigots.
How much warming should we expect in the next 100 years?
Well, we will probably surpass 560 ppm of CO2. Even if you believe that the greenhouse effect is responsible for all long-term warming, we have already realized something like 1/2 (40-75%, depending on the details of your calculation) of the greenhouse effect attributed to the CO2 doubling from 280 ppm to 560 ppm. It has led to 0.6°C of warming. It is not a hard calculation that the other half is thus expected to lead to an additional 0.6°C of warming between today and 2100.
Other derivations based on data that I consider rationally justified lead to numbers between 0.3°C and 1.4°C for the warming between 2000 and 2100. Clearly, one needs to know some science here. Laymen who are just interested in this debate but don't study the numbers by technical methods are likely to offer nothing else than random guesses and prejudices, regardless of their "ideological" affiliation in the climate debate.
When Richard Black quotes some uneducated people who are climate skeptics, it just shows that he is not being fair and he is spreading propaganda. The real problem with the global warming orthodoxy is that some of the craziest opinions about a coming catastrophe are heard from the most powerful alarmists. When we want to show that the alarmists are not quite sane, we don't have to pick the stupidest representative on the street. We don't have to humiliated Alexander Ač all the time. Chiefs of their institutes at NASA and Nobel prize winners will do the job, too. This is the real difference here.
Clearly, there can't be any consensus about precise values of the climate sensitivity simply because no accurate calculation of this quantity exists. Once again, if a large group has a consensus about the precise value, it is inevitably a religious group.
Now, would 1 Celsius degree of warming be a catastrophe?
This is a question in which skeptics probably agree once again. During the 20th century, the temperatures may have increased by 0.6°C. Not only we can say that it has caused no catastrophe. In fact, it seems that it has caused no visible problems at all. Our world is richer, more fertile, healthier than it was 100 years ago. By extrapolating this observation to the 21st century, there is absolutely no reason to think that a hypothetical additional warming would cause some big trouble.
Can the climate get out of control?
Once again, skeptics will say that probably not. But there is no rigorous proof. No one can say these things for certain. Catastrophes and instabilities are unlikely because they haven't occurred for billions of years even though our planet has tried many eras that were much more extreme than our times. So skeptics generally find theories about instabilities, tipping points, points of no return, and so forth very awkward. But they won't burn someone at stake because it can't be rigorously proven. They will just think that the person who propagates fear is not quite sensible.
Are the oceans more important than the solar activity?
Again, I don't know. If they are at least comparable, we would have to define the question more accurately anyway. Turbulent phenomena in the ocean surely play some role. 1998 was the warmest year mainly because of a huge El Nino. No doubts about it. But once again, there can't be any consensus here because we don't fully understand dynamics of these things. Even if we did, the system is so complex that we would have to carefully define two quantities that we are comparing. The question above is just too vague.
Is CO2 regulation a good idea?
Again, probably all skeptics will answer No, regardless of their belief about the previous questions. CO2 regulation is extremely expensive and it has extremely negligible impact on the climate. Moreover, we are not really sure whether the sign of this impact is positive or negative.
If I summarize. I think that the main difference between climate realists, also referred to as skeptics, and climate fanatics is not a different choice of some technical quantity such as the climate sensitivity. The main point in which they differ is their attitude to the society, freedom, and knowledge. Climate realists think that questions like that must be looked at in a very calm, balanced way, and each scientist must be free and independent to reach any conclusions that seem to be implied by the available evidence. They think that the complexity of complex systems must be acknowledged and uncertainties should never be masked. Moreover, the link between scientific conclusions and policies is extremely indirect and requires some people to answer many additional questions whose character is not related to climatology. Climate realists think that we must first answer these questions and then we can perhaps use the answers to influence policymaking.
On the other hand, climate fanatics think that these are moral questions that must be looked at in a very irrational way. Only one kind of an answer must be promoted or allowed. Policies must be created before the evidence is available and evidence must be tweaked to agree with the desired policies. Scientific conclusions must be deliberately exaggerated, cherry-picker, and oversimplified to achieved "sacred" goals. Everyone must be forced to believe the same thing and all infidels and heretics must be ostracized. The desired policies of CO2 regulations are good even if the climate apocalypse doesn't exist - even Marx himself believed these things - which justifies any amount of attacks and lies about the skeptics and the climate itself.
These two groups don't differ in technical details of science. They differ in their basic thinking about man, society, nature, freedom, truth, and science.
And that's the memo.
Berkovits, Vafa: proving AdS/CFT
Berkovits and Vafa have a paper that I choose to be the most interesting paper on the arXiv today. They may have made steps towards a full proof of the perturbative part of the most famous example of the AdS/CFT correspondence, namely between the N=4 gauge theory on one side and the AdS5 x S5 background of the type IIB string on the other side.
Some aspects of their construction are well-known to me but others are new. Among the aspects we have realized for some time, we can mention the fact that the relevant worldsheet description of the type IIB string is similar to the pure spinor language of Berkovits. It is an A-model which is a quotient of the U(2,2/4) supergroup and its maximal bosonic subgroup. Another aspect that I have believed for a few years is that the proof is based on a reduction of the two-dimensinoal worldsheet to Feynman diagrams, by "erasing" most of the information inside disk-shape regions included in the Feynman diagram.
What seems new, and what I am not able to fully check at this moment, is their statement that the proof could be analogous to the Ooguri-Vafa proof of a similar but topological Gopakumar-Vafa duality. The key feature that Berkovits and Vafa claim to be shared is that the worldsheet theory has a new Coulomb branch in this case much like it had in the topological case. The pieces of the worldsheet that are found in this Coulomb branch are interpreted as the faces of the Feynman diagrams while the boundaries between them are its propagators and vertices. As the 't Hooft coupling goes to zero, the regions found in the Higgs branch shrink and become the propagators and vertices.
Clearly, Cumrun Vafa is the first person who would be expected to suggest such a picture - in fact, I have heard such hints from him some time ago - and people who are less topological than he is, which surely includes your humble correspondent but most likely also all other humans on this planet, with a possible exception of Edward Witten and hypothetically also Marcos Mariňo (and with apologies to Aganagič, Gopakumar, Ooguri, Saulina, and others), face greater hurdles in trying to follow the details here.
Communism, capitalism, and environment
Environment in the Czech Republic: A Positive and Rapid Change
The black triangle
Monday, November 12, 2007 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Trillions for CO2 regulation & propagation of guilt
Two years ago, some people criticized the Kyoto counter in the sidebar because it assumes that the Kyoto protocol costs USD 150 billion per year which seemed too high to the champions of CO2 regulation. Well, times are changing. Almost no one would raise such a childish criticism today because most people realize that the actual costs of CO2 regulation are much higher.
The Guardian explains that according to Nicholas Stern's calculations, a U.S. climate bill would cost USD 212 billion per year while its EU counterpart would cost USD 164 billion. Add a few more countries that could also participate and you are well above USD half a trillion.
These are huge numbers that are completely comparable to trade surpluses and deficits of various countries. And as you know, these surpluses and deficits are no details or small perturbations of the economies. Their small changes add whole percentage points to the GDP growth. One percent of the world's GDP is a lot. If you assume that companies and societies invest and increase their capacities only from the "last" resources after everything else is subtracted, you would conclude that the annual costs around one percent of the GDP are equivalent to a reduction of the GDP growth by one percentage point.
Such a percentage point makes a huge difference. For many developed countries with the growth potential about 2 percent, it means that their growth rate is reduced to one-half. For countries that would otherwise grow a bit more slowly than at the one-percent annual rate, it means recession. Even the countries that used to have a three-percent GDP growth might switch to a two-percent regime, changing the time needed to double their GDP from 25 years to 35 years.
And we are still talking about CO2 policies that will have no discernible and demonstrable effect on the climate.
The article in the Guardian argues that one must carefully divide the CO2 emissions into the "politically correct ones", for example those in India, and the "luxurious ones", a point repeated by The Independent and others. Let me ask a rhetorical question: do you think that such a classification of CO2 emissions can be scientifically derived from physics or climatology? These things show that this whole hysteria about a "dangerous climate change" is politically driven. The greenhouse effect and all these things are just cheap tricks that are blown out of proportion in order to justify completely different things that those people actually care about.
But as Václav Klaus has emphasized, no country in the world is safe. If these CO2-regulating proposals win in some countries, their proponents will gain self-confidence and propagate the policies to ever broader set of countries.
Propagation of guilt
The Wall Street Journal explains that China, the country that generates the highest amount of CO2 emissions, might start to blame its rising CO2 emissions on the Western buyers. This question about the propagation of "guilt" is another important aspect of this whole debate. Imagine, for a little while, that CO2 emissions are harmful. Who is responsible for the Chinese emissions? Is it the buyers?
One of the comparative advantages of Asia is that a marketplace can be unified with a train station. Isn't it practical?
Well, the transactions between the Chinese producers and the Western buyers is arguably a contract that benefits both parties, much like most other contracts. But when the product is already completed, the hypothetical harm to the environment has already been made. It is very hard to realize this obvious fact in the case of CO2 that really causes no harm. But if you replace CO2 by mercury, you won't have any doubts who is guilty.
The Chinese products usually don't contain dangerous concentrations of mercury which makes it OK for consumers to buy them. In the same way, the products don't emit additional CO2 - except for cars that are made in China. So I find it obvious that if someone were guilty, it is the Chinese producers who could have used clean technologies or produce something completely different. One simply can't be viewed as a criminal for buying a legitimate product from a criminal, if you want me to amplify this language.
Of course that one can feel somewhat bad for having something - or anything - to do with some bad guys. But one of the principles of an enlightened modern society is that guilt simply cannot propagate in this way. For example, you shouldn't be held responsible for your parents' being killers even though you have had relationships of many kinds with your parents.
Guilt for well-defined sins must be localized to those who are really responsible. Indirect relations between these people and others cannot become a justification for the government to control those other people because if it became one, the government could control everyone. Why? Because everyone is indirectly connected to some people who do bad things, either more directly or less directly.
There are many other subtleties associated with these ambitious projects to regulate the carbon cycle. But if the goal were really to reduce total CO2 emissions, it is clear that their price would have to be universal for the whole planet, much like the price of oil. This conclusion might be controversial for those who mix science and economics with politics and religion. But a universal price would be necessary for these policies to regulate the net emissions instead of just moving them from one place to another.
Let me emphasize. These considerations are hypothetical in character because I don't believe that any CO2 regulation is rational.
Incidentally, the coal-oil price ratio is about 5 times smaller than what it was a decade ago. If this situation persists, I think that people, companies, and societies start to realize that. Someone should work on modern versions of gadgets that burn coal - for example 21st century steam engines in cars. ;-)
Thanks to Benny Peiser for the links.
Lord Rayleigh born 165 years ago
John William Strutt, 3rd Baron Rayleigh, was born on November 12th, 1842. Even though this white male aristocrat ;-) received his 1904 Nobel prize for the discovery of argon and geologists know him for Rayleigh-Lamb waves, most physicists would almost certainly think of the Rayleigh scattering.
This scattering is the main reason why the sky is blue: short-wavelength electromagnetic waves (e.g. blue photons) are scattered much more strongly which is why they arrive from random directions of the sky rather than the Sun. Rayleigh scattering is a good description when the impurities are much smaller than the wavelength of the light. Their calculable effect on radiation increases with the fourth power of the frequency.
This is probably not the best example but believe me - the sky is usually blue.
Two students of Lord Rayleigh were called Thomson and both of them were great physicists who won a Nobel prize. J.J. Thomson discovered isotopes, the mass spectrometer, and especially the electron for which he received the 1906 prize. George Paget Thomson shared the 1937 prize with Clinton Davisson for their observation of electron diffraction, a major experimental pillar of wave mechanics. If you want to be sure that the world of great physicists was really small, notice that George Paget Thomson was a son of J.J. Thomson. ;-)
Sunday, November 11, 2007 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Hugh Everett: 77th birthday
Hugh Everett was born on November 11th, 1930. He proposed the many-worlds interpretation of quantum mechanics and because no one cared, he left physics right after his PhD. More precisely, the term "many-worlds interpretation" comes from DeWitt who interpreted Everett in this way in 1971.
Later, he applied Lagrange multipliers in the commercial sector and earned some big bucks. As a chain smoker and drinker, he died at the age of 51. However, he believed in quantum immortality so the death was probably not such a big issue. His daughter, Elizabeth, suffered from schizophrenia and committed suicide in 1996, claiming that she was moving into a parallel universe to spend time with her dad.
Snow returns to Pilsen
So far, the amount of snow is not high enough to make it beautiful. Instead of photographs, I include a videoclip "Snow" from Richard Müller, a Slovak singer. The song is in Czech.
The scientific content of the song is small - we are so fresh and clean when it is snowing around. ;-) But it looks pretty.
Klaus: Merkel is the new 5-year planner
Abstract from the Czech Press Agency
Full text in German
Another interview in German for DPA about his book
Reuters news story
Interview of the President of the Czech Republic for the Wirtschaftswoche
Mr. President, German chancellor Angela Merkel fights for climate protection during her state visits throughout the world. She finds listeners in all countries except for yours. Why?
The unfair and irrational debate on global warming annoys me. The topic is increasingly turning into the fundamental ideological conflict of our times.
Has Mrs Merkel been caught into an ideology?
She probably thinks about these ideas. That surprises me. Because as a trained physicist, she should be undoubtedly able to test controversial hypotheses. But it also shows that this is not about science. The movement for the protection of the atmosphere embodies a new ideology. Surprisingly, it is espoused by Mrs Merkel who herself lived in socialist society. But she should know the risks associated with those ideologies that are directed against freedom.
Do you consider the chancellor to be a savior of the world?
I don't want to analyze Ms Merkel. The utopians are those who want to improve the world. However, politicians may find utopias to be an excellent thing because these politicians may start to talk about the distant future and avoid their everyday business. Such politicians are "escapists" because they want to escape reality. The issue of climate change is ideally suited for this purpose because we can spend 50 or even 100 years in the future by developing visions - while voters remain unable to control the consequences.
What are they escaping?
Politicians flee away from the emptiness of their own imagination. They have no ideas rich in content that could fill the present.
Does this also apply to the U.S. President George W. Bush who has apparently also warmed up to the climate debate?
I have talked about this topic with Bush several times. During our last meeting in the context of the U.N. high climate event in September, he asked me: "Václav, where is your book? I look forward (laughs)." As many Americans, he views the topic a bit more pragmatically. Americans have never been truly interested in utopias.
In your book, "Blue, Not a Green Planet", you only describe the environmentalists, as you call them, vaguely. Who are those conspirators whom you find so dangerous?
The climate debate itself deserves a sociological analysis. The politicians come first; they use the climate for the reasons explained above. Then we see the journalists who use the issue as a free ticket for a catchy theme on the title page. And finally the climate researchers only act to benefit and to maximize their profit by looking for subjects with the most promising funding situation.
Serious and prestigious researchers are among those who attack you. Are all of them opportunistic small minds?
Let's take for example the United Nations report on the climate. The presidium of the Intergovernmental Panel on Climate Change (IPCC) decides on what is in it. People like IPCC chairman Rajendra Pachauri may have been scientifically active in the past, but since then they have become bureaucrats. These people published their last journal article years ago. Today they work on policymaking. And among the real scientists, there are many who can't offer any new approaches. They simply follow the mainstream.
One can analyze scientists ad hominem. But if there is a critic with a legitimate criticism, why is he not heard?
Whatever the climatologists find incompatible with the so-called consensus is even not included in the U.N. climate report. Every day, I receive letters from all around the world in which scientists disagree with the prevailing opinion but no one wants to listen to or print their hypotheses. They are simply unfashionable.
You seem to suppose that the climate research is being censored.
You know, the whole thing is very familiar to me. After the Warsaw Pact troops intervened to terminate the Prague Spring, I was dismissed from the Czechoslovak Academy of Sciences as an enemy of Marxism. In the 1970s, I couldn't write any articles on economics.
You are trained as en economist, not a climate researcher - are you able to judge the scientific debate?
As an unemployed economist, I had a job in the State Bank of Czechoslovakia. We had the first computer over there. My task was to work on statistical and econometric models and against my will, I became busy with things that are important and relevant for climatology. Climatology is not one of the fields of physics and chemistry where a controlled experiment can be repeated a thousand times. It deals with data and hypotheses which can either be accepted or not. It works with time series that require statistical analysis.
Do you therefore distrust the method of climate researchers?
I have played with similar models for years. In hundreds or thousands of similar equations, I could always see that a slight change of a parameter or the addition of another parameter may radically change the outcome of complex models. That is why I am very critical about this methodology.
Do you flatly disagree that climate is changing?
No, of course not. The fact is that the climate is changing but every child knows that. There have to be no Nobel prize winners or a professor at the Potsdam Institute for Climate Impact Research. Of course, humans also play a role. But the crucial question is: How big is the influence of people on this process? The dispute is about orders of magnitude. Is the induced temperature change nonzero in the third, fourth, or fifth digit after the decimal point? This is a serious question that we must answer. And there is no consensus.
You say that the environmentalists such as the former U.S. Vice-President Al Gore threaten the freedom of thought. It is easy to argue against it. Who would be against freedom? What do you actually mean?
It is hard to answer in a few sentences. I have both political as well as economic and scientific freedom in mind. It is important that we don't lose either of them. Communism was another version of this ideology that placed something else as a "sacred" value above freedom. Environmentalism follows the same logic. First, the climate, then comes freedom followed by prosperity. Such priorities are wrong. For me, freedom is an important value. We Czechs have some experience with a lack of freedom. We sensitively and perhaps oversensitively respond to the threats to freedom - including those that the people in Western Europe don't understand too well.
The European Union has set - with the approval by the Czech government - ambitious climate targets. Your views make you totally lonely.
I am not alone. But I do find the current situation in Europe and the U.S. somewhat tragic. During the recent climate change conference in New York, my speech was the only one that criticized the climate policies. I didn't hear applause. Only after the dinner, many heads of state came to me and congratulated me. "There must have been someone to tell it," they said. One already probably needs political courage to speak against the policy of climate.
Who has thanked you?
I can't give you the names. It wouldn't have the right effect.
You argue that the economy and technological progress has the capacity to solve all problems resulting from climate change. What makes you so sure?
I didn't say the economy, I mean the market! This difference is fundamental. I believe in the market. Throughout my life, I have studied the economy in all of its manifestations, including communism. Plans vs market, external control vs spontaneity - these have been the eternal debates since Adam Smith. Why am I so confident? Because of my life experience. I have seen governments being mistaken hundreds of times. The market is not perfect, but its shortcomings are slight in comparison with the mistakes governments make. I lived in the regime of the planned economy - I consider the 50-year long plans of Angela Merkel just as misleading as the former five-year-plans.
What do you think about emissions trading? If carbon dioxide gets a price, the forces of the market will operate freely.
That's nonsense. This is a fraud by climatologists and environmentalists. Only fake economists could say what you did. This is about dirigism and not a free market. This method only pretends to be market-friendly. Emissions trading is just a game that looks like a market and as a classical liberal, I disagree with it.
There are entrepreneurs who earn money with the help of the environment. Germany has become the market leader in environmental technologies. It seems that the environment and the entrepreunerial spirit fit together wonderfully.
It is completely appropriate when entrepreneurs earn money by their effort to save energy. All of us should be thrifty regarding the energy, after all. Something else happens when entrepreneurs make profits out of alternative technologies. Transactions involving solar and wind energy are only possible because of the high subsidies paid for by the governments. These companies thus have political objectives and they don't play according to the rules of the free market.
No one doubts that we need traffic signs. Without minimal rules, chaos would threaten whole societies. Don't we need a couple of warning signs for the environment as well?
It depends on whether we talk about the environment or climate change. I have nothing against laws that protect ponds against waste disposal. But the environment protection laws, especially those in the EU, now go too far. But in this case we at least know what are the negative consequences of our actions or sins, if you wish. When the lake is polluted, it becomes contaminated. On the other hand, one cannot see how large and important the human influence on climate change is. It is an equation with too many unknowns - I am against climate restrictive laws and other forms of dirigism.
Václav Klaus, Wirtschaftswoche, November 10th, 2007 |
ae88e7ed78778145 | Picture of a black hole
Strathclyde Open Access research that creates ripples...
Discover more...
Modulational instability of dust envelope waves with grain and charge distribution
A reasonable normalization for a dusty plasma with many different species of dust grains is adopted. By applying a reductive perturbation technique to the equations governing a dusty plasma with N different species of dust grains, a nonlinear Schrödinger equation (NSLE) is derived that governs the modulation of dust-acoustic waves. The effect of dust size and charge distribution on the modulational instability of these waves is studied. If there are positively charged dust grains, which is a possibility suggested by experimental results, the envelope soliton solutions to the NLSE may be different from the ones associated with a dusty plasma containing mono-sized dust grains. The instability properties may also be different. In the former case the instability region depends on the percentage of electrons residing on the dust grains. In particular, if the number of electrons residing on the dust grains is small enough, the envelope waves are unstable. |
ca6aa59644568f22 | Quantum tunnelling
From Wikipedia, the free encyclopedia
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Quantum tunnelling or tunneling (see spelling differences) refers to the quantum mechanical phenomenon where a particle tunnels through a barrier that it classically could not surmount. This plays an essential role in several physical phenomena, such as the nuclear fusion that occurs in main sequence stars like the Sun.[1] It has important applications to modern devices such as the tunnel diode,[2] quantum computing, and the scanning tunnelling microscope. The effect was predicted in the early 20th century and its acceptance, as a general physical phenomenon, came mid-century.[3]
Tunnelling is often explained using the Heisenberg uncertainty principle and the wave–particle duality of matter. Purely quantum mechanical concepts are central to the phenomenon, so quantum tunnelling is one of the novel implications of quantum mechanics.
Quantum tunnelling was developed from the study of radioactivity,[3] which was discovered in 1896 by Henri Becquerel.[4] Radioactivity was examined further by Marie and Pierre Curie, for which they earned the Nobel Prize in Physics in 1903.[4] Ernest Rutherford and Egon Schweidler studied its nature, which was later verified empirically by Friedrich Kohlrausch. The idea of the half-life and the impossibility of predicting decay was created from their work.[3]
Friedrich Hund was the first to take notice of tunnelling in 1927 when he was calculating the ground state of the double-well potential.[4] Its first application was a mathematical explanation for alpha decay, which was done in 1928 by George Gamow and independently by Ronald Gurney and Edward Condon.[5][6][7][8] The two researchers simultaneously solved the Schrödinger equation for a model nuclear potential and derived a relationship between the half-life of the particle and the energy of emission that depended directly on the mathematical probability of tunnelling.
After attending a seminar by Gamow, Max Born recognised the generality of tunnelling. He realised that it was not restricted to nuclear physics, but was a general result of quantum mechanics that applies to many different systems.[3] Shortly thereafter, both groups considered the case of particles tunnelling into the nucleus. The study of semiconductors and the development of transistors and diodes led to the acceptance of electron tunnelling in solids by 1957. The work of Leo Esaki, Ivar Giaever and Brian Josephson predicted the tunnelling of superconducting Cooper pairs, for which they received the Nobel Prize in Physics in 1973.[3]
Introduction to the concept[edit]
Animation showing the tunnel effect and its application to STM microscope
Quantum tunnelling through a barrier. The energy of the tunnelled particle is the same but the amplitude is decreased.
Quantum tunnelling falls under the domain of quantum mechanics: the study of what happens at the quantum scale. This process cannot be directly perceived, but much of its understanding is shaped by the macroscopic world, which classical mechanics cannot adequately explain. To understand the phenomenon, particles attempting to travel between potential barriers can be compared to a ball trying to roll over a hill; quantum mechanics and classical mechanics differ in their treatment of this scenario. Classical mechanics predicts that particles that do not have enough energy to classically surmount a barrier will not be able to reach the other side. Thus, a ball without sufficient energy to surmount the hill would roll back down. Or, lacking the energy to penetrate a wall, it would bounce back (reflection) or in the extreme case, bury itself inside the wall (absorption). In quantum mechanics, these particles can, with a very small probability, tunnel to the other side, thus crossing the barrier. Here, the ball could, in a sense, borrow energy from its surroundings to tunnel through the wall or roll over the hill, paying it back by making the reflected electrons more energetic than they otherwise would have been.[9]
The reason for this difference comes from the treatment of matter in quantum mechanics as having properties of waves and particles. One interpretation of this duality involves the Heisenberg uncertainty principle, which defines a limit on how precisely the position and the momentum of a particle can be known at the same time.[4] This implies that there are no solutions with a probability of exactly zero (or one), though a solution may approach infinity if, for example, the calculation for its position was taken as a probability of 1, the other, i.e. its speed, would have to be infinity. Hence, the probability of a given particle's existence on the opposite side of an intervening barrier is non-zero, and such particles will appear on the 'other' (a semantically difficult word in this instance) side with a frequency proportional to this probability.
An electron wavepacket directed at a potential barrier. Note the dim spot on the right that represents tunnelling electrons.
Quantum tunnelling in the phase space formulation of quantum mechanics. Wigner function for tunnelling through the potential barrier U(x)=8e^{-0.25 x^2} in atomic units (a.u.). The solid lines represent the level set of the Hamiltonian H(x,p) = p^2 / 2 + U(x) .
The tunnelling problem[edit]
The wave function of a particle summarises everything that can be known about a physical system.[10] Therefore, problems in quantum mechanics center around the analysis of the wave function for a system. Using mathematical formulations of quantum mechanics, such as the Schrödinger equation, the wave function can be solved. This is directly related to the probability density of the particle's position, which describes the probability that the particle is at any given place. In the limit of large barriers, the probability of tunnelling decreases for taller and wider barriers.
For simple tunnelling-barrier models, such as the rectangular barrier, an analytic solution exists. Problems in real life often do not have one, so "semiclassical" or "quasiclassical" methods have been developed to give approximate solutions to these problems, like the WKB approximation. Probabilities may be derived with arbitrary precision, constrained by computational resources, via Feynman's path integral method; such precision is seldom required in engineering practice.
Related phenomena[edit]
There are several phenomena that have the same behaviour as quantum tunnelling, and thus can be accurately described by tunnelling. Examples include the evanescent wave coupling (the application of Maxwell's wave-equation to light) and the application of the non-dispersive wave-equation from acoustics applied to "waves on strings". Evanescent wave coupling, until recently, was only called "tunnelling" in quantum mechanics; now it is used in other contexts.
These effects are modelled similarly to the rectangular potential barrier. In these cases, there is one transmission medium through which the wave propagates that is the same or nearly the same throughout, and a second medium through which the wave travels differently. This can be described as a thin region of medium B between two regions of medium A. The analysis of a rectangular barrier by means of the Schrödinger equation can be adapted to these other effects provided that the wave equation has travelling wave solutions in medium A but real exponential solutions in medium B.
In optics, medium A is a vacuum while medium B is glass. In acoustics, medium A may be a liquid or gas and medium B a solid. For both cases, medium A is a region of space where the particle's total energy is greater than its potential energy and medium B is the potential barrier. These have an incoming wave and resultant waves in both directions. There can be more mediums and barriers, and the barriers need not be discrete; approximations are useful in this case.
Tunnelling occurs with barriers of thickness around 1-3 nm and smaller,[11] but is the cause of some important macroscopic physical phenomena. For instance, tunnelling is a source of current leakage in very-large-scale integration (VLSI) electronics and results in the substantial power drain and heating effects that plague high-speed and mobile technology; it is considered the lower limit on how small computer chips can be made.[12]
Radioactive decay[edit]
Radioactive decay is the process of emission of particles and energy from the unstable nucleus of an atom to form a stable product. This is done via the tunnelling of a particle out of the nucleus (an electron tunnelling into the nucleus is electron capture). This was the first application of quantum tunnelling and led to the first approximations.
Spontaneous DNA mutation[edit]
Spontaneous mutation of DNA occurs when normal DNA replication takes place after a particularly significant proton has defied the odds in quantum tunnelling in what is called "proton tunnelling"[13] (quantum biology). A hydrogen bond joins normal base pairs of DNA. There exists a double well potential along a hydrogen bond separated by a potential energy barrier. It is believed that the double well potential is asymmetric with one well deeper than the other so the proton normally rests in the deeper well. For a mutation to occur, the proton must have tunnelled into the shallower of the two potential wells. The movement of the proton from its regular position is called a tautomeric transition. If DNA replication takes place in this state, the base pairing rule for DNA may be jeopardised causing a mutation.[14] Per-Olov Lowdin was the first to develop this theory of spontaneous mutation within the double helix (quantum bio). Other instances of quantum tunnelling-induced mutations in biology are believed to be a cause of ageing and cancer.[citation needed]
Cold emission[edit]
Cold emission of electrons is relevant to semiconductors and superconductor physics. It is similar to thermionic emission, where electrons randomly jump from the surface of a metal to follow a voltage bias because they statistically end up with more energy than the barrier, through random collisions with other particles. When the electric field is very large, the barrier becomes thin enough for electrons to tunnel out of the atomic state, leading to a current that varies approximately exponentially with the electric field.[15] These materials are important for flash memory and for some electron microscopes.
Tunnel junction[edit]
A simple barrier can be created by separating two conductors with a very thin insulator. These are tunnel junctions, the study of which requires quantum tunnelling.[16] Josephson junctions take advantage of quantum tunnelling and the superconductivity of some semiconductors to create the Josephson effect. This has applications in precision measurements of voltages and magnetic fields,[15] as well as the multijunction solar cell.
A working mechanism of a resonant tunnelling diode device, based on the phenomenon of quantum tunnelling through the potential barriers.
Tunnel diode[edit]
Diodes are electrical semiconductor devices that allow electric current flow in one direction more than the other. The device depends on a depletion layer between N-type and P-type semiconductors to serve its purpose; when these are very heavily doped the depletion layer can be thin enough for tunnelling. Then, when a small forward bias is applied the current due to tunnelling is significant. This has a maximum at the point where the voltage bias is such that the energy level of the p and n conduction bands are the same. As the voltage bias is increased, the two conduction bands no longer line up and the diode acts typically.[17]
Because the tunnelling current drops off rapidly, tunnel diodes can be created that have a range of voltages for which current decreases as voltage is increased. This peculiar property is used in some applications, like high speed devices where the characteristic tunnelling probability changes as rapidly as the bias voltage.[17]
The resonant tunnelling diode makes use of quantum tunnelling in a very different manner to achieve a similar result. This diode has a resonant voltage for which there is a lot of current that favors a particular voltage, achieved by placing two very thin layers with a high energy conductance band very near each other. This creates a quantum potential well that have a discrete lowest energy level. When this energy level is higher than that of the electrons, no tunnelling will occur, and the diode is in reverse bias. Once the two voltage energies align, the electrons flow like an open wire. As the voltage is increased further tunnelling becomes improbable and the diode acts like a normal diode again before a second energy level becomes noticeable.[18]
Tunnel field-effect transistors[edit]
A European research project has demonstrated field effect transistors in which the gate (channel) is controlled via quantum tunnelling rather than by thermal injection, reducing gate voltage from ~1 volt to 0.2 volts and reducing power consumption by up to 100×. If these transistors can be scaled up into VLSI chips, they will significantly improve the performance per power of integrated circuits.[19]
Quantum conductivity[edit]
While the Drude model of electrical conductivity makes excellent predictions about the nature of electrons conducting in metals, it can be furthered by using quantum tunnelling to explain the nature of the electron's collisions.[15] When a free electron wave packet encounters a long array of uniformly spaced barriers the reflected part of the wave packet interferes uniformly with the transmitted one between all barriers so that there are cases of 100% transmission. The theory predicts that if positively charged nuclei form a perfectly rectangular array, electrons will tunnel through the metal as free electrons, leading to an extremely high conductance, and that impurities in the metal will disrupt it significantly.[15]
Scanning tunnelling microscope[edit]
The scanning tunnelling microscope (STM), invented by Gerd Binnig and Heinrich Rohrer, allows imaging of individual atoms on the surface of a metal.[15] It operates by taking advantage of the relationship between quantum tunnelling with distance. When the tip of the STM's needle is brought very close to a conduction surface that has a voltage bias, by measuring the current of electrons that are tunnelling between the needle and the surface, the distance between the needle and the surface can be measured. By using piezoelectric rods that change in size when voltage is applied over them the height of the tip can be adjusted to keep the tunnelling current constant. The time-varying voltages that are applied to these rods can be recorded and used to image the surface of the conductor.[15] STMs are accurate to 0.001 nm, or about 1% of atomic diameter.[18]
Faster than light[edit]
It is possible for spin zero particles to travel faster than the speed of light when tunnelling.[3] This apparently violates the principle of causality, since there will be a frame of reference in which it arrives before it has left. However, careful analysis of the transmission of the wave packet shows that there is actually no violation of relativity theory. In 1998, Francis E. Low reviewed briefly the phenomenon of zero time tunnelling.[20] More recently experimental tunnelling time data of phonons, photons, and electrons have been published by Günter Nimtz.[21]
Mathematical discussions of quantum tunnelling[edit]
The following subsections discuss the mathematical formulations of quantum tunnelling.
The Schrödinger equation[edit]
The time-independent Schrödinger equation for one particle in one dimension can be written as
-\frac{\hbar^2}{2m} \frac{d^2}{dx^2} \Psi(x) + V(x) \Psi(x) = E \Psi(x) or
\frac{d^2}{dx^2} \Psi(x) = \frac{2m}{\hbar^2} \left( V(x) - E \right) \Psi(x) \equiv \frac{2m}{\hbar^2} M(x) \Psi(x) ,
where \hbar is the reduced Planck's constant, m is the particle mass, x represents distance measured in the direction of motion of the particle, Ψ is the Schrödinger wave function, V is the potential energy of the particle (measured relative to any convenient reference level), E is the energy of the particle that is associated with motion in the x-axis (measured relative to V), and M(x) is a quantity defined by V(x) - E which has no accepted name in physics.
The solutions of the Schrödinger equation take different forms for different values of x, depending on whether M(x) is positive or negative. When M(x) is constant and negative, then the Schrödinger equation can be written in the form
\frac{d^2}{dx^2} \Psi(x) = \frac{2m}{\hbar^2} M(x) \Psi(x) = -k^2 \Psi(x),\;\;\;\;\;\; \mathrm{where} \;\;\; k^2=- \frac{2m}{\hbar^2} M.
The solutions of this equation represent traveling waves, with phase-constant +k or -k. Alternatively, if M(x) is constant and positive, then the Schrödinger equation can be written in the form
\frac{d^2}{dx^2} \Psi(x) = \frac{2m}{\hbar^2} M(x) \Psi(x) = {\kappa}^2 \Psi(x), \;\;\;\;\;\; \mathrm{where} \;\;\; {\kappa}^2= \frac{2m}{\hbar^2} M.
The solutions of this equation are rising and falling exponentials in the form of evanescent waves. When M(x) varies with position, the same difference in behaviour occurs, depending on whether M(x) is negative or positive. It follows that the sign of M(x) determines the nature of the medium, with positive M(x) corresponding to medium A as described above and negative M(x) corresponding to medium B. It thus follows that evanescent wave coupling can occur if a region of positive M(x) is sandwiched between two regions of negative M(x), hence creating a potential barrier.
The mathematics of dealing with the situation where M(x) varies with x is difficult, except in special cases that usually do not correspond to physical reality. A discussion of the semi-classical approximate method, as found in physics textbooks, is given in the next section. A full and complicated mathematical treatment appears in the 1965 monograph by Fröman and Fröman noted below. Their ideas have not been incorporated into physics textbooks, but their corrections have little quantitative effect.
The WKB approximation[edit]
The wave function is expressed as the exponential of a function:
\Psi(x) = e^{\Phi(x)}, where \Phi''(x) + \Phi'(x)^2 = \frac{2m}{\hbar^2} \left( V(x) - E \right).
\Phi'(x) is then separated into real and imaginary parts:
\Phi'(x) = A(x) + i B(x), where A(x) and B(x) are real-valued functions.
Substituting the second equation into the first and using the fact that the imaginary part needs to be 0 results in:
A'(x) + A(x)^2 - B(x)^2 = \frac{2m}{\hbar^2} \left( V(x) - E \right).
To solve this equation using the semiclassical approximation, each function must be expanded as a power series in \hbar. From the equations, the power series must start with at least an order of \hbar^{-1} to satisfy the real part of the equation; for a good classical limit starting with the highest power of Planck's constant possible is preferable, which leads to
A(x) = \frac{1}{\hbar} \sum_{k=0}^\infty \hbar^k A_k(x)
B(x) = \frac{1}{\hbar} \sum_{k=0}^\infty \hbar^k B_k(x),
with the following constraints on the lowest order terms,
A_0(x)^2 - B_0(x)^2 = 2m \left( V(x) - E \right)
A_0(x) B_0(x) = 0.
At this point two extreme cases can be considered.
Case 1 If the amplitude varies slowly as compared to the phase A_0(x) = 0 and
B_0(x) = \pm \sqrt{ 2m \left( E - V(x) \right) }
which corresponds to classical motion. Resolving the next order of expansion yields
\Psi(x) \approx C \frac{ e^{i \int dx \sqrt{\frac{2m}{\hbar^2} \left( E - V(x) \right)} + \theta} }{\sqrt[4]{\frac{2m}{\hbar^2} \left( E - V(x) \right)}}
Case 2
If the phase varies slowly as compared to the amplitude, B_0(x) = 0 and
A_0(x) = \pm \sqrt{ 2m \left( V(x) - E \right) }
which corresponds to tunnelling. Resolving the next order of the expansion yields
\Psi(x) \approx \frac{ C_{+} e^{+\int dx \sqrt{\frac{2m}{\hbar^2} \left( V(x) - E \right)}} + C_{-} e^{-\int dx \sqrt{\frac{2m}{\hbar^2} \left( V(x) - E \right)}}}{\sqrt[4]{\frac{2m}{\hbar^2} \left( V(x) - E \right)}}
In both cases it is apparent from the denominator that both these approximate solutions are bad near the classical turning points E = V(x). Away from the potential hill, the particle acts similar to a free and oscillating wave; beneath the potential hill, the particle undergoes exponential changes in amplitude. By considering the behaviour at these limits and classical turning points a global solution can be made.
To start, choose a classical turning point, x_1 and expand \frac{2m}{\hbar^2}\left(V(x)-E\right) in a power series about x_1:
\frac{2m}{\hbar^2}\left(V(x)-E\right) = v_1 (x - x_1) + v_2 (x - x_1)^2 + \cdots
Keeping only the first order term ensures linearity:
Using this approximation, the equation near x_1 becomes a differential equation:
\frac{d^2}{dx^2} \Psi(x) = v_1 (x - x_1) \Psi(x).
This can be solved using Airy functions as solutions.
\Psi(x) = C_A Ai\left( \sqrt[3]{v_1} (x - x_1) \right) + C_B Bi\left( \sqrt[3]{v_1} (x - x_1) \right)
Taking these solutions for all classical turning points, a global solution can be formed that links the limiting solutions. Given the 2 coefficients on one side of a classical turning point, the 2 coefficients on the other side of a classical turning point can be determined by using this local solution to connect them.
Hence, the Airy function solutions will asymptote into sine, cosine and exponential functions in the proper limits. The relationships between C,\theta and C_{+},C_{-} are
C_{+} = \frac{1}{2} C \cos{\left(\theta - \frac{\pi}{4}\right)}
C_{-} = - C \sin{\left(\theta - \frac{\pi}{4}\right)}
With the coefficients found, the global solution can be found. Therefore, the transmission coefficient for a particle tunnelling through a single potential barrier is
where x_1,x_2 are the 2 classical turning points for the potential barrier.
See also[edit]
1. ^ Serway; Vuille (2008). College Physics 2 (Eighth ed.). Belmont: Brooks/Cole. ISBN 9780495554752.
2. ^ Taylor, J. (2004). Modern Physics for Scientists and Engineers. Prentice Hall. p. 234. ISBN 013805715X.
3. ^ a b c d e f Razavy, Mohsen (2003). Quantum Theory of Tunneling. World Scientific. pp. 4, 462. ISBN 9812564888.
4. ^ a b c d Nimtz; Haibel (2008). Zero Time Space. Wiley-VCH. p. 1.
5. ^ Gurney, R. W.; Condon, E. U. (1928). "Quantum Mechanics and Radioactive Disintegration". Nature 122: 439. Bibcode:1928Natur.122..439G. doi:10.1038/122439a0.
6. ^ Gurney, R. W.; Condon, E. U. (1929). "Quantum Mechanics and Radioactive Disintegration". Phys. Rev 33 (2): 127–140. Bibcode:1929PhRv...33..127G. doi:10.1103/PhysRev.33.127.
7. ^ Interview with Hans Bethe by Charles Weiner and Jagdish Mehra at Cornell University, 27 October 1966 accessed 5 April 2010
8. ^ Friedlander, Gerhart; Kennedy, Joseph E.; Miller, Julian Malcolm (1964). Nuclear and Radiochemistry (2nd ed.). New York: John Wiley & Sons. pp. 225–7. ISBN 978-0-471-86255-0.
9. ^ "Quantum Tunneling Time". ASU. Retrieved 2012-01-28.
10. ^ Bjorken and Drell, "Relativistic Quantum Mechanics", page 2. Mcgraw-Hill College, 1965.
11. ^ Lerner; Trigg (1991). Encyclopedia of Physics (2nd ed.). New York: VCH. p. 1308. ISBN 0895737523.
12. ^ "Applications of tunneling". Simon Connell 2006.
13. ^ Matta, Cherif F. (2010). Quantum Biochemistry: Electronic Structure and Biological Activity. Weinheim: Wiley-VCH.
14. ^ Majumdar, Rabi (2011). Quantum Mechanics: In Physics and Chemistry with Applications to Bioloty. New Delhi: PHI Learning.
15. ^ a b c d e f Taylor, J. (2004). Modern Physics for Scientists and Engineers. Prentice Hall. p. 479. ISBN 013805715X.
16. ^ Lerner; Trigg (1991). Encyclopedia of Physics (2nd ed.). New York: VCH. pp. 1308–1309. ISBN 0895737523.
17. ^ a b Krane, Kenneth (1983). Modern Physics. New York: John Wiley and Sons. p. 423. ISBN 0471079634.
18. ^ a b Knight, R. D. (2004). Physics for Scientists and Engineers: With Modern Physics. Pearson Education. p. 1311. ISBN 0321223691.
19. ^ Ionescu, Adrian M.; Riel, Heike (2011). "Tunnel field-effect transistors as energy-efficient electronic switches". Nature 479 (7373): 329–337. Bibcode:2011Natur.479..329I. doi:10.1038/nature10679.
20. ^ Low, F. E. (1998). "Comments on apparent superluminal propagation". Ann. Phys. (Leipzig) 7 (7–8): 660–661. Bibcode:1998AnP...510..660L. doi:10.1002/(SICI)1521-3889(199812)7:7/8<660::AID-ANDP660>3.0.CO;2-0.
21. ^ Nimtz, G. (2011). "Tunneling Confronts Special Relativity". Found. Phys. 41 (7): 1193–1199. arXiv:1003.3944. Bibcode:2011FoPh...41.1193N. doi:10.1007/s10701-011-9539-2.
Further reading[edit]
External links[edit] |
c8a83dccf1a9199d | is consistent
Consistent histories
In quantum mechanics, the consistent histories approach is intended to give a modern interpretation of quantum mechanics, generalising the conventional Copenhagen interpretation and providing a natural interpretation of quantum cosmology. The theory is based on a consistency criterion that then allows probabilities to be assigned to histories of a system so that the probabilities for each history obey the rules of classical probability while being consistent with the Schrödinger equation.
According to this interpretation of quantum mechanics, the purpose of a quantum-mechanical theory is to predict probabilities of various alternative histories.
A homogeneous history H_i (here i labels different histories) is a sequence of propositions P_{i,j} specified at different moments of time t_{i,j} (here j labels the times). We write this as:
H_i = (P_{i,1}, P_{i,2},ldots,P_{i,n_i})
and read it as "the proposition P_{i,1} is true at time t_{i,1} and then the proposition P_{i,2} is true at time t_{i,2} and then ldots". The times t_{i,1} < t_{i,2} < ldots < t_{i,n_i} are strictly ordered and called the temporal support of the history.
Inhomogeneous histories are multiple-time propositions which cannot be represented by a homogeneous history. An example is the logical OR of two homogeneous histories: H_i vee H_j.
These propositions can correspond to any set of questions that include all possibilities. Examples might be the three propositions meaning "the electron went through the left slit", "the electron went through the right slit" and "the electron didn't go through either slit". One of the aims of the theory is to show that classical questions such as, "where are my keys?" are consistent. In this case one might use a large number of propositions each one specifying the location of the keys in some small region of space.
Each single-time proposition P_{i,j} can be represented by a projection operator hat{P}_{i,j} acting on the system's Hilbert space (we use "hats" to denote operators). It is then useful to represent homogeneous histories by the time-ordered tensor product of their single-time projection operators. This is the history projection operator (HPO) formalism developed by Christopher Isham and naturally encodes the logical structure of the history propositions. The homogeneous history H_i is represented by the projection operator
hat{H}_i = hat{P}_{i,1} otimes hat{P}_{i,2} otimes cdots otimes hat{P}_{i,n_i}
This definition can be extended to define projection operators that represent inhomogeneous histories too.
An important construction in the consistent histories approach is the class operator for a homogeneous history:
hat{C}_{H_i} := T prod_{j=1}^{n_i} hat{P}_{i,j}(t_{i,j}) = hat{P}_{i,1}hat{P}_{i,2}cdots hat{P}_{i,n_i}
The symbol T indicates that the factors in the product are ordered chronologically according to their values of t_{i,j}: the "past" operators with smaller values of t appear on the right side, and the "future" operators with greater values of t appear on the left side. This definition can be extended to inhomogeneous histories as well.
Central to the consistent histories is the notion of consistency. A set of histories { H_i} is consistent (or strongly consistent) if
operatorname{Tr}(hat{C}_{H_i} rho hat{C}^dagger_{H_j}) = 0
for all i neq j. Here rho represents the initial density matrix, and the operators are expressed in the Heisenberg picture.
The set of histories is weakly consistent if
for all i neq j.
If a set of histories is consistent then probabilities can be assigned to them in a consistent way. We postulate that the probability of history H_i is simply
operatorname{Pr}(H_i) = operatorname{Tr}(hat{C}_{H_i} rho hat{C}^dagger_{H_i})
which obeys the axioms of probability if the histories H_i come from the same (strongly) consistent set.
As an example, this means the probability of "H_i OR H_j" equals the probability of "H_i" plus the probability of "H_j" minus the probability of "H_i AND H_j", and so forth.
The interpretation based on consistent histories is used in combination with the insights about quantum decoherence. Quantum decoherence implies that only special choices of histories are consistent, and it allows a quantitative calculation of the boundary between the classical domain and the quantum domain.
In some views the interpretation based on consistent histories does not change anything about the paradigm of the Copenhagen interpretation that only the probabilities calculated from quantum mechanics and the wave function have a physical meaning. In order to obtain a complete theory, the formal rules above must be supplemented with a particular Hilbert space and rules that govern dynamics, for example a Hamiltonian.
In the opinion of others this still does not make a complete theory as no predictions are possible about which set of consistent histories will actually occur. That is the rules of CH, the Hilbert space, and the Hamiltonian must be supplemented by a set selection rule.
The proponents of this modern interpretation, such as Murray Gell-Mann, James Hartle, Roland Omnès, Robert B. Griffiths, and Wojciech Zurek argue that their interpretation clarifies the fundamental disadvantages of the old Copenhagen interpretation, and can be used as a complete interpretational framework for quantum mechanics.
In Quantum Philosophy, Roland Omnès provides a less mathematical way of understanding this same formalism. The consistent histories approach can be interpreted as a way of understanding which sets of classical questions can be consistently asked of a single quantum system, and which sets of questions are fundamentally inconsistent, and thus meaningless when asked together. It thus becomes possible to demonstrate formally why it is that the questions which Einstein, Podolsky and Rosen assumed could be asked together, of a single quantum system, simply cannot be asked together. On the other hand, it also becomes possible to demonstrate that classical, logical reasoning often does apply, even to quantum experiments – but we can now be mathematically exact about the limits of classical logic.
See also
• R. Omnès, Understanding Quantum Mechanics, Princeton University Press, 1999. Chapter 13 describes consistent histories.
• R. Omnès, Quantum Philosophy, Princeton University Press, 1999. See part III, especially Chapter IX.
• R. B. Griffiths, Consistent Quantum Theory, Cambridge University Press, 2003.
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b0bec8ebce2b06c4 | Wednesday, March 21, 2007
Merriam's Quantum Relativity
Paul Merriam posted a paper called Quantum Relativity: Physical Laws Must be Invariant Over Quantum Systems in which he puts forth a conceptual strategy for understanding how a relational interpretation addresses the foundational issues of quantum mechanics. Please see this prior post for more background. What follows is a summary and attempted interpretation of what I found to be key aspects of the paper. The usual caveats are in place: my summaries may be not only incomplete (including omission of formalisms) but also misleading due to errors in interpretation. Please read the paper to judge.
The paper starts with a section which discusses why decoherence does not solve the foundational issues of QM. Since I believe this is generally acknowledged (see this recent blog post from Matt Leifer; an old blog post of mine is here), I’ll just focus on the most important part of this discussion. Recall that one of the perceived shortcomings of the relational interpretation of QM revolves around the question of how two or more interacting systems come to “choose” the same basis. Merriam says that decoherence has a “change of basis” problem of its own.
To see this, Merriam returns to the “Wigner’s friend” framework and replaces "Wigner" with the "environment" to create a decoherence version of the scenario. Relative to the environment E, the experimenter (called A) and the system he or she is measuring (S) are in superposition and evolve according the Schrödinger picture. Decoherence would lead to the selection of relatively stable “classical” appearances of the observable which is the basis of the measurement. But suppose A decides to measure a different observable of S (change of basis). Decoherence takes place over a period of time (decoherence time); this time depends on many factors, but the “change of basis” is a problem for the time between zero and the decoherence time. (Decoherence is not measurement).
Next Merriam discusses (repeating the arguments of his older paper) the issues highlighted by the Wigner’s friend setup, arguing again that the quantum state describes a system relative to another system. Quantum mechanics is an intransitive theory.
The next section is titled “Quantum Relativity”. So having acknowledged the perspectivist nature of QM, what’s the next step? When considering two quantum systems: “The essential point of this paper is that since both systems physically exist they are both valid coordinate frames from which the laws of physics must hold. Quantum mechanics is as valid in S as it is in A.” If A describes S in terms of a superposition across some measurement basis, then S will describe A as starting out in a corresponding superposition. When A observes (measures) S to be in some eigenstate, “S must also observe A to be in some corresponding eigenstate…”
The key point is brought out by the word “must” here and in the title of the paper. The conceptual hurdle we are jumping here is as follows: if QM is valid from the point of view of all “quantum systems” (including everything from electrons to physicists), then when they interact they necessarily select the consistent basis for interaction. The basis problem is solved by asserting that basis choices must match if QM is to be valid from all points of view.
Merriam believes this conceptual leap has consequences analogous to special relativity. The next passage (see p. 6) looks at the formalism of the Schrödinger equation from A’s and S’s perspective and wonders how they can be consistent if the mass is so different in the two cases. But he notes the values for length or distance between the two quantum observations do not have to have the same numeral values in both systems. If distance is scaled to the relationship of the masses, then it is possible to create a transformation from the superposition of S as described by A to that of A described by S. There can be a group of such transformations for any number of systems. Merriam derives a transformation constant in analogy to the role the speed of light c plays in relativistic transformation.
Merriam also speculates about that one could extend the idea to include gravity by taking the equivalence of gravitational force and acceleration to be relative to the local quantum reference system. He suggests the shape a quantum version of Einstein’s equation would take. I will skip for now further discussion of this idea and a section on how gauge invariance might be impacted, since I think the key concept is in place with the analogue to special relativity.
Key to special relativity is the postulate that physical laws valid from one reference frame should be form-invariant when translated to another frame. To review, we assume that QM gives a valid physical description from the point of view of a system, and each quantum system forms a physically valid coordinate frame. Note that systems only share a reference frame when they interact. We should be able to translate the state of a system S which is in superposition relative to system A to the state of A relative to S. Again, this only works if we stipulate that if an interaction takes place, the “basis choice” is necessarily consistent from both perspectives.
Friday, March 16, 2007
Exploring the Borderlands
Recent books on atheism and religion have been the focus of much debate recently, which I think is a good thing. It’s no surprise that the debate is dominated by traditional religious believers on the one hand, and those who hold to a traditional materialist strain of atheism on the other. Of course, there is a wide, if seemingly less populated, territory between these views. I think the truth lies in the border regions.
If one is a realist, as I am, about first-person experience and the existence of some degree of freedom, then materialism is inadequate. On the other hand, one’s worldview must be shaped by valid inferences from the success of science. Because of this, I find traditional supernatural entities and interventions highly implausible, and have in the past characterized my own worldview has an enriched or expanded version of naturalism.
My more recent ruminations on modal realism and abstract entities have led me to consider that my realist commitments may actually require a necessary ground of possibilities underlying and penetrating our contingent concrete world. While at the end of the day labels aren't important, it seems as if a commitment to the idea that reality extends beyond our world in this way may get me expelled from the naturalist club.
I’m very reluctant to name this necessary existent “God”, since that unavoidably summons up a cluster of attributes and associations which go far beyond my commitments. But there is no getting around the fact that I may be moving into the vicinity of theism.
Monday, March 12, 2007
Priority Monism
[UPDATE: 25 Sept.2009: Links fixed, but note the post refers to an earlier draft of the paper]
As a follow-up to the last post, I want to very briefly take note of Jonathan Schaffer’s noble attempt to argue that the fundamental (ontologically prior) level is the whole rather than the parts – “priority monism”.
The draft paper “Monism: the Priority of the Whole” includes a fairly lengthy discussion of a historical context in which the case for monism has mostly gone unappreciated (it is often caricatured and dismissed as the position that there exists exactly one thing). He takes some time explicating the idea that both the whole and the parts exist, one of these must be prior, and the choice of either the whole or the parts is an exclusive and exhaustive list of options. Then it is “game on” to see which prevails.
There are four sections to the argument over priority. Two of these I consider a tie: the argument over which comports best with common sense, and which option better explains the apparent heterogeneity of the world (he’s right to say that saying pluralism explains heterogeneity begs the question).
The next section asks what fits best with science. Here, I think Schaffer makes a mistake. He invokes the idea of entanglement from quantum mechanics and infers that the whole world is entangled, making reference to a wave function for the entire universe. In my opinion, this is wrong. The entire world would be entangled only from a perspective standing outside the universe. There is no wave function for the entire universe. The interactions (measurements) between the many quantum systems in the world constitute concrete reality, and the whole of the concrete world is the relational network of these many interactions.
The last section asks which view on priority best deals with the possibility of the world being made of “gunk”, which is stuff with no proper parts (or to put it another way, stuff which is infinitely divisible). Schaffer references a couple of scientific theories and speculations that physical entities might be infinitely divisible. Here I think the existence of the Planck scale is actually good evidence of a limit to divisibility, so again his attempt to invoke science doesn't succeed.
I think that if we’re speaking of our concrete world, then the parts are prior to the whole. The possibility is open, however, that there is a holistic non-concrete ground of possibilia which supports the parts, but this would be a different discussion.
Monday, March 05, 2007
Must there be a Ground-Floor Turtle?
I have a strong intuition that there is a fundamental level of reality which ultimately grounds the phenomena of the world. This intuition forms a basis for preferring certain philosophical arguments over others. For instance, take the cosmological argument. In one traditional formulation, the argument goes something like this: every effect has a cause, and if you follow the chain backwards in time, there must be a first cause. I’ve never felt this argument was very forceful – what’s wrong with an infinite chain of causes? Now, however, if you recast the argument as saying that the contingent facts of the world ultimately and necessarily depend on a fundamental fact or collection of facts, then suddenly I start nodding my head affirmatively. There can’t be an infinite chain of contingent facts depending on other contingent facts, can there? Ontological priority seems to need a starting point more urgently than temporal priority. In the famous expression invoked by Ross P Cameron in his recent paper on this topic (found via OPP): it can’t be turtles all the way down, right?
Our desire for explanations seems to drive the intuition. If an entity is shown to depend on something else, it is thought to be explained. We want this search for explanation to find an ending point in terms of ultimate constituents. In our world, we seem indisputably to encounter composite things which seem comprised of parts; this drives our search for reductionist explanations. I guess it is possible to think that perhaps the “ceiling” rather than the “floor” is fundamental; perhaps the whole of the universe is the fundamental thing and the parts ontologically depend on the whole. Now, this seems counterintuitive to me: if we start with a whole, why should there be any parts? In any case, the direction of dependence is probably less important for this discussion than the idea that there is some fundamental level.
In his paper, Cameron asks whether there is a good argument for the truth of this intuition that there cannot be an unending chain of ontological dependence. Can we, for instance, argue that if there were no fundamental level grounding other entities, then nothing would be real? Cameron concludes that this would essentially be restating the intuition, rather than providing an argument. He considers a couple of other strategies in the paper and finds no satisfactory argument. On the other hand, he doesn’t see any good arguments against the intuition either. In fact the search for a metaphysical argument for the intuition may be seen to parallel the search for a deeper and deeper ontological level: you have to start somewhere, don’t you? Why not with an intuition? He notes as an example that Leibniz never argues for the Principle of Sufficient Reason, it’s just his starting point. Now, one can’t thereby defeat a skeptic who doesn’t share the intuition, but at the end of the day I don't find that the skeptics and deflationists of the world provide very good metaphysical explanations themselves.
Cameron says we can justify the intuition against infinitely descending chains of dependence by appeal to theoretical utility. We can give better explanations for entities if we identify an ultimate ontological basis in a collection of independent entities. This may be reason enough. He notes that this won’t convince someone who thinks the search for metaphysical explanation is misguided to begin with. On the other hand he says: ”…if you believe in metaphysical explanation you should believe it bottoms out somewhere.” He ends the paper by noting that given the pragmatic way the use of the principle is being justified, we should be modest about holding forth about the necessity of its truth.
Also interesting in this context is David Chalmers' recent paper: Ontological Anti-Realism (see blog post with links). His support for anti-realism in the paper (most of which is devoted to a mapping out of the terrain of meta-ontological stances) holds out the possibility of an exception for realism about the fundamental level. Jonathan Schaffer, in his commentary on the paper argues that Chalmers’ framework actually requires realism about the fundamental level. If Schaffer’s arguments are right, it seems to help to bolster the case that if you want to pursue metaphysical explanations, you need to be a realist about the fundamental level. |
b744765fd9664ce9 | Take the 2-minute tour ×
I am using Mathematica to construct a matrix for the Hamiltonian of some system. I have built this matrix already, and I have found the eigenvalues and the eigenvectors, I am uncertain if what I did next is correct: I took the normalized eigenvectors, placed them in matrix form, and did matrix multiplication with the basis set of solutions.
Let me try to be more precise since I am not sure I am using the right language when mentioning the basis solutions. In the problem we are using the set of solutions of the particle in a box model as our basis. I can increase the number of basis elements in the calculation of the matrix of the Hamiltonian (which amounts to doing $<\psi_n|H \psi_k>$ over a specified range of $n$ and $k$) in order for some of my smallest eigenvalues to begin to converge. Once I have this $H$ matrix built, and that I see that my eigenvalues are converging to some degree, I take the eigenvectors of the $H$ matrix, format them to be in matrix form, and multiply them by the set of basis solutions.
I hope that makes things clearer.
share|improve this question
I'm not sure what you are asking; eigenfunctions are a type of eigenvector, in that they satisfy the eigenvalue equation. If your eigenvectors are functions, then you already have your eigenfunctions. – KDN Feb 21 '13 at 18:26
@KDN, the eigenvectors I find are just numbers, it is my understanding that the eigenfunctions should be a linear combination of the basis solutions with the eigenvectors I found as coefficients. – user17338 Feb 21 '13 at 18:46
I think I see the confusion; The eigen values are just numbers. The eigenfunctions are the eigenvectors of the operator. In the expression $A \Psi_n = \lambda_n \Psi_n$, the $\lambda_n$ (the numbers) are the eigenvalues, and the $\Psi_n$ are the eigenfunctions, which are also the eigenvectors of $A$. – KDN Feb 21 '13 at 19:35
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1 Answer
up vote 2 down vote accepted
If $\bf{v}$ is an eigenvector of the matrix $\bf{H}$ (where the ith row and jth column of $\bf{H}$ is $<\psi_i|H|\psi_j>$) with eigenvalue $\lambda$, i.e.
$$\bf{H} \cdot \bf{v} = \lambda \cdot \bf{v}$$
the function (which is the one you are looking for)
$$ \varphi = \bf{v} \cdot \bf{\psi} = \sum_j \bf{v}_j \cdot \psi_j$$
is an 'eigenfunction' (solution of the Schrödinger equation) of the Hamiltonian corresponding to $\bf{H}$ because for all $i$:
$$ <\psi_i|H|\varphi> = \sum_j \bf{v}_j \cdot <\psi_i|H|\psi_j> = \sum_j \bf{v}_j \cdot \bf{H}_{ij} = (\bf{H} \cdot \bf{v})_i = (\lambda \cdot \bf{v})_i$$
Assuming your set of basis functions is orthonormal, i.e. $<\psi_i|\psi_j>= \delta_{ij}$ one can rewrite the above expression as:
$$<\psi_i|H|\varphi> = \sum_j \delta_{ij} \cdot \lambda \cdot \bf{v}_j = \lambda \cdot \sum_j \bf{v}_j <\psi_i|\psi_j> = \lambda \cdot <\psi_i|\sum_j \psi_j> = \lambda \cdot <\psi_i|\varphi>$$
because this holds for all $i$
$$H|\varphi> = \lambda \cdot |\varphi>$$
You say that you put the eigenvector $\bf{v}$ in Matrix form and then multiply it with the vector of basis functions to obtain the function $\varphi$. In fact it should be more like a 'dot product' but if you put the numbers of the eigenvector onto the diagonal (and leave zeros off the diagonal), that should be equivalent.
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0c20597e82e5c77d | Mathematics for Chemistry/Print version
From Wikibooks, open books for an open world
< Mathematics for Chemistry
Jump to: navigation, search
Table of contents[edit]
1. Introduction
2. Number theory
3. Functions
4. Units and dimensions
5. Statistics
6. Plotting graphs
7. Complex numbers
8. Trigonometry
9. Vectors
10. Matrices and determinants
11. Differentiation
12. Integration
13. Some useful aspects of calculus
14. Enzyme kinetics
15. Some mathematical examples applied to chemistry
16. Tests and exams
17. Further reading
1. This book was initially derived from a set of notes used in a university chemistry course. It is hoped it will evolve into something useful and develop a set of open access problems as well as pedagogical material.
For many universities the days when admission to a Chemistry, Chemical Engineering, Materials Science or even Physics course could require the equivalent of A-levels in Chemistry, Physics and Mathematics are probably over for ever. The broadening out of school curricula has had several effects, including student entry with a more diverse educational background and has also resulted in the subject areas Chemistry, Physics and Mathematics becoming disjoint so that there is no co-requisite material between them. This means that, for instance, physics cannot have any advanced, or even any very significant mathematics in it. This is to allow the subject to be studied without any of the maths which might be first studied by the A-level maths group at the ages of 17 and 18. Thus physics at school has become considerably more descriptive and visual than it was 20 years ago. The same applies to a lesser extent to chemistry.
Quantitative methods in chemistry[edit]
There are several reasons why numerical (quantitative) methods are useful in chemistry:
• Chemists need numerical information concerning reactions, such as how much of a substance is consumed, how long does this take, how likely is the reaction to take place.
• Chemists work with a variety of different units, with wildly different ranges, which one must be able to use and convert with ease.
• Measurements taken during experiments are not perfect, so evaluation and combination of errors is required.
• Predictions are often desired, and relationships represented as equations are manipulated and evaluated in order to obtain information.
1. ==Numbers==
For more details on this topic, see Number Theory.
Real numbers come in several varieties and forms;
• Integers are whole numbers used for counting indivisible objects, together with negative equivalents and zero, e.g. 42, -7, 0
• Rational numbers can always be expressed as fractions, e.g. 4.673 = 4673/1000.
• Irrational numbers, unlike rational numbers, cannot be expressed as a fraction or as a definite decimal, e.g. and
• Natural numbers are integers that are greater than or equal to zero.
It is also worth noting that the imaginary unit and therefore complex numbers are used in chemistry, especially when dealing with equations concerning waves.
The origin of surds goes back to the Greek philosophers. It is relatively simple to prove that the square root of 2 cannot be a ratio of two integers, no matter how large the integers may become. In a rather Pythonesque incident the inventor of this proof was put to death for heresy by the other philosophers because they could not believe such a pure number as the root of 2 could have this impure property.
(The original use of quadratic equations is very old, Babylon many centuries BC.) This was to allocate land to farmers in the same quantity as traditionally held after the great floods on the Tigris and Euphrates had reshaped the fields. The mathematical technology became used for the same purpose in the Nile delta.
When you do trigonometry later you will see that surds are in the trigonometric functions of the important symmetrical angles, e.g. and so they appear frequently in mathematical expressions regarding 3 dimensional space.
The notation used for recording numbers in chemistry is the same as for other scientific disciplines, and appropriately called scientific notation, or standard form. It is a way of writing both very large and very small numbers in a shortened form compared to decimal notation. An example of a number written in scientific notation is
with 4.65 being a coefficient termed the significand or the mantissa, and 6 being an integer exponent. When written in decimal notation, the number becomes
Numbers written in scientific notation are usually normalised, such that only one digit precedes the decimal point. This is to make order of magnitude comparisons easier, by simply comparing the exponents of two numbers written in scientific notation, but also to minimise transcription errors, as the decimal point has an assumed position after the first digit. In computing and on calculators, it is common for the ("times ten to the power of") to be replaced with "E" (capital e). It is important not to confuse this "E" with the mathematical constant e.
Engineering notation is a special restriction of scientific notation where the exponent must be divisible by three. Therefore, engineering notation is not normalised, but can easily use SI prefixes for magnitude.
Remember that in SI, numbers do not have commas between the thousands, instead there are spaces, e.g. , (an integer) or . Commas are used as decimal points in many countries.
Consider a number , where is the base and is the exponent. This is generally read as “ to the ” or “ to the power of ”. If then it is common to say “ squared”, and if then “ cubed”. Comparing powers (exponentiation) to multiplication for positive integer values of n , it can be demonstrated that
, i.e. four lots of added together
, i.e. multiplied by itself four times.
For , the result is simply . For the result is .
Order of operations[edit]
When an expression contains different operations, they must be evaluated in a certain order. Exponents are evaluated first. Then, multiplication and division are evaluated from left to right. Last, addition and subtraction are evaluated left to right. Parentheses or brackets take precedence over all operations. Anything within parentheses must be calculated first. A common acronym used to remember the order of operations is PEMDAS, for "Parentheses, Exponents, Multiplication, Division, Addition, Subtraction". Another way to remember this acronym is "Please Excuse My Dear Aunt Sally".
Keep in mind that negation is usually considered multiplication. So in the case of , the exponent would be evaluated first, then negated, resulting in a negative number.
Take note of this example:
, let x=5.
If evaluated incorrectly (left-to-right, with no order of operations), the result would be 16. Three plus five gives eight, times two is 16. The correct answer should be 13. Five times two gives ten, plus three gives 13. This is because multiplication is solved before addition.
Partial fractions[edit]
Partial fractions are used in a few derivations in thermodynamics and they are good for practicing algebra and factorisation.
It is possible to express quotients in more than one way. Of practical use is that they can be collected into one term or generated as several terms by the method of partial fractions. Integration of a complex single term quotient is often difficult, whereas by splitting it up into a sum, a sum of standard integrals is obtained. This is the principal chemical application of partial fractions.
An example is
In the above must equal since the denominators are equal. So we set first to +1 giving . Therefore B = -1/2. If we set instead , therefore . So
We can reverse this process by use of a common denominator.
The numerator is , so it becomes
which is what we started from.
So we can generate a single term by multiplying by the denominators to create a common denominator and then add up the numerator to simplify. A typical application might be to convert a term to partial fractions, do some calculus on the terms, and then regather into one quotient for display purposes. In a factorised single quotient it will be easier to see where numerators go to zero, giving solutions to , and where denominators go to zero giving infinities.
A typical example of a meaningful infinity in chemistry might be an expression such as
The variable is the energy E, so this function is small everywhere, except near . Near a resonance occurs and the expression becomes infinite when the two energies are precisely the same. A molecule which can be electronically excited by light has several of these resonances.
Here is another example. If we had to integrate the following expression we would first convert to partial fractions:
let then
let then
therefore the expression becomes
Later you will learn that these expressions integrate to give simple expressions.
Polynomial division[edit]
This is related to partial fractions in that its principal use is to facilitate integration.
Divide out
like this
3x - 7
x + 1 ) 3x2 -4x -6
3x2 +3x
0 -7x -6
-7x -7
So our equation becomes
This can be easily differentiated, and integrated. If this is differentiated with the quotient formula it is considerably harder to reduce to the the same form. The same procedure can be applied to partial fractions.
Substitutions and expansions[edit]
You can see the value of changing the variable by simplifying
This is an example of simplification. It would actually be possible to differentiate this with respect to either or using only the techniques you have been shown. The algebraic manipulation involves differentiation of a quotient and the chain rule.
Evaluating gives
Expanding this out to the s and s would look ridiculous.
Substitutions like this are continually made for the purpose of having new, simpler expressions, to which the rules of calculus or identities are applied.
A reader has identified this chapter as an undeveloped draft or outline.
You can help to develop the work, or you can ask for assistance in the project room.
Functions as tools in chemistry[edit]
The quadratic formula[edit]
In order to find the solutions to the general form of a quadratic equation,
there is a formula
(Notice the line over the square root has the same priority as a bracket. Of course we all know by now that is not equal to but errors of priority are among the most common algebra errors in practice).
There is a formula for a cubic equation but it is rather complicated and unlikely to be required for undergraduate-level study of chemistry. Cubic and higher equations occur often in chemistry, but if they do not factorise they are usually solved by computer.
Notice the scope or range of the bracket.
Notice here that the variable is a concentration, not the ubiquitous .
Units and dimensions[edit]
1. ==Units, multipliers and prefixes==
It is usually necessary in chemistry to be familiar with at least three systems of units, Le Système International d'Unités (SI), atomic units as used in theoretical calculations and the unit system used by the experimentalists. Thus if we are dealing with the ionization energy, the units involved will be the Joule (J), the Hartree (Eh, the atomic unit of energy), and the electron volt (eV).
These units all have their own advantages;
• The SI unit should be understood by all scientists regardless of their field.
• The atomic unit system is the natural unit for theory as most of the fundamental constants are unity and equations can be cast in dimensionless forms.
• The electron volt comes from the operation of the ionization apparatus where individual electrons are accelerated between plates which have a potential difference in Volts.
An advantage of the SI system is that the dimensionality of each term is made clear as the fundamental constants have structure. This is a complicated way of saying that if you know the dimensionality of all the things you are working with you know an awful lot about the mathematics and properties such as scaling with size of your system. Also, the same system of units can describe both the output of a large power station (gigaJoules), or the interaction of two inert gas atoms, (a few kJ per mole or a very small number of Joules per molecule when it has been divided by Avogadro's number).
In SI the symbols for units are lower case unless derived from a person's name, e.g. ampere is A and kelvin is K.
SI base units
Name Symbol Quantity
metre m length
kilogram kg mass
second s time
ampere A electric current
kelvin K thermodynamic temperature
candela cd luminous intensity
mole mol amount of substance
Derived units used in chemistry
Quantity Unit Name Symbol
Area m2
Volume m3
Velocity m s-1
Acceleration m s-2
Density kg m-3
Entropy J mol-1 K-1
Force kg m s-2 newton N
Energy N m joule J
Pressure N m-2 pascal Pa
Frequency s-1 hertz Hz
Approved prefixes for SI units
Prefix Factor Symbol
atto 10-18 a
femto 10-15 f
pico 10-12 p
nano 10-9 n
micro 10-6 μ
milli 10-3 m
centi 10-2 c
deci 10-1 d
kilo 103 k
mega 106 M
giga 109 G
tera 1012 T
peta 1015 P
exa 1018 E
Note the use of capitals and lower case and the increment on the exponent being factors of 3. Notice also centi and deci are supposed to disappear with time leaving only the powers of 1000.
Conversion factors[edit]
The , (sometimes call caret or hat), sign is another notation for to the power of. E means times 10 to the power of, and is used a great deal in computer program output.
An approximation of how much of a chemical bond each energy corresponds to is placed next to each one. This indicates that light of energy 4 eV can break chemical bonds and possibly be dangerous to life, whereas infrared radiation of a few cm-1 is harmless.
• 1 eV = 96.48530891 kJ mol-1 (Near infrared), approximately 0.26 chemical bonds
• 1 kcal mol-1 = 4.184000000 kJ mol-1 (Near infrared), approximately 0.01 chemical bonds
• 1 MHz = 0.399031E-06 kJ mol-1 (Radio waves), approximately 0.00 chemical bonds
• 1 cm-1 = 0.01196265819 kJ mol-1 (Far infrared), approximately 0.00 chemical bonds
Wavelength, generally measure in nanometres and used in UV spectroscopy is defined as an inverse and so has a reciprocal relationship.
There is the metre, the Angstrom (10-10 m), the micron (10-6 m), the astronomical unit (AU)and many old units such as feet, inches and light years.
The radian to degree conversion is 57.2957795130824, (i.e. a little bit less than 60, remember your equilateral triangle and radian sector).
Dipole moment[edit]
1 Debye = 3.335640035 Cm x 10-30 (coulomb metre)
Magnetic Susceptibility[edit]
1 cm3 mol-1 = 16.60540984 JT-2 x 1030 (joule tesla2)
Old units[edit]
Occasionally, knowledge of older units may be required. Imperial units, or convert energies from BTUs in a thermodynamics project, etc.
In university laboratory classes you would probably be given material on the Quantity Calculus notation and methodology which is to be highly recommended for scientific work.
A good reference for units, quantity calculus and SI is: I. Mills, T. Cuitas, K. Homann, N. Kallay, K. Kuchitsu, Quantities, units and symbols in physical chemistry, 2nd edition, (Oxford:Blackwell Scientific Publications,1993).
Greek alphabet[edit]
Unit labels[edit]
The labelling of tables and axes of graphs should be done so that the numbers are dimensionless, e.g. temperature is ,
and energy mol / kJ etc.
This can look a little strange at first. Examine good text books like Atkins Physical Chemistry which follow SI carefully to see this in action.
The hardest thing with conversion factors is to get them the right way round. A common error is to divide when you should be multiplying, also another common error is to fail to raise a conversion factor to a power.
1 eV = 96.48530891 kJ mol-1
1 cm-1 = 0.01196265819 kJ mol-1
To convert eV to cm-1, first convert to kJ per mole by multiplying by 96.48530891 / 1. Then convert to cm-1 by multiplying by 1 / 0.01196265819 giving 8065.540901. If we tried to go directly to the conversion factor in 1 step it is easy to get it upside down. However, common sense tell us that there are a lot of cm-1s in an eV so it should be obviously wrong.
1 inch = 2.54 centimetres. If there is a surface of nickel electrode of 2 * 1.5 square inches it must be 2 * 1.5 * 2.542 square centimetres.
To convert to square metres, the SI unit we must divide this by 100 * 100 not just 100.
Dimensional analysis[edit]
The technique of adding unit labels to numbers is especially useful, in that analysis of the units in an equation can be used to double-check the answer.
An aside on scaling[edit]
One of the reasons powers of variables are so important is because they relate to the way quantities scale. Physicists in particular are interested in the way variables scale in the limit of very large values. Take cooking the turkey for Christmas dinner. The amount of turkey you can afford is linear, (power 1), in your income. The size of an individual serving is quadratic, (power 2), in the radius of the plates being used. The cooking time will be something like cubic in the diameter of the turkey as it can be presumed to be linear in the mass.
(In the limit of a very large turkey, say one the diameter of the earth being heated up by a nearby star, the internal conductivity of the turkey would limit the cooking time and the time taken would be exponential. No power can go faster / steeper than exponential in the limit. The series expansion of goes on forever even though gets very small.)
Another example of this is why dinosaurs had fatter legs than modern lizards. If dinosaurs had legs in proportion to small lizards the mass to be supported rises as length to the power 3 but the strength of the legs only rises as the area of the cross section, power 2. Therefore the bigger the animal the more enormous the legs must become, which is why a rhino is a very chunky looking version of a pig.
There is a very good article on this in Cooper, Necia Grant; West, Geoffrey B., Particle Physics: A Los Alamos Primer, ISBN 0521347807
Making tables[edit]
Definition of errors[edit]
For a quantity the error is defined as . Consider a burette which can be read to ±0.05 cm3 where the volume is measured at 50 cm3.
• The absolute error is
• The fractional error is ,
• The percentage error is %
Combination of uncertainties[edit]
In an experimental situation, values with errors are often combined to give a resultant value. Therefore, it is necessary to understand how to combine the errors at each stage of the calculation.
Addition or subtraction[edit]
Assuming that and are the errors in measuring and , and that the two variables are combined by addition or subtraction, the uncertainty (absolute error) may be obtained by calculating
which can the be expressed as a relative or percentage error if necessary.
Multiplication or division[edit]
Assuming that and are the errors in measuring and , and that the two variables are combined by multiplication or division, the fractional error may be obtained by calculating
Plotting graphs[edit]
The properties of graphs[edit]
The most basic relationship between two variables and is a straight line, a linear relationship.
The variable is the gradient and is a constant which gives the intercept. The equations can be more complex than this including higher powers of , such as
This is called a quadratic equation and it follows a shape called a parabola. High powers of can occur giving cubic, quartic and quintic equations. In general, as the power is increased, the line mapping the variables wiggles more, often cutting the -axis several times.
Plot between -3 and +2 in units of 1.
Plot between -4 and +1 in units of 1.
Plot between -5 and +4 in units of 1.
Complex numbers[edit]
1. ==Introduction to complex numbers==
The equation:
Does not factorise
without complex numbers does not exist. However the number behaves exactly like any other number in algebra without any anomalies, allowing us to solve this problem.
The solutions are .
is an imaginary number. is a complex number.
Two complex numbers are added by .
Subtraction is obvious: .
Division can be worked out as an exercise. It requires as a common denominator. This is , (difference of two squares), and is .
This means
In practice complex numbers allow one to simplify the mathematics of magnetism and angular momentum as well as completing the number system.
There is an apparent one to one correspondence between the Cartesian plane and the complex numbers, . This is called an Argand diagram. The correspondence is illusory however, because say for example you raise the square root of to a series of ascending powers. Rather than getting larger it goes round and round in circles around the origin. This is not a property of ordinary numbers and is one of the fundamental features of behaviour in the complex plane.
Plot on the same Argand diagram
(Answers -2 plus or minus 5i, 3/2 plus or minus 2i, i(-1 plus or minus root 2)
2 important equations to be familiar with, Euler's equation:
and de Moivre's theorem:
Euler's equation is obvious from looking at the Maclaurin expansion of .
To find the square root of we use de Moivre's theorem.
so de Moivre's theorem gives
Check this by squaring up to give .
The other root comes from:
de Moivre's theorem can be used to find the three cube roots of unity by
where can be .
Put , and
This is the difference of two squares so
Similarly any collection of th roots of 1 can be obtained this way.
Another example is to get the expressions for and without expanding .
Remember Pascal's Triangle
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
Separating the real and imaginary parts gives the two expressions. This is a lot easier than
Use the same procedure to get
and .
1. ==Free Web Based Material from UK HEFCE==
There is a DVD on trigonometry at Math Tutor.
Trigonometry - the sin and cosine rules[edit]
Trigonometric triangle
In the following trigonometric identities, , and are the lengths of sides in a triangle, opposite the corresponding angles , and .
• The sin rule
• The cosine rule
• The ratio identity
Trigonometric identities[edit]
Remember this is a consequence of Pythagoras' theorem where the length of the hypotenuse is 1.
The difference of two angles can easily be generated by putting and remembering and . Similarly, the double angle formulae are generated by induction. is a little more complicated but can be generated if you can handle the fractions! The proofs are in many textbooks but as a chemist it is not necessary to know them, only the results.
Identities and equations[edit]
Identities and equations look very similar, two things connected by an equals sign. An identity however is a memory aid of a mathematical equivalence and can be proved. An equation represents new information about a situation and can be solved.
For instance,
is an identity. It cannot be solved for . It is valid for all . However,
is an equation where .
If you try and solve an identity as an equation you will go round and round in circles getting nowhere, but it would be possible to dress up into a very complicated expression which you could mistake for an equation.
Some observations on triangles[edit]
Check you are familiar with your elementary geometry. Remember from your GCSE maths the properties of equilateral and iscoceles triangles. If you have an iscoceles triangle you can always dispense with the sin and cosine rules, drop a perpendicular down to the base and use trig directly. Remember that the bisector of a side or an angle to a vertex cuts the triangle in two by area, angle and length. This can be demonstrated by drawing an obtuse triangle and seeing that the areas are .
The interior angles of a polygon[edit]
Remember that the interior angles of a -sided polygon are n * 180 -360,
For benzene there are six equilateral triangles if the centre of the ring is used as a vertex, each of which having an interior angle of 120 degrees. Work out the the angles in azulene, (a hydrocarbon with a five and a seven membered ring), assuming all the C-C bond lengths are equal, (this is only approximately true).
1. ==Free Web Based Material from HEFCE==
There is a DVD on vectors at Math Tutor.
Imagine you make a rail journey from Doncaster to Bristol, from where you travel up the West of the country to Manchester. Here you stay a day, travelling the next morning to Glasgow, then across to Edinburgh. At the end of a day's work you return to Doncaster. Mathematically this journey could be represented by vectors, (in 2 dimensions because we are flat earthers on this scale). At the end of the 2nd journey (D-B) + (B-M) you are only a short distance from Doncaster, 50 miles at 9.15 on the clockface. Adding two more vectors, (journeys) takes you to Edinburgh, (about 250 miles at 12.00). Completing the journey leaves you at a zero vector away from Doncaster, i.e. all the vectors in this closed path add to zero.
Mathematically we usually use 3 dimensional vectors over the 3 Cartesian axes , and .
It is best always to use the conventional right handed axes even though the other way round is equally valid if used consistently. The wrong handed coordinates can occasionally be found erroneously in published research papers and text books. The memory trick is to think of a sheet of graph paper, is across as usual and up the paper. Positive then comes out of the paper.
A unit vector is a vector normalised, i.e. multiplied by a constant so that its value is 1. We have the unit vectors in the 3 dimensions:
so that
The hat on the i, j, k signifies that it is a unit vector. This is usually omitted.
Our geographical analogy allows us to see the meaning of vector addition and subtraction. Vector products are less obvious and there are two definitions the scalar product and the vector product. These are different kinds of mathematical animal and have very different applications. A scalar product is an area and is therefore an ordinary number, a scalar. This has many useful trigonometrical features.
The vector product seems at first to be defined rather strangely but this definition maps onto nature as a very elegant way of describing angular momentum. The structure of Maxwell's Equations is such that this definition simplifies all kinds of mathematical descriptions of atomic / molecular structure and electricity and magnetism.
A summary of vectors[edit]
The unit vectors in the 3 Cartesian dimensions:
a vector is:
Vector magnitude[edit]
A constant times a vector[edit]
Vector addition[edit]
Vector subtraction[edit]
Scalar Product[edit]
Notice that if this reduces to a square.
If A and B have no common non-zero components in , and the value is zero corresponding to orthogonality, i.e. they are at right angles. (This can also occur by sign combinations making zero. corresponding to non axis-hugging right angles.)
Vector product[edit]
The minus sign on the middle term comes from the definition of the determinant, explained in the lecture. Determinants are defined that way so they correspond to right handed rotation. (If you remember our picture of going round the circle, as one coordinate goes up, i.e. more positive, another must go down. Therefore rotation formulae must have both negative and positive terms.) Determinants are related to rotations and the solution of simultaneous equations. The solution of simultaneous equations can be recast in graphical form as a rotation to a unit vector in -dimensional space so therefore the same mathematical structures apply to both space and simultaneous equations.
Matrices and determinants[edit]
1. ==Simultaneous linear equations==
If we have 2 equations of the form we may have a set of simultaneous equations. Suppose two rounds of drinks are bought in a cafe, one round is 4 halves of orange juice and 4 packets of crisps. This comes to 4 pounds 20. The thirstier drinkers at another table buy 4 pints of orange juice and only 1 packet of crisps and this comes to 6 pounds 30. So we have:
If you plot these equations they will be simultaneously true at and .
Notice that if the two rounds of drinks are 2 pints and 2 packets of crisps and 3 pints and 3 packets of crisps we cannot solve for the prices! This corresponds to two parallel straight lines which never intersect.
If we have the equations:
If these are simultaneously true we can find a unique solution for both and .
By subtracting the 2 equations a new equation is created where has disappeared and the system is solved.
Substituting back gives us .
This was especially easy because had the same coefficient in both equations. We can always multiply one equation throughout by a constant to make the coefficients the same.
If the equations were:
things would go horribly wrong when you tried to solve them because they are two copies of the same equation and therefore not simultaneous. We will come to this later, but in the meantime notice that 3 times 8 = 4 times 6. If our equations were:
we can still solve them but would require a lot of algebra to reduce it to three (2x2) problems which we know we can solve. This leads on to the subject of matrices and determinants.
Simultaneous equations have applications throughout the physical sciences and range in size from (2x2)s to sets of equations over 1 million by 1 million.
Practice simultaneous equations[edit]
Notice that you can solve:
because it breaks down into a (2x2) and is not truly a (3x3). (In the case of the benzene molecular orbitals, which are (6x6), this same scheme applies. It becomes two direct solutions and two (2x2) problems which can be solved as above.)
The multiplication of matrices has been explained in the lecture.
but cannot exist. To be multiplied two matrices must have the 1st matrix with the same number of elements in a row as the 2nd matrix has elements in its columns.
where the s are the elements of .
Look at our picture of and as represented by a unit vector in a circle. The rotation of the unit vector about the -axis can be represented by the following mathematical construct.
In two dimensions we will rotate the vector at 45 degrees between and :
This is if we rotate by +45 degrees. For and . So the rotation flips over to give . The minus sign is necessary for the correct mathematics of rotation and is in the lower left element to give a right handed sense to the rotational sign convention.
As discussed earlier the solving of simultaneous equations is equivalent in some deeper sense to rotation in -dimensions.
Matrix multiply practice[edit]
i) Multiply the following (2x2) matrices.
ii) Multiply the following (3x3) matrices.
You will notice that this gives a unit matrix as its product.
The first matrix is the inverse of the 2nd. Computers use the inverse of a matrix to solve simultaneous equations.
If we have
In matrix form this is....
In terms of work this is equivalent to the elimination method you have already employed for small equations but can be performed by computers for simultaneous equations.
(Examples of large systems of equations are the fitting of reference data to 200 references molecules, dimension 200, or the calculation of the quantum mechanical gradient of the energy where there is an equation for every way of exciting 1 electron from an occupied orbital to an excited, (called virtual, orbital, (typically equations.)
Finding the inverse[edit]
How do you find the inverse... You use Maple or Matlab on your PC but if the matrix is small you can use the formula...
Here Adj A is the adjoint matrix, the transposed matrix of cofactors. These strange objects are best described by example.....
This determinant is equal to: 1 ( 1 x 1 - 1 x (-1)) - (-1) ( 2 x 1 - 1 x 3) + 2 ( 2 x (-1) - ( 1 x 3) each of these terms is called a cofactor.
This thing gives the sign alternation in a form mathematicians like even though it is incomprehensible.
Use the determinant
to solve the simultaneous equations on page 47 by the matrix inverse method. The matrix corresponding to the equations on p47.2 is:
1 -1 2 6
2 1 1 = 3
3 -1 1 6
The cofactors are
2 1 -5
-1 -5 -2
-3 3 3
You may find these 9 copies of the matrix useful for
striking out rows and columns to form this inverse....
1 -1 2 1 -1 2 1 -1 2
2 1 1 2 1 1 2 1 1
3 -1 1 3 -1 1 3 -1 1
1 -1 2 1 -1 2 1 -1 2
2 1 1 2 1 1 2 1 1
3 -1 1 3 -1 1 3 -1 1
1 -1 2 1 -1 2 1 -1 2
2 1 1 2 1 1 2 1 1
3 -1 1 3 -1 1 3 -1 1
These are the little determinants with the -1 to the (n-1) factors
and the value of the determinant is -9.
The transposed matrix of cofactors is
2 -1 -3
1 -5 3
-5 -2 3
So the inverse is
2 -1 -3
-1/9 X 1 -5 3
-5 -2 3
Giving a solution
2 -1 -3 6 1
-1/9 X 1 -5 3 X 3 = -1
-5 -2 3 6 2
This takes a long time to get all the signs right. Elimination by subtracting equations is MUCH easier. However as the computer cannot make sign mistakes it is not a problem when done by computer program.
The following determinant corresponds to an equation which is repeated three times giving an unsolvable set of simultaneous equations.
Matrix multiplication is not necessarily commutative, which in English means does not equal all the time. Multiplication may not even be possible in the case of rectangular rather than square matrices.
I will put a list of the properties and definitions of matrices in an appendix for reference through the later years of the course.
Determinants and the Eigenvalue problem[edit]
In 2nd year quantum chemistry you will come across this object:
You divide by and set to equal to get:
Expand this out and factorise it into two quadratic equations to give:
which can be solved using
Simultaneous equations as linear algebra[edit]
The above determinant is a special case of simultaneous equations which occurs all the time in chemistry, physics and engineering which looks like this:
This equation in matrix form is and the solution is .
This is a polynomial equation like the quartic above. As you know polynomial equations have as many solutions as the highest power of i.e. in this case . These solutions can be degenerate for example the orbitals in benzene are a degenerate pair because of the factorisation of the polynomial from the 6 Carbon-pz orbitals. In the 2nd year you may do a lab exercise where you make the benzene determinant and see that the polynomial is
from which the 6 solutions and the orbital picture are immediately obvious.
The use of matrix equations to solve arbitrarily large problems leads to a field of mathematics called linear algebra.
Matrices with complex numbers in them[edit]
Work out the quadratic equation from the 3 determinants
They are all the same! This exemplifies a deeper property of matrices which we will ignore for now other than to say that complex numbers allow you to calculate the same thing in different ways as well as being the only neat way to formulate some problems.
1. ==Free web-based material from HEFCE==
There is a DVD on differentiation at Math Tutor.
The basic polynomial[edit]
The most basic kind of differentiation is:
There are two simple rules:
1. The derivative of a function times a constant is just the same constant times the derivative.
2. The derivative of a sum of functions is just the sum of the two derivatives.
To get higher derivatives such as the second derivative keep applying the same rules.
One of the big uses of differentiation is to find the stationary points of functions, the maxima and minima. If the function is smooth, (unlike a saw-tooth), these are easily located by solving equations where the first derivative is zero.
The chain rule[edit]
This is best illustrated by example: find given
Let and .
Now and
So using the chain rule we have
Differentiating a product[edit]
Notice when differentiating a product one generates two terms. (Terms are mathematical expression connected by a plus or minus.) An important point is that terms which represent physical quantities must have the same units and dimensions or must be pure dimensionless numbers. You cannot add 3 oranges to 2 pears to get 5 orangopears. Integration by parts also generates an extra term each time it is applied.
Differentiating a quotient[edit]
You use this to differentiate .
Differentiate with respect to
Notice we have .
Evaluate the inner brackets first.
a, b and c are constants. Differentiate with respect to .
Harder differentiation problems[edit]
Differentiate with respect to :
Differentiate with respect to
Differentiate with respect to
Using differentiation to check turning points[edit]
is the tangent or gradient. At a minimum is zero. This is also true at a maximum or an inflection point. The second gradient gives us the nature of the point. If is positive the turning point is a minimum and if it is negative a maximum. Most of the time we are interested in minima except in transition state theory.
If the equation of is plotted, is is possible to see that at there is a third kind of point, an inflection point, where both and are zero.
Plot between -4 and +3, in units of 1. (It will speed things up if you factorise it first. Then you will see there are 3 places where so you only need calculate 5 points.) By factorising you can see that this equation has 3 roots. Find the 2 turning points. (Differentiate once and find the roots of the quadratic equation using . This gives the position of the 2 turning points either side of zero. As the equation is only in it has 3 roots and 2 maxima / minima at the most therefore we have solved everything. Differentiate your quadratic again to get . Notice that the turning point to the left of zero is a maximum i.e. and the other is a minimum i.e. .
What is the solution and the turning point of .
Solve , by factorisation.
(The 3 roots are -3,0 and +2.
Solutions are and , i.e. -1.7863 and 1.1196.
There are 3 coincident solutions at , , at 0 so this is an inflection point.
The roots are 0, 1 and -1.
1. ==Free Web Based Material from HEFCE==
There is a DVD on integration at Math Tutor.
The basic polynomial[edit]
This works fine for all powers except -1, for instance the integral of
is just
-1 is clearly going to be a special case because it involves an infinity when and goes to a steep spike as gets small. As you have learned earlier this integral is the natural logarithm of and the infinity exists because the log of zero is minus infinity and that of negative numbers is undefined.
The integration and differentiation of positive and negative powers[edit]
1/3 x*x*x x*x 2x 2 0 0 0 0
1/3 x*x*x x*x 2x 2 ? ? ? ?
I(x) H(x) G(x) F(x) ln(x) 1/x -1/(x*x)
Here I, H, G and F are more complicated functions involving .
You will be able to work them out easily when you have done more integration. The thing to notice is that the calculus of negative and positive powers is not symmetrical, essentially caused by the pole or singularity for at .
Logarithms were invented by Napier, a Scottish Laird, in the 17th-century. He made many inventions but his most enduring came from the necessity of doing the many long divisions and trigonometric calculations used in astronomy. The Royal Navy in later years devoted great time and expense to developing logarithm technology for the purposes of navigation, leading to the commissioning of the first mechanical stored program computer, which was theoretically sound but could not be made by Charles Babbage between 1833 and 1871.
The heart of the system is the observation of the properties of powers:
This means that if we have the inverse function of we can change a long division into a subtraction by looking up the exponents in a set of tables. Before the advent of calculators this was the way many calculations were done.
Napier initially used logs to the base for his calculations, but after a year or so he was visited by Briggs who suggested it would be more practical to use base 10. However base is necessary for the purposes of calculus and thermodynamics.
Integrating 1/x[edit]
This is true because is our function which reduces or grows at the rate of its own quantity.
This is our definition of a logarithm.
Integrating 1/x like things[edit]
Just as
therefore by the chain rule
Examples of this are:
As the integral of is so the differential of is so
is just a constant so so
This can also be done by the chain rule
What is interesting here is that the 5 has disappeared completely. The gradient of the log function is unaffected by a multiplier! This is a fundamental property of logs.
Some observations on infinity[edit]
Obviously is .
nd are undefined but sometimes a large number over a large number can have defined values. An example is the of 90 degrees, which you will remember has a large opposite over a large hypotenuse but in the limit of an infinitesimally thin triangle they become equal. Therefore the is 1.
Definite integrals (limits)[edit]
Remember how we do a definite integral
where is the indefinite integral of .
Here is an example where limits are used to calculate the 3 areas cut out by a quartic equation:
We see that is a solution so we can do a polynomial division:
x3 -x2 -2x
x-1 ) x4 -2x3 -x2 +2x
x4 -x3
-x3 -x2
-x3 +x2
-2x2 +2x
-2x2 +2x
So the equation is which factorises to
Integration by substitution[edit]
where .
Simple integration of trigonometric and exponential Functions[edit]
i.e. the integral of .
Integration by parts[edit]
This is done in many textbooks and Wikipedia. Their notation might be different to the one used here, which hopefully is the most clear. You derive the expression by taking the product rule and integrating it. You then make one of the into a product itself to produce the expression.
(all integration with respect to . Remember by
( gets differentiated.)
The important thing is that you have to integrate one expression of the product and differentiate the other. In the basic scheme you integrate the most complicated expression and differentiate the simplest. Each time you do this you generate a new term but the function being differentiated goes to zero and the integral is solved. The expression which goes to zero is .
The other common scheme is where the parts formula generates the expression you want on the right of the equals and there are no other integral signs. Then you can rearrange the equation and the integral is solved. This is obviously very useful for trig functions where ad infinitum.
also generates itself and is susceptible to the same treatment.
We now have our required integral on both sides of the equation so
Integration Problems[edit]
Integrate the following by parts with respect to.
Actually this one can be done quite elegantly by parts, to give a two term expression. Work this one out. Expanding the original integrand by Pascal's Triangle gives:
2 3 4 5 6 7 8
x + 7 x + 21 x + 35 x + 35 x + 21 x + 7 x + x
The two term integral expands to
2 3 4 5 6 7 8 9
1/2 x + 7/3 x + 21/4 x + 7 x + 35/6 x + 3 x + 7/8 x + 1/9 x - 1/72
So one can see it is correct on a term by term basis.
If you integrate you will have to apply parts 7 times to get to become 1 thereby generating 8 terms:
7 6 5 4 3
-x cos(x) + 7 x sin(x) + 42 x cos(x) - 210 x sin(x) - 840 x cos(x) +
2520 x sin(x) + 5040 x cos(x) - 5040 sin(x) + c
(Output from Maple.)
Though it looks nasty there is quite a pattern to this, 7, 7x6,7x6x5 ------7! and sin, cos, -sin, -cos, sin, cos etc so it can easily be done by hand.
Differential equations[edit]
First order differential equations are covered in many textbooks. They are solved by integration. (First order equations have , second order equations have and .)
The arbitrary constant means another piece of information is needed for complete solution, as with the Newton's Law of Cooling and Half Life examples.
Provided all the s can be got to one side and the s to another the equation is separable.
This is the general solution.
Typical examples are:
by definition of logs.
This corresponds to:
The Schrödinger equation is a 2nd order differential equation e.g. for the particle in a box
It has taken many decades of work to produce computationally efficient solutions of this equation for polyatomic molecules. Essentially one expands in coefficients of the atomic orbitals. Then integrates to make a differential equation a set of numbers, integrals, in a matrix. Matrix algebra then finishes the job and finds a solution by solving the resultant simultaneous equations.
The calculus of trigonometric functions[edit]
There are many different ways of expressing the same thing in trig functions and very often successful integration depends on recognising a trig identity.
but could also be
(each with an integration constant!).
When applying calculus to these functions it is necessary to spot which is the simplest form for the current manipulation. For integration it often contains a product of a function with its derivative like where integration by substitution is possible.
Where a derivative can be spotted on the numerator and its integral below we will get a function. This is how we integrate .
We can see this function goes to infinity at as it should do.
Integration by rearrangement[edit]
Take for example:
Here there is no function producted in with the powers so we cannot use substitution. However there are the two trig identities
Using these we have
so we have two simple terms which we can integrate.
The Maclaurin series[edit]
We begin by making the assumption that a function can be approximated by an infinite power series in :
By differentiating and setting one gets
Sin, cos and can be expressed by this series approximation
Notice also works for negative .
When differentiated or integrated generates itself!
When differentiated generates .
By using series we can convert a complex function into a polynomial, and can use for small .
In actual fact the kind of approximation used inside computer programs is more like:
These have greater range but are much harder to develop and a bit fiddly on the calculator or to estimate by raw brain power.
We cannot expand this way because is
. However can be expanded.
Work out the series for .
The factorials you have seen in series come from repeated differentiation. also has a statistical meaning as it is the number of unique ways you can arrange objects.
is 1 by definition, i.e. the number of different ways you can arrange 0 objects is 1.
In statistical thermodynamics you will come across many factorials in expressions such as:
Factorials rapidly get unreasonably large: 6! = 720, 8! = 40320 but 12! = 479001600 so we need to divide them out into reasonable numbers if possible, so for example .
Stirling's approximation[edit]
Also in statistical thermodynamics you will find Stirling's approximation:
This is proved and discussed in Atkins' Physical Chemistry.
How can you use series to estimate . Notice that the series for
converges extremely slowly. is much faster because the
denominator becomes large quickly.
Trigonometric power series[edit]
Remember that when you use and
that x must be in radians.....
Calculus revision[edit]
integrate x to the power of x with respect to x
1. Differentiate , with respect to . (Hint - use the chain rule.)
2. Differentiate . (Chain rule and product rule here.)
3. Differentiate . (Hint - split it into a sum of logs first.)
4. Integrate . (Hint - use integration by parts and take the expression to be differentiated as 1.)
1. It is just . Bring a out of each term to simplify to .
2. .
3. - therefore it is 4 times the derivative of .
4. You should get by 1 application of parts.
Some useful aspects of calculus[edit]
1. ==Limits==
Many textbooks go through the proper theory of differentiation and integration by using limits. As chemists it is possible to live without knowing this so we might well not have it as an examinable topic. However here is how we differentiate sin from 1st principles.
As for small this expression is .
Similarly for
This is equal to .
Numerical differentiation[edit]
You may be aware that you can fit a quadratic to 3 points, a cubic to 4 points, a quartic to 5 etc. If you differentiate a function numerically by having two values of the function apart you get an approximation to by constructing a triangle and the gradient is the tangent. There is a forward triangle and a backward triangle depending on the sign of . These are the forward and backward differentiation approximations.
If however you have a central value with a either side you get the central difference formula which is equivalent to fitting a quadratic, and so is second order in the small value of giving high accuracy compared with drawing a tangent. It is possible to derive this formula by fitting a quadratic and differentiating it to give:
HCl r-0.02
sigma (iso) 32.606716 142.905788 -110.299071
HCl r-0.01
sigma (iso) 32.427188 142.364814 -109.937626
HCl r0 Total shielding: paramagnetic : diamagnetic
sigma (iso) 32.249753 141.827855 -109.578102
HCl r+0.01
sigma (iso) 32.074384 141.294870 -109.220487
HCl r+0.02
sigma (iso) 31.901050 140.765819 -108.864769
This is calculated data for the shielding in ppm of the proton in HCl when the bondlength is stretched or compressed by 0.01 of an Angstrom (not the approved unit pm). The total shielding is the sum of two parts, the paramagnetic and the diamagnetic. Notice we have retained a lot of significant figures in this data, always necessary when doing numerical differentiation.
Exercise - use numerical differentiation to calculated d (sigma) / dr and d2 (sigma) / dr2 using a step of 0.01 and also with 0.02. Use 0.01 to calculate d (sigma(para)) / dr and d (sigma(dia)) / dr.
Numerical integration[edit]
Wikipedia has explanations of the Trapezium rule and Simpson's Rule. Later you will use computer programs which have more sophisticated versions of these rules called Gaussian quadratures inside them. You will only need to know about these if you do a numerical project later in the course. Chebyshev quadratures are another version of this procedure which are specially optimised for integrating noisy data coming from an experimental source. The mathematical derivation averages rather than amplifies the noise in a clever way.
Enzyme kinetics[edit]
1. Mathematics for Chemistry/Enzyme kinetics
Some mathematical examples applied to chemistry[edit]
1. ==Variable names==
The ubiquitous is not always the variable as you will all know by now. One problem dealing with real applications is sorting out which symbols are the variables and which are constants. (If you look very carefully at professionally set equations in text books you should find that there are rules that constants are set in Roman type, i.e. straight letters and variables in italics. Do not rely on this as it is often ignored.)
Here are some examples where the variable is conventionally something other than .
1. The Euler angles which are used in rotation are conventionally and not the more usual angle names and . The rotation matrix for the final twist in the commonest Euler definition therefore looks like
2. The energy transitions in the hydrogen atom which give the Balmer series are given by the formula is just a single variable for the energy the tilde being a convention used by spectroscopists to say it is wavenumbers, (cm-1). The H subscript on has no mathematical meaning. It is the Rydberg constant and is therefore in Roman type. is known very accurately as 109,677.581 cm-1. It has actually been known for a substantial fraction of the class to make an error putting this fraction over a common denominator in examination conditions.
3. In the theory of light is used for frequency and not surprisingly for time. Light is an oscillating electric and magnetic field therefore the cosine function is a very good way of describing it. You can also see here the use of complex numbers. Using the real axis of the Argand diagram for the electric field and the imaginary axis for the magnetic field is a very natural description mathematically and maps ideally onto the physical reality. Here we are integrating with respect to and , the operating frequency if it is a laser experiment is a constant, therefore it appears on the denominator in the integration. In this case we can see a physical interpretation for the integration constant. It will be a phase factor. If we were dealing with sunlight we might well be integrating a different function over in order to calculate all of the phenomenon which has different strengths at the different light frequencies. Our integration limits would either be from zero to infinity or perhaps over the range of energies which correspond to visible light.
4. This example is a laser experiment called Second Harmonic Generation. There is an electric field , frequency and a property constant . is a fundamental constant. We have an intense monochromatic laser field fluctuating at the frequency , (i.e. a strong light beam from a big laser). Therefore the term contributes to the polarization. We know from trigonometric identities that can be represented as a cosine of the double angle Therefore the polarization is In this forest of subscripts and Greek letters the important point is that there are two terms contributing to the output coming from which multiplies the rest of the stuff. In summary we have is equal to where everything except the trig(t) and trig(2t) are to some extent unimportant for the phenomenon of doubling the frequency. and differ only in a phase shift so they represent the same physical phenomenon, i.e. light, which has phase. (One of the important properties of laser light is that it is coherent, i.e. it all has the same phase. This is fundamentally embedded in our mathematics.)
van der Waals Energy[edit]
The van der Waals energy between two inert gas atoms can be written simply as a function of
Notice that the term is positive corresponding to repulsion. The term is the attractive term and is negative corresponding to a reduction in energy. A and B are constants fitted to experimental numbers.
This function is very easy to both differentiate and integrate. Work these out. In a gas simulation you would use the derivative to calculate the forces on the atoms and would integrate Newton's equations to find out where the atoms will be next.
Another potential which is used is:
This has 1 more fittable constant. Integrate and differentiate this.
The is called a Lennard-Jones potential and is often expressed using the 2 parameters of an energy and a distance .
is an energy. Set the derivative of this to zero and find out where the van der Waals minimum is. Differentiate again and show that the derivative is positive, therefore the well is a minimum, not a turning point.
A diatomic potential energy surface[edit]
Interaction energy of argon dimer. The long-range part is due to London dispersion forces
In a diatomic molecule the energy is expanded as the bond stretches in a polynomial. We set . At the function is a minimum so there is no term.
Whatever function is chosen to provide the energy setting the 1st derivative to zero will be required to calculate . The 2nd and 3rd derivatives will then need to be evaluated to give the shape of the potential and hence the infra-red spectrum. is usually modelled by a very complicated function whose differentiation is not entered into lightly.
A one-dimensional metal[edit]
A one-dimensional metal is modelled by an infinite chain of atoms 150 picometres apart. If the metal is lithium each nucleus has charge 3 and its electrons are modelled by the function
which repeats every 150 pm. What constant must this function be multiplied by to ensure there are 3 electrons on each atom? (Hint... integrate between either and or -75pm and +75pm according to your equation. This integral is a dimensionless number equal to the number of electrons, so we will have to multiply by a normalisation constant.)
Here we have modelled the density of electrons. Later in the second year you will see electronic structure more accurately described by functions for each independent electron called orbitals. These are subject to rigorous mathematical requirements which means they are quite fun to calculate.
Kepler's Laws[edit]
Another physics problem but a good example of a log-log plot is the radius and time period relations of the planets.
This data is dimensionless because we have divided by the time / distance of the earth. We can take logs of both.
Mercury Venus Earth Mars Jupiter Saturn
r 0.3871 0.7233 1 1.524 5.203 9.539
T 0.2408 0.6152 1 1.881 11.86 29.46
Mercury Venus Earth Mars Jupiter Saturn
log10r -0.4122 0 0.9795
log10T -0.6184 0 1.4692
Try a least squares fit on your spreadsheet program. Using the Earth and Saturn data: (which is extremely bad laboratory practice, to usejust two points from a data set!)
so and
This is Kepler's 3rd law. If you use either a least squares fit gradient or the mercury to saturn data you get the same powers. We have got away with not using a full data set because the numbers given are unusually accurate and to some extent tautological, (remember the planets go round in ellipses not circles!).
Newton's law of cooling[edit]
is the excess temperature of a cooling body over room temperature (20oC say). The rate of cooling is proportional to the excess temperature.
This is a differential equation which we integrate with respect to to get
The water is heated to C and room temperature is C. At the beginning and , so
After 5 minutes the water has cooled to C.
so so
by the definition of logarithms. This gives the plot of an exponential decay between 80 and 20oC.
So after 10 minutes C. After 20 minutes C. After 30 minutes C.
Bacterial Growth[edit]
2 grams of an organism grows by 1/10 gram per day per gram.
This is a differential equation which is solved by integration thus
When we have 2 grams so
For the sample to double in mass
Half life calculations are similar but the exponent is negative.
Partial fractions for the 2nd order rate equation[edit]
In chemistry work you will probably be doing the 2nd order rate equation which requires partial fractions in order to do the integrals.
If you remember we have something like
Put the right-hand side over a common denominator
This gives
By setting x to 3 we get 1 = -B (B=-1). Setting x = 0 and B = -1
1 = 3A -2 (A=+1)
Check 1 = 3 -x -2 +x true...
noting the sign changes on integrating 1/(2-x) not 1/x.
Tests and exams[edit]
1. ==A possible final test with explanatory notes==
This test was once used to monitor the broad learning of university chemists at the end of the 1st year and is intended to check, somewhat lightly, a range of skills in only 50 minutes. It contains a mixture of what are perceived to be both easy and difficult questions so as to give the marker a good idea of the student's algebra skills and even whether they can do the infamous integration by parts.
(1) Solve the following equation for
It factorises with 3 and 5 so : therefore the roots are -5 and +3, not 5 and -3!
(2) Solve the following equation for
Divide by 2 and get .
This factorises with 2 and 5 so : therefore the roots are 5 and -2.
(3) Simplify
Firstly so it becomes .
(4) What is
64 = 8 x 8 so it also equals x i.e. is , therefore the answer is -6.
(5) Multiply the two complex numbers
These are complex conjugates so they are minus x i.e. plus 25 so the total is 34.
(6) Multiply the two complex numbers
The real part is -25 plus the . The cross terms make and so the imaginary part disappears.
(7) Differentiate with respect to :
Expand out the difference of 2 squares first.....collect and multiply....then just differentiate term by term giving:
This needs the product rule.... Factor out the ....
This could be a chain rule problem.......
or you could take the power 2 out of the log and go straight to the same answer with a shorter version of the chain rule to:.
(13) Perform the following integrations:
must be converted to a double angle form as shown many times.... then all 3 bits are integrated giving .......
Apart from , which goes to , this is straightforward polynomial integration. Also there is a nasty trap in that two terms can be telescoped to .
(15) What is the equation corresponding to the determinant:
The first term is the second and the 3rd term zero. This adds up to .
(16) What is the general solution of the following differential equation:
where A is a constant..
(17) Integrate by parts:
Make the factor to be differentiated and apply the formula, taking care with the signs... .
(18)The Maclaurin series for which function begins with these terms?
It is ....
as partial fractions.
It is .....
(20) What is in terms of sin and cos
This is just Euler's equation.....
so one disappears to give ... .
50 Minute Test II[edit]
(1) Simplify
(2)What is
(3) Solve the following equation for
(4) Solve the following equation for
(5) Multiply the two complex numbers
(6) Multiply the two complex numbers
(7) The Maclaurin series for which function begins with these terms?
(8) Differentiate with respect to :
where k is a constant.
where A is a constant.
(14) Perform the following integrations:
(16) What is the equation belonging to the determinant \begin{vmatrix} x & 0 & 0\\ 0 & x & i \\ 0 & i & x \\ \end{vmatrix} = 0</math>
(17) What is the general solution of the following differential equation:
(18) Integrate by any appropriate method:
(19) Express
as partial fractions.
(20) What is in terms of sin and cos.
50 Minute Test III[edit]
(1) Solve the following equation for
(2) What is
(3) The Maclaurin series for which function begins with these terms? |
9d078c4393a6a1e7 | "natural theology > notes > 1999 > 14 February 1999 "
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vol VII: Notes
[Sunday 14 February 1999 - Saturday 20 February 1999]
[BOOK, DB 50]
Sunday 14 February 1999
[page 157]
Monday 15 February 1999
. . . Steering the ship of state. CYBERNETICS. Sum up the steering of a ship a la NORBERT. Wiener
POLYTHEISM -> MONOTHEISM (Pope, Jesus, God, Spirit) -> POLYTHEISM (we are all gods with all these powers)
The MOMENTUM OF THE SHIP OF STATE is a product (function) of its multiplicity.
Once again I have to assume that you have a parent / child / boyfriend / girlfriend/ friend/ somebody who has done enough cybernetics to decode / expand this outrageously precise representation of something deep in modern technology to you with all the passion, excitement and intelligence that it deserves. I take it that this sort of stuff is bread and butter to all the mathematically cultured intelligentsia that purports to advise my governments, so I am allowed to speak in code for sheer want of space.
I notice that our little rat faced PM wants to put God (big G) in the preamble of our constitution and play down the Aboriginal.
The reason, moderation and learning of the English Church was beautifully done but poorly founded. Darwin, as the fathers of the Church knew, totally cut the ground out from under them, but like all those bomb factories shut down
[page 158]
by the end of the cold war, they continue to go through the motions more than a century later because they know nothing else.
The ship of state navigates through a state space which is immensely complex.
This complex state space is the environment of the state and includes the natural world, including all those highly evolved animals called citizens, and also includes all the other states (eg SA, Australia, the US, the UN, etc etc etc)
Our minds (here by mind I mean all internal information processing) are formed by our environment.
Junkies: take the drug for what it is, a painkiller for personalities rather too sensitive for the environment in which they find themselves.
Tuesday 16 February 1999
[page 160]
Wednesday 17 February 1999
An entity can be represented as a transformation (computation / process). My passage through life is manifested by a track of transformation in my environment, like grub tracks radiating
[page 161]
from oviposition under bark.
Looking for a line through space - back to the generalized geodesic. We seek the line of extremal action, the path through heaven. But how do we know how we are eon the right tack? Seek some model to justify the course taken.
We seek the path we want our children to follow - the path of maximum entropy ? complexity / life.
Force: increasing entropy - CANTOR - decreasing entropy E-THEOREM Cantor, Khinchin
KILL = Decrease entropy? Life is a maximum entropy state (if we count our entropy right?)
Maximise entropy for given dynamic realization of the system. Still cannot see it, but it is there. A few hundred more IQ points would surely help. Has to be some measure theoretical link between transfinite numbers.
[page 162]
Just talking ad lib here, on the monkey principle that if we try every possible sentence, we will find the right one (through variation and selection).
Thursday 18 February 1999
Friday 19 February 1999
The moving equilibrium - moving target requires feed forward (prediction) to maintain stability. Simple reaction leads to hunting, overshooting etc.
Cybernetic symmetries in the transfinite network.
Physics is an account of the symmetries in the transfinite network.
Speed up theorems (Goedel) and Cantor's principle of finitism (Skolem Lowenheim theorem?)
Structures: Planetary disc
Newton's law founds cellular automata called gravitating systems.
[page 163]
Static ACTUS PURUS is a CONTRADICTION. GOD MUST GROW if GOD IS ALIVE. Theology models god; here we have a new growing model of go, TNN (transfinite neural network)
The entry of a luscious woman into my life <++> the entry of my life into a luscious woman.
Particle moves in space. Space moves in particle. She enters my environment (me) I enter her environment (her)
A sequence or motif in DNA specifies an action to be performed by the resulting protein, ie it specifies a tool and the proteins of a cell are a set of tools spanning life.
TECHNOLOGY: A spanning set of tools a) tools for life b) tools for culture or civilization.
Release of tools in to the environment causes execution of the tooled activity whenever substrate(s) are present in the warm (random) environment
[page 164]
of the tool.
So the release of lego into the environment of children causes the evolution of lego structures, as car factories cause cars. But car factories do not make roads.
An implemented (realized) equation is a CONSTRAINT on the Universe.
Technology = TOOL
Economics = MARKET
Saturday 20 February 1999
The trouble with Xian god is male, simple and univocal. So now we introduce quantum mechanics, superposition, complexity,
[page 165]
femininity. Newton's linear superposition - the parallelogram of forces.
Central question: Neural network, humanity and superposition.
Theology: private /public experience. Drug is public externally applied chemical that changes private experience.
Addiction is disordered desire, outside the control of the whole system. One should not cancel a desire because some might become addicted to it.
A living god must grow. God became congealed in the notion that actus purus means realization of all possibility.
Possibility is as real as action. Possibility and action are PEERS, not an ORDERED ST. POTENCY IS CREATED BY ACTION and ACTION is driven and defined by POTENCY.
[page 166]
On the measure of justice (GOVERNANCE)
Bandwidth as a measure of JUSTICE.
Further reading
Beale, R, and T Jackson, Neural Computing: An Introduction, Adam Hilger 1991 Jacket: '... starts from basics and goes on to cover all the most important approaches to the subject. ... The capabilities, advantages and disadvantages of each model are discussed as are possible applications of each. The relationship of the models developed to the brain and its functions are also explored.'
Dirac, P A M, The Principles of Quantum Mechanics (4th ed), Oxford UP/Clarendon 1983 Jacket: '[this] is the standard work in the fundamental principles of quantum mechanics, indispensible both to the advanced student and the mature research worker, who will always find it a fresh source of knowledge and stimulation.' (Nature)
Feynman, Richard P, and Robert B Leighton, Matthew Sands, The Feynman Lectures on Physics (volume 3) : Quantum Mechanics, Addison Wesley 1970 Foreword: 'This set of lectures tries to elucidate from the beginning those features of quantum mechanics which are the most basic and the most general. ... In each instance the ideas are introduced together with a detailed discussion of some specific examples - to try to make the physical ideas as real as possible.' Matthew Sands
Khinchin, A I, Mathematical Foundations of Information Theory (translated by P A Silvermann and M D Friedman), Dover 1957 Jacket: 'The first comprehensive introduction to information theory, this book places the work begun by Shannon and continued by McMillan, Feinstein and Khinchin on a rigorous mathematical basis. For the first time, mathematicians, statisticians, physicists, cyberneticists and communications engineers are offered a lucid, comprehensive introduction to this rapidly growing field.'
Pais, Abraham, Inward Bound: Of Matter and Forces in the Physical World, Clarendon Press, Oxford University Press 1986 Preface: 'I will attempt to describe what has been discovered and understood about the constituents of matter, the laws to which they are subject and the forces that act on them [in the period 1895-1983]. . . . I will attempt to convey that these have been times of progress and stagnation, of order and chaos, of belief and incredulity, of the conventional and the bizarre; also of revolutionaries and conservatives, of science by individuals and by consortia, of little gadgets and big machines, and of modest funds and big moneys.' AP
Schwinger, Julian, and (editor), Selected Papers on Quantum Electrodynamics, Dover 1958 Jacket: In this volume the history of quantum electrodynamics is dramatically unfolded through the original words of its creators. It ranges from the initial successes, to the first signs of crisis, and then, with the stimulus of experimental discovery, the new triumphs leading to an unparalleled quantitative accord between theory and experiment. In terminates with the present position in quantum electrodynamics as part of the larger subject of theory of elementary particles, faced with fundamental problems and future prospect of even more revolutionary discoveries.'
van der Waerden, B L, Sources of Quantum Mechanics, Dover Publications 1968 Amazon Book Description: 'Seventeen seminal papers, dating from the years 1917-26, in which the quantum theory as wenow know it was developed and formulated. Among the scientists represented: Einstein,Ehrenfest, Bohr, Born, Van Vleck, Heisenberg, Dirac, Pauli and Jordan. All 17 papers translatedinto English.'
Wiener, Norbert, Cybernetics or control and communication in the animal and the machine, MIT Press 1996 The classic founding text of cybernetics.
Alan Turing, On Computable Numbers, with an application to the Entscheidungsproblem, 'The “computable” numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. Although the subject of this paper is ostensibly the computable numbers, it is almost equally easy to define and investigate computable functions of an integral variable or a real or computable variable, computable predicates, and so forth. The fundamental problems involved are, however, the same in each case, and I have chosen the computable numbers for explicit treatment as involving the least cumbrous technique.' back
Apple Inc., Mac Dev Center:Mac OS X Technology Overview: Mac OSX System Overview, 'A Layered Approach The implementation of Mac OS X can be viewed as a set of layers. At the lower layers of the system are the fundamental services on which all software relies. Subsequent layers contain more sophisticated services and technologies that build on (or complement) the layers below.' back
Dynamics (physics) - Wikipedia, Dynamics (physics) - Wikipedia, the free encyclopedia, 'In the field of physics, the study of the causes of motion and changes in motion is dynamics.' back
FIFA (Federation International de Football Association), FIFA - Laws of the Game, 'On 1 July 2009, the new Laws of the Game, modified at the 123rd Annual General Meeting of the International Football Association Board (IFAB) in Newcastle, Northern Ireland on 28 February 2009, came into force.' back
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I'm slightly confused as to answer this question, someone please help:
Consider a free particle in one dimension, described by the initial wave function $$\psi(x,0) = e^{ip_{0}x/\hbar}e^{-x^{2}/2\Delta^{2}}(\pi\Delta^2)^{-1/4}.$$
Find the time-evolved wavefunctions $\psi(x,t)$.
Now I know that since it is a free particle we have the hamiltonian operator as $$H = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2},$$ which yields the energy eigenfunctions to be of the form $$\psi_E(x,t) = C_1e^{ikx}+C_2e^{-ikx},$$ where $k=\frac{\sqrt{2mE}}{\hbar}$, and the time evolution of the Schrödinger equation gives $$\psi(x,t)=e^{-\frac{i}{\hbar}Ht}\psi(x,0)$$ but the issue I face is what is the correct method to find the solution so that I can then calculate things such as the probability density $P(x,t)$ and the mean and the uncertainty (all which is straight forward once I know $\psi(x,t)$.
In short - how do I find the initial state in terms of the energy eigenfunctions $\psi_E(x,t)$ so that I can find the time evolved state wavefunction.
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More on Gaussian wave packets: physics.stackexchange.com/search?q=Gaussian+wave+packet – Qmechanic May 16 '13 at 23:11
2 Answers 2
For a free particle, the energy/momentum eigenstates are of the form $e^{i k x}$. Going over to that basis is essentially doing a Fourier transform. Once you do that, you'll have the wavefunction in the momentum basis. After that, time-evolving that should be simple.
Hint: The fourier transform of a Gaussian is another Gaussian, but the width inverts, in accordance with the Heisenberg uncertainty principle. The phase and the mean position will transform into each other -- that is a little more subtle and you need to work it out.
Also have a look at http://en.wikipedia.org/wiki/Wave_packet.
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Some broadly applicable background might be in order, since I remember this aspect of quantum mechanics not being stressed enough in most courses.
[What follows is very good to know, and very broadly applicable, but may be considered overkill for this particular problem. Caveat lector.]
What the OP lays out is exactly the motivation for finding how an initial wavefunction can be written as a sum of eigenfunctions of the Hamiltonian - if only we could have that representation, the Schrödinger equation plus linearity get us the wavefunction for all time.
As Siva alludes to, this amounts to finding how a vector (our wavefunction) looks in a particular basis (the set of eigenfunctions of any Hermitian operator is guaranteed to be a basis). In general, one does this by taking inner products with the basis vectors, and the reasoning is as follows.
We know the set of vectors $\{\lvert \psi_E \rangle\}$ (yes, I'm using Dirac notation here - it's a good thing to get used to), where $E$ is an index ranging over (possibly discrete possibly continuous) energies, forms a basis for the space of all wavefunctions. Therefore, there must be complex numbers $c_E$ such that $$ \lvert \psi \rangle = \sum_E c_E \lvert \psi_E \rangle, $$ where $\lvert \psi \rangle$ is our initial wavefunction. If there are infinitely many energies, the sum has infinitely many terms. If there is a continuum of energies, it is an integral.1
Now the problem is clearly one of finding the coefficients $c_E$. To do that, we take inner products with the basis vectors, one by one, where presumably our energy basis is orthonormal. Pick a generic, unspecified basis element $\lvert \psi_{E'} \rangle$. Then we have $$ \langle \psi_{E'} \vert \psi \rangle = \sum_E c_E \langle \psi_{E'} \vert \psi_E \rangle = \sum_E c_E \delta_{E'E} = c_{E'}. $$ Whether the delta function is of the Kronecker or Dirac variety depends on whether the "sum" is a sum or an integral.
Here then we have our formula for coefficients, which reads (after removing the primes), $$ c_E = \langle \psi_E \vert \psi \rangle. $$ How does one go about solving this. At this point, it is okay to switch out of abstract vector notation and go into the position basis. We can do this with the somewhat cryptic yet awesome-sounding spectral resolution of the identity in, say, the position basis: $$ c_E = \langle \psi_E \vert I \vert \psi \rangle = \int_{-\infty}^\infty \langle \psi_E \vert x \rangle \langle x \vert \psi \rangle \ \mathrm{d}x. $$ Here $\langle \psi \vert x \rangle \equiv \psi(x)$ is just your wavefunction, expressed in more familiar terms.2 Furthermore, as you have hopefully been told, the correct inner product at play here introduces a complex conjugation if you switch the ordering, so $$ \langle \psi_E \vert x \rangle = \langle x \vert \psi_E \rangle^* \equiv \psi_E^*(x). $$
You now have enough to evaluate the coefficients $c_E$ for any initial problem given any orthonormal basis arising from a Hamiltonian. Given the free-particle form of $\psi_E(x)$ you can see that this process will essentially be a Fourier transform, so if you keep your wits about you you don't even need to do any messy integrals at all. Furthermore, depending on what is ultimately desired, the position basis may not be the most suitable basis for this problem, but doing a few problems the hard way builds character if nothing else.
1 Math aside: Countable infinities are not a big deal, since one of the assumptions of quantum mechanics is that our vector space isn't just a fancy inner product space, but also a really fancy Hilbert space. Then well-behaved linear combinations of wavefunctions, even countably infinitely many, well converge to perfectly well-defined wavefunctions. Justifying the integral is trickier, but it can be done.
2 Yes, this is the connection between Dirac notation and traditional "probability density as a function of space" notation students often learn first. Abstract kets become functions of position only when "bra-ed" with a generic position basis element.
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Molecular orbital diagram
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Molecular orbital diagram
A molecular orbital diagram, or MO diagram, is a qualitative descriptive tool explaining chemical bonding in molecules in terms of molecular orbital theory in general and the linear combination of atomic orbitals (LCAO) molecular orbital method in particular.[1][2][3] A fundamental principle of these theories is that as atoms bond to form molecules, a certain number of atomic orbitals combine to form the same number of molecular orbitals, although the electrons involved may be redistributed among the orbitals. This tool is very well suited for simple diatomic molecules such as dihydrogen, dioxygen, and carbon monoxide but becomes more complex when discussing even comparatively simple polyatomic molecules, such as methane. MO diagrams can explain why some molecules exist and others do not. They can also predict bond strength, as well as the electronic transitions that can take place.
• History 1
• Basics 2
• s-p mixing 3
• Diatomic MO diagrams 4
• Dihydrogen 4.1
• Dihelium and diberyllium 4.2
• Dilithium 4.3
• Diboron 4.4
• Dicarbon 4.5
• Dinitrogen 4.6
• Dioxygen 4.7
• Difluorine and dineon 4.8
• Dimolybdenum and ditungsten 4.9
• MO energies overview 5
• Heteronuclear diatomics 6
• Triatomic molecules 7
• Carbon dioxide 7.1
• Water 7.2
• References 8
• External links 9
Qualitative MO theory was introduced in 1928 by Robert S. Mulliken[4][5] and Friedrich Hund.[6] A mathematical description was provided by contributions from Douglas Hartree in 1928[7] and Vladimir Fock in 1930.[8]
Molecular orbital diagrams are diagrams of molecular orbital (MO) energy levels, shown as short horizontal lines in the center, flanked by constituent atomic orbital (AO) energy levels for comparison, with the energy levels increasing from the bottom to the top. Lines, often dashed diagonal lines, connect MO levels with their constituent AO levels. Degenerate energy levels are commonly shown side by side. Appropriate AO and MO levels are filled with electrons by the Pauli Exclusion Principle, symbolized by small vertical arrows whose directions indicate the electron spins. The AO or MO shapes themselves are often not shown on these diagrams. For a diatomic molecule, an MO diagram effectively shows the energetics of the bond between the two atoms, whose AO unbonded energies are shown on the sides. For simple polyatomic molecules with a "central atom" such as methane (CH
) or carbon dioxide (CO
), a MO diagram may show one of the identical bonds to the central atom. For other polyatomic molecules, an MO diagram may show one or more bonds of interest in the molecules, leaving others out for simplicity. Often even for simple molecules, AO and MO levels of inner orbitals and their electrons may be omitted from a diagram for simplicity.
In MO theory molecular orbitals form by the overlap of atomic orbitals. Because σ bonds feature greater overlap than π bonds, σ and σ* bonding and antibonding orbitals feature greater energy splitting (separation) than π and π* orbitals. The atomic orbital energy correlates with electronegativity as more electronegative atoms hold their electrons more tightly, lowering their energies. MO modelling is only valid when the atomic orbitals have comparable energy; when the energies differ greatly the mode of bonding becomes ionic. A second condition for overlapping atomic orbitals is that they have the same symmetry.
MO diagram hydrogen
MO diagram for dihydrogen. Here electrons are shown by dots.
Two atomic orbitals can overlap in two ways depending on their phase relationship. The phase of an orbital is a direct consequence of the wave-like properties of electrons. In graphical representations of orbitals, orbital phase is depicted either by a plus or minus sign (which has no relationship to electric charge) or by shading one lobe. The sign of the phase itself does not have physical meaning except when mixing orbitals to form molecular orbitals.
Two same-sign orbitals have a constructive overlap forming a molecular orbital with the bulk of the electron density located between the two nuclei. This MO is called the bonding orbital and its energy is lower than that of the original atomic orbitals. A bond involving molecular orbitals which are symmetric with respect to rotation around the bond axis is called a sigma bond (σ-bond). If the phase changes, the bond becomes a pi bond (π-bond). Symmetry labels are further defined by whether the orbital maintains its original character after an inversion about its center; if it does, it is defined gerade, g. If the orbital does not maintain its original character, it is ungerade, u.
Atomic orbitals can also interact with each other out-of-phase which leads to destructive cancellation and no electron density between the two nuclei at the so-called nodal plane depicted as a perpendicular dashed line. In this anti-bonding MO with energy much higher than the original AO's, any electrons present are located in lobes pointing away from the central internuclear axis. For a corresponding σ-bonding orbital, such an orbital would be symmetrical but differentiated from it by an asterisk as in σ*. For a π-bond, corresponding bonding and antibonding orbitals would not have such symmetry around the bond axis and be designated π and π*, respectively.
The next step in constructing an MO diagram is filling the newly formed molecular orbitals with electrons. Three general rules apply:
• The Aufbau principle states that orbitals are filled starting with the lowest energy
• The Pauli exclusion principle states that the maximum number of electrons occupying an orbital is two, with opposite spins
• Hund's rule states that when there are several MO's with equal energy, the electrons occupy the MO's one at a time before two electrons occupy the same MO.
The filled MO highest in energy is called the Highest Occupied Molecular Orbital or HOMO and the empty MO just above it is then the Lowest Unoccupied Molecular Orbital or LUMO. The electrons in the bonding MO's are called bonding electrons and any electrons in the antibonding orbital would be called antibonding electrons. The reduction in energy of these electrons is the driving force for chemical bond formation. Whenever mixing for an atomic orbital is not possible for reasons of symmetry or energy, a non-bonding MO is created, which is often quite similar to and has energy level equal or close to its constituent AO, thus not contributing to bonding energetics. The resulting electron configuration can be described in terms of bond type, parity and occupancy for example dihydrogen 1σg2. Alternatively it can be written as a molecular term symbol e.g. 1Σg+ for dihydrogen. Sometimes, the letter n is used to designate a non-bonding orbital.
For a stable bond, the bond order, defined as
\ \mbox{Bond Order} = \frac{(\mbox{No. of electrons in bonding MOs}) - (\mbox{No. of electrons in anti-bonding MOs})}{2}
must be positive.
The relative order in MO energies and occupancy corresponds with electronic transitions found in photoelectron spectroscopy (PES). In this way it is possible to experimentally verify MO theory. In general, sharp PES transitions indicate nonbonding electrons and broad bands are indicative of bonding and antibonding delocalized electrons. Bands can resolve into fine structure with spacings corresponding to vibrational modes of the molecular cation (see Franck–Condon principle). PES energies are different from ionisation energies which relates to the energy required to strip off the nth electron after the first n − 1 electrons have been removed. MO diagrams with energy values can be obtained mathematically using the Hartree–Fock method. The starting point for any MO diagram is a predefined molecular geometry for the molecule in question. An exact relationship between geometry and orbital energies is given in Walsh diagrams.
s-p mixing
In molecules, orbitals of the same symmetry are able to mix. As the s-p gap increases (Cg and 1πu MO levels in homonuclear diatomics between N2 and O2.
Diatomic MO diagrams
The smallest molecule, hydrogen gas exists as dihydrogen (H-H) with a single covalent bond between two hydrogen atoms. As each hydrogen atom has a single 1s atomic orbital for its electron, the bond forms by overlap of these two atomic orbitals. In figure 1 the two atomic orbitals are depicted on the left and on the right. The vertical axis always represents the orbital energies. Each atomic orbital is singly occupied with an up or down arrow representing an electron.
MO diagram of dihydrogen
Application of MO theory for dihydrogen results in having both electrons in the bonding MO with electron configuration 1σg2. The bond order for dihydrogen is (2-0)/2 = 1. The photoelectron spectrum of dihydrogen shows a single set of multiplets between 16 and 18 eV (electron volts).[9]
The dihydrogen MO diagram helps explain how a bond breaks. When applying energy to dihydrogen, a molecular electronic transition takes place when one electron in the bonding MO is promoted to the antibonding MO. The result is that there is no longer a net gain in energy.
Bond breaking in MO diagram
Dihelium and diberyllium
Dihelium (He-He) is a hypothetical molecule and MO theory helps to explain why dihelium does not exist in nature. The MO diagram for dihelium looks very similar to that of dihydrogen, but each helium has two electrons in its 1s atomic orbital rather than one for hydrogen, so there are now four electrons to place in the newly formed molecular orbitals.
MO diagram of dihelium
The only way to accomplish this is by occupying the both the bonding and antibonding orbitals with two electrons, which reduces the bond order ((2−2)/2) to zero and cancels the net energy stabilization. However, by removing one electron from dihelium, the stable gas-phase species He+
ion is formed with bond order 1/2.
Another molecule that is precluded based on this principle is diberyllium. Beryllium has an electron configuration 1s22s2, so there are again two electrons in the valence level. However, the 2s can mix with the 2p orbitals in diberyllium, whereas there are no p orbitals in the valence level of hydrogen or helium. This mixing makes the antibonding 1σu orbital slightly less antibonding than the bonding 1σg orbital is bonding, with a net effect that the whole configuration has a slight bonding nature. Hence the diberyllium molecule exists (and has been observed in the gas phase).[10] It nevertheless still has a low dissociation energy of only 59 kJ·mol−1.[10]
MO theory correctly predicts that dilithium is a stable molecule with bond order 1 (configuration 1σg2u2g2). The 1s MOs are completely filled and do not participate in bonding.
MO diagram of dilithium
Dilithium is a gas-phase molecule with a much lower bond strength than dihydrogen because the 2s electrons are further removed from the nucleus. In a more detailed analysis both the 1σ orbitals have higher energies than the 1s AO and the occupied 2σ is also higher in energy than the 2s AO (see table 1).
The MO diagram for diboron (B-B, electron configuration 1σg2u2g2u2u2) requires the introduction of an atomic orbital overlap model for p orbitals. The three dumbbell-shaped p-orbitals have equal energy and are oriented mutually perpendicularly (or orthogonally). The p-orbitals oriented in the x-direction (px) can overlap end-on forming a bonding (symmetrical) σ orbital and an antibonding σ* molecular orbital. In contrast to the sigma 1s MO's, the σ 2p has some non-bonding electron density at either side of the nuclei and the σ* 2p has some electron density between the nuclei.
The other two p-orbitals, py and pz, can overlap side-on. The resulting bonding orbital has its electron density in the shape of two lobes above and below the plane of the molecule. The orbital is not symmetric around the molecular axis and is therefore a pi orbital. The antibonding pi orbital (also asymmetrical) has four lobes pointing away from the nuclei. Both py and pz orbitals form a pair of pi orbitals equal in energy (degenerate) and can have higher or lower energies than that of the sigma orbital.
In diboron the 1s and 2s electrons do not participate in bonding but the single electrons in the 2p orbitals occupy the 2πpy and the 2πpz MO's resulting in bond order 1. Because the electrons have equal energy (they are degenerate) diboron is a diradical and since the spins are parallel the compound is paramagnetic.
MO diagram of diboron
In certain diborynes the boron atoms are excited and the bond order is 3.
Like diboron, dicarbon (C-C electron configuration:1σg2u2g2u2u4) is a reactive gas-phase molecule. The molecule can be described as having two pi bonds but without a sigma bond. [11]
The bond order for dinitrogen (1σg2u2g2u2u4g2) is three because two electrons are now also added in the 3σ MO. The MO diagram correlates with the experimental photoelectron spectrum for nitrogen.[12] The 1σ electrons can be matched to a peak at 410 eV (broad), the 2σg electrons at 37 eV (broad), the 2σu electrons at 19 eV (doublet), the 1πu4 electrons at 17 eV (multiplets), and finally the 3σg2 at 15.5 eV (sharp).
MO treatment of dioxygen is different from that of the previous diatomic molecules because the pσ MO is now lower in energy than the 2π orbitals. This is attributed to interaction between the 2s MO and the 2pz MO.[13] Distributing 8 electrons over 6 molecular orbitals leaves the final two electrons as a degenerate pair in the 2pπ* antibonding orbitals resulting in a bond order of 2. As in diboron, when these unpaired electrons have the same spin, this type of dioxygen called triplet oxygen is a paramagnetic diradical. When both HOMO electrons pair with opposite spins in one orbital, this other oxygen type is called singlet oxygen.
MO diagram of dioxygen
The bond order decreases and the bond length increases in the order O+
(112.2 pm), O
(121 pm), O
(128 pm) and O2−
(149 pm).[13]
Difluorine and dineon
MO diagram of difluorine
In difluorine two additional electrons occupy the 2pπ* with a bond order of 1. In dineon Ne
(as with dihelium) the number of bonding electrons equals the number of antibonding electrons and this compound does not exist.
Dimolybdenum and ditungsten
MO diagram of dimolybdenum
Dimolybdenum (Mo2) is notable for having a sextuple bond. This involves two sigma bonds (4dz2 and 5s), two pi bonds (using 4dxz and 4dyz), and two delta bonds (4dx2 − y2 and 4dxy). Ditungsten (W2) has a similar structure.[14][15]
MO energies overview
Table 1 gives an overview of MO energies for first row diatomic molecules calculated by the Hartree-Fock-Roothaan method, together with atomic orbital energies.
Table 1. Calculated MO energies for diatomic molecules in Hartrees [16]
H2 Li2 B2 C2 N2 O2 F2
g -0.5969 -2.4523 -7.7040 - 11.3598 - 15.6820 - 20.7296 -26.4289
u -2.4520 -7.7032 -11.3575 -15.6783 -20.7286 -26.4286
g -0.1816 -0.7057 -1.0613 -1.4736 -1.6488 -1.7620
u -0.3637 -0.5172 -0.7780 -1.0987 -1.4997
g -0.6350 -0.7358 -0.7504
u -0.3594 -0.4579 -0.6154 -0.7052 -0.8097
g -0.5319 -0.6682
1s (AO) -0.5 -2.4778 -7.6953 -11.3255 -15.6289 -20.6686 -26.3829
2s (AO) -0.1963 -0.4947 -0.7056 -0.9452 -1.2443 -1.5726
2p (AO) -0.3099 -0.4333 -0.5677 -0.6319 -0.7300
Heteronuclear diatomics
In heteronuclear diatomic molecules, mixing of atomic orbitals only occurs when the electronegativity values are similar. In carbon monoxide (CO, isoelectronic with dinitrogen) the oxygen 2s orbital is much lower in energy than the carbon 2s orbital and therefore the degree of mixing is low. The electron configuration 1σ21σ*222σ*242 is identical to that of nitrogen. The g and u subscripts no longer apply because the molecule lacks a center of symmetry.
In hydrogen fluoride (HF), the hydrogen 1s orbital can mix with fluorine 2pz orbital to form a sigma bond because experimentally the energy of 1s of hydrogen is comparable with 2p of fluorine. The HF electron configuration 1σ2224 reflects that the other electrons remain in three lone pairs and that the bond order is 1.
Triatomic molecules
Carbon dioxide
Carbon dioxide, CO
, is a linear molecule with a total of sixteen bonding electrons in its valence shell. Carbon is the central atom of the molecule and a principal axis, the z-axis, is visualized as a single axis that goes through the center of carbon and the two oxygens atoms. For convention, blue atomic orbital lobes are positive phases, red atomic orbitals are negative phases, with respect to the wave function from the solution of the Schrödinger equation.[17] In carbon dioxide the carbon 2s (−19.4 eV), carbon 2p (−10.7 eV), and oxygen 2p (−15.9 eV)) energies associated with the atomic orbitals are in proximity whereas the oxygen 2s energy (−32.4 eV) is different.[18]
Carbon and each oxygen atom will have a 2s atomic orbital and a 2p atomic orbital, where the p orbital is divided into px, py, and pz. With these derived atomic orbitals, symmetry labels are deduced with respect to rotation about the principal axis which generates a phase change, pi bond (π)[19] or generates no phase change, known as a sigma bond (σ).[20] Symmetry labels are further defined by whether the atomic orbital maintains its original character after an inversion about its center atom; if the atomic orbital does retain its original character it is defined gerade,g, or if the atomic orbital does not maintain its original character, ungerade, u. The final symmetry-labeled atomic orbital is now known as an irreducible representation.
Carbon dioxide’s molecular orbitals are made by the linear combination of atomic orbitals of the same irreducible representation that are also similar in atomic orbital energy. Significant atomic orbital overlap explains why sp bonding may occur.[21] Strong mixing of the oxygen 2s atomic orbital is not to be expected and are non-bonding degenerate molecular orbitals. The combination of similar atomic orbital/wave functions and the combinations of atomic orbital/wave function inverses create particular energies associated with the nonbonding (no change), bonding (lower than either parent orbital energy) and antibonding (higher energy than either parent atomic orbital energy) molecular orbitals.
Water (H
) is a bent molecule (105°) with C2v molecular symmetry. The oxygen atomic orbitals are labeled according to their symmetry as a1 for the 2s orbital and b1 (2px), b2 (2py) and a1 (2pz) for the three 2p orbitals. The two hydrogen 1s orbitals are premixed to form a a1 (σ) and b2 (σ*) MO.
Molecular orbital diagram of water
C2v E C2 σv(xz) σv'(yz)
A1 1 1 1 1 z x2, y2, z2
A2 1 1 −1 −1 Rz xy
B1 1 −1 1 −1 x, Ry xz
B2 1 −1 −1 1 y, Rx yz
Mixing takes place between same-symmetry orbitals of comparable energy resulting a new set of MO's for water:
• 2a1 MO from mixing of the oxygen 2s AO and the hydrogen σ MO. Small oxygen 2pz AO admixture strengthens bonding and lowers the orbital energy.
• 1b2 MO from mixing of the oxygen 2py AO and the hydrogen σ* MO.
• 3a1 MO from mixing of the oxygen 2pz AO and the hydrogen σ MO. Small oxygen 2s AO admixture weakens bonding and raises the orbital energy.
• 1b1 nonbonding MO from the oxygen 2px AO (the p-orbital perpendicular to the molecular plane).
In agreement with this description the photoelectron spectrum for water shows a sharp peak for the nonbonding 1b1 MO (12.6 eV) and three broad peaks for the 3a1 MO (14.7 eV), 1b2 MO (18.5 eV) and the 2a1 MO (32.2 eV).[22] The 1b1 MO is a lone pair, while the 3a1, 1b2 and 2a1 MO's can be localized to give two O−H bonds and an in-plane lone pair.[23] This MO treatment of water does not have two equivalent rabbit ear lone pairs.[24]
Hydrogen sulfide (H2S) too has a C2v symmetry with 8 valence electrons but the bending angle is only 92°. As reflected in its PE spectrum as compared to water the 5a1 MO (corresponding to the 3a1 MO in water) is stabilised (improved overlap) and the 2b2 MO (corresponding to the 1b2 MO in water) is destabilized (poorer overlap).
1. ^
2. ^ Organic Chemistry, Third Edition, Marye Anne Fox, James K. Whitesell, 2003, ISBN 978-0-7637-3586-9
3. ^ Organic Chemistry 3rd Ed. 2001, Paula Yurkanis Bruice, ISBN 0-13-017858-6
4. ^ Mulliken, R. (1928). "The Assignment of Quantum Numbers for Electrons in Molecules. I". Physical Review 32 (2): 186.
5. ^ Mulliken, R. (1928). "Electronic States and Band Spectrum Structure in Diatomic Molecules. VII. P2→S2 and S2→P2 Transitions". Physical Review 32 (3): 388.
6. ^ Hund, F. Z. Physik 1928, 51, 759.
7. ^ Hartree, D. R. Proc. Cambridge. Phil. Soc. 1928, 24, 89
8. ^ Fock, V. Z. Physik 1930, 61, 126
9. ^ .hydrogen @ PES database
10. ^ a b Keeler, James; Wothers, Peter (2003). Why Chemical Reactions Happen.
11. ^ Shaik, S., Rzepa, H. S. and Hoffmann, R. (2013), One Molecule, Two Atoms, Three Views, Four Bonds? . Angew. Chem. Int. Ed., 52: 3020–3033. doi:10.1002/anie.201208206
12. ^ Bock, H.; Mollere, P. D. (1974). "Photoelectron spectra. An experimental approach to teaching molecular orbital models". Journal of Chemical Education 51 (8): 506.
13. ^ a b Modern Inorganic Chemistry William L. Jolly 1985 ISBN 0-07-032760-2
14. ^ doi:10.1002/anie.200603600
15. ^
16. ^ Lawson, D. B.; Harrison, J. F. (2005). "Some Observations on Molecular Orbital Theory". Journal of Chemical Education 82 (8): 1205.
17. ^ Housecroft, C. E.; Sharpe, A. G. (2008). Inorganic Chemistry (3rd ed.). Prentice Hall. p. 9.
18. ^ "An Introduction to Molecular Orbitals". Jean & volatron. ""1993"" ISBN 0-19-506918-8. p.192
22. ^ Levine, I. N. (1991). Quantum Chemistry (4th ed.). Prentice-Hall. p. 475.
23. ^ Jochen Autschbach (2012). "Orbitals: Some Fiction and Some Facts".
24. ^ Laing, Michael (1987). "No rabbit ears on water. The structure of the water molecule: What should we tell the students?". Journal of Chemical Education 64: 124.
External links
• MO diagrams at Link
• MO diagrams at Link
• Molecular orbitals at Link
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03253a18b02c3425 | Quantum theory of observation/The forest of destinies
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The arborescence of the destinies of an ideal observer[edit]
An ideal observer is defined as a physical system capable of performing a succession of ideal measurements (see 2.2) and memorizing their results. Formally it can be considered as a collection of ideal measuring instruments, isolated from their environment except at predetermined times when they detect what they need to detect.
The are the instants of the observations. At each instant , the ideal observer performs the measurement associated with the observable (see 5.2) . An ideal observer is thus defined by the sequence of the . The operate on the space of states of the observer's environment, that is, of the whole universe except the observer itself.
Ideal here must be understood in the same sense as in ideal measurement. It is not, of course, an ideal of virtue, but only a theoretical fiction, simplified with respect to reality, but sufficiently similar to help us understand it.
An ideal observer can not forget. Of course real observers (living or mechanical) often forget what they first memorized. But in general the information has not been completely lost, it has only become inaccessible to them. If we complete the real observer with a physical memory which keeps all the information which it forgets, we get a system which looks more like an ideal observer.
To the above hypotheses is added a principle of ideal communication between ideal observers. When an observer A directly observes another observer B, the pointer states of B are always eigenstates of the observation by A. In this way, when A observes B, it merely copies the information memorized by B. By observing each other, therefore by communicating, the ideal observers can then share information about a reality common to their respective relative worlds (see 4.7).
A complete destiny of an ideal observer is defined by the sequence of the observation results at the instants . It determines a succession of quantum states of the observer. The first state at the initial instant just before the first measurement is the product of the initial states of all measuring instruments. The second state is where is the pointer state of the result . The th state just before the th measurement is :
A destiny is either a complete destiny, or only a segment of a complete destiny.
The destinies of an ideal observer form a tree. The foot of the tree is the initial state of the ideal observer. Between two observations, the tree grows without dividing its branches. When an observation occurs, a branch divides into as many branches as there are measurement results whose probability is non-zero.
In the model of the ideal observer, two branches which have separated can not join again, because the ideal observers keep the memory. They can not have many pasts because they can not memorize several pasts which contradict each other.
A more general model of an observer could be defined using the general theory of measurement (cf. chapter 5). We must then reason not on state vectors but on density operators. It is a little more complicated and it leads essentially to the same conclusions.
The theory of ideal observers, as defined here, is abstract and general. It makes no assumption about the space in which the observers are plunged, nor on the rest of its content. The three-dimensional space can be introduced by taking very localized quantum states as basis states.
The tree of multiple destinies of an observer does not deploy its branches in three-dimensional space but in the abstract space of quantum states of the observer. If these are located, if only in an approximate way, their destinies are also localized. The trees of multiple destinies then deploy their branches in space-time, making them always grow in the direction of the future.
The incompatibility of quantum measurements prevents two observers from simultaneously making two incompatible measurements on the same observed system. If two observers interact simultaneously with a third system, knowledge of the interactions between each observer and the third system is not sufficient to determine the result. One must reason as if it were a collision between three quantum systems. Hence it is not an ideal measurement.
Absolute destiny of the observer and relative destiny of its environment[edit]
The initial state of the observer and its environment is a state of the Universe. At later times the states of the Universe are determined by unitary evolution operators. They are usually entangled states between the observer and its environment. Thus, each state of the observer is associated with a relative state of its environment. An initial state of the Universe and a destiny of an ideal observer are therefore sufficient to determine the succession of relative states of the environment, which can be identified with the destiny of the environment relative to this destiny of the observer.
It can be said of the observer's destiny that it is absolute, in the sense that it is not relative to the destiny of another observer.
The probabilities of destinies[edit]
The Born rule enables to assign probabilities to the various destinies of an observer.
The probability of a measurement result depends only on the state of the environment relative (see 4.5) to the observer just before the th measurement:
where is the projector on the subspace of the eigenstates of .
is the relative state of the environment just after the measurement of .
In this way, with the initial state of the environment and evolutionary operators one can attribute a probability to all the destinies of an ideal observer. The same probabilities can be attributed to the relative destinies of its environment.
The incomposability of destinies[edit]
A destiny of an ideal observer A and a destiny of another ideal observer B are composable when the information memorized by one can be copied by the other. It is not required that it be copied, only that it can be copied. But at the end of the destiny of A it is necessary that all the observations of B can be communicated to A for their destinies to be composable. The destinies of two observers are composable when they can agree on a common reality. The probability of an encounter between two composable destinies is never zero.
Two destinies are incomposable when they are not composable. Incomposable destinies are definitely separated. They will never meet. This book introduces the neologism of incomposability because incompatibility already has another meaning in quantum physics (see 2.7). If the destinies of two ideal observers contain mutually contradictory results of observation then they are incomposable. The probability of an encounter between two incomposable destinies is always null.
The separation between two incomposable destinies is a specifically quantum separation, very different from spatial separation. When two destinies are separated quantumly, the impossibility of an encounter is definitive, even if they are in the same place (see 4.9). Two incomposable destinies will never be able to interact. When two destinies are separated spatially without being quantumly separated, they only have to come together in space to interact and to unite in this way.
We can define the incomposability in a more formal, less intuitive and mathematically more convenient way. Formally, all ideal observers can be combined by tensor product into a single ideal observer. The sequences of the observers are used to define a new sequence for the observer which unites them all. Each destiny of the total observer determines a single destiny for each of the observers thus united. Two destinies of two observers are composable if there exists at least one destiny of the total observer, of non-zero probability, which determines them both. They are otherwise incomposable.
The superposition (see 1.1) and the incomplete discernability (see 2.6) of states, the incompatibility of measurements (see 2.7), the entanglement of parts (see 4.1), the relativity of states (see 4.3) , the decoherence through entanglement (see 4.17), the selection of pointer states (see 5.4) and the incomposability of destinies are the main concepts, specifically quantum, without classical analogues, which enable to understand the physical meaning of the Schrödinger equation, or equivalently, of the formalism of unitary operators.
The growth of a forest of destinies[edit]
When observers do not interact in any way, either directly by observing each other or indirectly through a quantum system in their environment, their trees of destinies grow independently. For this to happen, each one has to observe different objects which are completely separated, in the quantum sense, from objects observed by others, that is, they are not entangled with them.
When two observers interact, directly or indirectly, they intertwine the branches of their trees of destinies, a little like Philemon and Baucis. One can thus see the multiple destinies of many interacting observers like a growing forest whose trees intertwine their branches. To represent a quantum evolution, the growth of such a forest must respect very strict rules of selection of possible intertwinings.
When the communication between two observers is ideal, each branch of one separates from all the branches of the other with which it becomes incomposable.
Two ideal observers A and B can also interact through a third quantum system C in their environment. It is not necessarily an ideal communication.
Suppose that A observes a system C which is then observed by B.
If A and B make the same measurement on C and if the latter is in one of this measurement eigenstates, then the branches do not multiply, A and B obtain the same result, and they intertwine their branches as if there had been an ideal communication of this result. If C is not in a proper state of the measurement, the branches of A first, then those of B, multiply after the measurement on C, and they entangle as if there had been ideal communication of obtained result.
If the observables of the measurements of A and B are incompatible (see 2.7), the results obtained by A can not be identified with those obtained by B. In this case, the entanglement between the branches can not be determined by Matching of results. If, for example, C is not a proper state of the measure by A while being a proper state of the measure by B, the branches of A first, then those of B, multiply after the measurement on C, but the branches of A which were composable with those of B before the measurement of C remain composable. The interaction via C does not introduce new constraints of incomposability between the destinies of A and B.
There are thus essentially two ways for two trees to intertwine their branches when two ideal observers interact. If they observe each other or if they measure the same observable of a third system, then they entangle their branches by matching the results. If the interaction does not lead to the sharing of the same information then they entangle their branches without discrimination.
Before the first interaction between A and B, direct or through a third system, all the destinies of one are composable with all the destinies of the other. Subsequent interactions introduce constraints of incomposability, prohibitions of meeting between destinies, as soon as A and B observe each other or make compatible measurements on a third system C. The growth of the forest is therefore accompanied by a a process of differentiation, of separation between trees, similar to cerebral maturation. Initially, in the early years of life, connections between neurons are very little differentiated and each neuron is connected to many others. Most of these connections disappear over time.
To speak of the growth of a forest of destinies is only one way of describing the solutions of the Schrödinger equation when applied to systems of ideal observers. It is a question of describing mathematical solutions which result from the simple assumptions which have been made. It is not a delusional imagination but a calculation of the consequences of mathematical principles.
Virtual quantum destinies and Feynman paths[edit]
The initial states and the pointer states of the measuring instruments which define an ideal observer determine, by tensor product, the pointer states of the observer itself. The selection of the pointer states of the measuring instruments (see 5.4) also selects the base of the pointer states of the ideal observer.
When a quantum system is not a macroscopic measuring instrument or an ideal observer, no pointer state basis is privileged (see 5.5). One can still define multiple destinies by arbitrarily choosing one of its bases of states. But there is no reason to think that these destinies are real, because the states which define them are not, in general, states by which the system really passes. In reality it is in a superposition of these states or in a state entangled with the environment. This is why this book call them virtual quantum destinies.
When the are instants of time and the , states of a system S indexed by the same index , The sequence of the is a Feynman path.
The are the evolution operators of S between and .
The probability amplitude associated with the Feynman path is by definition:
is the initial time and is the final time. is an initial state of S and a final state.
The probability amplitude of the evolution from to is:
The is an orthonormal basis of states of S. With it determines a set of Feynman paths from to . contains all the paths where the are always chosen among the . If is an element of , is its intermediate state at the instant .
We have :
where and for all in .
In other words, the probability amplitude of an evolution between an initial state and a final state is the sum of all the probability amplitudes of "all the paths" that connect these two states. This is the finite version of Feynman paths integrals (Feynman & Hibbs 1965).
Proof: knowing that , we have
A destiny of an observer is defined by a succession of quantum states at defined instants, like a Feynman path. As David Deutsch does not distinguish destiny from Feynman path , he suggests, surprisingly, that Feynman paths integrals could serve to prove the existence of multiple worlds (Deutsch 1997). To be properly defined, multiple worlds must be considered as worlds related to observers, who have multiple destinies. The state of one of these worlds is a state of the environment (the Universe except the observer) relative to a state of an observer.
A destiny of an observing system is real. The results of observation are really obtained. They are part of a destiny which really exists. Feynman paths can not be real destinies, because the intermediate states must not be observed in order to integrate probability amplitudes and not probabilities (see 4.18). If Feynman paths were real destinies, probabilities would have to be summed up.
Another fundamental reason prevents the identification of Feynman paths with real destinies. They would attribute very many pasts to the same present state. Feynman paths do not form a tree structure because they can converge as easily as they diverge. A quantum state on a Feynman path is a point of convergence of many paths that would define as many pasts if they were real destinies. This property of convergence of virtual destinies is important to make use of the parallelism of quantum computation, but it seems obviously excluded for real destinies, which in general seem to have a single past.
The parallelism of quantum computation and the multiplicity of virtual pasts[edit]
Consider a system with two qubits which interact in such a way that the first acts on the second without being affected in return, when we reason in the base {}. Their interaction is thus described by the operator :
where is any function which describes the effect of the first qubit on the second, and .
If the system is initially prepared in the state , we obtain :
If we obtain :
If we obtain :
The final state or of the first qubit thus reveals whether or not it always has the same effect on its partner.
One can analyze this quantum computation by distinguishing two virtual destinies of the first qubit, that in which it passes into the state immediately after the initial preparation, the other where it passes into the state . The operator determines the evolution of these two destinies in parallel. In a pictorial way it can be said that the first qubit lives two destinies in which it may or may not have the same effect on its partner. These two destinies finally converge on the same state or . If it is , the qubit has done the same effect in its past virtual destinies, if it is it has done a different effect. By having several virtual pasts, the first qubit enables to reap the fruits of the parallelism of quantum computation.
This example has general value (Deutsch 1985). Quantum computation always enables to calculate all the values of a function in a single step. If, for example, it has 100 qubits of memory for the data register, a quantum computer can compute in parallel and in one step (a thousand of billions of billions of billions approximately) values of any function. But the difficulty is to reap the fruits of this parallelism. It is necessary to observe a state which results from all the virtual destinies which occur in parallel, thus a state which has many virtual pasts.
Can we have many pasts if we forget them?[edit]
Two different real destinies of the same ideal observer can never converge on one state because an ideal observer never forgets and because it can not retain contradicting memories. But if it forgot, could it have several pasts, like the qubit in Deutsch's quantum algorithm above?
For a parallel quantum computation to provide a result it is necessary that the computer be protected against the decoherence through entanglement with its environment. If such a decoherence occurs, everything happens as if the parallel virtual destinies were observed by the environment. In this case, one has to sum not the amplitudes but the probabilities to calculate the probability of the final result (see 4.18). The states and would occur with the same probability regardless of the values of the function . And there would be no longer any reason to assert that they have two virtual pasts.
As we are constantly subjected to decoherence by interference with our environment, everything happens as if we were constantly observed by the environment. When we forget, the lost information is not completely lost. The environment always keeps a trace of it. This is why two really lived destinies can not converge on the same state in which they are superposed, even if in this state we have forgotten what could distinguish them.
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Sunday, August 9, 2015
Quantumology is the belief that quantum action describes all force and that gravity is a discrete quantum force. Quantumology necessarily begins with some kind of universal particle, like a discrete aether, and the decay of discrete aether then defines all force. What this means is that the photon diploe exchange that defines charge force also then defines a photon pair as the monopole-quadrupole force of gravity. Photon pairs as monopole-quadrupoles then bond neutral matter particles for quantum gravity and are scaled versions of the photon dipole emissions that bond charged particles.
A very simple way to scale gravity force from charge dipole force is to wrap the universe onto itself and let the ratio of the time delay of the atom to the time delay of the universe scale gravity. In other words, the charge dipole force acts locally between the charges of an atom as well as globally as a monopole-quadrupole force when the universe wraps onto itself in time. Gravity force is simply charge force scaled by the ratio of the time delay of an atom with the time delay of the universe as a pulse in time. This simple statement of unification is completely consistent with mass-energy equivalence, Lorentz invariance, gravitational radiation, and many of the other precepts of general relativity. This simple way of unifying gravity and charge force is not yet accepted by mainstream science.
However, the notions of discrete aether, matter exchange, and time delay are much more general that the notions of continuous space, motion, and time as axioms. Continuous space and motion are not congruent between gravity and charge forces and that incongruence precludes unification within the limits of continuous space and time. Instead of continuous space and motion, unification necessitates a pair of conjugates that are congruent and compatible for both charge and gravity forces.
Even though continuous space and motion are very intuitive and deeply embedded into our consciousness, the notions of continuous space and motion are not a priori axioms for all action. Discrete matter and time delay as the proper conjugate quantum operators apply even beyond the current limits of continuous space and motion, which bound more typical conjugates of space and momentum. Space and momentum still have the same meanings and utility for many predictions of action, but for both very large and very small scales, there are no expectation values for space and momentum. Time, for example, has a fundamental two dimensional representation instead of a single continuous dimension of spacetime and time reflects the nature of the boson aether pulse that is the universe.
Things happen to objects of matter in the universe because of the actions of both gravity and charge and we think of gravity and charge as being very different, but in fact they are simply different manifestations of the same force of aether decay at much different scale. The scale ranges from the time delay of the atom to the overall time delay the universe aether pulse. While charge force is a result of the boson matter decay of the universe, gravity force is a result of the fermion decay of microscopic matter. While the universe is mostly boson aether, it is fermion matter that makes up common objects.
The action of the earth's gravity creates stone from cooling inner molten magma and it is the microscopic charges of stone's atoms and molecules that hold those stones together. The much weaker action of gravity is only evident in holding those stones and us to earth's surface, but gravity is what makes earth earth. Someone building a stone wall depends on gravity not only to keep them and the stone wall bound to earth, that gravity also compresses and slightly heats stones in the actions of building a stone wall. That very slight heating of the stone is part of the gravity force of earth and leads to much greater heating of the inner earth.
Action is both what forms objects like stones from atoms and action is how we form objects like stone walls from stone. In both cases, smaller moments of matter come together to form larger objects. The heat and pressure of earth's gravity makes stone while people gather those stones and make stone walls on earth’s surface for some purpose. The gravitational bond between the stones in the wall and the earth heats the stones up very slightly on earth's surface and it is that radiative and conductive cooling that results in the bonding that we call gravitational compression.
Gravity describes how most things of common experience happen and simply depends on mass action, like the action of a deterministic path of an apple falling from a tree. Gravity results in a very deterministic cause and effect universe where it appears that all action results in only local effects. Our notions of space and momentum emerge from the actions of gravity on objects that we sense.
Charge describes how the microscopic actions of atoms and molecules of matter objects happen with quantum matter with both phase and amplitude. Quantum charge is how the apple grew on the tree in the first place and quantum charge released the apple from the tree into gravity mass action. Charge results in a wavelike and probabilistic universe that allows the matter wave amplitude of one object to affect the matter wave amplitude of another object instantaneously across the universe. As a result, both philosophy and science therefore have very different interpretations of the very different natures of gravity and charge actions.
Quantumology is the belief that gravity is just a scaled version of charge force and that quantum of gravity force is a coherent photon pair as a monopole-quadrupole. Although mainstream science and general relativity are not consistent with this view of quantum gravity, the decay of discrete aether and time delay are consistent with quantum gravity.
Charge bonds involve matter exchange between objects while gravity bonds also involve matter exchange between objects and the universe. Motion in the universe emerges from a change in an object’s inertial mass as equivalent energy and it is that exchange of aether that we call object momentum. Changes in an object’s inertial mass or kinetic energy define an object’s action for a given frame of reference while gains and losses of mass as impulse change object momentum. Although motion is a very common way to define momentum in space, the dimensionless ratio of velocity squared to the speed of light squared in ppb is embodied in the dimensionless Lorentz factor.
The equivalence of matter and energy means that velocity and acceleration are equivalent to changes in inertial mass. The dimensionless Lorentz factor impacts space, matter, and time even while most object action involves gains and loses of ordinary matter as impulsive momentum, which typically overwhelm changes in inertial mass. What we call the fields of charge or gravity force are actually matter exchanges among objects that result in acceleration and changes in object velocities. Charge and gravity fields are potential matter, which is the rate of change of inertial matter in time and is that proper matter that comes into existence as velocity or kinetic matter from an inertial frame. In matter time, fields in space are simply a manifestation of the exchange of matter between objects and those matter exchanges are the forces or accelerations of potential matter.
The decay of all universe matter with time, mdot, is in fact a fundamental principle of matter time and is the determinant of both gravity and charge actions, just at very different scales. This decay constant is simply a restatement of charge and gravity forces as cross sections and is equivalent to the dimensionless universal decay of all matter, αdot, at 0.255 ppb/yr. For charge force, αdot applies to the electron mass as the fundamental fermion while for gravity force, αdot applies to the gaechron mass as the fundamental boson, which is some 1e-39 times less than the electron mass.
Currently science uses two somewhat inconsistent theorets to separately predict the gravity and quantum futures of objects in time. This patchwork approach actually works very well for predictions of action within certain scales, but mainstream science yearns to describe gravity as part of a unified quantum action that includes both charge and gravity.
Gravity action is what holds us to the earth as well as what holds the earth in orbit around the sun and gravity action holds the rest of the greater universe together as well. So, gravity action is the way that we predict how objects move for much of our very deterministic and causal and chaotic reality here on earth and gravity action is how we measure the billions of years of our universe time delay. We have come to know gravity action as general relativity but still gravity action scales with the mass distribution of objects and gravity does not depend on exactly what the matter is.
Gravity action in matter time is very simply related to the binding of objects to the boson matter of the universe. Just like the quantum bonds of electrons to nuclei, the quantum bonds of atoms to the universe boson matter result in the attraction between neutral objects that we call gravity. Gravity is a quantum excitation that involves correlated pairs of photons as a mono-quadrupole time and for most common gravity action, quadrupole time is equivalent to proper time, τ. This approximation does not account for any quantum exchange effects, where the exchange of identical particles leads to an additional quantum gravity binding energy.
Our microscopic reality, though, is bound with charge and quantum action and, unless an object is very massive, gravity action is not much of a factor at all. In contrast to gravity action, quantum action is very dependent on the exact nature of matter amplitude and phase. Matter amplitude and phase are part of the quantum action that determines the nature of the bonds that hold an object’s matter together.
For example, an atom of hydrogen bonds much differently with another hydrogen atom as compared to a different element like oxygen. Oxygen bonds to two hydrogens and forms the water of our earth and comets. In contrast to charge action, the predictions of gravity action do not really need the details of atoms and bonds and amplitude and phase as long as we know a object's density and mass. However, at larger and smaller scale, the natures of quantum amplitude and phase do indeed impact gravity bonds.
Gravity and quantum actions represent somewhat inconsistent theorets or realities for science, but somehow we know that there is a relationship. General relativity is basically the gravity action that is what holds us to the earth and holds the sun in the galaxy and all galaxies to the universe and is very intuitive and deterministic. Each effect of gravity has a cause and that cause is local to that effect. In contrast to gravity action, quantum action depends on both matter amplitude and phase and not just mass. An extra phase coherence between objects links not only local object actions, but also correlates nonlocal object actions as well.
One of the more notable aspects of relativity is the statement of equivalence of energy and mass, E = mc2, with the proportionality of the speed of light squared and indeed quantum action has adopted that same principle as well. Just this simple matter-energy equivalence (MEE) explains much about both gravity and quantum action since all motion increases the inertial mass for each object proportional to its velocity squared, which is the kinetic energy of motion. Somehow an object gains and loses extremely small amounts of matter simply by changing its velocity.
Another notable result of relativity is the fact that the speed of light for an object does not depend on object velocity, which is a direct result of the equivalence of mass and energy and further results in a dilations of space and time associated with any motion as velocity and acceleration. When it comes to explaining the anomalous precession of Mercury about the sun or the bending of starlight by sol, the proportionality of energy and matter explains about one-half of such observations and the dilation of space and time explains the other half.
While the mass-energy equivalence principle is completely consistent with the formulation of a quantum gravity in matter time, the distortion of a continuous space by velocity and acceleration represents a little bit of a problem for any discrete quantum gravity. This is because dilation of continuous space is a result of gravity and so a particle that carries gravity force would therefore dilate space and alter the particle, which further dilates space, and so on. With discrete matter and time delay, spatial dilation is the result of action in discrete matter and time delay and not a result of gravity per se. While the distortion of continuous space and time with motion is definitely a part of our reality, this distortion is where there is a strain between gravity and quantum actions.
The question comes down to whether or not there is a continuous deterministic and predictable path for an object through space time. In general relativity, gravity distorts space and time and that is what results in a continuous deterministic path as a straight line in continuous 4-D space time. However, it is possible with mass-energy equivalence to have the same dilation of time and along with discrete changes in inertial matter, have the same path emerge for that object. In this reinterpretation, spatial dilation then emerges from the action of discrete matter and time delay and the result is what we call motion. What we imagine as action in space is really first of all an action or change of discrete matter with time delays and then only secondarily do continuous motions and dilations of continuous space emerge.
With discrete matter and time delay, a continuous spatial dilation emerges from the gravity action of an object in discrete matter time and spatial dilation therefore does not therefore cause action or motion in space. With this approach, quantum gravity becomes a straightforward result of action in matter time. While charge force is the exchange of photon dipoles between electrons and nuclei, gravity force is exchange of complementary photon pairs as mono-quadrupoles between the neutral matter and the boson matter of the universe.
The stress-energy tensor of GR then more properly emerges from a mono-quadrupole time and is not an a priori axiom. In quantum gravity, it is the mono-quadrupole time operator and its tensors that provide a proper time for each action from the two time dipoles of the rest and moving frames. For most common actions, the quantum time quadrupole is largely identical to proper time. However, for certain very massive and very small objects, there is a quantum exchange that enhances the gravitational bond. Gravity objects bind to each other by means of exchange of time quadrupoles.
Quantum action is largely about the behavior of coherent microscopic matter and is much less intuitive than gravity action at all scales. Quantum action depends on matter or mass just like gravity but quantum action also depends on something called phase and coherence and charge amplitude, properties of matter that have no relevance in general relativity. The interference effects of light are due to light’s phase and amplitude and so light shows polarization and partial reflection as a result. Yet these coherent effects occur for all objects of matter, not just for light. Neutral matter can show polarization and neutral matter can show partial reflection as well.
The basic equation of motion for quantum action is the Schrödinger equation for discrete matter,
which is a proportionality between the amplitude and phase for a matter wave of the future,
and the amplitude and phase of a matter wave of the present,
This is what is called a differential equation in time and is an action equation that describes how a matter wave changes over time, both in mass and phase. In this equation, mR represents the photon exchange energy that binds an electron to a proton to make hydrogen and is the mass equivalent of the Rydberg energy. There is an infinity of excited states for hydrogen whose energies emerge as spectral lines that converge to a finite ionization energy, which is called the Rydberg energy.
The integral form of the Schrödinger equation for discrete matter is
and shows that matter waves are also proportional to their integration over time, which is their action over time. That proportionality is the ratio of a binding energy, mR, and Planck’s constant and of course a phase factor, -i, which means that the action of an object is somehow orthogonal to its matter in time.
There is an infinity of excited states for hydrogen whose energies as discrete spectral lines converge to a finite value that is the hydrogen ionization or Rydberg energy. There are two solutions to each Schrödinger equation; an inner charge solution involving the charged electron along with an outer gravity solution involving discrete aether. The inner solution has photon dipole exchange that binds electrons to the nuclei of atoms and the outer solution involves pairs of complementary emitted photons that bind neutral atoms to the outer boson aether of the universe.
Matter waves scale with the square root of mass in matter time while the more typical wavefunctions of quantum mechanics are just dimensionless phase as probability amplitude. This means that the integral of a matter wave over all time is an action that results in the measurable property that we call mass. Matter waves are the moments of matter that make up all objects and sensation is the exchange of the matter waves of our senses with the matter waves of an object being sensed. In the parlance of quantum action, a matter wave or wavefunction collapses as a product of each exchange between us and an object and that collapse is the sensation that we imagine as the mass or some other property of an object.
We might see light from an object, feel the object, hear it, smell it, or even taste it. What we sense of an object alone is not the matter wave itself, but the product of the object matter wave with our own sensory matter waves. Sensation is an exchange of both amplitude and phase with objects in a bonding action that we imagine as reality. The discrete exchange of matter actually bonds us to objects with a quantum action that necessarily occurs in discrete quantum steps with discrete quantum states. This bonding action involves our whole body and not just our sensory organs.
A journey from point A to point B involves a series of steps or quantum jumps as an object exchanges discrete aether with other objects in order to get around the universe, successively bonding and conflicting with the matter waves of objects in order to move. Matter waves show action under the influence of operators and those actions result in discrete changes in object matter over time.
Time delay waves also show action, but now as a function of a quasi-continuum of matter. A journey from matter state A to matter state B involves a series of quantum jumps as an object exchanges time delays with other objects. While objects exists with discrete time delays, time is a quasi-continuum that depends on the very large number of quantum jumps of matter particles.
A continuum force like gravity in general relativity does not show the discrete states of quantum gravity but rather shows continuous motion from point A to point B. Continuous motion in space is a very natural and intuitive concept that is not how objects move in discrete matter and time delay. In fact, motion in continuous space results in serious conundrums like the Zeno’s paradox of an infinity of points and quantum action of whole particles resolves Zeno’s paradox but at the expense of a different interpretation for continuous macroscopic gravity action in the universe.
Gravity in matter time is a quantum action that binds atom pairs to the boson aether of the universe, which is discrete gaechron. The complementary photon pairs emitted from the charge actions of electron bonds for two atoms are the light that objects emit from charge and are the gravity force bonds between atoms and molecules as well. Emitted light represents the complementary outer state for the inner binding states of each atom and molecule and emitted light is the exchange that binds the matter waves of atoms and molecules with each other as the matter waves of the universe. Because we see light, we imagine emitted photons on trajectories through the void of space. In fact, emitted photons represent complementary changes in matter states that we call charge and gravity action.
There is a photon dipole exchange that binds an electron to a proton to form a hydrogen atom and such a mass defect is the Rydberg energy for hydrogen as well as binding atoms to each other with further energies and further shared electrons. That same charge force defect represents an equivalent photon pair exchange with the boson aether of the universe that is the gravity force that binds the hydrogen atom to the universe. The dephasing of discrete aether results in what we call gravity force and by scaling discrete aether exchange by the ratio of electron mass to discrete aeither, discrete aether decay is then what we call charge force as well. The light that we see from the stars at night represents a discrete aether exchange that binds the electrons and protons as well as atoms into stars and stars into the galaxy as well as the galaxy into the very fabric of the cosmos.
Although science expects a new particle called a graviton to be the exchange particle of gravity force, with the scaling of photon pairs in discrete matter, there is no new gravity particle. Rather, it is the universal dephasing of discrete boson aether that determines both gravity and charge forces and the photon is the basic exchange particle for both gravity and charge forces. Whereas photon exchange between the electron and proton represents charge force, photon pairs exchange between the electron and discrete aether represents gravity force. Thus, the ratio of the gaechron particle of discrete aether to the electron mass represents the 1e39 scaling between gravity and charge force cross sections.
Quantum action is often called odd although quantum action has been extraordinarily successful for virtually all predictions of action. However, quantum predictions are always probabilistic and uncertain and sometimes matter waves show correlated and coherent effects that entangle different locations in space. Even for a highly local matter wave action there is still some quantum uncertainty, which bothers many people. Since quantum phase can persist between two objects across the universe, the observation of one object phase seems to determine the other object phase instantaneously. So when that quantum uncertainty involves locations across the universe, people get even more uncomfortable and bothered.
And yet quantum action does not violate any causal principles, rather quantum action simply refines those causal principles to include matter wave phase, amplitude, and coherence as well as mass as the product of two matter waves. The phase or coherence of a matter wave is a property of an object that we do not directly experience and so it is less intuitive than just the mass of an object, which is the square of its amplitude and does not carry phase information.
There are many different ways of describing the issues of quantum nonlocality and entanglement, but basically it comes down to a set of fundamental differences between quantum and gravity notions of space and motion. Quantum motion involves both the wave amplitude and phase of an object, while gravity motion involves only the mass of an object, i.e. the product of two matter waves, and so gravity action for mainstream science does not involve or entangle matter wave phase and amplitudes between objects at all.
Objects follow certain action principles where action is the integral or sum total of an object’s matter over time. Any macroscopic object is the product of a very large number of actions over time and objects continually gain and lose discrete aether as a part of their existence in the universe. Our intuition typically represents action as some kind of spatial displacement of an object, but it is the discrete aether exchanges of an object in time that better represent quantum action instead of motion. Discrete matter exchanges occur as quantum action and are the action we see as motion for an object in space.
Einstein first recognized that both event and action times are equivalent to spatial displacements and his general relativity shows how gravity action dilates matter, space, and time in a continuous four dimensional spacetime. Objects that gain inertial mass from their potential matter we interpret as a relative motion in space and that mass gain affects the space and action time between objects as well.
There are, however, different ways to interpret the dilation of matter, space, and time, with quantum gravity and therefore with a pure quantum action. Objects are in constant discrete aether exchange with other objects and it is from the gained inertial mass from other objects that object motion in space emerges. However, in general relativity the trajectory of an object follows a determinate geodesic path determined by gravity. If rather the distortion of space is a result of the gravity actions of that object, the same principles apply but now with a complementary quantum action for both gravity and charge.
An object like a rocket ship gains velocity and momentum by ejecting matter with the mass impulse of some kind of burning fuel and the action of the burning fuel propels the rocket in the opposite direction by its equivalent momentum. However, the relative motions of both ship and fuel actually are a result of much smaller gains in inertial masses, discrete aether, as equivalent kinetic energy by the matter-energy equivalence principle.
In other words, even while we imagine that the total rest mass of rocket and fuel does not change due to exchange of equivalent and opposite momentum, in fact, it is the the very small changes in the inertial masses of both rocket and ejected fuel that results in their respective motions. In a strict sense, then, what causes motion in space is the increase in inertial masses of two objects with equal and opposite momentum by exchange of discrete aether. Both objects increase in mass proportionately with their velocities squared relative to a rest frame and this matter increase comes from the potential matter as energy that was embedded into the chemical and gravity and nuclear bonds of the fuel.
The quantum action of discrete matter and time delay, which along with action, are the three axioms that close our universe. An action equation predicts the future of an object as discrete exchanges of matter with other objects over time. Quantum gravity predicts a large number of possible futures for macroscopic objects, but quantum action for macroscopic objects involves much greater scale than the local actions of gravity.
While there are a large number of possible futures for an object undergoing quantum action, including nonlocal futures, under gravity action of mainstream science, there is only one possible future for an object. This difference of action principles goes for the same object and the same reality and leads to interminable scientific and philosophical discourse about which action actually better describes an object’s possible future. Gravity and quantum actions are largely consistent with each other in common experience, but the two actions can represent irreconcilable futures for certain very large or very small objects. For example, until science reconciles gravity with quantum action, there is simply no way to definitively address the mystery of quantum gravity nonlocality.
The single future of gravity action in GR is consistent with a reality that is deterministic and local. Local effects always have local causes and this is the reality that we normally experience with gravity. Gravity is a continuous and infinitesimal force with a pesky microscopic singularity centered on each particle of matter and so there is no coherence for an object between two different locations in space. Since gravity action is the basis of our intuition for macroscopic objects in everyday life, we therefore have a very strong expectation that local actions only correlate to other local effects. We know that two ballistic particles from a source can arrive simultaneously at very different locations along separate paths A and B. However, a single matter wave can propagate along both paths A and B and yet only appear as a single particle at A or B. Note the appearance at A is coherent and correlated with no appearance at B, but neither causes one nor the other to occur.
Our intuition and experience, after all, are both largely based on an intuition of gravity action and so we greatly favor gravity action and mass as bases for predictions. Gravity action is usually very predictable since after all, what goes up, must come down. For gravity force, there is no allowance for the coherence of a single matter wave across the time delay of the universe.
Phase coherence can make it seem like the appearance of an object in one place causes its absence in another place, or that the absence of an object in another place causes the object appearance in the one place. Coherence has many effects, but quantum action does not violate any causal principle. Quantum action simply includes phase along with amplitude and a source and so better represents the actions of the entire universe, including actions at very small and very large scales.
A quantum universe consists of objects simultaneously located everywhere in the universe as amplitudes of matter waves. What provides us with the sensation of an object in one place and on one path is the time and phase that separates that object from other objects. It is an object’s incoherence with all of its other possibilities as a matter wave that we sense as a local object in time and space. While some of the many possible futures of an object from quantum action are nonlocal, the issues with quantum nonlocality and entanglement are fundamentally related to the many very different possible futures or phases for quantum action.
Quantum action is perfectly causal, but unfortunately quantum action is just sometimes not very intuitive since quantum can involve phase and coherence among objects in different places. We find it hard to accept how a perfectly real and observable ballistic object could ever be a matter wave that has both an amplitude and phase and magically disappears from one place due to destructive interference and then equally magically reappears in a completely different place due to constructive interference of those same amplitudes.
Worse yet, objects as matter waves can actually exist as a possibility in more than one Cartesian location until it finally interacts with another object at one place or the other, i.e., the matter wave collapses or dephases. And yet our quantum reality shows that matter has both amplitude and phase and therefore matter will show the many nonintuitive effects of coherency and interference.
It is particularly confusing when explanations of quantum action give macroscopic objects like people and cats the coherent attributes of microscopic matter. Coherent matter behaves so differently from incoherent matter that comparisons between coherent and incoherent macroscopic matter can result in very confusing allegories. Although it is possible for macroscopic matter to show coherence, the dephasing times for any macroscopic object are typically very short unless the objects are very massive neutron stars or black holes.
Until science unites charge and gravity into a common quantum action for all objects, there will continue to be confusion and strong differences of opinion about the nature of quantum action versus gravity action. For example, given similar charge and gravity forces for a coherent object, quantum action shows interference effects due to superposition but gravity only predicts ballistic collisions between objects. We have an intuition and life experience with macroscopic matter and gravity action that is very difficult to reconcile with the reality of microscopic matter and quantum action.
Light is a rather unusual form of matter and a photon of light on a trajectory in space is also the exchange particle that binds charged particles together. An exchange of a photon dipole between an electron and proton represents the dipolar charge force that stabilizes a hydrogen atom dipole, which is the basis of quantum electrodynamics and is well accepted by science.
That emitted photon pair is then the binding force for gravity, but this is not a common understanding. For one thing, charge is a dipole force while gravity is a mono-quadrupole force and so it is not clear how a dipolar photon with spin = 1 and plus/minus amplitudes can result in mono-quadrupole gravity with spin = {2, 0, -2} and quadrupolar amplitudes. The radiative cooling of hydrogen at the CMB created photon pairs that are a quadrupole attractive force called gravity. Since there is a pair of photons for every two neutral atoms to the universe, it is that mono-quadrupole pair that is responsible for gravity force.
In order for a neutral atom to form from charged electrons and protons, the neutral atom must emit or otherwise radiate its dipole charge binding energy as a complementary photon. That emitted photon is equal to the atom’s binding energy, which is the Rydberg energy for hydrogen, for example. There actually can be and are many photon emissions and absorptions of various energies and so this description just simplifies that complexity into one single event pair.
Each pair of neutral atoms emits a pair of photons at creation and those photon matter waves have complementary spin and polarization. While the dipole force between these particles and the photons progressively cancels out over time, the mono/quadrupole force persists as a tensor. Thus gravity force behaves as the quadrupole tensor of a coherent photon pair with spin = 0 and is a single particle with physical dimensions that literally define the age of the universe.
There is just one future for gravity action in general relativity and that one future is still consistent with our deterministic intuition. General relativity dilates or distorts continuous matter, space, and time with gravity action and there are many strange results of general relativity having to do with time dilation, simultaneity, and frames of reference. But while distant objects far away from a gravity action do not affect a local gravity action very much, the ratio of hydrogen’s time dipole to the time dipole of the universe is the scaling between gravity and charge forces.
In contrast to the determinism of gravity action in GR, there are actually a large number of possible futures for the same action as a quantum time quadrupole. The Rydberg photon emitted from hydrogen at creation is the exchange with the universe that binds each hydrogen atom to the boson matter of the universe. The time delay of that bond is coherent with that of the electron around the proton. The photon exchange between the universe and each pair of such atoms binds each atom to the universe matter and therefore to each other as well. It is then the shrinkage of the universe about those atom’s center of mass that represents what we interpret as the binding force of gravity between these two hydrogens.
Therefore the binding energy for hydrogen is the sum of the binding energy of the electron and proton along with a second term that is the binding energy of the atom with the discrete aether of the universe. In a strict sense, the binding matter of the electron and proton of an atom scales to the binding matter of that atom to the universe. Since they are equal and opposite in sign, their sum is zero and that result is an example of the Taylor-DeWitt equation. Even though their energies are equal and opposite, charge and gravity matter waves are quite different.
Whatever future actions occur for atoms in their many possible futures, their center’s of action and the gravity action that goes along with these centers persists. As matter evolves into heavier elements in star fusion engines, there are additional light and energy exchanges between those heavier elements and the universe and this additional action matter means that matter bonds in more complex ways to the universe just as matter bonds in more complex ways with different elements. The nature of gravity force actually increases over time just as the universe of matter shrinks or dephases and it is the overall shrinkage of the universe that is the origin of all force.
Quantum mechanics represents matter as the two dimensions of amplitude and phase. Thus a particle on a trajectory in space represents the matter of an object as a wave in a spectrum of matter waves across all space and time. A classic example of the wave nature of light is a series of strong and weak intensities, fringes, that is an interference pattern. An equally classic example of the particle nature of light as photons is the photoelectric effect where a photon of some minimum energy results in ejection of an electron from a metal surface.
The wave nature of light results in a pattern of light and dark fringes due to a coherent action from a single source between two or more possible paths for a source’s photons. This coherence can be the result of any number of means but the typical experiment is with two slits and the resultant diffraction of a light source.
However, each peak of intensity of the fringe pattern comprises a large number of measurable single photon events from the source. We want very badly that each of those photons journeyed ballistically along straight line paths from the source to the pattern and are disappointed to learn that there is not a single ballistic path for any single photon. Rather, each photon journeys as a matter wave with a wavelike trajectory on multiple paths to the interference pattern.
We are further disappointed to learn that this fringe pattern could persist over the dimensions of the universe. That is, the photon that we detect right here right now that come to us from a source may have also possibly been on a different path, somewhere very far away connecting some other object to the same source at the same time distance away. Since the photon wave journeyed across the universe somehow on its way to us right here we presume that its journey was ballistic as a particle.
When we record the photon right here, right now, we know for certain that the photon was here now and therefore not ever anywhere else. But the moment before we measured the photon here now, there had been a possibility that that same photon as a wave would have occurred somewhere else in the universe and therefore not here. Our intuition, though, tells us that photons that emanate from a source do so in a continuous ballistic manner and those photons are on continuous ballistic paths. The quantum truth is that it is photon matter waves that emanate from a source, and a photon matter wave is not yet a ballistic photon localized in space.
This seems like a funny result since when we see a photon, we know that the photon came from the image of a source that we imagine behind the photon and so we imagine a ballistic Cartesian journey in a more or less straight line from the source to our eye. If the source is incoherent, we imagine that it shines equivalently in all directions, but still imagine each light wave as a ballistic photon particle.
This is how we imagine objects in our Cartesian minds and a quantum action as a wave goes against the deterministic intuition of our ballistic gravity action. This does not mean that the photon did not exist before its wave dephased from the source, rather it means that the photon existed as a matter wave with both amplitude and phase and not as a ballistic particle.
What gives? Why can an object appear to be in more than one place as a matter wave prior to its interaction with another object at a different location? And what about the recoil momentum of the source? The ballistic action of a photon leaving a source means a recoil of equal and opposite momentum of the source since that is our experience with the ballistics of firing a bullet from a gun. A gun immediately recoils with the bullet momentum and does not wait until the bullet hits a target. In other words, the bullet does not remain coherent with the gun from which it discharged for very long and so the ballistic path of the bullet is a single path from the source.
However, a bullet is really not an apt analogy for a photon as a matter wave.
A different perspective provides different information about an object and while that information from a different perspective is in principle knowable, we cannot ever know about an object from every possible perspective. We can never observe all of the different perspectives of an object, but still that lack of knowledge does not represent anything that is fundamentally unknowable.
The path of a photon through space, however, can represent information that is fundamentally unknowable. A matter wave is necessarily a superposition of states and so we can only know the result from say two possibilities, A and B, by seeing the photon along path A. However, we can only then conclude that the photon’s amplitude wave included path A and we cannot know that the photon was ballistic on path A. The photon may or may not have existed as a matter wave superposition on A and B even though we can still use the photon location at A or B to know the direction of the source.
A single photon event does not tell us very much about a source and we typically depend on many more than thousands of photons to locate a source image with any precision.
A photon and its source can remain coherent with each other and that coherency will persist until some kind of dephasing action occurs with another object. An action with another object can dephase either the photon or the source and if that happens, the photon becomes ballistic. A subsequent action between an object and the photon, such as reflection, polarization, diffraction, refraction, etc., in effect creates a new source and a new phase relationship with the photon.
Actually we readily accept some degree of time and spatial uncertainty for events as long as the uncertainites are local to an object or action. But it really distresses our causal nature when there are large spatial gaps between an object’s possibilities, i.e., when the fringe patterns of quantum interference are really large. It simply is not possible to assign ballistic trajectories to photons with anything more than a probability.
We as quantum beings are in a quantum universe and only have relational experiences with objects by exchange of matter. Yet we imagine from those limited relations a ballistic Cartesian existence outside of our quantum mind with well-defined objects that we recognize from past experience. While a Cartesian object has a single ballistic trajectory in space and time, there are many possible futures for a relational object with which we are in direct contact and so we exchange our own matter waves with that of an object. Quantum events and actions reveal that there is a relational dimension in our quantum existence, even though we normally only imagine a Cartesian world of objects from our relational experiences with those objects.
It is from our relational experience with an object that we project its Cartesian or ballistic reality and so that is the dilemma of existence. It is only possible for us to experience an object through our relations with an object’s matter waves, but we then imagine a ballistic Cartesian existence in our mind that represents that object on a trajectory in the space outside of our mind.
We can prepare a coherent state that represents a particle’s matter wave amplitude at two places across the universe from each other with different phases. However, once the particle interacts with an object in one place or the other, that action can dephase or collapse that matter wave and therefore localize the matter wave to a particle in that one place.
The background matter of the universe, whatever you want to call it, is mostly what defines the universe and there is necessarily a coherence in time for any matter action. The phase of an action of a particle defines the location and direction of the particle journey and so a particle reality occurs in just one location. A particle amplitude, though, goes into and out of existence as its matter wave oscillates in time, in principle for the whole time of the universe. And a particle as a matter wave at a given moment also varies in the matter spectrum of the universe, in principle involving all of the matter in the universe.
One way to unite gravity and charge force is by the principles of discrete matter and time delay. In discrete matter time, light is the exchange particle that is responsible for both charge and gravity forces. Light binds charges together into an atom with a single photon and light also binds atoms to the universe with photon pairs as an exchange that binds atoms to each other with gravity.
In much of our experience, particles are well localized and that means particles are dephased and incoherent and ballistic in both the time and matter of the universe. In quantum parlance, this is what we know as our Cartesian reality, where particles and objects all seem to behave ballistically and independently. If a particle is on a trajectory through space, that trajectory represents a continuum of displacements along that trajectory.
However, a particle as a coherent matter wave manifests itself with additional possible futures in both proper and action times of the universe. While charge force is a local exchange on the dimensions of an atom, gravity force is the stabilization of that atom with a photon exchange that occurs on the dimensions of the universe. A coherent charge state binds each atom with a coherent gravity state due to an emitted photon wave, a wave that has 2π symmetry. Gravity force, though, is a result of two complementary photon waves, which are the exchanges of photons on the much larger time and matter dimensions of the universe and therefore have a 4π symmetry.
In effect, gravity force is therefore coherent with charge force and the action of light scales both gravity and charge forces by the matter and time dimensions of the universe. The photon, electron, and proton of each atom are in an action that binds the atom together while a complementary emitted photon wave exchanges with discrete aether and binds atoms to each other through the universe of matter.
Coherent gravitational states are therefore possible, but only with very simple gravitational matter. The boson accretion that we call a black hole, for example, is an example of highly coherent gravitational matter.
In principle, a gravity beamsplitter as shown in the figure at right prepares small objects like atoms or molecules into a superposition of coherent gravity states. Two identical massive bodies like the earth and moon orbit each other around a center of mass as in the figure. Two much smaller and identical objects, A and B, are in orbits that intersect at a gravitational Lagrange point between the earth and moon.
It appears that any gravitational Lagrange point can result in generating coherent gravity matter states for small objects on different orbits. Moreover, two stars that are equidistant from a third star result in a similar degeneracy that results in a coherent matter wave resonance that affects all three stars. Such matter waves perturb the underlying discrete boson aether of the universe and so matter waves affect both charge and gravity actions in complementary ways.
Coherent matter states in the universe have the same proper times relative to a source event, even though they are widely separated in action time. While a matter wave can remain coherent with a source for a very long time, that does not mean that a particle’s existence is uncertain; it does mean that a particle’s state or future is uncertain.
There is a conflict between the ballistic Cartesian existence for an object that we typically project with our mind and the relational existence that actually binds us to the matter waves of objects with matter exchange. These two dimensions of existence represent the dual aspects of our quantum reality as well as the duality of Descartes’ and other philosophies. In our ballistic Cartesian experience, existence has one meaning; an object that exists does so right here and right now as part of a proper existence.
In our relational experience, the matter waves we exchange with objects only represent possible futures. When we exchange discrete aether waves, we in essence share or exchange both matter and phase with objects in the wavelike realm of quantum exchange, and existence of quantum matter waves means something more than Cartesian ballistic existence. The relational aether wave exchange that binds us to an object means that the object becomes a part of us and we become a part of the object, even though we only sense some small fraction of that matter wave exchange.
When we exchange matter waves with an object, we call that experience, and there is always a period of both matter exchange as well as phase coherence between two objects. Any residual coherence between us and the object can result in a further relational component beyond a mass change and is a quantum entanglement that is beyond the typical ballistic Cartesian experience of action and reaction that we imagine. Note that Cartesian and relational dimensions of experience are really both part of a dual quantum reality.
We can and do imagine and know that there are other possible futures for any event that we experience. In particular, an action can dephase a photon from its source in which case the photon becomes ballistic. But as long as a photon remains coherent with its source, a matter wave binds not only the photon to the source, but to other objects as well at the same time distance from the source.
The photon could have a single ballistic future or it could have the many possible matter wave futures that entangle it with other objects. It is the other possible nonlocal and unknowable futures that somehow bother our causal ballistic natures. We want to place each object that we experience on a single ballistic Cartesian trajectory that is continuous from an origin to a destiny. Our intuition does not have much patience for the seemingly endless waves of quantum coherency that entangle local aether waves with other aether waves on other trajectories in the universe.
A photon that remains coherent with the action of its source has different possible futures from a photon that has dephased from its source. A photon that has dephased from its source has a single ballistic future much like any macroscopic object. All macroscopic objects, though, continually emit and absorb light and particles with incoherent phases and so a macroscopic objects’ decoherence times can be quite short. Simple quantum objects like photons, though, can retain coherence with their sources across the universe.
We are very comfortable with the causal notion of directional coherence and expect that a single point of an object emits photons in a single direction. When we see a photon from such a point on an object, we know the direction from which it came and our quantum logic does not change that truth. Where we have trouble is in imagining a single photon event that also has a transverse phase coherence as a matter wave that is perpendicular to the photon direction from a source. Transverse phase coherence means that a photon amplitude travels as a coherent wave in different possible directions at the same time even though the photon will only be absorbed by another matter wave in one particular location or phase.
There are actually two dimensions to time and our two dimensional time along with two dimensional matter represents a total of four dimensions in matter time. Given a π/2 or perpendicular phase relationship between matter and time, these four matter time dimensions reduce to three; matter, time, and phase. Time’s two dimensions include a proper time and an action time and matter’s two dimensions likewise include proper matter and action matter.
Our proper time is relative to the CMB in our 371 km/s velocity inertial frame. Action time is that associated with velocities of common experience, perhaps all of several meters per second and so action time represents displacements that are orders of magnitude less than the displacement of proper time.
Proper matter describes our galaxy as it moves at 550 km/s with respect to the CMB and rotates at 200 km/s, while our sun moves at 220 km/s, about 20 km/s faster than the galaxy rotates. These actions all make up the proper matter that results from our 371 km/s proper motion with respect to the CMB while our action matter is what occurs at lower scale.
Earth rotates about the sun at 30 km/s and spins about its axis at 0.47 km/s while we travel down the freeway at 0.027 km/s and walk around at about 0.001 km/s. Matter is likewise two dimensional with one dimension being the proper matter of our comoving frame of reference in the universe. The second matter dimension is the action matter of common experience that we call kinetic and potential energies.
Each atom of the universe forms as bound charges in a quantum exchange of light and other bosons that complements a gravitational quantum exchange orbit of that atom with the gaechron matter of the universe. We like to imagine a ballistic orbit for gaechron around an atom through space just as we like to imagine an electron in a ballistic orbit around a proton. But the atom-gaechron orbit is through time and quantum phase and not through space just as the electron orbit is through time and quantum phase as well.
While continuous space and motion are very useful ways to imagine the universe, continuous space and motion do not always represent either electron-proton states or atom-aether states very well. In addition to the time of this atom-aether orbit, there is a quantum phase angle between time and matter and for typical action, and it is from matter and time and from that phase angle that we project what we call space.
For any pair of atom-universe bonds, the shrinkage of the universe aether is the gravity force by which atoms appear to attract each other. In fact, the shrinkage of the universe is responsible for both charge and gravity force, just at very different scale. Eventually, these gravitational accretions of fermionic matter evolve from hydrogen into other elements in stars and that nucleosynthesis releases more action matter. A portion of the total energy and luminosity or action matter of each galaxy derives from nucleosynthesis and that action matter eventually ends up as large boson accretions known as black holes.
The formation of protons and electrons from the aether of the early universe results in a light that is the integrated CMB luminosity at 2.7 K, very much colder than the 70-80 F that people prefer. Once stars begin to fuse hydrogen into other elements, there is enough action matter to reionize hydrogen as well as to begin to fuse matter into excited states of the universe. And this reionization is an additional source of energy that then contributes to an overall universe energy balance.
Suppose you see an object along path A, if the object was at some incremental displacement as A – ds the previous moment, then the object was ballistic and its action was local. There are objects that exist in a superposition of quantum states, {A, B, C, …} and such an object can distribute around the universe according to some prior coherent quantum action. Note that a ballistic object actually also follows that same quantum logic, but a ballistic object has dephased and no longer coherent with its source.
The action of a beamsplitter creates coherency between the two paths A and B and some kind of magic occurs at the beamsplitter that makes 50% of photons disappear by destructive interference at both A and B. The ballistic Cartesian interpretation is that the beamsplitter reflects 50% of the photons as particles to A and transmits 50% to B and although this answer is technically wrong, it is good enough for many applications. If all you need is a one-way mirror or a grayed window or sunglasses to block sunlight, you really do not need to know much about single photon coherence. Thus our ballistic Cartesian reality does work fairly well for most predictions of action, even for those quantum actions with quantum devices like sunglasses.
We often lack knowledge about the appearance of an object even though that object exists as a single state and its appearance is in principle knowable. We can also lack knowledge about the state of an object, but if the object does exist in a single state, that single state is in principle knowable as well and not subject to quantum entanglement. When an object or image is a superposition of two coherent amplitudes, though, a single state is not yet realized and therefore not even knowable in principle. The object or image will not appear until we or other objects dephase the amplitudes from each other and a single state occurs.
Using logic to test quantumology tries to get a more graphic description of nonlocality. Remember, though, that quantum logic is already quite rigorous since it is based on math. It is rather the word descriptions of quantum logic that somehow fail to convince our common ballistic intuition of the principle of coherency. Our language is full of loopholes and conundrums and logic itself is often thwarted by the words that confuse meaning.
You say A is B or A is not B, but of course, we have a lot of examples of words that provide ambiguous meaning even to simple logic statements. Nothing is true, but if that is correct, it means that nothing is not true as well. The universe is finite, and if that is true, it would mean that the universe is not finite as well. Everything is finite and if that is true, nothing is finite since nothing is a part of everything. If there is anything that is really true, it is that nothing is really true. But if nothing is true, then anything is not true as well.
Is matter real? Is time real? Is action real? What is matter and why is matter the way that it is? What is time and why is time the way that it is? What is action and why is action the way that it is? Why does the world exist?
Thinking is being, but thinking is in our mind and being is not in our mind, and if that is all true, thinking is not being.
One very significant issue with quantum versus gravity actions is in the definition of consciousness. Unless there is a way to express conscious choice in the context of quantum action, there will always be those who believe that conscious choice is an illusion of the chaos of a ballistic determinism. Usually the reasoning goes that all action in the world is actually deterministic, but the world it is also just really, really very complicated and so we can never hope to know all of that complexity and chaos.
In a world of chaotic determinism, while it seems like we have free choice, this is just an illusion and the truth is that we just have more choices that we can ever possibly know about. However, philosophers who take this position then need to stipulate that there is still a need for personal responsibility and morality. In a deterministic universe, it is not clear that anyone is really responsible for their actions. After all, action and behavior are simply the some total of their genes and experiences up until that point.
All choice comes down to a binary decision between action and inaction at some threshold of a neural action potential and since quantum probability determines the neural action potential as it does all action of the universe, quantum probability also governs choice. Circumstances at the time of a choice predetermine most choices that we make and so in that sense, even binary decisions are not random. Each set of circumstances determines the threshold of action, but at the threshold of each action/inaction there is a distribution of quantum possibilities and a superposition of action and inaction states. In particular, there are a number of even odds choices that we make that may still substantially change the path of our lives. Every action, then, is a quantum action and involves some superposition of states for some period after the action.
An aware matter algorithm is part of our consciousness is therefore an important part of what makes us us. While most actions have fairly predictable results, there are no perfectly predictable results of action, especially for the results of human actions. Given the free choice that is quantum action, we do have a responsibility for choosing moral action since we freely choose our path in life as part of our purpose.
What we know of as right and wrong and just and unjust is part of the purpose with which we journey in life from our origin to a destiny. We are not programmed to be good or evil, but we are free to choose our destiny despite any experience of our past.
Some of what happened in the past involved objects that persisted as amplitudes and never collapsed into intensities. What this means is not that these objects do not exist as one phase, rather it means that the objects persist with more than one possibility as matter amplitudes that still project into more than one spatial location in the present moment.
Continuous space and motion are really just the results of discrete matter and action and so space exists only as a result of discrete matter, time delay, and the action of matter exchange. What this means is that while space is a convenient and necessary way to imagine discrete matter and action, the notions of continuous space and motion are limited. Although we find it useful to remember space as an object of the past that contains objects of action, the universe exists as an object of matter and its matter spectrum is what actually exists. While we get confused by objects that appear to simultaneously exist in different places in space, the state of the universe matter spectrum at any past time is knowable. |
7c00fee65bd9fe42 | Born to be a Quantum Mechanic
Capricorn Research would not presume to understand Quantum Theory but it does show that on a subatomic level things are certainly not what they seem. In fact things can be in more than one place at the same time, and other things that we have always thought were different and separate from each other are in fact not. If this behaviour continued above the subatomic level, it would be very difficult to distinguish between one collection of atoms and another because they would be constantly moving and changing shape. People would be able to walk through walls, buildings and even each other without any problem at all. In fact people and anything else for that matter as separate entities could not really exist. A natural extension of Quantum Theory would say that we don’t exist, we only appear to because our brains tell us we do.
There always seemed to be an appropriate neatness in that the smallest phenomenon is so similar to the largest. Within an atom, electrons spin in orbits round the nucleus in the same way the planets spin round a Sun. Quantum Physicists have found one important difference, however. If an atom is heated these electrons could jump from one orbit to another without moving through the space in between therefore making a quantum leap. This would be tantamount to Mars instantaneously appearing in the orbit of Saturn and we all know what kind of problems that could cause.
So an electron could be in one place or another but nowhere in between. If this was translated above the atomic level anyone could suddenly disappear and appear somewhere else. It would mean that we could be at home and then at work instantly saving all that horrible commuting. The phrase ” Beam me up Scotty ” comes to mind.
Classical physics at the time as represented by Einstein believed that science could predict things with certainty. Quantum mechanics blows that idea out of the water. The original quantum physicists in the 1920s and 30s were truly radical people, the discoveries blew apart our conception of reality. Quantum mechanics was a science of Jupiter like adventure and discovery which expanded our thinking to a different plane altogether but also of Uranus, the rebellious planet of chaos and unpredictability and of breaking down the boundaries of convention.
So its interesting to note that the chart of one of the most famous quantum physicists, Max Born was utterly dominated by these two planets.
Max Born
Max Born had a stellium or multiple conjunction in Jupiter’s sign, Sagittarius. Sagittarius is the sign of the visionary, the explorer. Sagittarian journeys can be either of the body or of the mind, but the one thing they all have is a tendency to fire their arrows far and wide in a powerful search for knowledge and understanding. They have a broadminded approach to life, always interested in the big picture and have a strong desire to be free to explore wherever they wish to go. They are enthusiastic, and direct in their communications with others. They are frequently outspoken and will tell people what they think even if this upsets others. They can be tactless and blunt but their reasoning is that its important to be honest.
Born had the Sun, Mars, Mercury and Venus in the 10th house so it was inevitable that his career would take him to boldly go where no man had gone before. The Sun and Mars are in exact conjunction showing him to be a real pioneer. As if this wasn’t enough his Sun, Moon and Mars conjunction is part of a T Square in opposition to Jupiter.
This opposition points to an extremely powerful apex Uranus showing that Born was a real revolutionary in scientific thinking. So all that exploratory, pioneering Sagittarius / Jupiter energy is thrust onto Uranus. Uranus is independent, rebellious,inventive and provocative. It seeks to liberate us from our narrow, conditioned and conventional ways of seeing things. It can be uncomfortable and disruptive. If any planet was to take away the certainties of existence it would be Uranus, its far more likely to see the Universe as chaotic and disordered than any other planet. Born’s Uranus is made even more powerful given that it is his planetary ruler due to his Aquarius Ascendant.
The turning point of Born’s life came between 1901 and 1905 when Pluto was opposite his Sun. He began University in 1901 but moved to the University of Göttingen in 1904, where he found the three renowned mathematicians, Felix Klein, David Hilbert and Hermann Minkowski. Very soon after his arrival, Born formed close ties to the latter two men. From the first class he took, Hilbert identified Born as having exceptional abilities.
In 1905, Albert Einstein published his paper ‘ On the Electrodynamics of Moving Bodies ‘ about special relativity. Born was intrigued, and began researching the subject with Minkowski, and subsequently wrote his habilitation thesis on the Thomson model of the atom.
Quantum experiments showed that electrons would under certain conditions behave like waves rather than particles with the possibility of splitting and combining. This is another staggering discovery, until this point waves had been waves and particles had been particles, distinctly different things.
Another important contribution was made by Erwin Schrödinger, who looked at the problem using wave mechanics. Schrodinger thought that an electron could get spread out to become like a wave. This had a great deal of appeal to many at the time, as it offered the possibility of returning to deterministic classical physics.
In true Jupiter / Uranian style, Born would have none of this, as it ran counter to facts determined by experiment.
Born’s truly radical discovery said that the wave was not a spread out electron but a probability wave. He said the size of the wave at any location predicts the likelihood of the electron being found there. So the electron has many possibilities in terms of being at any place on the wave. He formulated the now-standard interpretation of the probability density function in the Schrödinger equation, and the probability thus calculated is sometimes called the “Born probability”. Born published his findings in July 1926 when Uranus was exactly opposite its natal position, recreating its relationship with the Sun, Moon, Mars and Jupiter. From an astrological perspective this would be the time in his life when he was at his most inventive, provocative and disruptive.
So Born found that we can’t predict where an electron is but Schrodinger’s equation can predict the likelihood of it being in any one place at one time compared to another. So all the rules of the Universe which is made up of atoms are governed by probability not certainty.
These probabilistic concepts were vigorously contested at the time by the original physicists working on the theory, and particularly by Einstein. In a letter to Born in December 1926, Einstein made his famous remark regarding quantum mechanics:
This famous quote that has been popularised as ‘ God doesn’t play dice ‘ brought a response from quantum physicist Niels Bohr ‘ Stop telling God what to do ‘.
Born’s discoveries have been the source of philosophical difficulties in the interpretations of quantum mechanics, topics that continue to be debated even today.
According to Quantum Mechanics the decision to measure the position of an electron will itself influence that position. When it is observed in a particular place the observation itself forces the electron to relinquish all of the other places that it could have been in at that moment. The act of measurement forces the electron to make a choice. So things only exist in any one place because we observe them to be there.
Einstein wasn’t keen on this – he said ‘ I like to think that the Moon is there even when I’m not looking at it. ‘ Einstein believed that Quantum Mechanics was not incorrect but incomplete.
To extend this concept to our own level of reality would suggest that nothing is definitely in any one space at any time. Other people are not really where they appear to be, its only our observation that makes them so.
Capricorn Research is tempted to test this theory with an experiment to kick a quantum physicist up the backside whilst not looking at him. There is no reason that it should hurt as in all probability he is not there anyway.
Another extraordinary discovery is the Quantum Concept of Entanglement – if two electrons come close enough together they absorb something of the nature of each other and they will continue to display this connection even if they go their separate ways over vast distances. So if you chose to measure one of these electrons establishing its position, you would also affect its partner no matter how far away it was. Recent experiments have verified that this works even when the measurements are performed 10,000 times more quickly than light could travel between them. According to quantum theory, the effect of measurement happens instantly.
These electrons can be zillions of miles away from each other and there’s no possibility of any forces or pulls operating between them and no way they can communicate with each other.
Of course Scientists are perfectly capable of getting their brains round this stuff when doing their own work, but not it seems when they come to dismiss the science of Astrology. A common objection is how can the planets affect individuals on earth, any gravitational or magnetic pull from even the Moon on an person would be less than that exerted by a double decker bus on the other side of the road.
Astrology doesn’t work like that, it works on the basis of symbolic entanglement. If entanglement can work between two electrons that come together in space, the same principle can surely work between phenomena that come together in time. As physicists are always telling us space and time are fundamentally the same thing. So if someone is born at a moment in time, the moment is the thing that brings everything in that moment together. So the positions of the planets become entangled with that individual and the relationship continues through time and space. The behaviour of that individual can be understood and even predicted by observing the entangled partner, in this case the position and behaviour of the planets.
Scientists will often dismiss Astrology as being stuck in the Middle Ages, but in fact its very way of working shows it to be a Quantum discipline.
As a quantum physicist Max Born might have thought the Universe unpredictable but because he had an apex Uranus Astrology would have predicted that he would think that. Pluto’s first transit to Born’s T Square predicted his first exposure to the subject of Relativity and set him on his journey. Uranus’s transit to the same pattern predicted his probability density function for which he eventually won the Nobel prize for Physics . Pluto’s second transit to his T Square actually predicted his death in 1970.
Max Born was literally born to destroy the certainties of science but Astrology can prove that the Universe is not as unpredictable as he thought.
The Turning Point in Your Life ?
Astrology and Celebrity – all in the timing
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04cc88071886a296 | טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים
M.Sc Thesis
M.Sc StudentBar-Ziv Uri
SubjectAccelerating Non Diffracting Acoustic Beams in Liquids:
Theory and Experiment
DepartmentDepartment of Physics
Supervisor ? 18? Mordechai Segev
Full Thesis textFull thesis text - English Version
Accelerating beams are recently attracting considerable research interest. They were first proposed in the context of quantum mechanics, where shape-preserving temporally-accelerating solutions of the Schrödinger equation were found. However, they have become an important research topic only three decades later when they were introduced into optics, in the form of Airy beams that "bend" in space. The field has attracted significant attention for several reasons: (i) the mathematical aesthetics of these beams; (ii) the search after similar phenomena in different fields of physics; and (iii) numerous promising applications. Indeed, soon after their first experimental demonstration, many applications were proposed and demonstrated, ranging from micromanipulation of particles, generation of curved plasma channels in the air, micromachining of curved surfaces, light sheet microscopy and 3D fluorescent imaging. In a different important domain, non-diffracting acoustic Bessel beams were demonstrated more than two decades ago, but until recently it remained a challenge to generate acoustic accelerating beams, that is, acoustic beams that bend in space. Recently, a pioneering work has demonstrated ultrasonic accelerating beams in gas (air), where an array of speakers with a specifically designed phase was used to launch acoustic bending beams in two and three dimensions. However, many applications of acoustics are in liquids. Examples range from ultrasound imaging and ultrasonic therapy in the medical arena, SONAR, hydrography and underwater communication, to ultrasonic detection of cracks in metallic components where the specimen is immersed in liquid to guarantee good coupling of the acoustic wave. Importantly, it has recently been suggested that acoustic half-Bessel beams could be used for micro-particle transport in liquids, but until now acoustic accelerating beams have never been demonstrated in liquids. Liquids are typically characterized by acoustic impedance which is orders of magnitude higher than in gas. As such, new methods, completely different than those used in gas, are required for generating such beams in liquids. Here, we demonstrate the first acoustic accelerating beam in a liquid: an underwater ultrasonic shape-preserving accelerating beam. The beam was generated by phase modulating a single projector using a tailored acoustic phase mask. The beam is propagating for a range in excess of 800 wavelengths, which are about 6 Rayleigh lengths, while preserving its shape and transversely accelerating. On top of many promising applications, such beams could provide new means to study non-linear interaction of acoustic beams. |
245a472469ebdcf2 | You are hereSZZL4
Course: Quantum and Statistical Physics
Department/Abbreviation: OPT/SZZL4
Year: 2020
Guarantee: 'prof. Mgr. Jaromír Fiurášek, Ph.D.'
Annotation: Atomic structure of a matter> Examples of structures of molecules and condensed systems Relation of observation of atoms and materials in a real and reciprocal space
Course review:
1. Basic experiments of quantum mechanics, de Brogli wave length, Stern-gerlach experiment, black body radiation, photoefect. 2. Formalism of quantum mechanics, representations. 3.Measurement in quantum mechanics, statistical interpretation, compatibility and dualism. 4. Dynamics of quantum systems, Schroedinger equation, Heisenberg equations. 5. Solutions for simple quantum mechanical systems: potential well, harmonic oscillator, spin and two level systems, laser, maser, NMR. 6. Schrödinger equation for hydrogen atom, centrifugal potential. 7. Symmetry in quantum mechanics, identical particles, parity and permutations. 8. Statistical mechanics and interpretation of thermodynamical laws, Phase space, probability distributions. 9. Microcanonical, canonicaland grand canonical statistical ensambles, partition function. 10. The first principle of thermodynamics, entropy and second principle of thermodynamics. Thermodynamical potentials. 11. Basic of quantum statistics, Bose- Einstein and Fermi distributions, Balck body radiation, Planck law. 12. Electron structure of the atoms. Atomic structure of a matter. Examples of structures of molecules and condensed systems. Electron structure of the systems of many atoms. Electrons in metals and semiconductors 13. Nuclei, basic a parameters and models, nuclear forces, meson fileds, electromagnetic, weak and strong interactions. Types of nuclear reactions, crossection and energy of nuclear processes, natural and artificial radioactivity. |
54e0358124790883 | Wonder Woman - S1E7
This episode - penned from the same guy who wrote a third of the Batman episodes. In this episode:
Mr. Brady has The Black Death
Dr. Bellows makes homemade earthquakes
And Wonder Woman does lots of math
So, what are we waiting for? Let's jump in.
Before we start, let me bring up a bit of confusion regarding the episode number designation. IMDb calls this Episode 8 because they count the pilot as Episode 1. I do not. Lynda Carter wasn't even WW in the pilot (it was Cathy Lee Crosby). Plus, the official series began seven months after the pilot; which puts the pilot even more in a world unto itself.
Mike Brady (Robert Reed) gets to play the villain in this episode - The Falcon... and he plays with a flair and panache the part deserves. This is post-Brady Bunch, and the rakish Reed is jaunting from one show to the next - from The Love Boat to McCloud to The Boy in the Plastic Bubble... and he's having a blast. The following year (1977) he'd be back with the Bradys in The Brady Bunch Variety Hour.
Our story begins with an Oscar Wilde-esque Falcon getting detained in a US airport. The lovely lady at his arm is Mikki McGoldrick (AKA Mikki Jamison) who just passed away in 2013 (obituary).
BTW the custom's agent in this scene is played by the daughter of Pat O'Brien, "Hollywood's Irishman in Residence". He was one of James Cagney's best friends, and starred with him in 9 films. He's the guy that said “win just one for the Gipper" (in Knute Rockne, All American).
Any clue on these equations? Are they for real, or gobbldygook? Inquiring minds want to know.
The Falcon escapes and steals his way into the laboratory of Professor Warren. Warren is played by Hayden Rorke, Dr. Bellows from I Dream of Jeannie.
The Falcon wants Warrens "Pluto File" - his secret for creating and prediction earthquakes. It's The Brady Bunch vs. I Dream of Jeannie. Who will win?
Yeoman Prince visits poor Dr. Bellows Professor Warren in the hospital - the old coot couldn't handle the trauma of a face-to-face with The Falcon.
It's filmed at Walter Reed Hospital. Lynda Carter was Miss USA in 1972 and toured the military hospital with Bob Hope visiting wounded soldiers just four short years before this episode was shot. Who knew that in '76 she'd be back, not as Miss USA, but as Wonder Woman. (insert dramatic music here)
Yeoman Prince notices The Falcon on a rooftop with a sniper rifle aimed at Professor Warren. She quickly changes into WW and deflects the bullets with her bracelets. But the ever elusive (and dashing rogue) Falcon escapes without a scratch.
I couldn't help but note that WW makes her twirly transformation right in front of the window The Falcon was aiming through. Seems he could've had a pretty easy head shot had he pulled the trigger.
Back at an undisclosed location, The Falcon plots his nefarious plan to "induce the world's first man-made earthquake". (insert Dr. Evil laughter here)
Actor Albert Stratton plays one of his forgettable henchmen, Charles Benson. Stratton hopped from one supporting TV role to next back in the day. He got a few prime gigs on the revived Perry Mason show and an episode of Quantum Leap. He's probably best remembered for a Star Trek: The Next Generation second season episode "The Outrageous Okona" (because you can always count on Trekkies to keep your memories alive).
In an interesting connection, this episode's director, Herb Wallerstein, directed four episodes of the original Star Trek.
Robert Reed was such a dandy.
And here's where shit starts getting a tad crazy with the story line. See if you can follow me:
Remember Mikka from the beginning? Well, she's got the Bubonic Plague. (You may need to read that sentence twice). The last case of The Bubonic Plague was in India, which just happens to be where The Falcon was before he came to the States. So he's a carrier!
Turns out, Benson is one of Professor Warren's fellow scientists... and he looks like Holy Hell. It doesn't take long for Steve Trevor to put two and two together and realize that, if Benson has The Bubonic Plague, he must have been in contact with The Falcon..... which can only mean Benson is a Nazi traitor!
I know, there's like a million mental leaps in order to come to that conclusion (maybe Benson just has a sinus infection); but the accusation is enough to make ol' Benson lose his marbles, pull a gun on Trevor, and get out of Dodge.
WW chases Bubonic Benson into a cannabis(?) field and uses her Lasso of Truth to get to the Falcon Connection.
So, if you're keeping track. The Falcon has not only an earthquake device, but also The Bubonic Plague. What's next, a nuclear bomb? Oh, snap...
Mike "The Falcon" Brady has plans to use The Pluto File to create an artificial earthquake around a nuclear reactor. The seismic activity will cause the reactor to go "boom" and The Falcon can go home to his Nazi brethren a conquering hero.
Note that "Pluto" was a genuinel WWII code-name used for the method of delivering oil during the D-Day landings (Pipe Line Under The Ocean = PLUTO). "Pluto" also could refer to the radioactive element, Plutonium, used to make these nuclear weapons. Who knew Wonder Woman had so many layers?
WW meets with the professor and the two combine their brains to solve a way to cool down the nuclear reactor. She actually schools the old scientist, and provides the solution for him. (Apparently, the solution is to just add water.... brilliant.)
This episode aired on Christmas Day 1976. Nothing says the holidays like natural disasters, nuclear meltdown, and biological warfare.
The Falcon bum rushes the lab, rightfully concluding that it would the professor who could stop this nuclear meltdown. But, he suddenly breaks down with The Bubonic Plague and can fight no more.
Professor Warren alerts the nuclear plant on how to stop the meltdown (just add water), and in a moment of parallelism, WW gets a glass of water for The Falcon.
And so it ends. Not one of the best episodes, but still none too shabby. It was written by Stanley Ralph Ross, an ordained minister who actually presided over the marriage of Burt Ward (Robin from Batman). Ross also directed that intro to Wide World of Sports we're all familiar with ("... and the agony of defeat")
And speaking of defeat, we learn that The Falcon didn't die from The Bubonic Plague, but is recovering in a maximum security prison.
The first WW pilot (with Cathy Lee Crorby) has no connection to the later series beyond the title.
Totally different concept and cast.
It's no more related than the 1960s promo reel with Linda Harrison ("Nova" in the first two Planet of the Apes flicks as Wonder Woman, done by William Dozier (Batman/Green Hornet).
Now, if they had integrated Crosby's pilot into the Lynda Carter series the way Star Trek incorporated "The Cage" into Classic Star Trek (maybe as an "alternate universe"), then you might have a case...
2. Linda Carter was Miss World 1972(-73), not USA. It surprised me to learn she was Miss Anything so I looked it up.
3. That's the Schrödinger equation on the blackboard, one of the workhorses of quantum mechanics.
4. who's looking at equations?????
5. Oh wow, this ep aired on Christmas day! you mean there was no lame reruns amid the hoilday shows? thoese were the days.
6. I think your field of marihuana is oleander (http://en.wikipedia.org/wiki/Nerium), which puts this scene somewhere warmer than Washington, DC.
7. I guess this episode didn't score any Steve Trevor knock-outs or WW being tied up.
8. I was never a fan of this episode. It is IMHO one of the few lemons in an otherwise great season.
9. What's more 1940's than over-the-ears feathered hair and sideburns? Great job, production team. |
9e28e539aa05e50c | BLOG 2019 Q1
House of Commons, 2019-03-29
What the f**k is going on?
Jonathan Pie
AR A comic rant that gets
to the rot at the core
of Austerity UK
The Times
Oliver Letwin
Sir Oliver Letwin
MP for West Dorset
Mein Urlaub
in Deutschland:
Michael Heseltine
Michael Heseltine
Michael Heseltine's
Saturday speech on Brexit
Bollocks to Brexit
Vladimir Putin
2019 March 31
Niall Ferguson
Britain and Japan have much in common. Both are densely populated island nations off the vast Eurasian landmass. Both were once mighty empires. Both are still quite rich. Both are constitutional monarchies. Yet while Britain today is in a state of acute political crisis, Japan seems a model of political stability.
You might think Japan has much bigger problems than Britain. The ratio of people over 65 to those of working age is 46%, the highest in the world. The gross public debt is now 238% of GDP, again highest in the world. Britain leads Japan in terms of innovation, economic and political freedom, ease of doing business, and even happiness.
Britain has embraced immigration. Japan has resisted it. There are now almost 1.3 million foreign workers in Japan, just 2% of the population. The figure for the UK is 13%. Perhaps conservatism is incompatible with immigration on this scale and the Brexit breakdown is a symptom.
2019 March 30
Article 13 Explained
New Scientist
The EU has issued a major new directive on copyright laws. Its Article 13 makes websites responsible for ensuring that content uploaded to their platforms does not breach copyright. Owners of websites where people can post content will be responsible for ensuring no unlicensed material appears.
To comply with Article 13, big platforms will need to ensure that any copyrighted material on their sites is licensed. The rules are intended to end music and video piracy online, ensure artists receive a fair payment for their work, and force tech giants to pay for content they aggregate.
Certain services are exempted, including non-profit sites, software development platforms, cloud storage services, and websites with less than €10 million annual turnover. Website owners are not required to install content monitoring software to detect copyright material.
EU member states must now pass legislation to implement the directive.
AR I trust this blog is not in new jeopardy.
2019 March 29
Meaningful Vote 2.5
BBC News, 1442 UTC
The House of Commons has rejected Theresa May's Withdrawal Agreement for a third time, this time by 286 votes for the deal and 344 against.
Past Caring
Nicholas Watt
An unnamed UK cabinet minister, when asked why Theresa May is holding another Brexit vote, said: "Fuck knows, I'm past caring."
2019 March 28
Brexit Deal Vote Tomorrow
BBC News, 1748 UTC
MPs will be asked to vote again on Brexit on Friday but only on part of the deal negotiated with the EU. They will vote on the withdrawal agreement but not the political declaration. This complies with House speaker John Bercow's ruling that the same deal cannot be introduced a third time.
Labour will not back the deal. Labour MPs called the new vote "extraordinary and unprecedented" and "trickery of the highest order" while shadow Brexit secretary Keir Starmer said: "We would be leaving the EU, but with absolutely no idea where we are heading. That cannot be acceptable."
The DUP say they will not back the deal and several ERG members refuse to back it. Boris Johnson said it was "dead" but he would reluctantly support it. The withdrawal agreement must be passed by close of play tomorrow to meet the EU requirement for a delay of B-day to 22 May.
A Way Forward
Oliver Letwin
The issue is whether parliament can come to a majority in favour of a way forward on Monday. MPs will be voting on the basis of seeing what happened last night. And either the prime minister will have got her deal through on Friday, in which case all this is unnecessary, or people will see that isn't going to happen by 12 April. Quite a lot of Tories who didn't vote for any of the options may then come round and say: OK, we'll choose among these options.
It's very difficult to translate how people vote the first time, when they don't know how other people are voting, to how they will vote when they can see how other people are voting, under new circumstances. Many of us think leaving without a deal on 12 April is not a good solution. But is parliament on Monday willing to come to a majority view about a way forward?
DUP Thwarts May Gambit
Financial Times
Theresa May has gambled her premiership to win support for her Brexit deal. She hoped to make a third attempt to pass her Brexit deal on Friday. But the Northern Irish DUP says it will continue to vote against it. Steve Baker and other ERG Brexiteers say they will also vote against it. On May's offer to resign, Baker said: "I'm consumed with a ferocious rage after that pantomime."
No No No No No No No No
The Guardian
In a series of indicative votes in the Commons, all eight proposed alternatives to the government's Brexit deal were defeated. The two closest:
A plan to negotiate a "permanent and comprehensive UK-wide customs union with the EU" in any Brexit deal, proposed by Conservative veteran Ken Clarke and others, was lost by 264 votes to 272.
A plan to require a second referendum to confirm any Brexit deal, proposed by Labour former foreign secretary Margaret Beckett, was lost by 268 votes to 295.
Oliver Letwin, who pushed to let MPs take control of the order paper for the votes, said the results were "disappointing" but hopes for more clarity after new votes on Monday.
AR Only the option that dare not speak its name remains: Revoke Article 50.
2019 March 27
May Vows To Quit
BBC News, 1943 UTC
Theresa May has promised Tory MPs she will resign as prime minister if they back her Brexit deal.
A smiling Boris Johnson says he will now back the deal.
AR Excited commentators are saying this is the biggest thing in British politics since the fall of Neville Chamberlain and the rise of Winston Churchill in May 1940.
European Citizens
Donald Tusk
We should be open to a long extension if the UK wishes to rethink its Brexit strategy. We cannot betray the 6 million people who signed the petition to revoke article 50, the 1 million people who marched for a people's vote, or the increasing majority of people who want to remain in the EU. They may feel they are not sufficiently represented by the UK parliament, but they must feel that they are represented by the European Parliament. Because they are Europeans.
AR I am by UK law a subject of the Crown and by EU law a citizen of the European Union. Losing my preferred citizenship but retaining my embarrassing subjection is utterly dismaying to me.
The Brexit Delusion
Martin Wolf
Brexiteers say the UK is going to take back control. This was the biggest delusion of all.
Control is different from sovereignty. The UK was already sovereign. Control is about power.
The EU is more powerful than the UK. For the EU, the UK market is important. For the UK, the EU market is vital.
The world contains three economic superpowers: the United States, the EU (without the UK), and China. These generated about 60% of global output in 2018. The UK contribution was 3%.
The UK is a trading nation and has no future as anything else. Markets all over the world cannot compensate for reduced access to the market of 450 million people on its doorstep.
The United States will impose hard terms in any bilateral bargaining with the UK. Both China and India will insist on UK acceptance of their terms. Australia, Canada, and New Zealand together contain fewer people than the UK.
Outside the EU, the UK will not have greater control over its global environment. Trade agreements are increasingly about regulatory standards. The UK will often have to align itself with the standards of others.
The UK will not take back control by leaving the EU.
2019 March 26
Brexit: Taking Back Control
Financial Times
Theresa May on Monday night risked losing control of Brexit, after MPs voted to seize control of the House of Commons timetable and test support for alternatives to her withdrawal deal. She had ordered her ministers to oppose the Letwin amendment.
Former Conservative minister Sir Oliver Letwin hoped his amendment would give parliament a chance to find a cross-party way forward on Brexit. Several senior ministers say there is a growing possibility that a general election might be needed to end the stalemate.
May warned of a protracted "slow Brexit" if an extension to the Article 50 process were agreed by the EU and the UK took part in European elections. She remains at loggerheads with the hardline ERG Brexiteers.
Letwin: "This is just the beginning of a very difficult process as we struggle to find consensus across the House."
AR Oliver Letwin is a former Cambridge philosopher. His prizewinning PhD thesis was on emotions and led to his 1987 book Ethics, Emotion and the Unity of the Self.
Grand Wizards of Brexit
The Jouker
A while back, we all had a good laugh at Jacob Rees-Mogg's European Research Group naming their elite team of lawyers the Star Chamber.
The Star Chamber was a court of inquisitorial and criminal jurisdiction in England that sat without a jury, used arbitrary methods, and imposed severe punishments. It was abolished in 1641.
May met leading ERG members at Chequers on Sunday for crisis talks on Brexit. The hard Brexit day-trippers failed to reach an agreement with her.
BBC reporter Laura Kuenssberg: "The 'Grand Wizards' (the new name for the Chequers day-trippers apparently) also had another meeting this morning .."
Grand Wizard was a title used for the leader of the Ku Klux Klan.
AR Kuenssberg later said it was just a nickname, but the Grand Wizards of ERG in their Star Chamber are no joke.
An Island Alone — No!
Michael Heseltine
Brexit is the biggest peacetime crisis we have faced. A no-deal Brexit could provoke a national emergency. The most sensible step would be to put the issue on hold, complete the negotiations, and then hold a referendum.
I dismiss with contempt the image of us as an island wrapped in a union jack, glorying in the famous phrase that captured, for so many, Winston Churchill's spirit of defiance in 1940: "Very well, alone." I was there. I saw our army evacuated, our cities bombed, our convoys sunk. Churchill did everything in his power to end this isolation. Alone was never Churchill's hope or wish: it was his fear.
Now, I look back over the years: 70 years of peace in Europe, 50 years of partnership between the UK and the rest of the EU. The fascists have gone from Spain and Portugal, the colonels from Greece. Now we have 28 democracies working together on a basis of shared sovereignty, achieving far in excess of what any one of us could individually. Never forget that it was the memories of Europe's war that laid the foundations of the European Union today.
Margaret Thatcher would have been appalled to see Britain excluded from the top table. Theresa May dashed across the Channel last week, only to be excluded from a meeting of our former partners, and presented with a take-it-or-leave-it offer. That is what the Brexiteers have done to our country: a national humiliation, made in Britain, made by Brexit.
Britain cannot run from today's global realities of a shrinking world menaced by terrorism, international tax avoidance, giant corporations, superpowers, mass migration, the rise of the far right, climate change, and a host of other threats. Against them, our duty is to build on our achievements in the areas of peace and security that the EU has given us, to maintain our trade access where it matters and to keep our place at the centre of the world stage.
We have a responsibility to hand over and pass on to a younger generation a country richer, more powerful, and safer than that which we ourselves inherited. And doing so in partnership with Europe is our destiny.
AR A great speech — Heseltine's finest hour.
2019 March 25
Brexit: Parliament Seizes Control
BBC News, 2248 UTC
By 327 votes to 300, MPs pass a motion as amended by Sir Oliver Letwin allowing the Commons to take control of the parliamentary agenda to hold indicative votes on Brexit options. The amendment was passed by 329 votes to 302. Three government ministers resigned to cast their votes.
Sylvie Kauffmann
Europe is under attack. For the United States, China, and Russia, Europe is a political and economic target.
Russia has been at work for some time. Moscow's efforts to undermine democratic processes and the cohesion of the EU are now part of the political landscape. In parallel, Russia is increasing its economic footprint in EU countries that are more welcoming than others.
Chinese president Xi Jinping wants to connect Europe to China economically. China has bought the port of Athens and some other gates to southern Europe. The Belt and Road Initiative has involved setting up an organization called 16+1 (16 European former Communist states, 11 of them EU members, plus China) to help them build infrastructure.
The United States has its own fight with China. In a normal world, Washington would have enrolled its European allies in its fight. But Trump America treats Europe either as a competitor or as a vassal.
Europe is a soft target, hampered by its complex politics. The Brexit chaos will leave a mark. Europeans must decide whether they wish to let their continent be cut up by competing big powers, or whether they want to regain their strength and control their own destiny.
French president Emmanuel Macron: "Europe is not just an economic market. It is a project."
European Struggles
Gideon Rachman
Last Saturday, Remainers protested in London and gilets jaunes again came out in Paris and other French cities. The previous weekend saw mass demonstrations by Catalan separatists in Madrid.
Britain's crisis is part of a wider pattern. Its vote to leave the EU in 2016 was swayed by the German refugee crisis of 2015. Radical Leavers have taken to wearing yellow vests, as in France. The independence referendum in Catalonia was inspired by the referendum in Scotland in 2014.
Europe is changing. Nationalist-populist governments are in power in Italy, Hungary, and Poland, and form part of the coalition government in Austria. The far right has also performed strongly in elections in France, Germany, and the Netherlands, and is making gains in Spain.
European leaders have to ask whether to cut Britain loose to discredit radical forces across the continent. But they risk deepening the crisis. Their decisions will affect the whole of Europe.
British Contagion
Nic Robertson, CNN
The British state is not faring well. The UK political establishment appears to be crumbling, as a pioneer of modern democracy flounders in archaic and arcane process. Attitudes are stiffening in Europe, as the EU resolves to protect European democracy from British contagion.
AR Economic inequality, democratic dysfunction, mass immigration — go figure.
2019 March 24
Mueller Finds No Trump-Russia Conspiracy
The New York Times
The investigation led by Robert S. Mueller III found that neither President Trump nor any of his aides conspired or coordinated with the Russian government's 2016 election interference, according to a summary made public by the attorney general.
The special counsel's team lacked sufficient evidence to establish that President Trump illegally obstructed justice but stopped short of exonerating Trump.
Putin's Russia
Andrew Higgins
Russian president Vladimir Putin sits atop a ramshackle system driven more by the calculations of competing bureaucracies and interest groups than by Kremlin diktats.
Ekaterina Schulmann: "This is not a personally run empire but a huge and difficult-to-manage bureaucratic machine with its own internal rules and principles. It happens time and again that the president says something, and then nothing or the opposite happens."
Russia today resembles not so much the Soviet state ruled by Stalin as the dilapidated autocracy of Russia in the early 19th century. Czar Nicholas I presided over corrupt bureaucracies that led Russia into a disastrous war in Crimea and let the economy stagnate.
Schulmann: "It is a great illusion that you just need to reach the leader and make him listen and everything will change. This is not how it happens."
In his annual state of the nation address last month, Putin stressed the need to let business people work freely. He admitted he had made the same demand in a previous address: "Unfortunately, the situation has not improved much."
Brexit: May Meets Rebels
The Guardian
UK prime minister Theresa May met with a group of senior Conservative rebels including Boris Johnson, Dominic Raab, Jacob Rees-Mogg, Steve Baker, and Iain Duncan Smith at her Chequers country retreat today.
Chancellor Philip Hammond: "I'm realistic that we may not be able to get a majority for the prime minister's deal, and if that is the case then parliament will have to decide not just what it's against but what it is for."
Article 50 Petition: 5M+
The Guardian
Brexit petition to revoke Article 50 exceeds 5 million signatures.
March, London, 2019-03-23
Photo: EPA
The crowd on Piccadilly
The Guardian: People's Vote Brexit rally draws 1 million marchers (2:32)
BBC News: People's Vote march to Westminster — sped up (1:30)
AR I was there too, alongside a million people aiming to send a message to
HM government. Whether it succeeds, only the next few weeks will tell.
Put it to the people
London, Saturday
K.K. Uhlenbeck
K.K. Uhlenbeck
2019 March 23
Trump Investigation
The New York Times
Special counsel Robert S. Mueller III has delivered his report to the US Justice Department.
2019 March 22
EU Lifeline
The Times
EU leaders give Theresa May 3 weeks to come up with an alternative Brexit plan if MPs reject her deal again. May now has an unconditional extension until April 12. If her deal is passed, she has until May 22 to pass legislation implementing Brexit.
Strategic Failure
Confirmation that a longer extension may still be on the table makes May's defeat more likely. Brexiteers will say no deal remains the default outcome. But defeat of her deal will not end this drama: Those holding out for a softer Brexit or no Brexit at all will hold out.
Ministers opposed to no-deal thought they had a commitment from May to seek a long extension if a deal had not been agreed, and then to hold indicative votes to find an alternative way forward. They now see she is ready to take the UK out of the EU with no deal.
MPs can seize control of events to vote on alternative strategies. That probably means resigning the whip. To prevent no-deal, parliament may need to find a new prime minister.
AR My former ministry had plans: Operation Yellowhammer was to start Monday, with thousands of troops on standby and reservists called up. Hard Brexit would trigger Operation Redfold, run from a crisis room in the nuclear bunker deep beneath Whitehall.
UK Political Breakdown
Gary Younge
The idea that Brexit has broken the UK gives too much credit to the Brexiteers. The two main trends in postwar electoral politics have been the decline in turnout and waning support for the two major parties. Brexit merely shows the UK system is bust.
Since the 2008 crash, most Western countries have seen electoral fracture, the demise of mainstream parties, a rise in nativism and bigotry, increased public protest, and general political dysfunction. The virus that drove the UK mad is on the loose.
AR Time to re-engineer Western democracy.
2019 March 21
"No deal for sure"
CNN, 1430 UTC
French President Emmanuel Macron: "In case of no vote — or no — I mean directly — it will guide everybody to a no deal for sure. This is it."
Brits Are EU Citizens Too
Timothy Garton Ash
More than 16 million British citizens voted for Britain to remain in the EU in 2016. European citizenship is at stake.
The UK contains three nations: England, Wales and Scotland, together with a part of a fourth, Ireland. The EU27 member states have been impressive in their solidarity with Ireland. But Scotland voted by a majority of 62% to 38% to remain in the EU.
Europe will lack the power to defend our shared interests and values in the world if Brexit goes ahead. Not harmonious cooperation but dissonance will almost certainly be the consequence.
Brexit On Hold
Oliver Wright, Henry Zeffman
UK prime minister Theresa May wrote to European Council president Donald Tusk asking for B-day to be delayed until June 30.
Tusk responded by making a short extension conditional on MPs approving her deal next week. A short extension can only last until May 23, the date of European Parliament elections, as the UK seats will then be redistributed among other member states.
May is likely to ask the Commons to vote on her deal again (MV3) on Monday. MPs may refuse. The last date the UK can opt to take part in the European Parliament elections is April 12.
"Nuke it from space"
BBC News
"Time to take [Article 50], bin it, set the bin on fire, kick it over, and nuke it from space
— we're done."
Dr Mike Galsworthy about the trending petition Revoke Article 50 and remain in the EU
AR Update 1504 UTC: Petition has 1,012,938 signatures and counting (site keeps crashing)
2019 Vernal Equinox
Theresa May, 2041 UTC
You want us to get on with it, and that is what I am determined to do.
AR She still hasn't given up.
Europe and China
Financial Times
The EU summit this week will focus on China. EU official: "While we were absorbed in our own crises for 10 years, the GDP of China soared and Trump was elected. We entered a different game."
The EU is China's largest trading partner, and China is the EU's second-largest, behind the US. In 2018, China accounted for about a fifth of EU goods imports and more than a tenth of its exports. Levels of Chinese direct investment in the EU have soared.
German economy minister Peter Altmaier says China's growing technological prowess shows Europe needs a new industrial strategy. Chinese investments in Germany raise fears in Berlin about sensitive areas of the economy. German Council on Foreign Relations director Daniela Schwarzer: "For a long time the business sector was highlighting the relationship with China as a bonus but they are now highlighting the cost of this kind of engagement. The debate now is risk minimization."
In the EU, 13 member states have signed endorsements of China's Belt and Road program. Northern member states call it opaque and strategically aggressive and say China can impose crippling debts on recipient states.
Chinese foreign minister Wang Yi: "Europe will surely keep its fundamental long-term interests in mind and pursue a China policy that is consistent, independent and forward-leaning. Overall China and Europe relations are in good shape. There are far more areas where we agree than disagree."
Abel Prize 2019
Norwegian Academy of Science and Letters
The Abel Prize for 2019 goes to Karen Keskulla Uhlenbeck of the University of Texas at Austin for her pioneering achievements in geometric partial differential equations, gauge theory, and integrable systems, and for the impact of her work on analysis, geometry, and mathematical physics.
Uhlenbeck developed tools and methods in global analysis, which are now in the toolbox of every geometer and analyst. Her work also lays the foundation for contemporary geometric models in mathematics and physics.
Her fundamental work in gauge theory is essential for the modern mathematical understanding of models in particle physics, string theory, and general relativity.
AR I'm awed. This is stuff I struggle with.
2019 March 19
Brexit Crisis
BBC News, 1612 UTC
Theresa May is writing to the EU to ask for Brexit to be postponed until 30 June with the option of a longer delay. A cabinet minister says there was "no agreement" in the cabinet this morning. Under current law the UK will leave the EU with or without a deal in 10 days.
Physics Beyond Higgs
Natalie Wolchover
In 2012, the Higgs boson materialized at the LHC, leaving open many mysteries about the universe.
We understand little about the Higgs field, or the moment in the early universe when it shifted from being zero everywhere into its current state. That symmetry-breaking event instantly rendered quarks and many other particles massive, which led them to form atoms and so on.
Perhaps Higgs symmetry breaking led to matter-antimatter asymmetry. Another question is whether the Higgs field is stable or could suddenly trigger vacuum decay. A growing bubble of true vacuum would swallow up the false vacuum we live in, obliterating everything.
A proposed supercollider would collide electrons and positrons with energies tuned to maximize their chance of yielding Higgs bosons, whose decays could be measured in detail. In phase two, it would collide protons, resulting in messier but much more energetic collisions.
We want to observe the triple Higgs coupling in which a Higgs boson decays into two of itself. The Standard Model predicts its value, so any measured deviations would signify new particles. Measuring the coupling would also pin down the shape of the Higgs field.
Should we invest billions of dollars in a machine that might simply sharpen up our knowledge?
AR Better to invest in this than in nuclear overkill.
Milky Way
Andrew Whyte / Sony
Milky Way viewed from the cliffs of the Dorset coast
Revoke remain rebuild
British Empire
IV. Reich
Brexit and Democracy
PDF: 2 pages
2019 March 18
Brexit: No Return
BBC News, 1557 UTC
Commons Speaker John Bercow has ruled out the government holding another vote on its previously rejected Brexit agreement if the motion remains substantially the same.
Brexit: No Delay
Stefan Kuzmany
Theresa May will probably ask the EU to give her more time. The EU-27 should refuse. We have enough problems without this farce called Brexit. The EU urgently needs reform. Letting the divided Brits remain, only to have them hinder progress, would be fatal. They must go.
European Nationalists Love Israel
Ivan Krastev
National populists in central Europe are fascinated with Israel and its right-wing prime minister.
Zionism mirrored the nationalistic politics in central and eastern Europe between the two world wars. European populists see Israel today as an ethnic democracy. It has preserved the heroic ethos of sacrifice for the nation that nationalists covet for their own societies.
Central and eastern Europeans see Israel as winning the population war by reversing demographic decline. At a time when the population of eastern Europe is shrinking fast, Israel is persuading diaspora Jews to return and convincing Israelis to have more children.
European populists agree with Yoram Hazony that the big political clash in world history is not between classes or nations but between nationalists who believe that the nation state is the best form of political organization and imperialists who push for universal empire.
Israel faces existential threats. The threats are real. Whereas the European states are in the EU.
AR Brexiteers see Israel as a model for Fortress UK: defiant, militarized, and tight on immigration.
Enola May
Peter Müller, Jörg Schindler
UK prime minister Theresa May is the main impediment to solving the Brexit mess.
Last week, May was humiliated by her own party once again. Parliament rejected her divorce deal for the second time, again by a huge majority. Whatever happens now is no longer up to her.
May has led her country, her party, and herself into a labyrinth. She has neither the power nor the ideas to find a way out. Now, for many, Brexit has become a vote of confidence in May herself.
May vacillated for months before defining her Brexit. And then she got it wrong. She set bold red lines, she uttered hollow phrases, and she miscalculated the kind of deal parliament would accept.
May said what matters is the "will of the people," but she was mostly thinking about her own party. To push Brexit over the finish line in a third vote this week, she is again looking to hard-liners.
May could still choose a different path.
AR To quote Prince Charles: "Really? You don't say."
British Science
Alice Gast
Science is one area where Britain is world-class. As Brexit and immigration checks loom, we must keep Britain attractive for scientists.
Breakthroughs in frontier science rely on EU collaborations. EU peers want continued frictionless partnership in the Horizon program.
We cannot afford to lose talent mobility in Brexit. UK universities attract the world's best scientists. Brexit Britain could lose them.
AR Science is universal: British science is an oxymoron.
2019 March 17
United Ireland
Timothy Egan
For going on three years now, Britain has taken a holiday from sanity. But from the depths of British bungling, hubris, and incompetence is emerging a St Patrick's Day miracle: the real chance of a united Ireland.
After more than 800 years, London's ruling reach in Northern Ireland may end with the whimpering last gasps of Brexit. Don't wait for Her Majesty's government to resolve the sovereignty issues holding up the divorce between Britain and the European Union. There is no solution.
What UK prime minister Theresa May calls "our precious union" is held together by 10 MPs representing the old hatreds of North Ireland: the DUP. Given a choice, a majority in Northern Ireland could well be persuaded to ditch what is left of Britain and form a single Irish nation.
This was all Britain's doing — a single Irish nation finally free of foreign rule.
2019 March 16
Mathematical Models
Patrick Honner
Mathematics has a long history of defying expectations and forcing us to expand our imaginations. So mathematicians strive for proof. Still, evidence is important and useful in mathematics.
The twin primes conjecture is an example. The twin primes conjecture is not the twin primes theorem, because no one has been able to prove it. Yet almost everyone believes it is true, because there is lots of evidence that supports it.
As we search for large primes, we continue to find extremely large twin prime pairs. The largest currently known pair of twin primes have nearly 400,000 digits each. We know that there are infinitely many pairs of primes that differ by no more than 246, but we still haven't proved the twin primes conjecture that there are infinitely many pairs of primes that differ by 2.
Mathematical models are used everywhere in science and can used to study mathematics itself. They are powerful tools that let us trade a problem we don't fully understand for one we have a better handle on. But we can never be certain that our model behaves enough like the thing we are trying to understand to draw conclusions about it.
Mathematicians know to be cautious when working with their models.
AR This recalls for me the book Proofs and Refutations by Imre Lakatos, which I read with pleasure in 1972, in which he espoused an evolutionary conception of mathematics (inspired by the philosophy of Karl Popper), and which I soon used to develop my own dialectical picture of logic, mathematics, and reality (inspired by the works of Hegel, Frege, Gödel et al.).
Dreams of Empire
James Meek
What may seem, rationally, to be dead, gone, and buried is actually still there, immanent, or hidden, or stolen. An empire. The past week has laid bare the crisis in British politics.
Leavers dream about the Britons who endured the Nazi siege of the early 1940s as "we" who feel bound to re-enact the slaying of a European dragon every few generations.
A subliminal empire persists in their dreaming. From Margaret Thatcher they take the credo that nationalism and borderless capitalism can easily coexist. This idea makes sense only if your country happens to control a global empire.
AR Empire 2.0, Commonwealth 2.0, Common Market 2.0 — all seek refuge in the past.
Common Market 2.0
Nick Boles
Next week the prime minister will hold a third "meaningful vote" on her deal. A third defeat is likely.
A Brexit compromise many MPs could support is Common Market 2.0: the UK would join Norway outside the EU but inside the single market.
The UK is already a member of the EEA, which covers the EU and EFTA. All it would need to do is secure consent to renew its EEA membership after it left the EU and join EFTA by the end of 2020.
Common Market 2.0 would leave the UK out of EU policies on agriculture, fishing, justice, defence, and foreign affairs, out of ECJ jurisdiction, and paying only for chosen programs and agencies.
The UK would have to accept the free movement of people, but with an emergency brake.
AR I could accept this as an alternative to EU membership.
2019 March 15
A Fourth Reich
Thomas Meaney
First Reich — God the Father and the Hebrews, Second Reich — Jesus and the Christians, Third Reich — the Nazis. More prosaically, the First Reich of the Holy Roman Empire under Charlemagne and the Second Kaiserreich secured by Bismarck led to the Third Reich under Hitler.
The Nazis deprecated the term "Third Reich" because it suggested a coming Fourth Reich. SPD intellectuals drafted a constitution for the Fourth Reich that would come about after the fall of Hitler. It would be dedicated, they said, to global democracy and the equality of peoples.
Since 1945, talk of a Fourth Reich offers perspective for calibrating the rise of the Alternative für Deutschland (AfD). There were several right-wing German parties in the postwar years. The Socialist Reich Party was founded in 1949 but soon banned by the fledgling Federal Republic.
Today, European critics view the European Union as a kind of Reich in thin disguise. The history of the European Union can be written as an origin story that begins with Hitler but was only realised in opposition to his aims. Europe is now too anglo to have patience with a German Reich.
AR A provocative train of thought, but worth a moment.
2019 March 14
Brexit: UK To Request Delay
BBC News, 1823 UTC
House of Commons passes the following motion by 412 votes to 202:
The government (1) will seek to agree with the EU an extension of the period specified in article 50;
(2) agrees that, if the house has passed a resolution approving May's deal by 20 March, then the government will seek to agree with the EU an extension of the period specified in article 50 for a period ending on 30 June for the purpose of passing the necessary EU exit legislation; and
(3) notes that, if the house has not passed a resolution approving the negotiated deal by 20 March, then it is highly likely that the EU would require a clear purpose for any extension, and that any extension beyond 30 June would require the UK to hold European Parliament elections in May 2019.
Amendment (h), calling for an extension to article 50 to allow time for a referendum on Brexit, was rejected by 334 votes to 85.
Amendment (i), calling for time next week for a debate that would start the process of allowing MPs to hold indicative votes on Brexit alternatives, was rejected by 312 votes to 314.
Amendment (e), calling for an extension to article 50 to provide parliamentary time for MPs to find a majority for a different approach to Brexit, was rejected by votes 318 to 302.
AR Some of these questions will be revisited as events unfold in the coming weeks.
Brexit and Democracy
Andy Ross
A democratic political system is a formalised way of enacting the will of the people. Since no individual politician can credibly claim to know the will of the people directly, the system forms a snapshot of that will by collecting the votes of the people and subjecting them to some simple procedure, such as counting, to assemble a pixelated image.
We can safely leave the technicalities of the pixelation process and the production of a snapshot to the political experts. Experience of many systems over many years has reduced the business, if not to an exact science, at least to a fine art. What remains is to evaluate the meaning and the importance of the portrait of the people that results.
Ask a stupid question, get a stupid answer: This piece of folk wisdom constrains the value of the individual pixels that depict the will of the people. A simple yes-no question generates black or white pixels from which only a grainy outline image can be extracted. On the other hand, a nuanced question will mean different things to different people.
Whatever the outcome, the paramount risk in a democratic system is that the image is taken as the reality. However flawed, the portrait becomes an icon, a sacred symbol toward which politicians must perform holy rites to appease their voters. The risk is that the people, thus venerated, develop an inflated sense of their own importance.
Traditional religion, for all its flaws, drummed humility into its followers, and a traditional monarchy drummed humility into its subjects. But a modern democracy invites its voters, or at least those of them who are on the winning side in a division, to imagine their sovereign will is supreme. This used to be condemned as the Christian sin of pride.
Self-will, in all its forms, is a dangerous spur to action. The momentary self of an individual person may prompt overindulgence of a vice such as gluttony or lechery, but the larger shared sense of self of an organised group of people, as in a political movement, can lead to catastrophic outcomes. History is awash with cautionary examples.
For this reason, in modern times, the nation states of Europe have organised themselves into a superordinate body, the European Union, that contains and shapes the sovereignty of its members and preserves a modicum of order between its peoples. Similarly, in earlier times, the different peoples on the British Isles organised themselves into the United Kingdom. In both cases, the aim was to limit and channel the expression of political self-will toward higher values or virtues that might better serve the common interest.
In recent years, the UK has found itself on a collision course with the EU. The titanic parliamentary juggernaut of the UK establishment, trailing a historic wake of martial and imperial glory, is now grinding disastrously against the massive continental iceberg into which the formerly fractious nations of Europe have frozen their animosities. The predicted outcome toward which all sober expectation converges is that the EU, for all its obvious flaws and weaknesses, will be less damaged by the collision than will the UK.
The bigger picture is worth pondering. The victory of democracy in 1945 led experts to conclude that politicians heeding the popular will, as expressed in democratic elections and parliaments, were stronger than dictators in more authoritarian systems who failed to carry the people with them on their political adventures. That conclusion has been allowed to decay in recent years into a lazy acceptance that populism, in which demagogues uphold relatively wild expressions of popular will for opportunistic reasons, is a valid way to continue the democratic tradition.
In Ancient Greek philosophy, the decay of democracy into populism was a precursor to tyranny: A populist leader channels the popular will by means that short-circuit the checks and balances of the usual democratic processes until that leader finally usurps the popular will and rules as a tyrant. For some observers, President Trump in America illustrates the early stages of this process. For others, the emergence across Europe, including Russia, of popular and increasingly authoritarian leaders reveals the same trend.
In the wider sweep of politics, it is worth remembering that democracy is a means, not an end. Individual people will this or that end in ways that can only be deconflicted in a system that balances the conflicting ends against each other, and democracy has proved to be a simple and robust mechanism to establish and deliver that balance. By contrast, an authoritarian system will prioritise one set of ends above all others and force the losers to swallow their pride and accept defeat, if not total ruin.
Populists on the path to tyranny tend to take a crudely pixelated image of the popular will and weaponise it against all opposition. Soon enough, the image becomes an abstract icon, like a cross on the shield of a crusader, and the people are praised in name only under the tyrant's rule. This is the road the Bolsheviks took in Soviet Russia when they established the dictatorship of the proletariat, first under Lenin and then under Stalin, before proceeding to ruin old Europe.
Applied to the collision between the UK and the EU, the drift from democracy to populism is evident in the aggressive sacralisation of the 17.4 million votes for the Leave cause in the 2016 referendum. That cartoon snapshot of the will of the people may be upheld as iconic, but like the 2005 Danish cartoon of Muhammad it serves more to divide than unite us. Times change, and reasonable people are not too proud to change their opinions to reflect new facts.
More specifically, UK parliamentarians have acted in genuflection to the 2016 icon without due appreciation of the need for a better portrait of the people. The 2017 general election offered no royal road for voters disaffected by the icon and thus deepened their disaffection. The obvious solution is to commission a new portrait.
Print version (2 pages)
The Guardian
UK papers, Thursday morning
EU gap
WWW @ 30
Sir Tim Berners-Lee invented
the World Wide Web
30 years ago.
AR And changed my life
— thanks, Tim!
From a video on how to
visualize quaternions
JooHee Yoon
2019 March 13
Brexit: No No Deal
BBC News, 1950 UTC
House of Commons passes motion to reject no-deal Brexit on 29 March by 321 votes to 278,
amended to reject a no-deal Brexit at any time (approved by 312 votes to 308), but
lacking the "Malthouse compromise" amendment (rejected by 374 votes to 164).
The motion and its amendments are expressions of feeling with no legal force.
AR Sterling rises on the news.
Brexit: On The Brink
The Times
Only 16 days before the UK is due to leave the EU, Theresa May's strategy for delivering an orderly departure lies in tatters.
May pursued an unsound strategy, misread her opponents in Brussels, and refused to be honest about the compromises and trade-offs that the rupture of relations with the EU was bound to entail. Instead she tried to conduct the negotiations by stealth, running down the clock on her cabinet, her party, parliament, and the public.
The result of all this dissembling has been a calamitous loss of trust. The prime minister long ago forfeited the trust of Brexiteers. She is not trusted by Remainers. Above all, she has forfeited the trust of the EU.
Last night May was forced to concede a free vote today on whether parliament should back leaving the EU without a deal. That is an admission that the government is no longer able to provide leadership at this time of crisis. On Thursday she will almost certainly have to offer another free vote on whether to extend Article 50.
The Conservative party may now decide that only a new leader can find a way forward.
AR Parliament has legislated for Brexit on March 29. Only a surprise plot twist can stop it.
ERG chair Jacob Rees-Mogg: "I think our expectations are that we will leave without a deal."
Remainers must force a surprise twist. The End of Days scenario is too grim to contemplate.
2019 March 12
Brexit: Titanic Defeat
BBC News, 1922 UTC
House of Commons rejects Theresa May's deal by 391 votes to 242.
Brexit: Avoidable Damage
The Times
Today MPs will be asked to cast what will almost certainly be the most important vote of their lives on a Brexit motion that they will have had just hours to assess.
Brexit is not just about economics. MPs will vote at a time of intense geopolitical volatility, when the unity of the western alliance has never looked less certain. How their decisions affect this instability should be uppermost in their minds.
The degree of fragmentation of the western alliance was scarcely imaginable when Britain voted in 2016 to quit the EU. New sources of tension between the allies are emerging almost daily.
President Trump will decide within weeks whether to launch a trade war with the EU. There are also multiple tensions within the EU itself, not least a new war of words between France and Italy.
There have been tensions between NATO members before. But they never undermined the strategic cohesion of the West. The last time the world faced such a geopolitical shift came with the fall of the Berlin wall and the collapse of the Soviet empire.
A no-deal Brexit would be a profound geopolitical shock. We cannot assume strategic and security partnerships are unaffected by economic relationships.
AR See my recent essay Ringlord.
AI Is Changing Science
Dan Falk
Machine learning and AI offer a new way of doing science. Generative modeling can help identify the most plausible theory among competing explanations for observational data, based solely on the data. This is a third way between observation and simulation.
A generative adversarial network (GAN) can repair images that have damaged or missing pixels and can make blurry images sharp. The GAN runs a competition: A generator generates fake data, while a discriminator tries to distinguish fake data from real data. As the program runs, both halves improve.
More broadly, generative modeling takes sets of data and breaks each of them down into a set of basic blocks in a latent space. The algorithm manipulates elements of the latent space to see how this affects the original data, and this helps uncover physical processes at work in the system. Generative modeling automates part of the process of science.
Perhaps future machines will discover physics or mathematics that the brightest humans alive cannot find on their own. Perhaps future science will be driven by machines that operate on a level we can never reach.
AR In 1988, in a Springer physics newsletter, I said simulation was a third way of doing science, between observation and theory.
2019 March 11
Brexit Showdown
The Observer
What happens this week is likely to prove decisive. If May loses the vote on her deal on Tuesday as expected, will parliamentarians rally round a referendum on the deal as the only realistic route out of this mess? If they don't, they will edge closer to the cliff edge and a binary choice between May's deal and no deal. And they will be entirely complicit in whatever follows.
AR Vote for a people's vote.
Animated Math
Grant Sanderson
3blue1brown centers around presenting math with a visuals-first approach. That is, rather than first deciding on a lesson then putting illustrations to it for the sake of having a video, almost all projects start with a particular visualization, with the narrative and storyline then revolving around it.
Topics tend to fall into one of two categories:
Lessons on topics people might be seeking out.
Problems in math which many people may not have heard of, and which seem really hard at first,
but where some shift in perspective makes it both doable and beautiful.
I think of the first category as motivating math by its usefulness, and the second as motivating math as an art form.
The YouTube channel
AR I've liked Grant's work for years.
2019 March 10
America vs China
The New York Times
By imposing tariffs on Chinese imports, President Trump created an opportunity to improve the US economic relationship with China. His decision to go it alone, rather than making common cause with longstanding allies, was ill advised, and his trade war has caused pain for many Americans.
The proper measure of any deal is whether it persuades China to curb its use of state subsidies, regulations, and various kinds of informal interference that limit the ability of American companies to sell goods and services in China, and help Chinese companies sell goods in the United States.
The United States has focused its demands on making it easier for American companies to operate in China. But the United States has failed in past efforts to hold China to its commitments. The risk is that Trump will accept a deal that allows him to claim a superficial triumph.
A Rogue President
James Kitfield
President Trump reportedly plans to transform America's alliances into a protection racket with a "cost plus 50" plan that would require allies to pay 150% of the cost of hosting US troops, with a good behavior discount for those countries willing to take their marching orders from Washington.
Former NSC staffer Kori Schake: "The question that dominated the Munich Conference was whether the United States would once again lead the Western democracies after Trump is gone, or whether the Europeans need to protect themselves further against a disruptive America."
Former US defense secretary Bill Cohen: "Why has Trump adopted an agenda that exactly replicates Vladimir Putin's bucket list? .. The President of the United States may well be compromised by the Russians, which I truly believe is the case. And he is unfit to serve."
2019 March 9
Neuroscience and Consciousness
Philip Ball
Consciousness is a hard problem in science. A new project funded by the Templeton World Charity Foundation aims to narrow the options for tackling it. Researchers will collaborate on how to conduct discriminating experiments.
Bernard Baars and Stanislas Dehaene suggest conscious behavior arises when we hold information in a global workspace within the brain, where it can be broadcast to brain modules associated with specific tasks. This view is called global workspace theory (GWT).
Christof Koch and Giulio Tononi say consciousness is an intrinsic property of the right kind of cognitive network. This is integrated information theory (IIT). IIT portrays consciousness as the causal power of a system to make a difference to itself.
Koch and Tononi define a measure of information integration, Φ, to represent how much a network as a whole can influence itself. This depends on interconnectivity of feedback.
Researchers have now designed experiments to test the different predictions of GWT and IIT. According to GWT, the neural correlates of consciousness should show up in parts of the brain including the parietal and frontal lobes. According to IIT, the seat of consciousness is instead likely to be in the sensory representation in the back of the brain.
Anil Seth thinks the Templeton project may be premature.
Animals Are Emotional
Frans de Waal
I believe we share all emotions with other species in the same way we share virtually every organ in our bodies with them. Like organs, the emotions evolved over millions of years to serve essential functions. Their usefulness has been tested again and again, giving them the wisdom of ages, and none is fundamentally new.
Open your front door and tell your dog that you are going out for a walk, then close the door and return to your seat. Your dog, who had been barking and wriggling with excitement, now slinks back to his basket and puts his head down on his paws. You have just witnessed both hope and disappointment in another species.
Whatever the difference between humans and other animals may be, it is unlikely to be found in the emotional domain.
AR Who doubts it?
Are We Alone?
Rebecca Boyle
Enrico Fermi said there are lots of stars and extraterrestrial life might be common, so we should get visitors. But where are they?
In a new paper, Jonathan Carroll-Nellenback, Jason Wright, Adam Frank, and Caleb Scharf model the spread of a settlement front across the galaxy, and find its speed is strongly affected by the motions of stars. A settlement front could cross an entire galaxy based just on the motions of stars, regardless of the power of propulsion systems.
The Fermi paradox does not mean ET life does not exist. The Milky Way may be partially settled, or intermittently so. The solar system may well be amid other settled systems and has just been unvisited for millions of years.
AR We are not alone.
Crew Dragon returns
NASA (1:07)
SpaceX Crew Dragon returns to Earth in Atlantic splashdown
"Trump is not forever.
Brexit is. Britain's youth
oppose it. A decision of
this import should be
grounded in reality."
Roger Cohen
Stop Brexit
2019 March 8
Quantum Computing
Katia Moskvitch
The Large Hadron Collider generated about 300 GB of data per second. To make sense of all that information, the LHC data was pumped out to 170 computing centers in 42 countries. This global collaboration helped discover the Higgs boson.
A proposed Future Circular Collider would create at least twice as much data as the LHC. CERN researchers are looking at the emerging field of quantum computing. The EU has pledged to give $1 billion to researchers over the next decade, while venture capitalists invested some $250 million in quantum computing research in 2018 alone.
Qubits can be made in different ways. Two qubits can be both in state A, both in state B, one in state A and one in state B, or vice versa, to give four probabilities. To know the state of a qubit, you measure it, collapsing the state. With every qubit added to its memory size, a quantum computer should get exponentially increased computational power.
Last year, Caltech physicists replicated the discovery of the Higgs boson by sifting through LHC data using a quantum computer based on quantum annealing. Dips in a landscape of peaks and valleys represent possible solutions and the system finds the lowest dips via quantum tunneling.
There are three other main approaches to quantum computing: integrated circuits, topological qubits, and ions trapped with lasers.
Quantum chips are integrated circuits with superconducting quantum gates. Each quantum gate holds a pair of qubits. The chip is supercooled to 10 mK to keep the qubits in superposition. A useful machine needs about 1,000 qubits with low noise and error correction to make up just one logical qubit. So far, we only have error correction for up to 10 qubits.
Topological qubits would be much more stable. The idea is to split a particle in two, creating Majorana fermion quasi-particles, so that one topological qubit is a logical one. Scaling such a device to thousands of logical qubits would be much easier.
Trapped ions show superposition effects at room temperature and each ion is a qubit. Researchers trap them and run algorithms using laser beams that write data to the ions and read it out by change the ion states. So far, the ion qubits are noisy.
Meanwhile, at CERN, the clock is ticking.
Theresa May
Next week MPs in Westminster face a crucial choice: whether to back the Brexit deal or to reject it. Back it, and the UK will leave the European Union. Reject it, and no one knows what will happen. We may not leave the EU for many months, we may leave without the protections that the deal provides. We may never leave at all.
AR Revoke, remain, repent, reform (UK and EU)
2019 March 7
To Poole
Drove from Amiens to Cherbourg, then enjoyed a stormy sea voyage from Cherbourg to Poole
2019 March 6
To Amiens
Drove from Gaiberg to Amiens, then enjoyed a fine dinner in that beautiful city
2019 March 5, Faschingsdienstag
Europe Renew!
Emmanuel Macron
Citizens of Europe, I am taking the liberty of addressing you directly ..
Never has Europe been in such danger. Brexit stands as the symbol of .. the trap that threatens the whole of Europe: the anger mongers, backed by fake news, promise anything and everything.
Europe is not just an economic market. It is a project .. European civilisation unites, frees, and protects us .. We need to .. reinvent the shape of our civilisation in a changing world.
Now is the time for a European renaissance .. I propose we build this renewal together around three ambitions: freedom, protection, and progress .. We need to build European renewal on these pillars .. In this Europe, the people will really take back control of their future.
The Brexit impasse is a lesson for us all. We need to escape this trap and make the forthcoming elections and our project meaningful .. Together we chart the road to European renewal.
AR Good — Britain needs a hero like Macron.
Russian Doll
Chelsea Whyte
Russian Doll is a dark comedy starring a woman stuck in a time loop. She dies, only to be resurrected in a new branch of the multiverse.
Nadia has to convince someone that she is reliving the same night. She meets Alan, who also keeps dying and reliving the same day, and sees their experience as a video game. Their inner lives continue as one linear experience while their bodies keep dying.
Nadia: "Time is relative to your experience. We've been experiencing time differently in these loops, but this tells us that somewhere, linear time as we used to understand it still exists."
AR Experienced time is the innermost bastion of consciousness. Its linearity through outer confusion (in this case sorted into a multiverse experience) is a criterion of rationality. To lose the thread is to lose your mind.
2019 March 4, Rosenmontag
Upside to Brexit?
Jochen Bittner
Brexit may have an upside. When its most globally minded member leaves, the EU must rethink its mission and vision.
The new global rivalry is between free and unfree market economies. China has decoupled personal freedom from freedom of innovation. With a GDP of close to $25 trillion, China is potentially the most powerful economy in world history.
For the first time in modern history, technological leadership is being assumed by a power unchecked by the democratic vote. China's legal tradition puts collective interests above individual rights.
China maintains a clear strategic outlook. Chinese Communists appear to have learned lessons from both the rise of the British Empire and the fall of the Soviet Union. Unlike Europe, China speaks with one voice, and expresses one vision.
The West needs a stronger alliance to compete. Brexit could force Britain and Europe to push back against China.
China vs Germany
Wolfgang Münchau
Germany is ambivalent about China. It needs Chinese technology. But Germany also worries about Chinese companies acquiring its technology.
Germany once saw China as an export market for machinery with which China would develop its industrial base. Today, China is becoming the senior partner in the relationship.
The two countries have a lot in common. Both are export-driven economies with large external savings surpluses. But German economic strategy is not nearly as consistent.
In Europe, macroeconomic policy, industrial policy, and foreign and security policy are run independently of each other. China has an integrated approach to policy.
The Europeans did not see this coming. Complacency is about to turn into panic.
Europe vs Brexit
Manfred Weber
The European way of life includes fundamental values and rules: the rule of law, democracy, independent media, the social market economy, and the equality of men and women.
Developments across Europe are shocking. Antisemitism is returning with a bang. The development of a European Islam rooted in our fundamental values has not been successful.
I am concerned that populists could become stronger in the European Parliament. Brexit shows what happens if you follow the simplistic answers presented by populists.
We have been negotiating with Britain for almost three years and we have hardly made any progress. I have little sympathy for a postponement that would simply prolong the chaos in London.
The participation of British voters in the EU election is inconceivable to me. I can't explain to people in Germany or Spain that people who want to leave the EU should be given a vote on its future.
The EU must reform its institutions, limit migration, and face up to the challenges presented by Donald Trump, such as a trade conflict. I can't let the British tragedy infect the rest of the EU.
Utterly, Utterly Stupid
Simon Wren-Lewis
What the UK is doing is utterly, utterly stupid, an act of self harm with no point, no upside.
The days when Leavers talked about the sunlit uplands are over. Instead there has emerged one justification for Brexit: the 2016 referendum. People voted for it, so it must be done.
Warnings from big business become an excuse to talk about WW2 again. The case for Leaving has become little more than xenophobia and nationalism.
The worst excuse not to hold a people's vote is that a second referendum would be undemocratic. Orwell must be turning in his grave.
2019 March 3
The Trump Narrative
Larry Jacobs
President Trump has been a magician in masterminding a narrative that he's going to stand up for America and he's not beholden to the swamp. This week put the lie to his narrative.
The collapse of the talks in North Korea has put the lie to his story that he had a historic accomplishment. There has not been a breakthrough, and Trump conceded the point and left.
Back home, the idea Trump is a beacon of truth was seriously damaged by what Michael Cohen said and the people he identified who will be brought forward to testify.
Cosmic Expansion
Dennis Overbye
A changing Hubble constant suggests dark energy might be increasing. To calibrate the Hubble constant, we use supernovas and variable stars whose distances we can estimate.
NASA HST results give 72 km/s per Mpc for the Hubble constant, and other results agree. But the ESA Planck map of the CMB predicts a Hubble constant of 67. We have a problem.
We can use quasar emissions to trace back the history of the cosmos nearly 12 billion years. The rate of cosmic expansion seems to deviate from expectations over that time.
The cosmos is now doubling in size every 10 billion years. String theory allows space to be laced with energy fields called quintessence that oppose gravity and could change over time.
In the coming decade, ESA mission Euclid and NASA mission WFIRST are designed to help solve the problem with the Hubble constant.
2019 March 2
Europawahl: Briten Raus
Europäischen Volkspartei (EVP) Spitzenkandidat Manfred Weber (CSU): "Eine Teilnahme der britischen Bürger an der Europawahl ist für mich undenkbar. Ich kann doch in Deutschland oder Spanien niemandem erklären, dass Bürger, die die EU verlassen wollen, noch mal wesentlichen Anteil daran nehmen sollen, deren Zukunft zu gestalten."
AR Weber hat sicherlich recht: Schmeißen die Briten raus!
The Brexit Mess
Sir Ivan Rogers
Four weeks before the Brexit deadline, the British political class is unable to come to any serious conclusion about what kind of Brexit they want.
The UK political elite has fractured in both parties. In British politics, unless you occupy the center you are finished. But the center has largely collapsed and populists have gained more influence.
Theresa May wants to reduce the numbers of people coming into the UK. Having started with her hardline position, every time she moves a little bit back, the right wing of her party cries betrayal.
I have worked with several prime ministers very closely, and none of them had a deep understanding how the EU works. The UK has always had a rather mercantile relationship with its neighbors.
European leaders spend too little time thinking about how the continent should look in future after Brexit. Assuming Brexit happens, German politicians must ask how we are going to work together.
Europeans need to tell the UK what they want. They will need to be told what degree of divergency the UK wants and why. This will take years.
AR Perhaps 40 years in the wilderness will teach Brits some manners.
2019 March 1
Fast Radio Bursts
Joshua Sokol
Fast radio bursts (FRBs) are millisecond-long blips of intense radio signals that pop up all over the sky. To explain them, we need an object that can emit lots of energy and a way to transform the energy into a bright radio signal.
FRBs may arise from a magnetar, a young neutron star that can emit charged particles into the surrounding clutter and create a shock wave, which beams a brief flash of radio waves into the universe. Some FRBs repeat at unpredictable intervals from dense regions of plasma with extreme magnetic fields. Each burst contains sub-bursts that shift from higher to lower frequencies.
In models of nuclear detonations, the shock fronts sweep up more gas as they expand outward. That extra weight slows down the shock, and because it slows, radiation released from the shock front shifts downward in frequency.
Flares from a magnetar run into particles emitted during previous flares. Where new ejecta meets older debris, it piles up into a shock, inside which magnetic fields soar. As the shock presses outward, the electrons inside gyrate around along magnetic field lines, and that motion produces a burst of radio waves. That signal then shifts from higher to lower frequencies as the shock slows.
If the model is correct, future FRBs should follow the same downward shift in frequency. They might show gamma-ray or X-ray emission and should live in galaxies that are producing fresh magnetars. When they repeat, they should take breaks from bursting after a major flare.
Coming soon: new data to help us explain them.
Andy, Rolf
Flying with Rolf Kickuth in his gyrocopter over Mannheim, Germany, 25 February
Moebius band
Independent Team
BBC News
Conservative party loses
3 MPs to the new group
formed by 7 Labour MPs.
An 8th Labour MP joins.
Team total so far: 11
Fortress Europe
2019 February 28
To Woods
Walk in the woods, alone with my thoughts, near Dudenhofen, west of Speyer
2019 February 27
To Schwetzingen
Coffee date in the sun with old friend Matthias Störmer in Schwetzingen Schlossplatz
2019 February 26
With Rolf to BASF press conference in Ludwigshafen — photo
2019 February 25
To Blue Sky
In gyrocopter from Mannheim City Airport to blue sky over the Rhine-Neckar region
2019 February 24
To Germany
In car from Amiens to my friends Angela and Rolf in Gaiberg, Germany
2019 February 23
To France
With car to Cherbourg, in car to Amiens, driving in warm sunlight
2019 February 22
The Dawn Of Time
New Scientist
Near the South Pole, the BICEP3 telescope captures light from the dawn of time. A few years ago, BICEP2 researchers thought they had found proof of cosmic inflation, but they made an error.
Up to 12 Ts ABB, the universe was a hot, dense soup of elementary particles. Then it cooled, atoms formed, and the cosmos became transparent. The CMB is made up of the first free photons.
Inflation explains the smooth distribution of galaxies in the universe. Tiny quantum fluctuations in the first moments ABB produced an uneven distribution of matter that was amplified as the cosmos expanded. Inflation smoothed out the bumps.
Tiny temperature variations in the CMB are largely consistent with the main inflationary models. We seek a more detailed appreciation of how CMB photons are polarized. The Planck telescope mapped this polarization with only limited sensitivity.
Inflation implies that turbulence in the fabric of the early universe made gravitational waves. These waves left a "B-mode" pattern in the CMB polarization. But any such polarization signals are far smaller than the fluctuations mapped by Planck.
The BICEP2 detector had 256 pixels but BICEP3 has 1280. Teamed with the Keck Array, the researchers began gathering data in 2016 and could soon detect the primordial B-mode signal.
A Theory Of Everything
New Yorker
Richard Feynman said there are multiple valid ways of describing many physical phenomena. In quantum theory, Feynman diagrams indicate the probabilities, or scattering amplitudes, of different particle-collision outcomes.
In 2013, Nima Arkani-Hamed and Jaroslav Trnka discovered a reformulation of scattering amplitudes that makes no reference to spacetime: The amplitudes of certain particle collisions are encoded in the volume of a geometric object: the amplituhedron.
Einstein's general theory of relativity weaves space and time into the 4D fabric of spacetime. The theory is incomplete, but it has a clean and compelling mathematical structure. To discover a deeper way of explaining the universe, you must jump to a totally different mathematical structure.
To Arkani-Hamed, theoretical physics is a matter of discovering questions. Calculating the volume of the amplituhedron is a question in geometry. The answer describes the behavior of particles without mentioning spacetime.
2019 February 21
Möbius Bands In Space
Moscow mathematician Olga Frolkina has proved that the Möbius band (a 2D loop with a half-twist) cannot be packed an uncountably infinite number of times into an infinite amount of 3D space.
The Möbius band is an example of a non-orientable manifold, a mathematical object on which you cannot fix a notion of inside and outside that will stay consistent as you travel around the space.
Objects such as disks and spheres can be tamely embedded into 3D space. Wild embeddings are trickier. An uncountable infinity of spheres and tori can be embedded into 3D space without overlap if the embeddings are tame but not if they are wild.
Uncountably many tamely embedded Möbius bands cannot fit in 3D space without intersecting each other. Frolkina proved this too for wildly embedded Möbius bands.
AR The higher-dimensional results are interesting too.
2019 February 20
Record German Export Surplus
The Times
Germany ran the world's largest trade surplus in 2018. German sales of goods and services overseas last year outstripped its imports by €249 billion. This was by far the widest in the world.
AR Make things people want to buy — sounds good to me.
Honda Abandons Brexit Britain
Financial Times
Japanese company Honda is closing its car plant in Swindon.
Access to the EU market caused global car companies to locate in the UK. Friction between the UK and the EU hinders their operations. The car industry involves massive economies of scale, and the supply chains cannot be confined to the UK.
Other car companies will cut production in the UK. Uncertain access to the EU market after Brexit is a reason. Whatever the final relationship between the UK and the EU, the tactic of running down the clock to March 29 carries costs.
Global car companies will tend to avoid Brexit Britain.
AR Japan will see Brexit Britain as a failing state.
British Labour Antisemitism Split
The Times
Labour MP Ruth George says the seven MPs who quit the party might be secretly funded by Israel. They are resigning over Jeremy Corbyn's handling of antisemitism in the party as well as Brexit.
AR Labour and antisemitism — fatal.
2019 February 19
Gideon Rachman
Islamophobia is now a central part of politics in most major capitals worldwide. And countries that were once seen as strongholds of moderate Islam are witnessing a rise in radical Islamism.
China has imprisoned more than a million Uighur Muslims in Xinjiang in mass internment camps. International slowness to protest may reflect an increasingly hostile attitude to Muslim minorities in other parts of the world.
India is governed by Hindu nationalists. BJP militants regard Islam as alien to India. About 1 in 7 of the Indian population is Muslim, but there was no Muslim among the 282 BJP MPs in 2014.
In America since 9/11, many more American civilians have fallen victim to school shootings than to Islamist terrorists, but anti-Muslim rhetoric by US politicians has become more pronounced.
In Europe, mass migration has produced a surge in support for nationalist and Islamophobic parties. Such parties are now in government in Hungary, Austria, Italy, and Poland.
In Turkey, secularists fear the president will Islamize their country.
In Pakistan, Islamists use blasphemy laws as a weapon.
A clash of civilizations is emerging.
AR I think monotheism needs an upgrade.
2019 February 18
Climate: Time to Panic
David Wallace-Wells
Last October, the UN Intergovernmental Panel on Climate Change released a report detailing climate effects at 1.5 K and 2 K of global warming. The report gave good reason for scientists worldwide to freak out. This is progress.
Alarmism and catastrophic thinking are valuable, for several reasons:
1 Climate change is a crisis because it is a looming catastrophe that demands an aggressive global response. The emissions path we are on today is likely to take us to 1.5 K of warming by 2040 and 2 K within decades after that. Many big cities in the Mideast and South Asia would become lethally hot in summer. Coastal cities worldwide would be threatened with inundation. Many millions of people would flee droughts, floods, and extreme heat. It is right to be alarmed.
2 Catastrophic thinking makes it easier to see the threat of climate change clearly. For years, we have pictured a landscape of possibilities that began with the climate as it exists today and ended with the pain of 2 K, the ceiling of suffering. In fact, it is almost certainly a floor. By far the likeliest outcomes for the end of this century fall between 2 K and 4 K of warming.
3 Complacency remains a much bigger political problem than fatalism. A national survey showed a majority of Americans were unwilling to spend even $10 a month to address global warming, and most drew the line at $1 a month. If we delay the decarbonization effort by another decade, we will have to cut emissions by some 9% each year. We have to get started now.
4 Our mental reflexes run toward disbelief in the possibility of very bad outcomes. Complacency is hard to shake. Cognitive biases distort and distend our perception of a changing climate. All the biases that push us toward complacency are abetted by our storytelling about warming.
Individual lifestyle choices are trivial compared with what politics can achieve. Buying an electric car is a drop in the bucket compared with raising car-emission standards sharply. Flying less is a lot easier if high-speed rail is an option. Politics is a moral multiplier.
Ben, Andy
Brexit: The Movie
Ben Aston and I discuss Brexit at Bournemouth University on February 14 (YouTube: 1 hr, 45 min, 39 sec)
UK in EU
2019 February 17
Brexit: May Could Lose
The Times
Theresa May might well lose the Commons vote on her Brexit deal on February 27. Cabinet sources fear this would let parliament seize control of the Brexit negotiations. On the same day, MPs are due to vote on an amendment by Yvette Cooper and Sir Oliver Letwin that would force May to ask Brussels for a delay to Brexit.
A cabinet minister: "We may lose Brexit altogether."
AR That loss would give me occasion to celebrate for the first time in three years.
2019 February 16
Global Security
The Guardian
At the Munich Security Conference, German chancellor Angela Merkel warned of a collapse of the international order into tiny parts: "Do we fall apart into pieces of a puzzle and think everyone can solve the question best for himself alone?"
On the Russian and American decision to cancel the 1987 INF treaty: "Disarmament is something that concerns us all, and where we would, of course, be delighted if such talks were held not just between the United States, Europe and Russia, but also with China."
Merkel said German defense spending is due to reach 1.5% by 2024 and that development spending in Africa also brought greater security: "We have to think in networked structures. The military component is one of them. We need NATO as a stability anchor in stormy times. We need it as a community of values."
Munich Security Conference
Martin Knobbe
Bundeskanzlerin Angela Merkel: "Wir leben in einem Zeitalter, in dem die Spuren des Menschen so tief in die Erde eindringen, dass auch die nachfolgenden Generationen sie sehen können .. Who will pick up the pieces? .. Nur wir alle zusammen."
Merkel erinnerte daran, dass sich auch Deutschland bewegen müsse, wenn Europa eine gemeinsame militärische Kultur entwickeln wolle. Wenn man mit Frankreich über gemeinsame Rüstungsprojekte rede, müsse man sich auch auf eine gemeinsame Politik bei den Rüstungsexporten einigen.
Brexit Strategy Failed
The Guardian
Theresa May will face a wall of resistance when she returns to Brussels next week. EU chief negotiator Michel Barnier told diplomats from EU member states her Brexit strategy had "failed" after her latest parliamentary defeat and that her strategy could not work.
An ambassador: "We have a major problem."
2019 February 15
Trump Emergency
The New York Times
President Trump is planning to take executive overreach to new heights. Cornered into accepting a budget deal that lacks the $5.7 billion in border wall funding he demanded, the president has a solution: Sign the bill while simultaneously declaring a national emergency that lets him shift funds and order the military to start building his wall.
White House press secretary Sarah Huckabee Sanders: "President Trump will sign the government funding bill, and as he has stated before, he will also take other executive action, including a national emergency, to ensure we stop the national security and humanitarian crisis at the border."
The influx of migrant families at the southern border does not constitute a national security crisis. There is a worsening humanitarian crisis, actively fueled by the policies of the administration. The suffering requires thoughtful policy adjustments, not a wall.
Confronted with this power grab, every lawmaker should be bellowing in alarm. Until recently, the threat of an imperial presidency was of grave constitutional concern to Republicans, who accused President Obama of misusing executive authority.
House speaker Nancy Pelosi: "Just think of what a president with different values can present to the American people."
Trump: "I have the absolute right to do national emergency if I want."
AR And the House has the right to do impeachment.
Quantum Foundations
Philip Ball
In 1925, Erwin Schrödinger wrote an equation to describe the wavy nature of quantum particles. The Schrödinger equation ascribes to a particle a wave function, φ, whose amplitude determines the particle's behavior.
In 1926, Max Born suggested interpreting the wave function in terms of probability. He said the amplitude of the wave function at position x is related to the probability of finding the particle at x in a measurement: The probability is given by the product φ*φ of the wave function φ with its complex conjugate φ*.
Lluís Masanes, Thomas Galley, and Markus Müller (MGM) show that the Born rule follows from basic postulates of quantum mechanics, given a few basic assumptions:
1 Quantum states are formulated as vectors.
2 So long as a particle is not measured, it evolves so as to preserve information.
3 How you group the parts of a system is irrelevant to a measurement outcome.
4 A measurement on a quantum system produces a unique outcome.
MGM do not assume the technical requirements of quantum mechanics directly. Instead, they derive them, like the Born rule, from the basic assumptions.
Adán Cabello assumes there is no underlying physical law that dictates measurement outcomes. Every outcome can happen if it is consistent with the outcome probabilities of different experiments. He shows that quantum measurement outcomes follow the Born rule as a matter of logic.
In both approaches, quantum theory is founded on simple postulates.
2019 February 14
Brexit: May Defeat
House of Commons, 1745 UTC
MPs have delivered a blow to Theresa May's authority by rejecting her motion by 303 votes to 258: Motion: "That this House welcomes the prime minister's statement of 12 February 2019; reiterates its support for the approach to leaving the EU expressed by this House on 29 January 2019 and notes that discussions between the UK and the EU on the Northern Ireland backstop are ongoing."
"When the chips are down, [Theresa May] will actually prefer to do what some of my esteemed colleagues prefer, and to head for the exit door without a deal, which the secretary of state informed us is the policy of Her Majesty's government in the event that her deal has not succeeded. That is terrifying fact."
Sir Oliver Letwin
AR Depose her.
Brexit: Insurmountable Impact
Financial Times
Dutch prime minister Mark Rutte says Britain is a diminished country after its vote for Brexit.
Rutte says companies are shifting offices and staff to The Netherlands from the UK: "Every businessman I speak to from the UK is saying they will cut investments, cut their business in the UK. It will have an insurmountable impact on the UK."
Rutte: "We have to realise we are not toothless, we have our means of power as the EU."
Brexit: National Crisis
Sir Nigel Sheinwald et al.
As former diplomats, our advice to Theresa May is that we should not leave the EU when we have no clarity about our final destination. We must seek to extend the Article 50 negotiating period.
Brexit has turned into a national crisis. There is no possible deal that will be a sensible alternative to the privileged one we have today as members of the EU with a seat at the table.
There is now a powerful argument to go back to the people and ask them whether they want the negotiated Brexit deal or would prefer to stay in the EU.
Sir Nigel Sheinwald, Lord Kerr, Lord Hannay, .. [total > 40]
Brexit: Dark Money
George Monbiot
The Brexit referendum was won with the help of widespread cheating. Both of the main leave campaigns were fined for illegal activities. But the government has so far failed to introduce a single new law in response to these events.
Since mid-January, Britain's Future has spent £125,000 on Facebook ads demanding a hard or no-deal Brexit. Britain's Future has no published address and releases no information about who founded it, who controls it, and who has been paying for these ads.
British rules governing funding for political parties, elections, and referendums are next to useless. They were last redrafted 19 years ago, when online campaigning had scarcely begun. The Electoral Commission has none of the powers required to regulate online campaigning.
The UK government wants to keep the system as it is.
How Brains Code Time
Jordana Cepelewicz
Marc Howard and Karthik Shankar have built a mathematical model of how the brain might encode time: As sensory neurons fire in response to an unfolding event, the brain maps the temporal component of that activity to a representation of the experience in a Laplace transform. The brain preserves information about the event as a function of a variable it can encode and then maps it back into an inverse Laplace transform to reconstruct a compressed record.
There are "time cell" neurons in the brain, each tuned to fire at certain points in a span of time, with different frequencies, to bridge time gaps between experiences. One can look at the firings and determine when a stimulus was presented from which cells fired. This is the inverse Laplace transform part of the model.
The medial and lateral entorhinal cortex provide input to the hippocampus, which generates episodic memories of experiences that occur at a particular time in a particular place. Albert Tsao knew the medial entorhinal cortex was responsible for mapping place and guessed the lateral entorhinal cortex harbored a signal of time.
Tsao examined the neural activity in the lateral entorhinal cortex of rats as they foraged for food. In the trials, the firing rates of the neurons spiked when the rat entered the box and decreased at varying rates as time passed. That activity ramped up again at the start of the next trial. In some cells, activity declined not only during each trial but throughout the entire experiment. Hundreds of neurons worked together to record the order of the trials and the length of each one.
Howard saw that the different rates of decay in the neural activity looked like a Laplace transform of time. His model can explain how we create and maintain a timeline of the past. That timeline could be of use not just to episodic memory in the hippocampus, but to working memory in the prefrontal cortex and conditioning responses in the striatum.
Howard is working on extending the theory to other domains of cognition.
Dresden 1945
On February 13-14, 1945, the city of Dresden was incinerated by British and American bombers
George Soros
George Soros
"That was another hard
round of negotiations"
Ultima Thule
has an odd shape
The Paradise Papers
revealed tax avoidance
on a global scale
In 1945, B-29 "Enola Gay"
delivered an atom bomb
to Hiroshima
"I've been wondering what
the special place in hell
looks like for people who
promoted Brexit without
even a sketch of a plan how
to carry it out safely."
Donald Tusk
Avraham Sutzkever
Avraham Sutzkever
2019 February 13
Germany on Brexit
Helene von Bismarck
Angela Merkel remains chancellor, and her priorities for the Withdrawal Agreement are shared widely across the German political spectrum: EU27 cohesion and single market integrity come first, the Anglo-German relationship second.
The EU is currently facing great challenges, such as EZ fragility, populism, and migration. Faced with a choice between punishing the UK for Brexit or punishing the EU for it, the German government will not hesitate. British attempts to persuade the German government to act as a broker for Britain within the EU27 are a waste of time.
2019 February 12
European Nightmare
George Soros
The European Union could go the way of the Soviet Union in 1991.
Europe needs to recognize the magnitude of the threat. In the elections for the European Parliament in May 2019, the present party system hampers those who want to preserve the founding values of the EU but helps those who want something radically different.
Germany and the Right
The dominant CDU/CSU alliance in Germany has become unsustainable. The AfD entry into the Bavarian parliament broke the purpose of the alliance. The current ruling coalition cannot be as robustly pro-European as it would be without the AfD threatening its right flank. The good news is that the Greens are rising and the AfD seems to have reached a peak. But the CDU/CSU commitment to European values is ambivalent.
UK and Brexit
The antiquated UK party structure prevents the popular will from finding proper expression. Both Labour and the Conservatives are internally divided, but both parties seem determined to deliver Brexit. The public is becoming aware of the dire consequences of Brexit, which could raise a groundswell of support for a referendum or for revocation of the Article 50 notification.
Italy and Immigration
The EU made a fatal mistake in 2017 by strictly enforcing the Dublin Agreement, which unfairly burdens countries like Italy where migrants first enter the EU. This drove Italy to the populists in 2018, leaving the pro-Europeans with no party to vote for. A similar reordering of party systems is happening in France, Poland, Sweden, and probably elsewhere.
Hungary and Nationalism
National parties at least have some roots in the past, but the trans-European alliances are entirely dictated by their party leaders. The European People's Party (EPP) is almost entirely devoid of principles, as demonstrated by its willingness to permit the continued membership of the Hungarian Fidesz party.
One can still make a case for preserving the EU in order radically to reinvent it. But the current EU leadership is reminiscent of the politburo when the Soviet Union collapsed.
Mobilize the masses to defend EU founding values!
AR The founding values of the EU are still a high point in human moral achievement. The challenge is to re-engineer their implementation to accommodate new technology (fake news in social media, high cost of medical advances, rise of global manufacturing chains, and so on) by radically transforming the party landscape. As Soros says, the Greens offer a beacon of hope.
2019 February 11
The Brexit Effect
Financial Times
Brexit has visibly depressed UK economic data for the year 2018. Most economists agree that Brexit has cost Britain 2 ± 0.5% of GDP, and higher inflation and lower growth since 2016 has reduced household incomes by 4.1%, an average of £1500 per household.
Brexit: High Noon?
Matthew d'Ancona
Theresa May proposes to return to the Commons by 27 February. That vote may be Brexit high noon. A delay until 25 March is unthinkable. No MP can take such a big decision so soon before D-Day.
Brexit: Extend Article 50
Gus O'Donnell
The British people does not have any real clarity about the future UK relationship with its closest neighbours. The political declaration was meant to set out a framework for a future relationship and clarify the general direction on such issues as membership of the single market or a customs union.
What has happened in recent days has erased these basic navigation points. A lack of clarity about Brexit that was once seen as unfortunate political necessity has been trumpeted as its chief political virtue. The Conservative party is united only around a set of ambiguous ideas that settle nothing.
What is being brushed over here is a fundamental choice about how the UK economy, society, and government will operate in years to come. It is irresponsible for any government to contemplate embarking on such a perilous journey as Brexit without giving voters any idea of the destination.
A better understanding of future customs arrangements, trade policy, immigration, and rules for businesses is essential for jobs, investment, and the work of government. Questions about all this are not details to be filled in at a later date. These are massively important to all British citizens.
Leaving now, with so much unclear and uncertain, is a recipe for further division and dysfunction in politics. If government and parliament cannot agree, they can hand the final decision back to the people in a new referendum. The UK must now seek an extension of the Article 50 timetable.
Brexit: "Halt Ze German Advance"
Tanja Bueltmann
English identity is in crisis. Germany has long since been the most prominent other. This has fueled the ongoing Brexit chaos and the deterioration of political discourse in the UK.
Brexit supporters have framed Brexit as a means to return to a more triumphal era: the idea of Empire 2.0, a Spitfire aircraft restored with government funds to fly around the world, a new Global Britain unshackled from Europe — all a retreat into fantasy.
Brexiteers are becoming increasingly shrill in their rhetoric, pushing the blame on to others. Germany is a frequent target, and now someone is putting up "Halt Ze German Advance" anti-EU billboards.
From Theodora Dickinson's incorrectly quoted Thatcher reference to Germans and the Holocaust, to Conservative MP Mark Francois' shameful words about "Teutonic arrogance" and "bullying" on live TV, to his fellow Conservative MP Daniel Kawczynski's lying tweet about the Marshall Plan, all these cases have two things in common: they ignore facts and abuse history.
With their cheap populist jingoism, these politicians and commentators are blowing up bridges built over decades. Underpinned by an entirely misplaced sense of exceptionalism and entitlement, they reveal that Brexit has nothing positive to offer.
If the only way you can define yourself and your role in the world is by talking contemptuously about another country and its people, abusing history, and distorting facts while expecting preferential treatment for yourself, well, to hell with you.
AR The last five words are my final flourish.
2019 February 10
Trump and Brexit
Simon Kuper
The Trump and Brexit projects have ended up remarkably similar. Both have broken down over the issue of a hard border with a neighbouring country. Both are flirting with a trade war. Neither looks able to pass any more legislation.
Anglo-American populism is a unique mixture of wronged superpower vengeance plus buccaneering capitalism. Here is the Trump-Brexit governing philosophy, as revealed in power:
Destroying the status quo might be better than the status quo.
▸ ▸ ▸ [+16 further points of similarity]
The revolution never compromises, not even with reality.
A Crazy Situation
Kenneth Clarke
Everyone is waiting for a miraculous solution. I have never seen such a crazy situation in all my life.
The prime minister is obsessed with keeping the Conservative party in one piece. The hardline Brexiteers have formed a party within the party. I would love to see them leave the party, but Theresa May is trying to keep them on side.
The Brexit debate has crippled our political system and distorted the usual process of politics.
A Secret State
Nick Cohen
Brexit is a war the British have declared on themselves. The only major European country to escape both communism and fascism, or occupation by the armies of Hitler or Stalin, has hard time taking the possibility of disaster seriously.
Britain has a hidden government, thinking the unthinkable in secret, to prevent voters realising the scale of the trouble they are in. Civil servants are eager to be part of the great Brexit game.
But the collapsing UK political system can no more provide the civil service with a clear direction than it can say where it will be this summer. This is not competent public administration.
Brexiteers' lack of concern for their fellow citizens borders on sociopathic. When Brexit fails, they will say it was betrayed by the Whitehall establishment.
Another Alcatraz
Mail on Sunday
The Danish island of Lindholm, home to a research station for animal diseases and a cemetery for infected carcasses, will become a fortress to dump rejected and criminal refugees.
Denmark is bitterly divided over migration. Danish prime minister Lars Lokke Rasmussen warns that migrant ghettos could fuel gang violence, and the Danish People's Party (DPP) calls for harsh policies to defend Danish values.
A €110 million plan to turn Lindholm into a holding pen for up to 125 unwanted arrivals, including convicted killers, will leave the migrants free to leave the island on its two ferries, one called Virus, so long as they check in daily with police. Protesting locals fear their peace will be ruined.
AR Brexit Britain is an island solution too.
2019 February 9
US Presidential Oversight
The New York Times
A president whose administration does not have the confidence of the people cannot govern effectively, or legitimately. Accountability is crucial to that confidence.
House speaker Nancy Pelosi: "It's not investigation; it's oversight. It's our congressional responsibility, and if we didn't do it, we would be delinquent in our duties."
The president should focus on the big picture. The public will feel much more confident in his leadership once some of the more disturbing questions have been answered.
Brexit Backstop Plan
Roland Alter
We have a disaster waiting to happen on 29 March.
The UK, Northern Ireland (NI), the Republic of Ireland (RI), and the EU all want to preserve peace in NI based on the Good Friday agreement. I propose a plan that tries to take the interests of four parties into account:
The UK says the backstop has to go. It undermines UK sovereignty, which has priority over peace
in NI. The UK wants freedom to negotiate FTAs, which is not possible under a customs union.
NI wants to avoid a negative economic impact.
The RI wants to preserve peace in NI.
The EU will stay loyal to the RI and protect the integrity of the single market.
My proposal drops the current backstop clause and gives NI citizens the right to vote to join the RI. By 2020, the UK and EU will have reached an agreement that either does or does not require a hard border. If it does, an NI referendum is held.
The vote takes the backstop decision away from the EU and gives it to NI citizens, thus respecting UK sovereignty. NI citizens could vote for Irish unity, but that genie is already out of the bottle. If they vote for a hard border, the problem arises first at the end of 2020.
This proposal could avoid disaster.
AR I dislike the idea that NI citizens get a benefit I don't, namely to vote again on whether they want to rejoin/stay in the EU. I want that too.
2019 February 8
Enola May
Tom Peck
No press conference for Theresa May in Brussels. All we got were some short strangulated barks of nothing, lasting about a minute, delivered into a microphone held by Laura Kuenssberg, to make clear, as only she can, that nothing has changed.
"I'm, erm, clear that I'm going to deliver Brexit. I'm going to deliver it on time. That's what I'm going to do for the British public."
Half the country doesn't want it delivered on time. They don't want it delivered at all. At some point, it's possible she'll work out she should never have pretended to be Winston Churchill, charged with some sacred mission to deliver Britain to its promised land. The promised land will be terrible. She knows it, but she can't extend her emotional range to acknowledge it.
2019 February 7
The State of the UK
Gina Miller
Just 50 days from now, the UK might be under martial law, experiencing shortages of foods and essential drugs, with a sharp economic downturn, and national security and public services drastically compromised.
Young people are deeply concerned about a no-deal Brexit. All of the Brexit options will be a disaster for the UK. None of them make a success of Brexit.
Theresa May, your deal is not the only one available to the UK. Accept the ambitions of the Tusk package. Leverage the fact the UK is an integral part of the EU.
Push for reforms on issues such as sovereignty, immigration, and economic governance. Accept publicly that to allow a no-deal departure from the EU by default would be the ultimate dereliction of duty and an unforgivable betrayal of future generations.
No viable alternative so far offers anything as advantageous as the deal the UK has right now with the EU. Restart the Tusk negotiations.
2019 February 6
The State of the Union
The New York Times
President Trump showed up with a standard list of broad policy aims. He brought up abortion and Syria and a "tremendous onslaught" of migrants on the southern border.
The spectacle evinced the true state of the union — fractured, fractious, painfully dysfunctional — as the president called for an end to "ridiculous partisan investigations": "If there is going to be peace and legislation, there cannot be war and investigation. It just doesn't work that way!"
Trump assailed Democratic leaders and repeatedly threatened to declare a national emergency if lawmakers didn't provide billions for his border wall.
AR Trump said he will meet Kim Jong Un again in Vietnam.
Black Honey
Jewish Review of Books
Avraham Sutzkever was born in Lithuania in 1913. He and his wife and son were in Vilna in 1941 when the Nazis conquered the city and murdered the baby boy. With horror compounding horror, Sutzkever obsessively wrote poems. From one:
The warm breath of a pile of dung
May become a poem, a thing of beauty
(translated by Benjamin Harshav)
Carlos Fraenkel
Massimo Pigliucci aims to bring Stoicism to modern life. He says we can develop a moral character and attain peace of mind by taking charge of our desires, by acting virtuously in the world, and by responding calmly to events we can't control.
Stoics said virtue is all we need. A virtuous person is happy under all circumstances. Everything that happens is part of a providential order, designed by a divine mind, and virtue consists in living in agreement with that order.
Einstein's God will no doubt appeal more to us than the divine mind. But Einstein's God doesn't care about anyone or anything. For us, improving our circumstances is better than searching for philosophical consolation.
AR Improving them — but how?
What Is Life?
Paul Davies
Life seems to fly in the face of the second law of thermodynamics. Erwin Schrödinger addressed this question.
Optimized information processing is key to the survival of living things. The genetic code is inscribed in DNA as sequences of the chemical bases A, C, G, and T. The information constructed from this alphabet is mathematically encrypted. To be expressed in an organism, it must be decoded and translated into the amino acid alphabet used to form proteins.
Living things have elaborate networks of information flow within and between cells. Gene networks control basic housekeeping functions and such processes as the development of an embryo. Neural networks provide higher-level management. Living organisms use these informational pathways for regulation and control.
Life = matter + information. The hard question is how chemicals can self-organize into complex systems that store information and process it using a mathematical code. We may need a new law or organizing principle that couples information to matter and links biology to physics.
Living cells are replete with nanomachines running the business of life. There are molecular motors and rotors and ratchets, honed by evolution to operate at close to perfect thermodynamic efficiency, playing the margins of the second law to gain a vital advantage.
Our brains contain voltage-gated ion channels that use information about incoming electrical pulses to open and close molecular shutters in the surfaces of axons, and so let signals flow through the neural circuitry. Working together, these channels give rise to cascades of signalling and information processing, as in computers.
Perhaps the transition from non-living to living is marked by a transformation in the organization of information. Treating information as a physical quantity with its own dynamics enables us to formulate laws of life. The whole of life is greater than the sum of its parts.
In quantum mechanics, a system such as an atom evolves according to the Schrödinger equation until a measurement collapses the wave function. This measurement cannot be defined locally but depends on the overall context.
Quantum biology may take us further.
RAF Tornado GR4
Ministry of Defence 2019
The Royal Air Force is retiring its Tornado strike aircraft after 40 years of front-line service. Image: RAF Tornado GR4
Rotary Club of Poole
Poole Bay
Poole Bay in winter
US Army
Pershing missiles
2019 Chinese New Year
I Become a Rotarian
Rotary Club of Poole
I was inducted today as a member in the Rotary Club of Poole. My introductory biography:
Andy Ross is a native European, born in Luton in 1949 and raised in Poole. He went to Poole Grammar School, where he won an award to read Physics at Exeter College in the University of Oxford. He graduated in PPE in 1972 and went on to earn three more degrees, one from the LSE and two more from Oxford in mathematical logic and scientific philosophy.
In 1977, he joined the Ministry of Defence in Whitehall as an administration trainee, only to find one year there was enough. After a year teaching English in Japan and a few years teaching maths and physics in London, he moved to Germany in 1987.
From 1987 to 1998, Andy worked as a physics and computer science editor at the academic publisher Springer in Heidelberg. Then from 1999 to 2009, he worked as a developer in the global software company SAP, where he contributed to courses at SAP University and wrote a book on a major new database development.
Also in 2009, Andy published his contributions to the emerging science of consciousness, where new ideas in artificial intelligence seem to promise a new age of machines with minds, on which he had spoken at international conferences in Europe and America over ten years. Then aged 60, he retired from SAP and wrote and self-published several more philosophical books.
In 2013, Andy returned from Germany to Poole and in 2014 joined the Conservatives, where he worked as a parliamentary assistant to the Poole MP, Sir Robert Syms. He also helped the current Poole Council to get elected in 2015 and has supported them ever since.
His Poole work continues with a new scheme to assist in the training and employment of talented young local people.
Brexit and Ireland
The Guardian
Brexit is at odds with the 1998 Good Friday agreement which sought to erase hard border between the north and south of Ireland. Theresa May's problem is that she has committed the UK to leaving the EU while respecting the peace deal.
London and Dublin have committed not to reintroduce border checkpoints. May's withdrawal agreement enshrines this in law as an insurance policy: If the UK left the EU without securing a deal, a backstop arrangement would allow for frictionless trade.
Fanatical Brexiteers were not bothered about the peace process. They saw in the insurance policy a devious mechanism to force Britain to march in lockstep with EU regulations. To reverse her government's historic Commons defeat, May agreed to replace the backstop.
In Belfast on Tuesday, the prime minister seemed to favour a revised backstop. MPs considering alternative arrangements were meeting in London as she spoke. All agree that the backstop must be a temporary measure, but no one wants to say so in law.
Northern Ireland voted to remain. Unionism risks defeating itself if it becomes too closely identified with Brexit. Fragmentation is by no means inevitable, but without a sense of common purpose and community it becomes possible.
Brexiteers Say No
The Times
EU top civil servant Martin Selmayr spent 90 minutes with members of the Brexit select committee last night and offered Britain a legal guarantee that it would not be trapped by the Irish backstop. But the Brexiteer MPs immediately rebuffed his offer.
Fool Britannia
Hari Kunzru
Britons "never, never, never shall be slaves," as Rule Britannia triumphantly puts it. The underside of nostalgia for an imperial past is a horror of finding the tables turned. For extreme Brexiteers, leaving the EU takes on the character of a victorious army coming home with its spoils.
Though imperial decline looms large in the imagination of Brexit, "the war" is crucial in structuring English feeling about the EU. The equation of a European superstate with a project of German domination is part of the map of English conservatism. To such people, the EU is just a stealthy way for the Germans to complete Hitler's unfinished business.
The English cult of heroic failure, exemplified by the charge of the Light Brigade and the evacuation from Dunkirk, suggests that the secret libidinal need of Boris Johnson, Jacob Rees-Mogg, Michael Gove, and their colleagues is actually for their noble project to fail in the most painful way possible, as an immolation on the altar of past glories.
The English seem unable to conceive of a relationship with Europe other than subjection or domination. They will try to regain the whip hand even if they have to immiserate the country to do it. For them, the principle of equal partnership on which the EU is predicated is not an option.
A Brutal End To Cool Britannia
Vincent Boland
Brexit is a retreat. It has profound strategic consequences for the UK and threatens to make it culturally more exclusionary.
British artists who wrote the soundtrack for European popular culture emerged when Britain was opening up to the world after its enforced postwar austerity and unleashing boundless creativity.
Brexiteers say Britain is leaving the EU, but not leaving Europe. In 2019, Europe is the EU. The UK political establishment has failed to accommodate itself to that fact.
Brexit will unleash demons. It will wound not only Britain but also Europe.
AR Brexit is foolish and uncool — obviously.
2019 February 4
End The War In Afghanistan
The New York Times
In September 2001, President George W. Bush went to war in Afghanistan: "Our war on terror begins with Al Qaeda, but it does not end there. It will not end until every terrorist group of global reach has been found, stopped and defeated."
More than 17 years later, the US military is engaged in counterterrorism missions in 80 nations. The price tag will reach about $6 trillion by the end of FY 2019. The war on terror has claimed an estimated half a million lives around the globe.
When Donald Trump ran for the White House, he promised to rein in overseas military adventurism and focus US resources on core strategic priorities. That retrenchment can start with Afghanistan: Withdraw NATO forces by the end of 2019.
Conservatives Will Not Be Forgiven
Andrew Rawnsley
Former cabinet minister Sir Oliver Letwin will support whatever Brexit deal Theresa May comes up with next, because if the UK crashes out of the EU without a deal and things turn bad, "my party will not be forgiven for many years".
Ministers, civil servants, and heads of government agencies who have responsibility for essential services and commerce are sweating fear. The people who would have to handle the consequences of Britain crashing out of the EU are very scared indeed.
Conservative prime ministers called the 2016 referendum and presided over the combination of tragedy and farce that has unfolded since. If Brexit goes horribly wrong, voters are going to blame the Conservatives.
Albion Through The Looking Glass
Matt Ross
UK politics has taken a running jump down the rabbit hole.
Having repeatedly insisted that her deal was the only one available, Theresa May caved to Brexiteers. The Brady amendment requires her to find "alternative arrangements" to replace the backstop.
May's deal would be worse for the UK than remaining in the EU. Ministers attempt to defend her deal as respecting "the will of the people" whereas holding another vote would be "undemocratic" and drive up support for the far right.
ERG Brexiteers seem to believe that, following a brief period of discomfort, No Deal would carry the UK to the sunlit uplands as a free sovereign state, with the pain dumped on EU27 shoulders. So Brexiteers run down the clock.
Diehard Remainers believe that May understands the chaos that would result from a disorderly exit and that MPs will agree on a new poll or a revocation of Article 50. So Remainers run down the clock.
May's strategy has been to scare people into backing her deal. And while the Brady amendment weakens perceptions of her as a credible negotiating partner, it kicks the can further down the road. So May runs down the clock.
A narrative is growing among Brexiteers that the EU is being unreasonable. Continental partners are sounding ever more impatient. Anger on both sides of the Channel is hindering agreement.
The UK political system is suffering a dissociative fugue.
AR The last two words are my diagnosis of what Matt called a nervous breakdown.
2019 February 3
Brexit: Queen Has Evacuation Plan
The Sunday Times
The Queen and other senior royals will be evacuated from London in the event of riots triggered by a no-deal Brexit, under secret emergency plan to rescue the royal family first drafted during the cold war, as the risk rises that things might turn ugly if the UK crashes out of the EU without a deal.
AR As numerous rich people emigrate, big overseas investors pull out, and the Queen plans to flee from London, Brexit Britain will become a prison or quarantine state. Millions of citizens will feel the pain if basics like food and medicines run out, and their agonies will lead to massive disruption.
Brexit Britain will become ripe for a radical socialist revolution or outlaw declarations of shariah neighbourhoods. The army is not big enough to maintain law and order in such such circumstances, opening up a serious risk of a meltdown of civil society, anarchy, and a descent into savagery.
2019 February 2
Russia Suspends INF Treaty
Following the US suspension of the Intermediate-Range Nuclear Forces Treaty, Russia has announced it is suspending it too.
Russian president Vladimir Putin: "The American partners have declared that they suspend their participation in the deal, we suspend it as well .. We have repeatedly, during a number of years, and constantly raised a question about substantiative talks on the disarmament issue, notably, on all the aspects. We see, that in the past few years the partners have not supported our initiatives."
AR Putin's dog in the White House has been seduced by Pentagon hawks into making a move that plays into the hands of Kremlin hawks who want to threaten Europe. NATO must work harder.
The Rise and Fall of British Politics
Jonathan Powell
A hundred years ago, Max Weber gave a lecture on the profession and vocation of politics. He admired the British system and the way its politicians and officials managed prosperity and stability in a working democracy. Britain was known as the cradle of democracy and decency.
Britain has now gone from being the most stable country in Europe to one of the least, from a country governed by a broad consensus to a society divided into two camps, and from a government that managed crises well to leaders who cannot even control their own parties.
The government's botched handling of Brexit was both a failure of political leadership and a failure of planning. Theresa May set red lines without real thought as to what might be achievable and walked into a deal where she sacrificed too much and failed to think through the consequences. She tried to play hardball.
The obsession with Brexit has prevented the government from addressing the deep problems that confront the UK. It has no time or energy to develop serious policies on how new technology impacts traditional jobs, or on the housing crisis for young people or the social care crisis for older people. For everything but Brexit, the country is on autopilot.
The concept of facts has disappeared. Not only the politicians but also the civil servants have failed. The result is a collapse of public confidence in the political system.
Max Weber warned about the dangers of professionalizing politics. In Britain that professionalization has reached a new level in this century. As a result, people now think they are governed by an elite that serves its own interests.
The tragedy of Brexit shows the need for a new kind of politics in Britain.
The Future of the Mind
Susan Schneider
I think about the fundamental nature of the mind and the nature of the self. If we have artificial general intelligence, I want to know whether it would be conscious or just computing in the dark.
We need to keep an open mind. If machines turn out to be conscious, we will be learning not just about machine minds but about our own minds. That could be a humbling experience for humans.
In a relatively short amount of time, we have managed to create interesting and sophisticated artificial intelligences. We already see tech gurus like Ray Kurzweil and Elon Musk talking about enhancing human intelligence with brain chips.
I see many misunderstandings in current discussions about the nature of the mind, such as the assumption that if we create sophisticated AI, it will inevitably be conscious. Many of the issues at stake here involve classic philosophical problems that have no easy solutions. Now that we have an opportunity to possibly sculpt our own minds, I believe that we need to dialogue with these classic philosophical positions about the nature of the self.
As we use neuroprosthetics or brain chips in parts of the brain that underlie conscious experience in humans, if those chips succeed and if we don't notice deficits of consciousness, then we have reason to believe that the microchips could underwrite consciousness. In principle, we could develop a synthetic consciousness.
I like living in that space of humility where we hit an epistemological wall.
AR David Chalmers speculated on replacing brain parts step by step with chips and exploring whether consciousness faded as a result.
2019 February 1
US Suspends INF Treaty
Secretary of state Mike Pompeo says the United States is suspending the Intermediate-Range Nuclear Forces Treaty. This pact with Russia has been a centerpiece of European security since the Cold War.
AR I vividly recall the nightmare of "theater" nuclear weapons in Europe that was ended by the INF treaty. The month the treaty came into force I moved to Germany to live and work there.
Neural Network Theory
Kevin Hartnett
Neural networks implement our most advanced artificial intelligence systems. Yet we have no general theory of how they work.
A neural network is made of neurons connected in various ways. We set its depth by deciding how many layers of neurons it should have. We set the width of each layer to reflect the number of different features it considers at each level of abstraction.
We also decide how to connect neurons within layers and between layers, and how much weight to give each connection. For image processing, convolutional neural networks have the same pattern of connections between layers repeated over and over. For natural language processing, recurrent neural networks connect neurons in non-adjacent layers.
A neural network with only one computational layer, but an unlimited number of neurons with unlimited connections between them, can perform any task, but is hard to train and computationally intensive. By increasing depth and decreasing width, you can perform the same functions with exponentially fewer neurons, but the functions set a minimum width for the layers.
For example, imagine a neural network tasked to draw a border around dots of the same color in an array of colored dots. It will fail if the width of the layers is less than or equal to the number of inputs. Each dot has two coordinates for its position. The neural network then labels each dot with a color and draws a border around dots of the same color. In this case, you need three or more neurons per layer to perform the task.
All this is not yet a general theory of neural networks.
Poole Harbour
Wintry outlook: Poole Harbour, January 31
"The EU including Ireland stands
by the withdrawal agreement,
including the protocol and
backstop relating to Ireland."
Leo Varadkar
"The backstop is part of the
withdrawal agreement, and
the withdrawal agreement is
not open for renegotiation."
Donald Tusk
Sterling falls as Cooper
amendment fails
2019 January 31
Trump Targets US Intelligence
President Donald Trump's assaults on US spy chiefs are shocking coming from a commander in chief.
The president's Twitter barrage over a global threat matrix produced by US intelligence agencies that contradicts his idiosyncratic worldview is no surprise. His habit of fashioning a truth that fits his personal prejudices and goals over an objective version of reality is well known. But consider:
His rejection of US intelligence assessments that Russia interfered in the 2016 election
His one-man flattery offensive toward Russian president Vladimir Putin
His claim to have ended the North Korean nuclear threat
His assertion that ISIS is "badly" beaten
His withdrawal from the nuclear deal with Iran
His plans to withdraw troops from Syria and cut the US garrison in Afghanistan
His constant undermining of NATO
In the realm of national security, Trump's approach can be deeply destructive.
May Must Be Stopped
Philip Stephens
Theresa May's mandate to rewrite the Irish backstop is worthless. The hardliners do not want an agreement. They want to run down the clock to a no-deal Brexit.
May faced a choice on Brexit. She could prioritise party unity by bowing to nationalists or she could try to build a wider coalition. Her red lines set her course.
The latest Faustian pact was a logical destination. EU leaders know a slender majority of 16 is not a sustainable negotiating mandate. They will not abandon the Irish government.
The British government can longer be trusted as a partner. There is nothing its EU partners can do to save the UK from its crash-out course. May must be stopped.
The Price Of Party Unity
The Guardian
Theresa May won a slim majority for revising her Brexit deal. Her trump was party unity. The EU has no appetite to change the backstop, but May set this aside.
The 27 EU member states whose interests are stitched into the withdrawal agreement will find her change of mind obnoxious. Governments around the world see an unreliable Britain.
The prime minister has appeased her party fanatics. The rightwing fringe of her party is hostile to the EU. Some Tory MPs welcome a disorderly Brexit as prelude to a blame game.
May has lost credibility and goodwill with EU27 partners. They know any concessions would simply provoke more demands. Appeasing the hardliners is worse than weakness.
2019 January 30
Brexit Brinkmanship
The Times
The EU sees an extension to the Article 50 exit process as certain but will set conditions on any extension. These must be agreed unanimously by all 27 European leaders.
One option is to offer the UK a 3-month extension to continue negotiations with the option of further such extensions until the end of the year. The next summit of European leaders that could agree an extension is on March 21, D-Day minus 8.
Brexit Disaster
Jonathan Freedland
History will damn the architects of Brexit. Imagine the day, years from now, when an inquiry probes the epic policy disaster that was Brexit and delivers its damning final report, concluding that this was a failure of the entire British political class.
Theresa May had repeated endlessly that her deal was the only deal on offer. Yet last night she urged MPs to vote for an amendment that trashes that deal. The Brady amendment demands replacement of the Irish backstop with alternative arrangements.
Cheering her on was a Conservative party declaring that they like the good bits of the deal but not the bad bits. This was the same logic the Brexiteers used for the referendum: Do you want to stay in the EU, with all its flaws, or would you like alternative arrangements?
MPs had the chance to prevent the national cataclysm of a no-deal crash-out last night and they refused to take it. They voted for a toothless amendment that does nothing to prevent it.
A handful of MPs are using every parliamentary wile they can to stop the country from slamming into the iceberg. But almost everyone else will be damned for their role in a saga that disgraces this country and its supposed leaders.
Closer To No Deal Than Ever
Anand Menon
What a pantomime! The prime minister joined her party in demanding changes to her deal. She backed an amendment instructing her to go back to Brussels and try harder.
Parliament proudly displayed its inability to decide. They don't want no deal but they do want changes to the Ireland backstop.
Tory tribalism is a consensus built on alternative arrangements to replace the backstop. The assumption is that the EU will cave in. So far there is little reason to believe the member states will do so.
The natural party of government votes for unicorns. The party in opposition is not quite sure what it wants. Hardcore Brexiteers pushed a narrative of EU intransigence.
Now we wait for May to go back to Brussels and seek to achieve in under a fortnight what she failed to achieve in a year of negotiations.
Start Panicking
James Ball
A major country stands on the brink of potential meltdown. If the UK does not find a way to support a deal either to exit the EU or to negotiate an extension on the Article 50 process, it is set to crash out of the EU on March 29.
Most coverage has focused on the initial chaos. A country that relies heavily on cross-channel shipping for food, medicines, manufacturing supplies, and more could see crossing fall by 75%−87% for six months, with almost no viable alternative routes to match anything like the lost capacity.
But the longer-term economic harm would be devastating. The midpoint estimates of such a crisis suggest a drop of close to 9% in GDP — a far, far deeper recession than the financial crisis of 2009 — and huge increases in unemployment, the cost of borrowing, plummeting value of the pound, and more. This damage could easily take a decade or more to repair.
The UK is perhaps the largest financial center in the world. More than a third of global foreign exchange trading operates out of London, as does a similar proportion of derivatives trading. Its banking sector is sized at more than €10 trillion, and it manages more than a third of European financial assets.
Britain is on the brink of economic crisis. Be very afraid. A no-deal exit could launch a global economic crisis.
2019 January 29
Brexit: Parliamentary Amendments
BBC News
Amendment A (Labour frontbench amendment, which rejects the idea of a no-deal Brexit and calls for a permanent customs union with the EU)
296 Ayes, 327 Noes
Amendment O (SNP/Plaid Cymru, notes that Scotland and Wales voted against Brexit and calls on government to extend Article 50 and reject no-deal Brexit)
39 Ayes, 327 Noes
Amendment G (Dominic Grieve, call for six extra sitting days to debate other business, such as another referendum or rejections of a no-deal Brexit)
301 Ayes, 321 Noes
Amendment B (Yvette Cooper, seeks binding legislation for an extension of Article 50 if withdrawal agreement not approved by 26 February)
298 Ayes, 321 Noes
Amendment J (Rachel Reeves, call to seek two-year extension to Article 50 process if no agreement between UK and EU approved by 26 February)
290 Ayes, 322 Noes
Amendment I (Dame Caroline Spelman, stating that the UK will not leave the EU without a deal — but only advisory, with no legislative force)
318 Ayes, 310 Noes
Amendment N (Sir Graham Brady, calling for the backstop to be replaced with alternative arrangements to avoid a hard border)
317 Ayes, 301 Noes
AR A victory for party tribalism over basic common sense.
Brexit: Malthouse Compromise
David Henig
The Conservatives' new "Malthouse compromise" stands no chance of being acceptable to the EU.
There seem to be three fundamentals to the plan: the extension of the implementation period to 2021, technological solutions to ensure no border infrastructure requirements on the island of Ireland, and an interim free trade agreement to be tabled immediately.
Technological solutions for the Irish border have been endlessly debated, but there is no border in the world outside of the EU where there are no physical checks. Only where both countries are in a customs union and have more or less the same product regulations backed by a common court have checks been eliminated.
The backstop is a firm red line for the commission. The EU does not trust the UK not to backslide from a vaguer commitment.
An interim free trade agreement would not help avoid border checks. The chances of the EU accepting one are close to zero.
2019 January 28
Trump Horror Movie
Michael Bloomberg
President Trump cannot be helped. The presidency is not an entry-level job. There is just too much at stake.
The longer we have a pretend CEO who is recklessly running this country, the worse it's going to be for our economy and for our security. This is really dangerous. It's like a bad horror movie: Instead of Freddy Krueger and the Nightmare on Elm Street, we've got Donald Trump and the Nightmare at 1600 Pennsylvania Avenue.
The government shutdown is one of the worst cases of incompetent management I have ever seen.
Take No-Deal Brexit Off Table
Financial Times
Westminster has the opportunity this week to end the damaging prospect of a no-deal Brexit. MPs will vote on Tuesday on a range of amendments to UK prime minister Theresa May's Brexit proposals. They must act now to avoid calamity.
The chief executive of Airbus was right to brand the negotiations a disgrace. Several other manufacturers caution that no deal would threaten jobs and investment. This is not Project Fear but an acknowledgment of reality.
The only way to gain the support of Brexiteers for May's deal is to rethink the Irish border backstop. But there is no sign that Brussels is willing to do so. Extending Article 50 is the only reasonable course of action.
The priority must be to avoid the risk of chaos. Parliament remains sovereign in the UK. MPs should use that sovereignty to act in the national interest.
The Conservative Party Is Becoming Repellent
Matthew d'Ancona
As a believer in fiscal discipline, strong defence, robust anti-terrorism measures, the Atlantic alliance, and the social liberalism of those who live in the here and now, I ought to be at ease with modern Conservatism. And I really am not.
The attack on "Teutonic arrogance" by Mark Francois MP is the tip of a nativist iceberg. Brexit has summoned the very worst demons that lurk in the Conservative psyche, liberating Tories to bellow nonsense about WW2. It has fatally compounded the party's demented fixation with immigration and distracted it from the true challenges of the 21st century.
In a crisis of this nature, the proper role of Conservatives should be to cut through the infantile rhetoric and show true statesmanship. Instead, we see a party cravenly fetishising the 2016 referendum as if no further expression of popular opinion on Brexit were possible, behaving as if the only thing that matters is to get out of the EU by 29 March.
Where are the Tories prepared to risk their careers and to say that the instruction given by the electorate in 2016 cannot be delivered in a way that does not do terrible harm to those same voters and their children?
DeepMind AI Beats Humans at StarCraft II
New Scientist
StarCraft II is sometimes called the Grand Challenge for artificial intelligence. Now DeepMind AI has defeated two top-ranked professional players at it, both by 5-0.
In StarCraft II, players control armies across a terrain. They build infrastructure, juggling short-term gain with long-term benefits, when they can't always see the full map.
DeepMind created five versions of their AI, called AlphaStar, and trained them on recordings of human games. The versions then played against each other in a league. AlphaStar played on a single GPU, but it was trained on 16 tensor processing units hosted in the Google cloud.
AlphaStar skills apply in other areas. StarCraft is like running a company or a logistic operation. It involves planning R&D and getting goods to the right place at the right time.
AR Memo to SAP: Try it.
The Open Society
Tim Hayward
"Why should we care about
what's going on over there?
The answer is that what
is going on 'over there'
affects us. If Africa loses,
Europe can't win."
Bono, Davos
"Those who've lived under
dictatorships say, 'please don't
give me a ministry of trust'.
But is it acceptable that
Mark Zuckerberg is the
minister of trust?"
Marietje Schaake MEP
Morning sun, Poole
2019 Holocaust Memorial Day
A Quiet Life
Jeremy Cliffe
The European project knows no higher ideal than calm good living. The EU and most of its states were born or reborn from the rubble of war and the traumas of totalitarianism. The opposite of horror and cataclysm is the quiet life.
This European dream is glimpsed not in luxurious ceremonies in Paris, Brussels, or Berlin but in well heated social housing blocks in Utrecht or Vienna, in comfortable houses with gardens on the outskirts of Barcelona or Prague, in safe streets and decent hospitals. The EU mission is to protect this comfortable European garden from outside threats.
The EU obsession with the quiet life also explains its weaknesses. It concentrates too little on global security or competitors outside its borders. It prioritizes averting losses above grasping opportunities. It generally values dull uniformity above dazzling difference.
Its Brexit talks with Britain illustrate these traits. Shortly after the referendum, EU leaders agreed that the risk of a fragmenting EU was the greatest danger to European life. Brexit will succeed insofar as it serves the pursuit and preservation of the comfortable European garden.
2019 January 26
The Danger Posed by China
George Soros
An unprecedented danger is threatening the survival of open societies. The danger is from the instruments of control that machine learning and artificial intelligence can put in the hands of repressive regimes.
In China, a centralized database to create a social credit system will evaluate people with algorithms that determine whether they pose a threat to the one-party state. The social credit system will treat people according to the interests of the state. The instruments of control developed by artificial intelligence give an inherent advantage to authoritarian regimes over open societies.
In an open society, the rule of law prevails as opposed to rule by a single individual, and the role of the state is to protect human rights and individual freedom. By contrast, authoritarian regimes use whatever instruments of control they possess to maintain themselves in power at the expense of those whom they exploit and suppress.
China has a Confucian tradition, according to which advisors of the emperor are expected to speak out when they strongly disagree with one of his actions or decrees, even that may result in exile or execution. The committed defenders of open society in China have mostly been replaced by younger people who depend on Xi Jinping for promotion. But a new political elite has emerged that is willing to uphold the Confucian tradition.
Xi presents China as a role model for other countries to emulate. His Belt and Road Initiative is designed to promote the interests of China, not the interests of the recipient countries. An effective American policy toward China must include a US response to the Belt and Road Initiative.
China wants to dictate rules and procedures that govern the digital economy by dominating the developing world with its new platforms and technologies. The combination of repressive regimes with IT monopolies endows those regimes with an advantage over open societies. The instruments of control pose a mortal threat to open societies.
China is wealthy, strong, and technologically advanced. Xi Jinping is the most dangerous opponent of open societies. We must pin our hopes on the Chinese people.
AR As a student Soros was influenced by Karl Popper, as I was, but his conception of open societies needs radical revision.
China shows the need for a new view to accommodate the priorities of development and the opportunities of new technology. Machine learning and so on force all of us to reconsider how human rights and the constraints of our planetary ecosystem can be reconciled. The Western liberal model of the state is verging on obsolescence.
As I see it, China offers a more positive model of how to proceed than Soros seems to think.
2019 January 25
Doomsday Threats
Bulletin of the Atomic Scientists
Dire as the present may seem, there is nothing hopeless or predestined about the future. But threats must be acknowledged before they can be effectively confronted.
Speech to the Sandringham Women's Institute
Queen Elizabeth II
"The continued emphasis on patience, friendship, a strong community focus, and considering the needs of others are as important today as they were when the group was founded all those years ago .. As we look for new answers in the modern age, I for one prefer the tried and tested recipes, like speaking well of each other and respecting different points of view, coming together to seek out the common ground, and never losing sight of the bigger picture."
AR Cited in the UK as front-page and TV news for its presumed relevance to Brexit. Who can doubt that the UK is in thrall to a quasi-religious cult of royalty?
2019 January 24
Brexit: No No Deal
Financial Times
UK prime minister Theresa May faces intense pressure to rule out a no-deal Brexit.
Chancellor Philip Hammond: "Not leaving would be seen as a betrayal of the referendum decision, but equally leaving without a deal would undermine our prosperity and would equally represent a betrayal of the promises that were made."
Work and pensions secretary Amber Rudd: "There is no doubt that No Deal would be bad for prosperity and bad for our security."
Business minister Richard Harrington: "I'm very happy if the prime minister decides I'm not the right person to do the business and industry job."
Airbus CEO Tom Enders: "Please don't listen to the Brexiteers' madness, which asserts that, because we have huge plants here, we will not move and we will always be here. They are wrong."
Siemens UK CEO Jürgen Maier: "The thing all of us won't be able to manage is a no-deal .. the writing is on the wall on what will happen to those factories in a decade."
Second Vote Now
Martin Wolf
Brexit is a disastrous course. The golden future awaiting "global Britain" is fantasy.
The UK faces a significant probability of crashing out in a disorderly exit, which would be hugely disruptive in the short run and costly in the long run. It would damage relations with the EU forever.
A slight victory in the referendum gives Brexiteers no right to drive the country over a cliff. Remainers have the right to ask parliament to consult the people again. Fanatics cannot dictate what Leave means.
Let us have a second vote.
Theresa May Will Accept Delaying Brexit
Daily Mail
Theresa May is privately resigned to having to delay Brexit if MPs vote for it next week, say allies. In public, she hit out at MPs seeking to extend article 50, saying this does not solve the issue.
Downing Street will not say whether the government will accept a bid to delay Brexit for up to nine months. The proposal would allow MPs to table legislation on February 5 to extend article 50. If it passed, May would have three weeks to win a vote on her Brexit deal in parliament before being required to seek an extension of article 50.
Labour and a number of Tories hint they will back the bid, which will be debated and voted on next Tuesday. Extending article 50 would require unanimous consent from the EU27.
Germans Owe Britain
Alexander von Schönburg
German politicians and pundits are urging chancellor Angela Merkel to harden her line toward the UK and not throw Theresa May a lifeline. French president Emmanuel Macron said Brexit was a British problem it would have to solve on its own.
Such blinkered and sour responses ignore the great debt that Germany and Europe owe the UK. It was Great Britain that first stood up to Hitler in 1939. Britain opened its doors to the thousands of Jewish refugees fleeing the Holocaust.
There would be no free Europe without Britain and its bloody sacrifice. But thanks to opposition by French president Charles De Gaulle, it took more than a decade for the UK to be accepted into the forerunner of the EU.
The driving force for welcoming UK membership was German chancellor Konrad Adenauer. In 1965, Queen Elizabeth II toured Germany. Adenauer quickly saw that Europe needed Britain precisely because Europe is a patchwork made up of vibrantly different nations.
We needed Britain 80 years ago and we need Britain now. For Europe, Britain is the real backstop. Without the UK, power in Europe will tend to gravitate ever more to the center.
We need to let the UK leave without punishing it. German statesman Otto von Bismarck said one should always win but never humiliate one's opponents. The best victory is the magnanimous one.
AR Victory. Brexit certainly feels like the bum's rush.
2019 January 23
Davos 2019
Aditya Chakrabortty
The plutocrats are terrified. The Davos organisers ask: "Is the world sleepwalking into a crisis?"
The last three decades have seen the political and economic elites hack away at our social scaffolding. It proved profitable, for a while, but now it threatens their own world. And still they block more taxes on wealth, more power for workers, companies not run solely to enrich their owners. The solutions to this crisis will not be handed down from a mountain top.
Three decades after Ronald Reagan, we laugh at: "I'm from the elite and I'm here to help."
Let The People Vote
Financial Times
The UK ship of state is steaming toward the iceberg of a no-deal Brexit. A catastrophe looms.
Parliament should legislate against a no-deal outcome and seek to extend the Article 50 process. MPs should then hold indicative votes to test support for other exit options.
The withdrawal process has been taken hostage by ideological extremists determined to reject the sole sensible version of an EU exit that is on the table. If parliament remains unable to back a new deal, a general election changes nothing.
Ask the public whether they still want Brexit. They deserve a chance to weigh the realities of departing against remaining in the EU under the existing terms: outside the EZ and Schengen zone and inside the single market.
A second plebiscite might deliver no clear outcome and might repeat the previous result. It would also be divisive. But leaving the EU will be divisive in any case.
Parliamentary gridlock is worse than a new plebiscite. The people must have their say.
Whitehall Hysteria
Rachel Sylvester
A sense of hysteria is spreading around Whitehall. Five government ministers:
M1 If the government doesn't agree to a second referendum I'll lose my seat.
M2 If a people's vote goes ahead we'll be thrown out by the voters.
M3 The only way out might be a general election.
M4 If we do that I'll definitely lose my seat.
M5 We're so fucked.
Battle for Britain
Melanie Phillips
Theresa May made a fundamental error. Britain is bitterly split down the middle between Brexiteers and Remainers. May wanted to deliver a Brexit deal that would bring both sides together.
On the issue of sovereign British independence, there can be no compromise. The UK is either out of the EU or it is in. May's deal is Brexit in name only and leads to surrender.
Westminster is currently heaving with plots. But there's no deadlock. The legally binding default position is that if no deal with the EU is struck, Britain will leave on March 29 without a deal.
Britain is a very special country. The countries of mainland Europe have a shallow understanding of national identity. Britain is an island nation with a distinct and separate identity.
Three nations see themselves as uniquely blessed: Britain, America, and Israel. All played a big role in bringing civilization to the world. But all three are beset from within by a subversive intelligentsia.
The idea of the modern nation state grew out of the Enlightenment. Britain was first into the Enlightenment and is first out. Brexit offers Britain its last chance to become itself again.
AR If this is not self-serving tosh I don't know what is. Britain is a cake baked with European ingredients, from its Celtic roots to its Viking and Norman conquerors, from its European monarchy to its Franco-Germanic language. Its separate identity is no more real than that of Germany or France, which have now seen the light and agreed to cooperate.
Coastal borders do not a nation make, not in a world of transcendent religion and universal science, and especially not in a world of air travel and nuclear missiles. A sovereign nation needs a rich culture, a strong heritage, and native genius. Europe was the cradle for the nation state and is well stocked with them.
America is a special case because it blends the cultures of Europe and other lands to form a uniquely powerful state. Israel is a special case because Jews have nursed a uniquely deep heritage over millennia. Britain is unique mainly because its maritime empire was the first whose outer perimeter shrank to an antipodal point on a finite planet.
Trump America is floundering as it confronts the global limits of its power in a world hosting rising competitors. The Israel of the Zionists is endangered because its history looks ever less special as our planetary civilization emerges. And Brexit Britain is doomed to suffer its proud sovereignty as an albatross in an emerging global self that rewards harmony between its national parts.
Melanie Phillips betrays her case by bewailing the intelligentsia. This outs her as a romantic populist. Britain, America, and Israel are transient forms in the fluid swarming of people in big history.
2019 January 22
Aachener Vertrag
Thomas Schmid
Frankreichs Präsident Emmanuel Macron und Bundeskanzlerin Angela Merkel unterzeichnen einen neuen Freundschaftsvertrag zwischen Frankreich und Deutschland.
Sowohl Merkel als auch Macron agieren aus einer Position der Schwäche. Das Brexit-Chaos vervollständigt den Eindruck: Kerneuropas Zustand ist desolat.
Brexiteers Feel Heat
Financial Times
UK prime minister Theresa May announced her new Brexit strategy on Monday: Plan B is Plan A.
But Brexiteers and the DUP are starting to feel the heat.
Some might back her plan if she can win more concessions in Brussels. May promised to go back to the EU to address the backstop, which Brexiteers hate and the DUP opposes. They are now under pressure to back her deal or lose Brexit.
A group of MPs aim to force May to extend the Article 50 exit process if no deal is in place by the end of February. Work and pensions secretary Amber Rudd says many ministers would quit if they were not allowed to support the move.
Brexiteers fear a delay could let parliament push for a second EU referendum. War gaming for a snap election is also under way. Either could lose Brexit.
Greenland Ice Melting
Oliver Milman
Greenland's ice is melting faster than we previously thought, new research finds. Its glaciers are dumping ever larger icebergs into the Atlantic, but the largest ice loss in the decade from 2003 occurred in SW Greenland, where surface ice is melting as temperatures rise, causing meltwater to flow into the ocean and push up sea levels.
PNAS paper lead author Michael Bevis: "Increasingly large amounts of ice mass are going to leave as meltwater, as rivers that flow into the sea. The only thing we can do is adapt and mitigate further global warming — it's too late for there to be no effect. This is going to cause additional sea level rise. We are watching the ice sheet hit a tipping point."
Greenland lost around 280 billion tons of ice per year between 2002 and 2016, and the ice melted four times faster in 2013 than in 2003. If the entire Greenland ice sheet, 3 km thick in places, were to melt, global sea levels would rise by 7 m and flood coastal cities.
Asteroids Hit Moon and Earth
Joshua Sokol
Some 290 million years ago, large asteroids began to rain down on Earth several times more frequently than before.
Collisions in the asteroid belt between Mars and Jupiter made rocky fragments in similar orbits. Solar heat nudged them into new positions and sometimes made them fall and hit Earth.
On Earth, most large impact craters are relatively young. Craters from over 650 million years ago vanished as Earth scoured off much of its crust. In the snowball Earth scenario, multiple waves of glaciation once encased the planet in ice. The glaciers pulverised several km of continental rock and dumped it into the ocean.
On the Moon, when an impactor digs a crater it scatters boulders around it. Over perhaps a billion years, a drizzle of micrometeorites hitting the Moon dissolve the boulders into finer soils called regolith. So the ratio of rocks to regolith in a crater tracks the time since the initial impact.
During the day, both boulders and regolith soak up heat. At night, the boulders hold on to this heat, glowing faintly with thermal radiation for hours, whereas the regolith cools quickly. The nighttime IR glow indicates the relative rockiness of a crater and thus its age.
Researchers used IR data to estimate the age of 111 big lunar craters. Since about 290 million years ago, big impacts have been 2.6 times more frequent than in the preceding 710 million years. The jump occurred around the same time as Earth's largest mass extinction, the Great Dying.
Super blood wolf moon
Daniel Monk / Bav Media
Super blood wolf moon over Northumberland
Nicola Jennings
the dodo
the dodo
MiG-31K with hypersonic
Kinzhal missile
Me at work today
Fog in Westminster:
Continent cut off
Green party
Space elevator
Chase Design Studios
2019 January 21
Brexit Options Narrowing
Tony Blair says a no-deal Brexit will not happen. The future UK relationship with the EU is so vague that Europeans must prepare for continuing struggle. Parliament is tied up, there is no consensus, and the UK should not even consider no deal: "A second vote remains the only solution."
Business Confidence Sinking
Financial Times
ICSA finds that 73% of FTSE 350 company secretaries predict their company will be damaged as a result of Brexit. Some 11% think global economic conditions are likely to improve in 2019, but only 2% think the UK economy will improve in 2019. The research was conducted at the end of 2018.
ICSA policy and research director Peter Swabey says a public lack of trust in business is reflected in political rhetoric: "This seems to have produced an own goal for the main UK political parties."
Rebel Spirit Rising
John Harris
Wetherspoons pub chain founder and chairman Tim Martin has been on the road since November, speaking at his pubs. His case for Brexit is unconvincing to the point of tedium. But the spectacle of the Brexit hardcore, many of them on the pints and riled to fury by interruptions from local Remoaners, is fascinating.
Terrible logic, combined with a certain stubborn ignorance, makes them insist that the only Brexit that matches what millions of people thought they were voting for in 2016 is a clean break. Their defiant rejection of all the warnings about falling off a cliff edge comes from the same performative "fuck you" as moved much of the original vote for leave.
The pub crowd wants drama and crisis. The romantic idea of a besieged Brexit Britain nobly trying to make its way without interference from Brussels, an island nation standing alone, could push UK politics over the edge.
2019 January 20
Brexit: Wer kann das Chaos noch verhindern?
ARD, 61:54 min
Über den Brexit diskutiert Anne Will mit Jean Asselborn, Sahra Wagenknecht, Norbert Röttgen,
Greg Hands und Kate Connolly
Big Data
Hugo Rifkind
Behavioral surplus is the data we give to tech companies over and above what they need. For example, if you have googled fridges, Google will try to sell you a fridge. It is the raw data for modeling human behavior.
At Google, the monetization of your surplus raised revenues from $86 million in 2001 to $347 million in 2002 to $3.5 billion in 2004. By 2016, Google and Facebook took more than a fifth of all advertising spending in the world.
Shoshana Zuboff goes beyond the idea that if you aren't the customer, you're the product. For her, we are more like elephants slaughtered for ivory: "You are not the product, you are the abandoned carcass."
Google and Facebook want to know everything about you. For them, the development of AI and VR is all about knowing anything you might ever do, and everywhere you might go, so as to sell you a Coke when you get there.
Regulating tech is like running around a burning house closing doors to rooms that won't exist in the morning. Breaking up Facebook and Google only means 50 rival companies will leech on your soul.
Brain Maps
Brains use a spatial coding scheme that may help us navigate many kinds of information, including sights, sounds, and abstract concepts.
The entorhinal cortex next to the hippocampus contains grid cells that make cognitive maps. Grid cells form a coordinate system by firing at regularly spaced positions in a hexagonal pattern. Different sets of grid cells form grids with larger or smaller hexagons, grids oriented in other directions, and grids offset from one another.
Together, such hexagons can map spaces with many dimensions. Grid code logic can apply to any structured data.
2019 January 19
The Dodo Is Dead
Fintan O'Toole
Brexit is a choice between two evils: the heroic but catastrophic failure of crashing out or the unheroic but less damaging failure of swapping first-class for second-class EU membership.
Brexit is not about the UK relationship with the EU. The drama has really served to displace a crisis of belonging. The visible collapse of the Westminster polity this week may be a result of Brexit, but it is also the result of the invisible subsidence of the political order over recent decades.
Brexit is the outward projection of an inner turmoil. An archaic political system had carried on while its foundations in collective belonging were crumbling. Brexit has forced the old system to play out its death throes in public. The spectacle is ugly, but it shows the need for radical change.
Britain can no longer pretend to be a settled and functioning democracy. The Westminster dodo is dead. The problem with British democracy is not the EU but the UK.
I Want My Continent Back
Polly Toynbee
Today the UK turns remain. Even if not a single person has changed their mind since the referendum, the demographic shift alone has done the heavy lifting. Enough old leavers have died and enough young remainers have come on to the electoral register to turn the tide.
This does not guarantee remain would win a referendum this year. People change their minds, though polls show the main movement is toward remain. But once a ferocious campaign gets under way no one knows what might swing opinion.
Theresa May's red lines have led to deadlock. Jeremy Corbyn's red line is no no-deal Brexit. Logic suggests the only answer that will rescue both party leaders is a people's vote.
AR Parliament has passed all the legislation to ensure that if no further action is taken, the UK leaves the EU on 29 March. No deal is yet agreed. If the government calls a general election now, parliament is dissolved and no new legislation can prevent a no-deal Brexit.
Mass civil disobedience can head off that catastrophe. People can boycott an election to protest against the cynical motive behind it. They need only pledge in advance not to participate in any general election called to run down the clock.
Only a reset of the article 50 deadline followed by a people's vote can break the deadlock. Proclaim these facts and urge parliament to prevent a default Brexit.
2019 January 18
Star Wars II
The Times
President Trump has announced the biggest push for US missile defense since Ronald Reagan's "star wars" plan with a pledge to test space-based weapons to defend America and attack its enemies. He announced 20 new ground-based interceptors in Alaska to detect and destroy incoming missiles.
Trump: "We are ordering the finest weapons in the world, that you can be sure of. Our goal is simple: to detect and destroy any missile launched against the United States anywhere, anytime, any place. The best way to keep America safe is to keep America strong."
President Putin has also unveiled new strategic weapons, including a hypersonic glide vehicle that can fly at Mach 20 and make sharp turns to avoid interception.
Europe Needs Britain
Norbert Röttgen
For decades the European Union has brought our peoples closer together. It has forged a European identity and lays the foundation for peace in Europe. Within this fruitful environment our nations have equally prospered.
A Brexit with no deal puts all these achievements at risk. Germans are deeply committed to a close relationship with the UK. We want you to stay. This sentiment is broadly shared in Germany.
After this week's parliamentary vote against the Brexit deal, there seems to be no majority in the House of Commons for any way forward. With only two options on the table, a hard Brexit or Remain, the British people may want to hold a second vote. If they need more time, I cannot imagine the EU will disagree.
Europe needs Britain, especially in these troubled times.
Sir, Without your great nation, this continent would not be what it is today: a community defined by freedom and prosperity. After the horrors of the second world war, Britain did not give up on us. We are grateful.
Should Britain wish to leave the European Union for good, it will always have friends in Germany and Europe. But Britons should equally know that we believe that no choice is irreversible. Our door will always remain open: Europe is home.
We would miss Britain. We would miss the British people, our friends across the Channel. We would miss Britain as part of the EU, especially in these troubled times. We want them to stay.
Annegret Kramp-Karrenbauer, Norbert Röttgen, and 29 other leading figures in Germany
2019 January 17
The Malign Incompetence of the British Ruling Class
Pankaj Mishra
Britain made a calamitous exit from its Indian empire in 1947 when it left India partitioned. The British exit from the EU is proving to be another act of moral dereliction by British rulers.
In a grotesque irony, imperial borders imposed in 1921 on Ireland have proved to be the biggest stumbling block for the Brexiteers chasing imperial virility.
People in Ireland are aghast over the aggressive ignorance of English Brexiteers. Business people everywhere are outraged by their cavalier disregard for the economic consequences of Brexit.
The malign incompetence of the Brexiteers was precisely prefigured during the rushed British exit from India in 1947. Up to one million people died in the botched partition.
The mention of Winston Churchill stiffens the spines of many Brexiteers today. Churchill, a fanatical imperialist, worked harder than any British politician to thwart Indian independence.
The rolling calamity of Brexit threatens bloodshed in Ireland, secession in Scotland, and chaos in Dover. Ordinary British people stand to suffer from the exit wounds.
2019 January 16
Putin Smiles at US and UK
Stephen Collinson
The news just keeps on getting better for Russian leader Vladimir Putin.
The United States and Britain, the two great English-speaking democracies that led Moscow's defeat in the Cold War, are undergoing political breakdowns.
In London, Theresa May suffered the worst defeat by a prime minister in UK parliamentary history.
The United States remains locked in its longest-ever government shutdown, thanks to the chaotic presidency of Donald Trump.
London and Washington are suffering the effects of populist revolts that erupted in 2016 and are now slamming into legislatures and breeding chaos.
The result is that Britain and the United States are all but ungovernable on important questions that confront both nations.
Putin has made disrupting liberal democracies a core aim of his rule, as he seeks to avenge the fall of the Soviet empire.
AR Victory Ringlord
EU Responds to UK Vote
The Times
European Commission president Jean-Claude Juncker: "I take note with regret of the outcome of the vote in the House of Commons this evening. The risk of a disorderly withdrawal of the United Kingdom has increased with this evening's vote. I urge the United Kingdom to clarify its intentions as soon as possible. Time is almost up."
European Council president Donald Tusk: "If a deal is impossible, and no one wants no deal, then who will finally have the courage to say what the only positive solution is?"
"Die britische Politik ist nach wie vor nicht bereit, die Konsequenzen der Brexit-Entscheidung anzunehmen, und deshalb kann sie auch nicht wissen, was sie eigentlich von der EU will. Aus diesem Grund ist es auch wenig sinnvoll, von der restlichen EU zu verlangen, auf die Regierung in London zuzugehen."
Markus Becker
Brexit Betrayal
Daniel Finkelstein
Brexiteers like the feeling that they have been betrayed by the political establishment. Well, a deal has been negotiated that would allow us to leave and you, the Brexiteers, defeated it.
It is we who have been betrayed. Those who faithfully and diligently tried to make Brexit happen smoothly and on time, even though we had our doubts, have been left high and dry.
The Brexiteer rebels are now saying Brexit could mean leaving without any trade deal, breaking the Good Friday agreement, failing to settle financially with our continental allies, and departing without a transition arrangement.
If they really believe the majority of voters support this burn-it-all-down Brexit, let's have another referendum and ask the electorate.
AR Withdraw Article 50 or delay its effect to give time for a vote.
2019 January 15
Brexit Deal Defeated
BBC News, 1939 UTC
The government is defeated on its proposed Brexit deal by a majority of 230.
The result of the vote is 202 in favour and 432 against.
Brexit: Think and Act Anew
Caroline Lucas
The crisis at the heart of UK democracy can no longer be ignored.
Russell Brand: "People saw a bright red button that said Fuck Off Establishment, and they pressed it."
The 2016 referendum result should tell us that people want hope. The Remain campaign was seen primarily as defending the status quo, with the political elite pulling the strings. The campaign utterly failed to inspire any kind of love for the EU as something worth defending.
The EU is the greatest international venture for peace, prosperity, and freedom in history. That astonishing achievement ought to be front and centre of the Brexit conversation. So too the social and environmental protections, and the remarkable gift of free movement, and the friendships across borders, the cultural opportunities, the life without fear, and the solidarity. To have reduced all that to an argument about the cost of a trolley load of shopping was a tragedy.
Decisions made in the EU affect us every day, so let's talk about EU politics. The EU must dismantle the habitual domination of corporate power over the will of citizens. Such reforms are long overdue, and we should advocate changing the EU at the same time as fighting to stay part of it.
Brexit laid bare the extent to which UK governance structures are derelict. The Palace of Westminster, Gothic, rat-infested, and crumbling into the Thames, has become a symbol of political decay. Parliamentary sovereignty needs to be better rooted in the people.
We the people must lead the way.
Brexit: Anfang vom Ende Europas?
Klaus Raab
Gibt es ein zweites Brexit-Referendum? Ein Misstrauensvotum gegen May? Tatsächlich einen ungeregelten Brexit? Wie lange werden die Staus zwischen London und Dover? Fordert die AfD irgendwann ein Referendum über den Verbleib Deutschlands in der EU?
Londoner Politologe Anthony Glees: "Mein Land ist in tiefstem Chaos." Einen "Irrsinn" nannte er den Brexit und forderte zwar, die Entscheidung zu respektieren, befand aber: "Weil eine Mehrheit etwas wählt, bedeutet das nicht, dass es richtig war."
Europäischen Volkspartei Spitzenkandidat Manfred Weber beklagte, dass man sich ein wenig hängen gelassen fühle von Großbritannien, was die Vermittlung von Ideen zur künftigen Zusammenarbeit angehe.
Beatrix von Storch (AfD): Man selbst wolle ein Bündnis souveräner Nationalstaaten, in dem Entscheidungen nach einer Kosten-Nutzen-Kalkulation getroffen würden. Ein riesiges Chaos gebe es in Großbritannien nicht.
AR Beware of cost-benefit analysis: The costs of a political union cannot be quantified or its benefits predicted.
2019 January 14
The Times
UK prime minister Theresa May today spoke to workers at a factory in Stoke-on-Trent. She warned Brexiteers that staying in the EU is now more likely than their goal of leaving with no deal.
She said parliament has a duty to implement the result of the 2016 referendum:
"If a majority had backed remain, the UK would have continued as an EU member state. No doubt the disagreements would have continued too, but the vast majority of people would have had no truck with an argument that we should leave the EU in spite of a vote to remain or that we should return to the question in another referendum."
AR I begin to sense a retreat once St Theresa has endured her Commons crucifixion tomorrow.
A Space Elevator
Kelly Oakes
The future of space travel may be an elevator shaft that can transport people and equipment directly into low Earth orbit. A Japanese corporation aims to develop a space elevator by 2050, and China has its sights set on building one by 2045.
The sky lobby of a space elevator needs to orbit at the same speed as Earth rotates, with its center of mass some 36 Mm over our heads in geostationary orbit (GSO). A successful space elevator will need to stretch much further, with a tether nearly 100 Mm long, extending well beyond GSO, and strong enough to lift satellites and astronauts into space.
Keeping the base of such an enormous structure safe, engineers suggest placing the Earth end of the tether near the equator, where hurricanes almost never form. To avoid collisions with space junk, it should ideally be anchored on a mobile ocean platform.
Researchers in Japan have sent a miniature space elevator to the International Space Station for release into orbit. Two 10 cm satellites tied together by an 11 m steel cable are passing a tiny climber back and forth while researchers monitor the system.
To install a full-size cable, put a satellite in orbit, carrying many Mm of wire, and have it dangle the wire down to Earth. Send a small mechanical climber up that wire carrying a second ribbon. Repeat this process with ever bigger climbers, adding more and more ribbons, to weave a sturdy cable leading from the ground to the orbiting cluster of climbers acting as a counterweight to keep the elevator's center of mass up in GSO.
Because all the mass below GSO pulls down on the elevator shaft, and all that above pulls it up, the tether will be in great tension. To hold it all together, we need a strong material. The top candidates are carbon nanotubes and graphene sheets.
International Space Elevator Consortium president Peter Swan: "When you can provide scalable, inexpensive and reliable access to space, capabilities emerge that will benefit those on Earth."
AR I guess this will take longer than its enthusiasts expect.
Elon Musk / SpaceX
Elon Musk reveals assembled Starship hopper prototype for suborbital VTOL tests
Universe Today
Starshot light sail
"I'm not saying I'm a Trump fan,
I'm just saying, it's bad in
America, but it's a thousand
times worse in Britain."
Jeremy Clarkson
Me Me Me
British army fights
to attract recruits
Jade Rabbit 2
Quantum error correction
for beginners
M.C. Escher
M.C. Escher
Fish in a tiling illustrate
hyperbolic geometry
EU vs Russia
Norbert Röttgen
Putin has made destabilization
of other countries the guiding
principle of his foreign policy
2019 January 13
Brexit: Case For People's Vote
Lord Kerr of Kinlochard
As the Brexit clock ticks on, the case for a people's vote grows stronger:
A clear majority of MPs opposes the government's proposed withdrawal deal. The government
is now braced for a heavy defeat on Tuesday evening.
Crashing out in March with nothing agreed is not the only option if the Commons votes down
the deal. MPs can block a no-deal Brexit.
Parliament can take back control. The government will have to bring forward a new plan
within three days. A people's vote amendment can be tabled.
The ECJ confirms that the UK has the absolute right to stop the Article 50 process, withdraw
its notice, and remain in the EU.
UK polls now show a consistent 8% lead in favour of staying in the EU. The lead is 16 to 26 points
for a choice between staying and either the government's deal or no deal.
Stop and Think
Sir John Major
Crashing out of the EU without a deal would be deeply harmful for the UK. Every single household would be worse off for many years to come. The only sensible course now is for the government to revoke article 50 and suspend any decision on departure.
2019 January 12
Trump Endgame
Alan J. Steinberg
Trump will not be removed from office by the constitutional impeachment process. Instead, he will use his presidency as a bargaining chip with federal and state authorities in 2019, agreeing to leave office in exchange for their not pursuing criminal charges against him.
Aside from all the legal nightmares facing Trump and his presidency, it appears virtually impossible for Trump to be reelected in 2020. The economy appears headed for a severe recession, as evidenced by the recent plunge in the stock market.
With his approval ratings in an abysmal state, and the forthcoming recession making it near impossible for Trump to stage a political recovery, it appears most likely that he will use the continuation of his presidency as a bargaining chip.
Brexit Endgame
Two of the biggest donors to the Brexit campaign say they now expect the UK to stay in the EU despite their campaign victory in the 2016 referendum.
Hedge fund manager Crispin Odey does not believe there will be a second referendum or that Brexit will happen either. He is now positioning for the pound to strengthen from $1.27 to about $1.34.
Billionaire Peter Hargreaves says the political establishment is determined to scuttle Brexit, first by asking the EU for an extension to the exit process and then by calling for a second referendum.
Beauty Shapes Evolution
Ferris Jabr
Numerous species have conspicuous, costly, and burdensome sexual ornaments. To reconcile such splendid beauty with a utilitarian view of evolution, biologists have favored the idea that beauty in the animal kingdom is an indicator of health, intelligence, and survival skills.
Charles Darwin proposed that ornaments evolved through sexual selection. Females choose the most appealing males according to their standard of beauty. In this way, beauty can be the glorious flowering of arbitrary preference, nothing to do with survival.
Two environments govern the evolution of sentient creatures: an external one, which they inhabit, and an internal one, which they construct. To understand evolution, we must uncover the hidden links between those two worlds.
Richard Prum says animals are agents in their own evolution. Dinosaurs originally evolved feathers, he says, because they found them beautiful. Birds transformed them into enviable adaptations, but they never abandoned their sense of style.
Environment constrains anatomy, which determines how a creature experiences the world, which generates adaptive and arbitrary preferences, which loop back to alter its biology, sometimes in maladaptive ways. In humans, many types of physical beauty and sexual desire have arbitrarily co-evolved without reference to health or fertility.
Flowers expose the futility of trying to contain beauty in a single theoretical framework. Early pollen-producing plants depended on the wind to spread their pollen and reproduce. But certain insects began to eat those pollen grains, inadvertently transporting them from one plant to another. Through a long process of co-evolution, plants and pollinators formed increasingly specific relationships, driving each other toward aesthetic and adaptive extremes.
Beauty evolves in a dialogue between perceiver and perceived.
AR Beauty is epiphenomenal to evolution by natural selection rather as consciousness is epiphenomenal to cognitive processing in a cerebral neuronet.
2019 January 11
Brexit: World Is Watching
Shinzo Abe
The world is watching the UK as it exits the EU.
Japan and the UK have been building a very strong partnership. For Japan, the UK is the gateway to the European market. It is the strong will of Japan to further develop this strong partnership.
We hope a no-deal Brexit will be avoided.
Brexit: Delay Withdrawal
James Blitz
When Theresa May puts her Brexit deal to the Commons next Tuesday she will face a major defeat. She must then return to the Commons within three days with a proposed Plan B. MPs can amend her new motion and debate alternative Brexit plans.
Eurasia Group managing director Mujtaba Rahman: "A three-month extension to Article 50 would now make sense."
Black Hole Stability Conjecture Solved
Daily Mail
Imperial College London physicist Gustav Holzegel has won the prestigious Blavatnik Award and a cash prize for calculating what happens when a black hole is perturbed. Solutions to the black hole stability conjecture had eluded physicists for decades.
From his award citation: "Professor Holzegel has pushed the frontiers of our understanding of the universe as outlined by the general relativity theory."
Holzegel proves that a perturbed black hole will settle down into a stable form. He proved a version of the conjecture for spherical black holes in 2016. His new work applies to any kind of black hole.
How Machines Think
Been Kim
Interpretability can mean studying a neural network with scientific experiments to understand the details about the model, how it reacts, and that sort of thing.
Interpretability for responsible AI means understanding just enough to safely use the tool. We can create that understanding by confirming that useful human knowledge is reflected in the tool.
Doctors using a machine-learning model for diagnosis will want to know that their own diagnostic knowledge is reflected in the model. A machine-learning model that pays attention to these factors is more understandable, because it reflects the knowledge of the doctors.
Prior to this, interpretability methods only explained what neural networks were doing in terms of input features. If you have an image, every single pixel is an input feature. But humans communicate with concepts.
Testing with Concept Activation Vectors (TCAV) does sensitivity testing. When you add a new diagnostic concept, you can output the probability of a certain prediction as a number between 0 and 1, its TCAV score. If the probability increases, it is an important concept to the model.
To validate the concept, a statistical testing procedure rejects the concept vector if it has the same effect on the model as a random vector. If your concept fails this test, the TCAV will tell you the concept looks unimportant to the model.
Humans are gullible. It is easy to fool a person into trusting something. The goal of interpretability for machine learning is to resist this, to tell you if a system is not safe to use.
Inherently interpretable models reflect how humans reason. Having humans in the loop, and enabling the conversation between machines and humans, is the crux of interpretability.
2019 January 10
Extraterrestrial Visitors
Avi Loeb
In October 2017, the PanSTARRS telescope in Hawaii detected an object moving so fast it must have come from beyond our solar system, making it the first visitor from outer space that we know of. We called it Oumuamua.
It was quite a mystery. Its brightness changes dramatically, suggesting a very strange shape. A sphere would always reflect the same amount of sunlight. Only a disk or a cigar-shaped body would flicker while rotating.
The more we found out about Oumuamua, the weirder it got. Its orbit differs significantly from an orbit shaped by solar gravitation. Some additional force was acting on it. If it was a comet, it might have emitted gases while flying past the Sun, but no comet tail was observed. And its rotation should have changed during outgassing, but this effect was not observed either.
The other force acting on Oumuamua is the pressure of the sunlight. Solar radiation could only have a visible effect if it is a very thin object, less than 1 mm thick. If Oumuamua is a randomly wandering object, every solar system would have to produce millions of billions of such objects. If it's not random, it could be a targeted mission, an artificial product, a light sail made by intelligent beings.
The Breakthrough Starshot project started when Yuri Milner asked me in 2015 if I would be willing to lead a project to send a probe to Proxima Centauri, 4 light years away from Earth. To get there within our lifetime, it would have to travel at 0.2c.
A light sail, accelerated by a powerful laser from Earth, seemed to be the only feasible way. The idea is to accelerate the probe with a 100 GW laser beam for a few minutes. With many small infrared lasers across an area of 100 ha, we could focus the beam up to about five times the distance to the Moon and accelerate the probe to 0.2c.
We plan a payload of about 1 g. The probe needs a camera and a navigation and communication device. And we need a sail material that almost completely reflects the incoming laser light. To receive the radio signal the probe sends back, we will need a big radio telescope on Earth. We can send a lot of probes into space once the launch system is constructed, because the expensive part is the infrastructure of the laser beam. The probes themselves will be relatively cheap.
Oumuamua is moving too fast for a rocket, but with a laser-driven light sail we could catch up with it. We only became aware of it after it had already left, but we will study the next visitor earlier and more thoroughly.
Crystal Balls
White dwarfs are stellar embers depleted of nuclear energy sources that cool over billions of years. These stars are supported by electron degeneracy pressure and reach core densities of 10 Gg/l.
A phase transition occurs during white dwarf cooling, leading to the crystalization of the non-degenerate C and O ions in the core. This releases latent heat and delays the cooling process by about a billion years. The cooling is further slowed by the liberation of gravitational energy from element sedimentation in the core.
Pier-Emmanuel Tremblay
2019 January 9
Trump: US Governance Crisis
The New York Times
President Trump is painfully out of his element. Two years in, he remains ill suited to the work of leading the nation. Governance clearly bores him, as do policy details both foreign and domestic. He has proved a poor judge of talent. He prefers grandstanding to negotiating, and he continues to have trouble with the whole concept of checks and balances.
Most of the electorate has grown weary of his outrages and antics.
Brexit: UK Government Failure
The Guardian
Brexit is not a policy prescription for UK problems. For many Tories, it is an attitude of mind, an amorphous resentment against the modern world.
There has been a collective floundering across the political spectrum. Britain is living through a period of national democratic failure. Badly framed referendums are a crude way of making democratic decisions and empower those who shout loudest.
Brexit has exposed the decrepit nature of UK constitutional arrangements and UK politics. Parliamentary sovereignty needs to be better rooted in the people. Other forms of debate are essential buttresses of the parliamentary process.
Britain should pause the article 50 process and put Brexit on hold. The government has failed, so we must go back to the people.
Brexit: Plan B in 3 Days
BBC News, 1436 UTC
MPs have voted to force the government to return to parliament with a plan B within three days if the withdrawal deal is rejected next week, by 308 votes to 297.
Brexit: End of Days Scenario
Daily Mail
Government calls a general election for April 4, parliament goes into recess, cutting off further debate or votes, and no-deal Brexit happens on March 29 while politicians are out campaigning.
AR The fixed-term parliaments act ensures that this can only happen after the house passes a vote of no confidence in the government, which is already an extreme scenario.
2019 January 8
Brexit: No No Deal
BBC News, 1921 UTC
MPs have backed an amendment to the Finance Bill that limits spending on no-deal Brexit preparations unless authorised by parliament, by 303 to 296 votes.
AR Relief — I trust this means no-deal Brexit is off the menu.
Brexit: The Uncivil War
The Times
James Graham's TV drama was rollickingly good entertainment, but it wasn't really the story of the Leave and Remain campaigns. It was the story of Dominic Cummings, brilliantly done by Benedict Cumberbatch.
Cummings was shown running rings intellectually around MPs and old-guard Brexiteers, basically delivering the Leave victory through vision and data mining, putting that £350 million for the NHS claim on the side of the bus, and devising the "Take Back Control" slogan.
Credit must go to Graham for making something about Brexit enjoyable. A final verdict from Cummings resonated: "It's all gone crap."
AR The drama did well to catch the way Cummings raised demons with the immigration issue to demolish a weak and complacent Remain campaign.
2019 January 7
Europe of Nations
Right-wing populists seek a Europe of Nations. In May 2019, they could gain enough seats in the European Parliament to turn back the clock on European integration.
Italian Lega party member and interior minister Matteo Salvini: "There are people who have betrayed the European dream. But we will give our blood and veins for a new Europe."
In Austria, FPÖ member and vice chancellor Heinz-Christian Strache cultivates relations with fellow populists across Europe: "We can save Europe."
Russian president Vladimir Putin's United Russia party has a cooperation agreement with the FPÖ and with right-wing parties in neighboring countries, such as Hungarian prime minister Viktor Orbán's Fidesz party.
There are two right-wing populist groups in the European Parliament, the Europe of Freedom and Direct Democracy (EFDD) and the Europe of Nations and Freedom (ENF), plus the more moderate groups European People's Party (EPP) and European Conservatives and Reformists (ECR).
Salvini wants to see refugees distributed more fairly across Europe. But the Polish Law and Justice (PiS) party is adamantly opposed, as is Orbán in Hungary. The European right wing is also divided on relations with Russia.
The Alternative for Germany (AfD) includes members who question the legitimacy of the German-Polish border. German parliament AfD co-leader Alice Weidel is critical of Italy: "Germany cannot become Italy's paymaster!"
Orbán stigmatizes refugees in alarmist terms. His government built a razor-wire border fence and demanded that the EU pay for it. He is fond of posing as the savior of the Christian West. His Fidesz party belongs to the EPP group in the European Parliament.
Hungary depends on EU subsidies adding up to €40 billion by the end of 2017. The same is true of Poland, led by the PiS party. Yet Poland and Hungary cause trouble in Brussels.
European nationalists say they are the people and will drive out the elites. From May, the ECR will lose British MEPs with Brexit, giving PiS a leading role in the group.
Lega, FPÖ, AfD, and Marine Le Pen's Rassemblement National could join the ENF group. Together they could make up over 20% of the European Parliament.
AR Without the calming presence of the UK, the EU could turn ugly.
The Times
The Brexit psychodrama goes on. It is difficult for outsiders to believe that the UK has put itself in this absurd and reckless position. Ireland must respond, because Brexit will affect it more severely than any other EU member.
Brexiteers have consistently failed to acknowledge that the UK departure from the EU will have negative consequences for other countries. Their ignorance of Irish history and of the special sensitivities of the border ought to shame them.
Jacob Rees-Mogg even had the effrontery last week to blame the Irish government for the bind in which the UK finds itself. The proper response is to reiterate why the backstop is an essential part of any withdrawal agreement.
Almost everybody on the island of Ireland, except the DUP and a cohort of other unionists, wishes the UK to remain in the EU.
AR Most other interested parties do too.
2019 January 6
China, Space Superpower
The Times
China is making a tremendous effort to become a space power.
Heritage Foundation China expert Dean Cheng: "In some ways the US has fallen behind China. You cannot be more glaring in your deficiency than having lost the ability to put a person into space. China is able to do this. Currently the US is not."
Space expert Namrata Goswami: "There has been a tendency to underplay Chinese achievements in space .. But they have shown that they are capable of doing very difficult feats away from Earth. That should be a wake-up call."
Goswami believes the Chinese have three long-term civilian goals in space:
To establish a permanent presence on the Moon
To use that presence as a base for deeper space exploration
To focus on asteroids as a possible source of raw materials
Cheng: "Where China is ahead is thinking about the military roles of space .. This week's achievement is the product of at least a decade of sustained funding and sustained programmatic stability."
British Imperial Nostalgia
Ishaan Tharoor
Imperial nostalgia shadows the push for Brexit. Brexiteers conjure visions of Britain restored to its former glory once free of the EU. Empire 2.0 will arise from new trade deals with Commonwealth countries.
The Sunday Times: "A people who within living memory governed a quarter of the world's land area and a fifth of its population is surely capable of governing itself without Brussels."
UK foreign secretary Jeremy Hunt: "Britain's post-Brexit role should be to act as an invisible chain linking together the democracies of the world, those countries which share our values and support our belief in free trade, the rule of law and open societies."
UK defense secretary Gavin Williamson: "This is our biggest moment as a nation since the end of the second world war, when we can recast ourselves in a different way. We can actually play the role on the world stage that the world expects us to play."
An anonymous Tory grandee: "We simply cannot allow the Irish to treat us like this. The Irish really should know their place."
A Good British Spanking
Nick Cohen
Margaret Thatcher cut government spending and whacked up interest rates during a recession. She defied the experts, who warned that her policies would deepen the recession, erode the UK industrial base, and threaten social and political stability.
Brexiteers tell voters a good spanking is worth it to restore national independence from Europe. Britain was never invaded by Hitler or Stalin, so they are careless about the risk of leaving the EU.
Solar Thermal Fuel
Researchers have developed a fluid that absorbs solar energy, holds it for months or years, and then releases it when needed. Such a solar thermal fuel can help replace fossil fuels.
As a pump cycles the fluid through transparent tubes, solar UV light excites its molecules into an energized state. The photons rearrange bonds among the C, H, and N atoms to convert norbornadiene into quadricyclane. Because the energy is trapped in strong chemical bonds, it is retained even when the fluid cools down.
To extract the stored energy, the activated fluid is passed over a cobalt-based catalyst. The quadricyclane molecules then transform back into norbornadiene. This heats the fluid from room temperature up to about 80 C.
The fluid and the catalyst are not consumed by the reactions, so the system can run in a closed loop. The researchers have run it though 125 cycles without significant degradation.
Calculations show the fluid can store up to 900 kJ/kg, about 2% as much as gasoline. The next step is to optimize its shelf life, energy density, and recyclability.
AR Thanks to Christian de Quincey for this.
2019 January 5
Quantum Spacetime
Natalie Wolchover
Qubits are superpositions of two states, |0⟩ and |1⟩. When qubits interact, their possible states become interdependent. The contingent possibilities proliferate exponentially as many qubits become more and more entangled in a quantum computation.
Physical qubits are error-prone. The tiniest disturbance causes them to undergo bit-flips that switch their chances of being |0⟩ and |1⟩ relative to the other qubits, or phase-flips that invert the relationship between their two states. For quantum computers to work, we need to protect the logic even when physical qubits get corrupted.
Quantum error-correcting codes exist, and they can theoretically push error rates close to zero. The best error-correcting codes can typically recover all of the encoded information from slightly more than half of the physical qubits.
There is evidence of a deep connection between quantum error correction and the nature of spacetime and gravity. We think spacetime and gravity somehow emerge from a quantum origin. This emergence works like a quantum error-correcting code, at least in anti-de Sitter (AdS) universes.
We live in a de Sitter universe with positive vacuum energy, but AdS space has negative vacuum energy, giving it a hyperbolic geometry. Spatial dimensions radiating away from a center gradually shrink down to the universe's outer boundary.
The interior of a 4D AdS spacetime is holographically dual to a quantum field theory on the 3D gravity-free boundary. Any point in the interior of AdS space could be constructed from slightly more than half of the boundary, just as in an optimal quantum error-correcting code. Such a code can be understood as a 2D hologram.
In 2015, Daniel Harlow, John Preskill, Fernando Pastawski, and Beni Yoshida (HaPPY) found another holographic code that captures more properties of AdS space. The code tiles space in 5-sided blocks, each one representing a single spacetime point. Think of the tiles as the fish in an Escher tiling.
In the HaPPY code and other holographic error-correcting schemes, everything in a region of the interior spacetime called the entanglement wedge can be reconstructed from qubits on an adjacent region of the boundary. Overlapping regions on the boundary have overlapping entanglement wedges, just as a logical qubit can be reconstructed from many different subsets of physical qubits.
Preskill: "It's really entanglement which is holding the space together. If you want to weave spacetime together out of little pieces, you have to entangle them in the right way. And the right way is to build a quantum error-correcting code."
Entanglement Weaves Spacetime
Jennifer Ouellette
Spacetime emerges from entangled nodes in a network. Entanglement is the thread that weaves the fabric of spacetime. Curved spacetime emerges naturally from entanglement in tensor networks via holography. Think of 2D program code for a virtual 3D game world. We live in the game space.
2019 January 4
Russia Chose Trump For Prez
Russia chose Donald Trump as the US presidential candidate who would be most advantageous to Moscow, and used online tactics to win him the presidency.
Former Mossad chief Tamir Pardo: "Officials in Moscow looked at the 2016 US presidential race and asked, 'Which candidate would we like to have sitting in the White House? Who will help us achieve our goals?' And they chose him. From that moment, they deployed a system [of bots] for the length of the elections, and ran him for president .. What we've seen so far with respect to bots and the distortion of information is just the tip of the iceberg. It is the greatest threat of recent years, and it threatens the basic values that we share."
The US Senate commissioned a report finding that Russia had used every social media tool available to influence the 2016 presidential election in favor of Trump. One described the Internet Research Agency (IRA), a Russian troll farm:
"Run like a sophisticated marketing agency in a centralized office environment, the IRA employed and trained over a thousand people to engage in round-the-clock influence operations, first targeting Ukrainian and Russian citizens, and then, well before the 2016 US election, Americans .. They reached 126 million people on Facebook, at least 20 million users on Instagram, 1.4 million users on Twitter, and uploaded over a thousand videos to YouTube."
AR I discuss the Russian role in my essay Victory.
Brexiteers Lied
Chris Patten
As 29 March 2019 gets closer, no deal is in sight that is acceptable to both Westminster and Brussels. The problem began before the 2016 referendum. The Leave campaign was rife with delusions and dishonesty.
UK prime minister Theresa May laid down red lines:
The UK will leave the EU and the single market and the customs union.
The UK will not accept any jurisdiction by the European Court of Justice.
The UK will end the freedom of European citizens to come to the UK.
The UK will not accept a hard border between Northern Ireland and Eire.
It is well-nigh impossible to negotiate an exit deal that meets them.
The Brexiteers lied. The costs of leaving the EU outweigh the benefits.
AR Putin nudged the UK toward Brexit with his support for Leave.
Britain Needs A Plan
Jochen Bittner
UK prime minister Theresa May is likely to fail with her proposed withdrawal agreement. As things stand, at 11:00:00 pm on 29 March 2019 the UK will be a part of the EU and at 11:00:01 it won't.
With no deal, trade relations between the UK and the EU revert to basic WTO rules. Neither side can treat the other more favorably than they treat other trade partners around the globe. A new customs regime would have to be installed. In the interim, confusion.
If May's agreement is voted down, parliament has to deal with the Brexit mess. Only Brextremists want a hard Brexit. MPs can vote to give the decision back to the people with a new referendum.
AR I explore the strategic cost of hard Brexit in my story Ringlord.
Turing Model Builds Patterns
In 1952, Alan Turing devised an elegant mathematical model of pattern formation. His theory outlined how endless varieties of stripes, spots, and scales could emerge from the interaction of two chemical agents, or morphogens. The mechanism lies behind the development of mammalian hair, the feathers of birds, and even the denticles that cover the skin of sharks.
Turing's reaction-diffusion mechanism requires two interacting agents, an activator and an inhibitor, that diffuse through tissue. The activator initiates a process and promotes its own production. The inhibitor halts both actions. The inhibitor spreads through tissue more quickly and prevents pockets of activation from spilling over. The pockets of activation can appear as patterns of dots or stripes.
A mathematical model of activator and inhibitor interactions can produce patterns that match those of developing shark skin, feathers, or hair. Reducing or blocking the expression of a gene and showing that the pattern disappears reveals which genes play roles in pattern production. Once such a pattern is set, other mechanisms form denticles, feathers, or other epithelial appendages.
AR Mathematics lets us explain the evolution of biological diversity.
Far side of Moon
Chang'e 4 lunar surface image, far side of Moon
Ultima Thule
New Horizons image of
Ultima Thule, a world
33 km long over
6 Tm away
2019 January 3
The Far Side of the Moon
China has successfully landed a rover on the far side of the Moon — a big first for its space program.
China National Space Administration (CNSA) landed the Chang'e 4 lunar probe at 0226 UTC today, in the South Pole−Aitken Basin. The rover made our first close-up image of the far side of the Moon.
This success is a landmark in human space exploration. Since the Chang'e 4 rover cannot communicate directly with ground control, China earlier launched a dedicated satellite orbiting the Moon to relay data from the rover to Earth.
The Chang'e 4 rover is 1.5 m long and about 1 m wide and tall, with two foldable solar panels and six wheels. It will conduct a number of tasks, including conducting a low-frequency radio astronomy experiment, observe whether plants will grow in the low-gravity environment, look for water or other resources, and study the interaction between solar winds and the lunar surface.
CNSA Lunar Exploration and Space Program Center deputy director Tongjie Liu:
"Since the far side of the Moon is shielded from electromagnetic interference from the Earth, it is an ideal place to research the space environment and solar bursts, and the probe can listen to the deeper reaches of the cosmos."
US Naval War College professor Joan Johnson-Freese:
"It is highly likely that with the success of Chang'e and the concurrent success of the human spaceflight Shenzhou program, the two programs will eventually be combined toward a Chinese human spaceflight program to the Moon."
NASA administrator Jim Bridenstine:
"This is a first for humanity and an impressive accomplishment!"
2019 January 2
Jovian Moon Io
Daily Mail
On 2018-12-21, NASA probe Juno cameras captured images of Jupiter's moon Io. When Juno was about 300 Mm from Io, JunoCam acquired three images of Io, all showing a volcanic plume near the terminator. JunoCam, the Stellar Reference Unit (SRU), the Jovian Infrared Auroral Mapper (JIRAM), and the Ultraviolet Imaging Spectrograph (UVS) observed Io for over an hour.
Juno principal investigator Scott Bolton: "We knew we were breaking new ground with a multi-spectral campaign to view Io's polar region, but no one expected we would get so lucky as to see an active volcanic plume shooting material off the moon's surface."
Financial Times
The decision in 2016 to renew the UK nuclear deterrent and build four new submarines at a cost of £31 billion could sink the UK defence budget. The total 2017/18 defence budget was £37 billion.
A funding gap of up to £15 billion in the MoD equipment program looms over the next decade. The gap stems mainly from Dreadnought, the nuclear deterrent submarine renewal program, described as "the ultimate guarantee" of UK security.
The cost of the nuclear deterrent includes maintaining the present fleet of V-boats, the Trident missile system, the British nuclear warheads, and building four new Dreadnought submarines. All this makes up a big fraction of the MoD budget.
BAE Systems is building the new submarines, which will be 153 m long and displace 17 200 tons. They will be powered by a new Rolls-Royce nuclear reactor, posing a big risk to the project, and will each have 12 Trident missile tubes.
Jane's Fighting Ships consultant editor Richard Scott: "You hear comments about the disproportionate impact the deterrent has on the overall defence budget."
AR Form a commission to review the strategic purpose of UK nuclear weapons.
2019 New Year's Day
Childhood's End
George Dyson
The digital revolution began when stored-program computers broke the distinction between numbers that mean things and numbers that do things. As computers proliferated, the humans providing instructions could no longer keep up.
The digital revolution has come full circle and the analog revolution has begun. To those seeking autonomy and control among machines, the domain of analog computing is the place to look. The next revolution is the assembly of digital components into analog computers.
Nature uses digital coding for processing sequences of nucleotides, but relies on analog coding and analog computing for intelligence and control. In analog computing, complexity resides in topology, not code. Analog computers embrace noise.
Analog computation can be implemented in solid state. Algorithms and digital simulations loom large in our culture but other forms of computation effectively control much of the world.
The successful social network is no longer a model of the social graph, it is the social graph. These new hybrid organizations are operating as analog computers on a global scale.
AR This is the global brain: Globorg
2018 Was Putin's Year
President Vladimir Putin of Russia had a good 2018.
In March, Putin sailed to a re-election victory, winning a fourth term by a handsome margin. The Kremlin even secured high turnout numbers to claim a broad mandate.
In June and July, Russia hosted the 2018 FIFA World Cup, a resounding triumph for Putin. More than 3 million fans attended matches in 12 stadiums across Russia. FIFA praised the Russia tournament as the best World Cup to date.
The games allowed Russia to show its side after years of political isolation and confrontation with the West following the 2014 annexation of Crimea and the imposition of economic sanctions.
Russia is still locked in a confrontation with the United States and the West. Its economy lost years of growth in the wake of the 2014 annexation of Crimea. New sanctions could further cripple the economy.
In September, Putin said the two suspects named by UK authorities over the poisoning of former Russian double agent Sergei Skripal and his daughter in Salisbury were not criminals and suggested the pair come forward. They did, and an investigative website revealed their real identities.
The World Cup provided a welcome respite to bad headlines. Visitors saw a country that could roll out the red carpet.
Putin: "Millions of people have changed their views on Russia. It is a big achievement."
AR Victory
Imaging a Black Hole
New Scientist
The Event Horizon Telescope (EHT) has made its first observations of the supermassive black hole at the center of our galaxy. Nine radio observatories around the world, including four in America and one in Antarctica, make up the EHT. Together they make a virtual telescope spanning the planet.
In April 2017, the EHT looked at two supermassive black holes: Sagittarius A* at the center of the Milky Way and the much more massive black hole at the center of galaxy M87. The images will be the first proof that black hole event horizons exist.
The observations could also help us formulate a theory of quantum gravity.
AR This could be good.
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c76780b71c57edbb | I'm having troubles with the assertion "(normalizable) wave-functions constitutes (projective) Hilbert space".
The standard argument I find for this seems to go as following: say $\Psi(\vec{x},t)$ is a quantum state of a spinless particle. Born rule asserts that for a given $\vec{x}_0, t_0$, the number $|\Psi(\vec{x}_0, t_0)|^2$ should be interpreted as the probability of detecting the particle in position $\vec{x}_0$ at time $t_0$. So $\Psi$ induces a probability density, and we should have $\int_V{|\Psi(\vec{x},t_0)|^2}dV=1$. So we're dealing with square integrable functions (w.r.t. to the Lebesgue measure over $R^3$), that form a Hilbert space (with the inner product $\langle\Psi_1,\Psi_2\rangle=\int_V{\Psi_1^*\Psi_2}dV$).
But... all that was for a fixed $t_0$. Actually, the result of an "inner product" $\langle\Psi_1(\vec{x},t),\Psi_2(\vec{x},t)\rangle$ is a function $t\mapsto z\in C$. In what sense do the "full" quantum states ($\Psi(\vec{x},t)$, as a functions of both space and time), form a Hilbert space?
It doesn't seem to me that considering direct-sums $\oplus_{t\ge0}\mathscr{H}_t$ or tensor-products $\otimes_{t\ge0}\mathscr{H}_t$ (where $\mathscr{H}_t$ is the Hilbert space of all the possible quantum states at time $t$) leads anywhere (the temporal restriction imposed over $\Psi$ by the Schrödinger equation complicates things, and the inner-products make no physical-sense to me in this context).
The approach I find somewhat sensible, is to rely on the fact that the said function $t\mapsto z\in C$ is actually constant (I think so at least, based on the continuity equation for probability currents), so we can construct an inner product by canonically assigning this constant complex value to $\langle\Psi_1(\vec{x},t),\Psi_2(\vec{x},t)\rangle$). But I couldn't find a hint for such construction anywhere, which leads me to the conclusion that I'm missing something very basic somewhere. What am I missing? What is, explicitly, the Hilbert space to which quantum states belong?
• $\begingroup$ Have a look at this answer of mine. You don't need a different Hilbert space for every time step, you give the time evolution in the Schrödinger picture as a map $\mathbb{R}\to\mathcal{H}$ into the space of time-independent states. If you wish, the space of time-dependent states is then $C^\infty(\mathbb{R},H)$. $\endgroup$ – ACuriousMind Oct 15 '15 at 11:08
The time evolution isn't random, it is unitary in the already given inner product.
So that time dependent function is already constant in the original inner product.
As an example, the Schrödinger equation evolution is unitary.
A Hilbert Space is different than a projective Hilbert space. You can start with $\mathbb C^k$ or with functions from $\mathbb R^{3n}$ into the tensor product of the individual single particle spin states. Then put an inner product on the spin states and then restrict to functions that are square integrable in the sense that the inner product on the slin states is a probability density on the configuration space $\mathbb R^{3n}$. That's the Hilbert space (equivalence classes of functions whose difference has L2 norm of zero).
OK, then you make a projective Hilbert space by doing one more quotient, first, remove zero, then say that two elements are in the same equivalence class if one is a nonzero complex scalar multiple of the other. And that set of equivalence class is the projective Hilbert Space.
Your Answer
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6b3ce92119d1a6a9 | Sunday Scribbles #31 - Bedtime Stories
Photobucket - Video and Image HostingOnce upon a time, in a Land filled with instant gratification, there lived an Urban Princess. The Princess was due at the Ball the next morning, but the excitement of it left her mind racing at night.
Each evening, the Princess lay her petite head upon the pillow, yet sleep eluded her.
Surely, she thought, I have been hexed by that powerful witch of a Department Manager.
Surely, she lamented, it is the fault of the President, who wages war for barrels of oil.
Surely, her mind cried out in the darkened room, I can not sleep because it is a conspiracy of the Liberal Media and Dick Cheney to keep me awake for days on end so that my blurry eyes can not guide my fair hand on Election Day.
(In this Land, it was right and proper to blame everyone else for your shortcomings.)
As she gazed at her popcorn ceiling, vowing to have contractors come out and replaster it sometime before next Summer, a gentle glow filled the room. A large, near translucent green butterfly had gently floated in through her open window while she had been contemplating. It fluttered above her head, sweetly humming a tune.
"My, but aren't you a gaudy, CG-gauzy thing?" she cried, reaching for her slipper.
"Wait!" the butterfly pleaded, ducking behind a half-finished glass of rum on the rocks. "I am magical!"
The Princess gritted her teeth and calculated how much force would be needed in order to squash the insignificant bug without damaging her Ethan Allen Tuscany nightstand. "Magic, my ass!"
"Fair Urban Princess, it is true!" it said, skittering away from the glass and hiding behind an empty package of Unisom. "I can give you and your restless mind the sleep you need. So you can finally enjoy a restful night and a fresh start! So from the time your head hits the pillow until the second your alarm clock sounds, you're getting the peaceful sleep you need."
"Hmmm?" the Princess questioned, a bit intrigued.
"I am designed to give you a restful night's sleep. It not only helps most people fall asleep quickly, I help you stay asleep all night long with fewer interruptions and you will wake up refreshed. I will not lose my effectiveness over time as shown in a 6-month study. Additionally, I am approved for long-term use. That is what makes me unique."
The Princess scratched her chin and contemplated the fact that she was having a conversation with a green-glowing CG-animated butterfly.
"You can feel quite good about taking me," the butterfly added, sensing the close of a sale. "When you're about to go to bed, simply swallow me with a bit of water and get ready to enjoy a restful night's sleep."
"Oh, alright," the Princess said, grateful to have a bit of magic to usher her into slumber.
"Important Safety Information!" babbled the Butterfly suddenly, so quickly that it nearly sounded as though he were an Auctioneer in his larval stage,
"I should only be taken immediately before bedtime. Be sure you have at least eight hours to devote to sleep before becoming active. You should not engage in any activity after taking me that requires complete alertness, such as driving a car or operating machinery. You should use extreme care when engaging in these activities the morning after taking me. Do not use alcohol while taking any sleep medicine. Most sleep medicines carry some risk of dependency. Do not use sleep medicines for extended periods without first talking to your doctor. Side effects may include unpleasant taste, headache, drowsiness and dizziness."
"Oh, alright!" moaned the Princess, too eager to capture a dream to bother with words that seemed as if spoken in fine print. "Shut up already and get in my mouth."
She quickly fell asleep moments later, and awoke feeling refreshed the next day. She dressed for the ball, jumped into her Mercedes ML320, and drove along the highway - only to realize that she was still partially asleep as her SUV kissed the guardrail and plummeted off a steep cliff.
The moral of this story: there is no magic pill that will solve all your problems. Treat the condition; do not simply medicate the symptom.
The End.
Read about other Bedtime Stories: Sunday Scribblings: #31
Pharmaceuticals and Caribou
Take that, GSK!
Aut? What are you on?
Friday the 13th: the Muse Roams Blogland.
Happy Friday the 13th to all of you. Friday is my blogging day, where I can sit back and read catch up on everyone's week. I thought it would be nice to share some of my faviortes with you today:
I have just come back from Roadchick's
Roadtrip, where I LMAO over her spewing pumpkin and Friday the 13th Follies. I'm down to the granny panties, but not desperate enough to do the Man Solution and turn things inside out (after first conducting a Sniff Test), at least as far as undergarments go. Better Half has brought only a few items up from the Dungeon Laundry Cell, so by my visual perspective of the upstairs closet, it would appear sniff tests might come into play this weekend. I'm certain (hoping, praying!) that there is more clean laundry downstairs.
Stopping by
To Love, Honor and Dismay, I saw an interesting article concerning "How not to ask your husband for help." Thank God for Better Half. I may have to motivate him from time to time, but he is not ashamed of running a vacuum or using cleaning liquids. I did crack up over Dr. Andrew's interview on Basil's Blog.
Paris Parfait has delighted my sense once again, and her photograph of La Giralda Cathedral takes my breath away!
I was shocked to hear that someone I knew passed away this week. Although JerryALT and I didn't chat often, I loved his insight into the Jewish faith. David Shelton gave a wonderful
tribute to him in his blog. David has ben working hard on his new book, The Rainbow Kingdom, and it is now available for preorders.
On that note, Michael sent me an autographed copy of his latest work, and I am working on a review of it for Amazon. If I can just get it completed, I will offer a copy to B&Noble. Michael's book can be found at your local book store, or you can order it
here. I will publish my review here once it is completed. Michael also mailed me a copy of his article, Kidnapped in Iraq, which is the story of peace activist James Loney and his partner, Dan. This atricle was published in the August 29, 2006 edition of The Advocate.
Lori~Flower had me grinning as she shared that mother's never cease to mother, even after we have left our roaring twenties. My mother would have done the same. Actually, now that I think about it, Mum never fails to give good advice at least once per phone call.
The Benedict Notes, by AnnieElf, set my heart soaring - the return of the Latin Mass. It's about time! I am probably one of the few people who really enjoys Latin, and the greatest beauty is hearing an entire Mass in that tongue. Many Americans will scorn it, to be certain, but they hardly have a say in it, especially as most of them don't bother to even learn what the Mass is about. They sit quietly and recite prayers and have no clue as to what they are supposed to be thinking as they pray. It is my opinion that the average American Catholic is a pod. Can you tell that I have never been a Vatican II fan? Annie's other blog has a lovely haiku about a blackbird, which I promptly printed.
Darren Naish had an interview on the BBC news, which I missed. He also explores the controversial origins of the family dog, and our views are very similar on that subject.
Finally (for today at least) I ended my reading with
Sunday Scribblings. This week's theme is #29 - If I could stop time. . . and I encourage all of you to check out their blog and post your own entry.
My Apologies
I'm sorry I have not made much of an effort to keep up my blog this week. I will probably lose a few readers over that.
It has been a bad month for me physically. The month itself, and the environment around me, is epically beautiful however, and I managed to get a few pictures of the season before the leaves are snatched from the trees by the hands on winter. I'll try to get them posted tomorrow.
As I write this, the first snowflakes are falling outside!
Other than that, I have not been able to do much. My apologies to you.
A Catfish Paprika Recipe Worthy of George Totin
What does one do with 4 pounds of fresh catfish? The packets were on sale yesterday, and for roughly $3, we would have the makings of a goof fish fry. We would have, that is, if Better Half wasn't too sore from doing yard work yesterday.
There are many things I can do well, but frying fish is not one of them. Better Half, the Southern Boy, can fry anything to perfection. He surely channels Paula Dean and Bobby Flay, if not Emeril. I woke up this morning anticipating spicy fried catfish and a side of winter squash.
Better Half woke up anticipating going back to bed.
What does one do with 4 pounds of fresh catfish? They improvise.
A family favorite in this house (handed down from my Father) is Chicken Paprika. It's a Hungarian dish, heavy on the paprika and sour cream; a true comfort food. I used to make this traditional dish every Father's Day, but since moving away from my parents, I have not bothered to whip up a batch (with the exception of their visit out here this past summer.) It is not that the recipe is difficult, or the ingredients too hard to come by. It simply reminds me of my Dad, and to make chicken paprika is to admit that it saddens me as he is not here in person to enjoy it with us. I have made the dish with chicken, beef and pork... and my mind pondered the possibilities of catfish. Would it work? Would it taste terrible? Could I add the squash to it? Oh, what the hell! Let's go.
Catfish Paprika Recipe
4 pounds fresh catfish, 1" cubes
1 yellow onion, diced small
1 Patty Pan squash, diced small
1 Tablespoon butter
Salt, to taste
Pepper, to taste (we use 1 tablespoon)
Paprika, ground, to taste (we use a whopping 1/8 cup or more!)
1 can low fat, no MSG chicken broth
1 cup sour cream
1 cup shell noodles, cooked
Melt butter on med-hi heat in a large skillet, then add diced onions, salt, pepper, and 1/2 the paprika. Cook until translucent. Add squash and fry for about 2 minutes, or until it begins to become tender. If things get too dry, you can add a tad more butter.
Add catfish to the pan, and stir fry for a few minutes, then add chicken broth and remaining paprika. Cook until catfish is done. Lower heat and add sour cream, a bit at a time, working it into the broth mixture. If you would like the sauce to be more hardy, you can thicken it with flour. Once sour cream is combined, add noodles. Your completed dish should have a medium pale orange color to it.
Enjoy! Share a hardy pot with friends and family.
I tired it... and I like it! The fish doesn't overpower, and the flavor blends well with the paprika and sour cream. Normally, this recipe would be done without squash (and you can substitute pork or chicken.) However, the squash added a bit of harvest aroma to it.
So here's to you, Dad! Wish you were here to savor Catfish Papikosh with us!
Sunday Scribblings #28 - An Assignment
This Sunday Scribblings was a tough one, as I'm sort of homebound this week - thanks to my crappy body.
As I can not write about any people that I observe (and writing about Better Half becomes too mundane for some of my readers), I'll draw you into Bold's world.
The Autumn air has chilled, and the leaves prepare to slip their bonds. The sun dances through them, and the canopy of the tree becomes a stained glass church. It is here that Bold dwells, the summer and autumn of this, his first year, a true test of his stock.
Bold is a rugged thing, a burley thick-bodied American Tree Sparrow (Spizela arborea), masquerading among the Chipping sparrows. His red cap is eternally tussled, bits of feather sticking up at odd angles as he pecks frantically at the harvest seed in the hanging feeder. When I first laid eyes on him, earlier this year, I thought him perhaps sickly, as no healthy bird would run about with such a poorly preened coating. Yet he remained, steadfast against all odds, the mutant Tree-Chipping Sparrow skulking amidst his beautiful cousins. He never offered a humble chirp, but always chose to announce his presence with a rather throaty CHURP, accompanied by the strangest dancing displays. After closer observation, I believe he was either lost, or else the two different species mated to produce him. If 'o' is a typical Chipping sparrow, then Bold waddles in with 'O'... larger, rounder, louder, and much much bolder. He drives even the largest of Ravens from his territory.
For some weeks, I have lost track of our house wens, cardinals, and chipping sparrows. I have not heard the haunting cry of our mourning doves in quite a while. I have not been able to sit on my front porch to enjoy their community as it draws together each day in celebration of bountiful food and water. I have keep my eyes opened, hoping for some small sign that my freakish little bird was still about. Bold was my companion, and my inspiration to keep fighting, no matter how heavily stacked against me the odds are.
Yesterday graced us with a heavy downpour of rain. Better Half was about to start the mower, and I had patted my hair into place and had ventured outside to keep him company. The rains began almost immediately. I grabbed the container of seed, urging Better Half to at least get that feast set up for our friends, and then tore open the seed cake packets for the mesh feeders. A flash of lightening sent a few lurking Chipping sparrows racing for the protection of the canopy of our large tree... and in that flash, I saw Bold.
He stood firm in the tree, his head cocked to one side as he waited for Better Half to resupply the hanging feeder. Rain and thunder be damned, for Mother Nature herself would not drive him from his perch. I shouted to Better Half and pointed, crying "Oh look, there's Bold" as the poor man did his best to get food in place while being drenched by the storm. Unfortunately, Better Half could not see Bold through his rain streaked glasses.
Bold has changed in the past few weeks. He is even larger, and more bedraggled in feather. He is every bit as lively, however, and offered a singular CHURP in gratitude for the free meal. His is an unquenchable spirit; each passing day means another chance to profit from the last moments of summer. His robust form skittered from twig to twig, and he regarded me momentarily before ducking out of sight behind a particularly large clump of leaves. I have never been able to capture him on film, yet his enthusiasm for life is etched upon my heart.
I struggled with insomnia until early this morning, . I had left the bedroom window open, preferring the feel of the crisp night air. Better Half let the dogs out around 7 am, and closed the door, allowing me the luxury of a warm bed sans any dogs, cats or other humans. In the still morning air, I heard a particularly pleasing CHURP, and lifted the blinds just so. I lay quietly, allowing the early sunshine to warm the air, and my eyes gazed into the depths of the maple tree. The CHURP came again, and I spotted Bold on a limb. He shook himself, and glistening droplets flung out from his plumage. He cocked his head and stared at me from one gleaming black eye, and then ducked his head under his wing to nibble at some small itch. When his head emerged, his cap was just as scruffy as ever, and I smiled silently and thought of how tussled my own hair must look.
What was he thinking at that moment? Was he already dwelling upon the bird feeder below, or perhaps he was testing his resolve to migrate to some distant place? I do not know, in honestly, but it seemed to me that he had come up to the very top of the tree just to check in on me, for he stayed quite a while. I closed my eyes and lay back, warmed by the occasional song he offered in lullaby.
The sun climbed higher, and the bright light dragged me from my groggy state. I got up quietly and began to shut the blind - and there Bold sat, still in the same place. I whispered "Good morning, Bold" and offered him a nod. He scratched his head lazily with his leg, tearing several spent downy feathers from his neck and chest in the process, and then gave a final CHURP in return. It was the bit of peace that I needed, and it ushered me into a deep sleep.
As I write this, I hear a familiar song creeping in through the cracked window in the office. My heart soars.
On Entropy, the Arrow of Time, and Anthropic bias
I am going to digress from my usual rambling to allow you a brief snapshot into what Better Half and I do while driving: we communicate. Talking is a lost art to many people. It is more than a method of conveying needs; it is the prime method whereupon we can convey thoughts, theory, and philosophical ideas. To dialog, to communicate what seems incommunicable, is divine.
This entire topic began when I purchased a cheap watch. I have owned many in my life, but seldom wear one. I tend to exist outside the ideals of the space-time continuum, as I ignore time as a dimension.
(In physics, spacetime is a mathematical model that combines three-dimensional space and one-dimensional time into a single construct called the space-time continuum, in which time plays the role of the 4th dimension. According to Euclidean space perception, our universe has three dimensions of space, and one dimension of time. By combining space and time into a single manifold, physicists have significantly simplified a good deal of physical theory, as well as described in a more uniform way the workings of the universe at both the supergalactic and subatomic levels.)
Time is a strange thing. We can have a perception of the passage of time, as things move along in a sequence - the sun rises, and the sun also sets. This is what most think of when they hear the word "time" itself - the Time of Day/Night (and you don't even need to know who Isaac Newton is!)
I hold closer to Immanuel Kant's view of time: time is part of the fundamental intellectual structure (together with space and number) within which we sequence events, quantify the duration of events and the intervals between them, and compare the motions of objects. In this view, time does not refer to any kind of entity that "flows", that objects "move through", or that is a "container" for events.
I simply couldn't care less when I wake up, when I go to bed, or when I eat breakfast. I do not keep a schedule that is set, as I set my own schedule and never seem to do things exactly the same from day to day. I lose track of time, not because I fail to pay attention to its passing, but because I have no need to bother with tracking it at all. Clocks assault my vision in just about every room, but how often have I bothered to actually glance at one simply for the desire to know what time of day it is? Hardly ever, unless my existence must suddenly grind itself back to a more mundane path due to the pressing need to coordinate my personal time with the synchronicity of the rest of the world (or to keep an appointment in time with a doctor or group.) Thus I exist, and thus Isaac Newton rolls over in his grave. Kant, I am sure, would applaud that there is at least one being who does not need to rely upon Newton's theories in order to maintain sanity. I am quite happy to exist without a schedule or the knowledge of "what time it is" right now.
Hence, I shrug at time. I am chronologically challenged, meaning that the time arrow does affect me mentally (although I do age) yet I see all things as relevant. I balk at the evidence of time's passing, for it means nothing. I am not immortal, yet my mortality is not hinged upon moving forward in time or in time's stagnation (for if time stagnates, then nothing moves forward, and the only option is to find out why, or hold on as we surf the event horizon and the effects of reverse of time back to the black hole of Antioch. Never mind. You had to be there - 19 years ago - in the singularity of that moment, for that joke to hit home as humor.)
Alright. I'll try to explain (and will borrow, heavily, from other sources!)
In the natural sciences, time’s arrow, or arrow of time as it is also known, is a term used to distinguish a direction of time on a four-dimensional relativistic map of the world - which can be determined by a study of organizations of atoms, molecules, and bodies. \
The thermodynamic arrow of time is provided by the Second Law of Thermodynamics, which says that in an isolated system entropy will only increase with time; it will not decrease with time. Entropy can be thought of as a measure of disorder; thus the Second Law implies that time is asymmetrical with respect to the amount of order in an isolated system: as time increases, a system will always become more disordered. This asymmetry can be used empirically to distinguish between future and past. (I won't delve into Chaos Theory here.)
The Second Law does not hold with strict universality: any system can fluctuate to a state of lower entropy (see the Poincaré recurrence theorem). However, the Second Law seems accurately to describe the overall trend in real systems toward higher entropy.
Certain subatomic interactions involving the weak nuclear force violate the conservation of parity, but only very rarely. According to the CPT Theorem, this means they should also be time irreversible, and so establish an arrow of time. Such processes should be responsible for matter creation in the early universe. To me, in my daily life, time follows that pathway perfectly. I can not undo what has been done. I can not reverse time to change things that would later become a pinnacle by which I gain the desire to change so that the pinnacle does not take place, therefore changing my own timeline infinitely as that pinnacle is reshaped and reformed with each attempt to rid myself of it (and should I remove it I remove the desire to return to that point in time, thereby it does happen... or does it? Parallel universes explode, and Mickey Mouse does the Mashed Potato on Elvis' grave.)
This arrow is not linked to any other arrow by any proposed mechanism, and if it would have pointed to the opposite time direction, the only difference would have been that our universe would be made of anti-matter rather than from matter. More accurately, the definitions of matter and anti-matter would just be reversed. Does it matter? Not to me, but that is because I am just weird. It effects me, as I can not escape the clutches of time itself.
That parity is broken so rarely means that this arrow only "barely" points in one direction, setting it apart from the other arrows whose direction is much more obvious.
Quantum evolution is governed by the Schrödinger equation, which is time-symmetric, and by wave function collapse, which is time irreversible. As the mechanism of wave function collapse is still obscure, it's not known how this arrow links to the others. While at the microscopic level, collapse seems to show no favor to increasing or decreasing entropy, some believe there is a bias which shows up on macroscopic scales as the thermodynamic arrow. According to the theory of quantum decoherence, and assuming that the wave function collapse is merely apparent, the quantum arrow of time is a consequence of the thermodynamic arrow of time. Geeks everywhere are wondering if I would touch up "the cat". I won't. I don't believe it exists, and I walk through it. I won't let anthropic bias hinder me.
"Anthropic bias" is a term coined by the philosopher Nick Bostrom, as an expression for the bias arising when "your evidence is biased by observation selection effects". This is, basically an extreme generalization of the confirmation bias and the cognitive bias, involving not only mind-set, memory and methodology, but the whole way in which one sees oneself as an entity investigating an environment. As the etymology of the term suggests (from the Greek word for "human being") Bostrom's main claim could be reduced to saying that being a human being itself constitutes a bias for, and consequently a hindrance to, objective observation. In my own pondering, I tend to take things from different perspectives, and I often forget that I am approaching things as a human being. I escape the bounds and limitations of time and space, disregard biological necessities, and "lose track" of time as a whole. I spend hours probing a forming hypothesis, testing it to see if it would withstand the beatings necessary to become theory. I cease to the be entity, and become that which I study, bit by bit, on a mental scale. I leave the realm of hard science and embrace philosophy, but science remains my grounding point as the laws of mathematics must always be applied.
Bostrom suggests a way out using what amounts to quasi-empirical methods, and I enjoy embracing his philosophy. In his book Anthropic Bias: observation selection effects in science and philosophy, Bostrom explores the implications of these for "polling, cosmology (how many universes are there?), evolution theory (how improbable was the evolution of intelligent life on our planet?), the problem of time's arrow (can it be given a thermodynamic explanation?), game theoretic problems with imperfect recall (how to model them?), traffic analysis (why is the "next lane" faster?)."
It has been suggested that the whole idea of an anthropic bias is irrefutable. How could a criticism, presumably made by a human being, against the theory of anthropic biases be conceived? If it is not possible to review it critically, the whole theory becomes a will-o'-the-wisp without any practical consequences for our human lives here on Earth. I can tell you that existing the way I do when I'm on a mental tangent is harmful, as the "real life" things that are critical are often ignored. To remove oneself, one must remove one's self. To remove one's self, one neglects others. Few people can so totally remove themselves and remain sane. Perhaps that is an indication that I am insane, yet do we base sanity upon how an individual reacts to his environment, or do we base it on that individuals ability to grasp reality? Even on a "tangent", I assure that I grasp reality for what it is. I simply choose to ignore that which is not immediately essential for me to complete by reasoning.
Another problem with the theory purporting the existence of a general anthropic bias, is that it sounds self-referentially inconsistent — If Nick Bostrom is a human being, and the anthropic principle is valid, then his observations will be biased; the anthropic principle is an observation made by Nick Bostrom; hence, either (α) Nick Bostrom is not a human being (or alternatively, knowledge of the anthropic principle was supernaturally revealed to him), or (β) the anthropic principle is itself anthropically biased, or (γ) at least one observation made by a human being (e.g. N.B.'s observation of the anthropic bias) is not biased (γ is a counterexample of the general anthropic principle, and all three alternatives (α, β, and γ) point to Bostrom's theory being poorly conceived.
Needless to say, this entire line of thinking stems from a conversation between Better Half and myself, whereupon we dialoged concerning what terminology would best apply to me as far as my attunement to time is concerned. Am I chronologically challenged, in regards to my complete ignorance of the actual time of day? Am I entropically hindered, as I throw the 4th dimension out the window on a daily basis? Perhaps we are socially challenged, Better Half and I. Perhaps other spouses discuss the kids, or groceries, or shoes? Perhaps they dwell upon stupid, mundane matters such as what to eat next Friday? Perhaps the only thing holding them together is the daily events that bind them, and their relationship goes stagnate as they attempt to keep cohesive as a pair by interjecting commentaries about how they think things should be when the sun rises? I don't know. Better Half and I have always had the ability to remove ourselves from the "mate" prospect, male and female, in order to explore the scientific and philosophical nature of things as a combined mind. That, dear readers, is why I married him. Time destroys, breaks down mountains and turns seas into deserts.
In time, relationships based solely upon sexual fulfillment fall by the wayside. When I chose Better Half, it was for his mind as well as his body. As we age, and as the arrow of time reminds us that we are indeed mortal, we will lose our bodies to the ravages of time, yet we have a bond that will keep cohesive for as long as our minds hold out. For those that are curious - I exist in a world that does not reply upon time, and my thread of connectivity to the real world is held by a being that is content to obey the laws of time: hence, I am grounded. |
167ca2d1f19fcff0 | Open main menu
In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give the same value of energy upon measurement. The number of different states corresponding to a particular energy level is known as the degree of degeneracy of the level. It is represented mathematically by the Hamiltonian for the system having more than one linearly independent eigenstate with the same energy eigenvalue.[1]:p. 48 In classical mechanics, this can be understood in terms of different possible trajectories corresponding to the same energy.
Degeneracy plays a fundamental role in quantum statistical mechanics. For an N-particle system in three dimensions, a single energy level may correspond to several different wave functions or energy states. These degenerate states at the same level are all equally probable of being filled. The number of such states gives the degeneracy of a particular energy level.
Degenerate states in a quantum system
The possible states of a quantum mechanical system may be treated mathematically as abstract vectors in a separable, complex Hilbert space, while the observables may be represented by linear Hermitian operators acting upon them. By selecting a suitable basis, the components of these vectors and the matrix elements of the operators in that basis may be determined. If A is a N × N matrix, X a non-zero vector, and λ is a scalar, such that , then the scalar λ is said to be an eigenvalue of A and the vector X is said to be the eigenvector corresponding to λ. Together with the zero vector, the set of all eigenvectors corresponding to a given eigenvalue λ form a subspace of Cn, which is called the eigenspace of λ. An eigenvalue λ which corresponds to two or more different linearly independent eigenvectors is said to be degenerate, i.e., and , where and are linearly independent eigenvectors.The dimensionality of the eigenspace corresponding to that eigenvalue is known as its degree of degeneracy, which can be finite or infinite. An eigenvalue is said to be non-degenerate if its eigenspace is one-dimensional. The eigenvalues of the matrices representing physical observables in quantum mechanics give the measurable values of these observables while the eigenstates corresponding to these eigenvalues give the possible states in which the system may be found, upon measurement. The measurable values of the energy of a quantum system are given by the eigenvalues of the Hamiltonian operator, while its eigenstates give the possible energy states of the system. A value of energy is said to be degenerate if there exist at least two linearly independent energy states associated with it. Moreover, any linear combination of two or more degenerate eigenstates is also an eigenstate of the Hamiltonian operator corresponding to the same energy eigenvalue.
Effect of degeneracy on the measurement of energyEdit
In the absence of degeneracy, if a measured value of energy of a quantum system is determined, the corresponding state of the system is assumed to be known, since only one eigenstate corresponds to each energy eigenvalue. However, if the Hamiltonian has a degenerate eigenvalue of degree gn, the eigenstates associated with it form a vector subspace of dimension gn. In such a case, several final states can be possibly associated with the same result , all of which are linear combinations of the gn orthonormal eigenvectors .
In this case, the probability that the energy value measured for a system in the state will yield the value is given by the sum of the probabilities of finding the system in each of the states in this basis, i.e.
Degeneracy in different dimensionsEdit
This section intends to illustrate the existence of degenerate energy levels in quantum systems studied in different dimensions. The study of one and two-dimensional systems aids the conceptual understanding of more complex systems.
Degeneracy in one dimensionEdit
In several cases, analytic results can be obtained more easily in the study of one-dimensional systems. For a quantum particle with a wave function moving in a one-dimensional potential , the time-independent Schrödinger equation can be written as
Since this is an ordinary differential equation, there are two independent eigenfunctions for a given energy at most, so that the degree of degeneracy never exceeds two. It can be proved that in one dimension, there are no degenerate bound states for normalizable wave functions. A sufficient condition on a piecewise continuous potential and the energy is the existence of two real numbers with such that we have .[3] In particular, is bounded below in this criterion.
Degeneracy in two-dimensional quantum systemsEdit
Two-dimensional quantum systems exist in all three states of matter and much of the variety seen in three dimensional matter can be created in two dimensions. Real two-dimensional materials are made of monoatomic layers on the surface of solids. Some examples of two-dimensional electron systems achieved experimentally include MOSFET, two-dimensional superlattices of Helium, Neon, Argon, Xenon etc. and surface of liquid Helium. The presence of degenerate energy levels is studied in the cases of particle in a box and two-dimensional harmonic oscillator, which act as useful mathematical models for several real world systems.
Particle in a rectangular planeEdit
Consider a free particle in a plane of dimensions and in a plane of impenetrable walls. The time-independent Schrödinger equation for this system with wave function can be written as
The permitted energy values are
The normalized wave function is
So, quantum numbers and are required to describe the energy eigenvalues and the lowest energy of the system is given by
For some commensurate ratios of the two lengths and , certain pairs of states are degenerate. If , where p and q are integers, the states and have the same energy and so are degenerate to each other.
Particle in a square boxEdit
In this case, the dimensions of the box and the energy eigenvalues are given by
Since and can be interchanged without changing the energy, each energy level has a degeneracy of at least two when and are different. Degenerate states are also obtained when the sum of squares of quantum numbers corresponding to different energy levels are the same. For example, the three states (nx = 7, ny = 1), (nx = 1, ny = 7) and (nx = ny = 5) all have and constitute a degenerate set.
Degrees of degeneracy of different energy levels for a particle in a square box:
1 1 2 1
2 2 8 1
3 3 18 1
Particle in a cubical boxEdit
In this case, the dimensions of the box and the energy eigenvalues depend on three quantum numbers.
Since , and can be interchanged without changing the energy, each energy level has a degeneracy of at least three when the three quantum numbers are not all equal.
Finding a unique eigenbasis in case of degeneracyEdit
If two operators and commute, i.e. , then for every eigenvector of , is also an eigenvector of with the same eigenvalue. However, if this eigenvalue, say , is degenerate, it can be said that belongs to the eigenspace of , which is said to be globally invariant under the action of .
For two commuting observables A and B, one can construct an orthonormal basis of the state space with eigenvectors common to the two operators. However, is a degenerate eigenvalue of , then it is an eigensubspace of that is invariant under the action of , so the representation of in the eigenbasis of is not a diagonal but a block diagonal matrix, i.e. the degenerate eigenvectors of are not, in general, eigenvectors of . However, it is always possible to choose, in every degenerate eigensubspace of , a basis of eigenvectors common to and .
Choosing a complete set of commuting observablesEdit
If a given observable A is non-degenerate, there exists a unique basis formed by its eigenvectors. On the other hand, if one or several eigenvalues of are degenerate, specifying an eigenvalue is not sufficient to characterize a basis vector. If, by choosing an observable , which commutes with , it is possible to construct an orthonormal basis of eigenvectors common to and , which is unique, for each of the possible pairs of eigenvalues {a,b}, then and are said to form a complete set of commuting observables. However, if a unique set of eigenvectors can still not be specified, for at least one of the pairs of eigenvalues, a third observable , which commutes with both and can be found such that the three form a complete set of commuting observables.
It follows that the eigenfunctions of the Hamiltonian of a quantum system with a common energy value must be labelled by giving some additional information, which can be done by choosing an operator that commutes with the Hamiltonian. These additional labels required naming of a unique energy eigenfunction and are usually related to the constants of motion of the system.
Degenerate energy eigenstates and the parity operatorEdit
The parity operator is defined by its action in the representation of changing r to -r, i.e.
The eigenvalues of P can be shown to be limited to , which are both degenerate eigenvalues in an infinite-dimensional state space. An eigenvector of P with eigenvalue +1 is said to be even, while that with eigenvalue −1 is said to be odd.
Now, an even operator is one that satisfies,
while an odd operator is one that satisfies
Since the square of the momentum operator is even, if the potential V(r) is even, the Hamiltonian is said to be an even operator. In that case, if each of its eigenvalues are non-degenerate, each eigenvector is necessarily an eigenstate of P, and therefore it is possible to look for the eigenstates of among even and odd states. However, if one of the energy eigenstates has no definite parity, it can be asserted that the corresponding eigenvalue is degenerate, and is an eigenvector of with the same eigenvalue as .
Degeneracy and symmetryEdit
The physical origin of degeneracy in a quantum-mechanical system is often the presence of some symmetry in the system. Studying the symmetry of a quantum system can, in some cases, enable us to find the energy levels and degeneracies without solving the Schrödinger equation, hence reducing effort.
Mathematically, the relation of degeneracy with symmetry can be clarified as follows. Consider a symmetry operation associated with a unitary operator S. Under such an operation, the new Hamiltonian is related to the original Hamiltonian by a similarity transformation generated by the operator S, such that , since S is unitary. If the Hamiltonian remains unchanged under the transformation operation S, we have
Now, if is an energy eigenstate,
where E is the corresponding energy eigenvalue.
which means that is also an energy eigenstate with the same eigenvalue E. If the two states and are linearly independent (i.e. physically distinct), they are therefore degenerate.
In cases where S is characterized by a continuous parameter , all states of the form have the same energy eigenvalue.
Symmetry group of the HamiltonianEdit
The set of all operators which commute with the Hamiltonian of a quantum system are said to form the symmetry group of the Hamiltonian. The commutators of the generators of this group determine the algebra of the group. An n-dimensional representation of the Symmetry group preserves the multiplication table of the symmetry operators. The possible degeneracies of the Hamiltonian with a particular symmetry group are given by the dimensionalities of the irreducible representations of the group. The eigenfunctions corresponding to a n-fold degenerate eigenvalue form a basis for a n-dimensional irreducible representation of the Symmetry group of the Hamiltonian.
Types of degeneracyEdit
Degeneracies in a quantum system can be systematic or accidental in nature.
Systematic or essential degeneracyEdit
This is also called a geometrical or normal degeneracy and arises due to the presence of some kind of symmetry in the system under consideration, i.e. the invariance of the Hamiltonian under a certain operation, as described above. The representation obtained from a normal degeneracy is irreducible and the corresponding eigenfunctions form a basis for this representation.
Accidental degeneracyEdit
It is a type of degeneracy resulting from some special features of the system or the functional form of the potential under consideration, and is related possibly to a hidden dynamical symmetry in the system. It also results in conserved quantities, which are often not easy to identify. Accidental symmetries lead to these additional degeneracies in the discrete energy spectrum. An accidental degeneracy can be due to the fact that the group of the Hamiltonian is not complete. These degeneracies are connected to the existence of bound orbits in classical Physics.
Examples of systems with accidental degeneraciesEdit
The Coulomb and Harmonic Oscillator potentialsEdit
For a particle in a central 1/r potential, the Laplace–Runge–Lenz vector is a conserved quantity resulting from an accidental degeneracy, in addition to the conservation of angular momentum due to rotational invariance.
For a particle moving on a cone under the influence of 1/r and r2 potentials, centred at the tip of the cone, the conserved quantities corresponding to accidental symmetry will be two components of an equivalent of the Runge-Lenz vector, in addition to one component of the angular momentum vector. These quantities generate SU(2) symmetry for both potentials.
Particle in a constant magnetic fieldEdit
A particle moving under the influence of a constant magnetic field, undergoing cyclotron motion on a circular orbit is another important example of an accidental symmetry. The symmetry multiplets in this case are the Landau levels which are infinitely degenerate.
The hydrogen atomEdit
In atomic physics, the bound states of an electron in a hydrogen atom show us useful examples of degeneracy. In this case, the Hamiltonian commutes with the total orbital angular momentum , its component along the z-direction, , total spin angular momentum and its z-component . The quantum numbers corresponding to these operators are , , (always 1/2 for an electron) and respectively.
The energy levels in the hydrogen atom depend only on the principal quantum number n. For a given n, all the states corresponding to have the same energy and are degenerate. Similarly for given values of n and l, the , states with are degenerate. The degree of degeneracy of the energy level En is therefore : , which is doubled if the spin degeneracy is included.[1]:p. 267f
The degeneracy with respect to is an essential degeneracy which is present for any central potential, and arises from the absence of a preferred spatial direction. The degeneracy with respect to is often described as an accidental degeneracy, but it can be explained in terms of special symmetries of the Schrödinger equation which are only valid for the hydrogen atom in which the potential energy is given by Coulomb's law.[1]:p. 267f
Isotropic three-dimensional harmonic oscillatorEdit
It is a spinless particle of mass m moving in three-dimensional space, subject to a central force whose absolute value is proportional to the distance of the particle from the centre of force.
It is said to be isotropic since the potential acting on it is rotationally invariant, i.e. :
where is the angular frequency given by .
Since the state space of such a particle is the tensor product of the state spaces associated with the individual one-dimensional wave functions, the time-independent Schrödinger equation for such a system is given by-
So, the energy eigenvalues are
where n is a non-negative integer. So, the energy levels are degenerate and the degree of degeneracy is equal to the number of different sets satisfying
which is equal to
Only the ground state is non-degenerate.
Removing degeneracyEdit
The degeneracy in a quantum mechanical system may be removed if the underlying symmetry is broken by an external perturbation. This causes splitting in the degenerate energy levels. This is essentially a splitting of the original irreducible representations into lower-dimensional such representations of the perturbed system.
Mathematically, the splitting due to the application of a small perturbation potential can be calculated using time-independent degenerate perturbation theory. This is an approximation scheme that can be applied to find the solution to the eigenvalue equation for the Hamiltonian H of a quantum system with an applied perturbation, given the solution for the Hamiltonian H0 for the unperturbed system. It involves expanding the eigenvalues and eigenkets of the Hamiltonian H in a perturbation series. The degenerate eigenstates with a given energy eigenvalue form a vector subspace, but not every basis of eigenstates of this space is a good starting point for perturbation theory, because typically there would not be any eigenstates of the perturbed system near them. The correct basis to choose is one that diagonalizes the perturbation Hamiltonian within the degenerate subspace.
Physical examples of removal of degeneracy by a perturbationEdit
Some important examples of physical situations where degenerate energy levels of a quantum system are split by the application of an external perturbation are given below.
Symmetry breaking in two-level systemsEdit
A two-level system essentially refers to a physical system having two states whose energies are close together and very different from those of the other states of the system. All calculations for such a system are performed on a two-dimensional subspace of the state space.
If the ground state of a physical system is two-fold degenerate, any coupling between the two corresponding states lowers the energy of the ground state of the system, and makes it more stable.
If and are the energy levels of the system, such that , and the perturbation is represented in the two-dimensional subspace as the following 2×2 matrix
then the perturbed energies are
Examples of two-state systems in which the degeneracy in energy states is broken by the presence of off-diagonal terms in the Hamiltonian resulting from an internal interaction due to an inherent property of the system include:
• Benzene, with two possible dispositions of the three double bonds between neighbouring Carbon atoms.
• Ammonia molecule, where the Nitrogen atom can be either above or below the plane defined by the three Hydrogen atoms.
• H+
molecule, in which the electron may be localized around either of the two nuclei.
Fine-structure splittingEdit
The corrections to the Coulomb interaction between the electron and the proton in a Hydrogen atom due to relativistic motion and spin-orbit coupling result in breaking the degeneracy in energy levels for different values of l corresponding to a single principal quantum number n.
The perturbation Hamiltonian due to relativistic correction is given by
where is the momentum operator and is the mass of the electron. The first-order relativistic energy correction in the basis is given by
where is the fine structure constant.
The spin-orbit interaction refers to the interaction between the intrinsic magnetic moment of the electron with the magnetic field experienced by it due to the relative motion with the proton. The interaction Hamiltonian is
which may be written as
The first order energy correction in the basis where the perturbation Hamiltonian is diagonal, is given by
where is the Bohr radius. The total fine-structure energy shift is given by
for .
Zeeman effectEdit
The splitting of the energy levels of an atom when placed in an external magnetic field because of the interaction of the magnetic moment of the atom with the applied field is known as the Zeeman effect.
Taking into consideration the orbital and spin angular momenta, and , respectively, of a single electron in the Hydrogen atom, the perturbation Hamiltonian is given by
where and . Thus,
Now, in case of the weak-field Zeeman effect, when the applied field is weak compared to the internal field, the spin-orbit coupling dominates and and are not separately conserved. The good quantum numbers are n, l, j and mj, and in this basis, the first order energy correction can be shown to be given by
, where
is called the Bohr Magneton.Thus, depending on the value of , each degenerate energy level splits into several levels.
Lifting of degeneracy by an external magnetic field
In case of the strong-field Zeeman effect, when the applied field is strong enough, so that the orbital and spin angular momenta decouple, the good quantum numbers are now n, l, ml, and ms. Here, Lz and Sz are conserved, so the perturbation Hamiltonian is given by-
assuming the magnetic field to be along the z-direction. So,
For each value of ml, there are two possible values of ms, .
Stark effectEdit
The splitting of the energy levels of an atom or molecule when subjected to an external electric field is known as the Stark effect.
For the hydrogen atom, the perturbation Hamiltonian is
if the electric field is chosen along the z-direction.
The energy corrections due to the applied field are given by the expectation value of in the basis. It can be shown by the selection rules that when and .
The degeneracy is lifted only for certain states obeying the selection rules, in the first order. The first-order splitting in the energy levels for the degenerate states and , both corresponding to n = 2, is given by .
See alsoEdit
1. ^ a b c Merzbacher, Eugen (1998). Quantum Mechanics (3rd ed.). New York: John Wiley. ISBN 0471887021.CS1 maint: uses authors parameter (link)
2. ^ Levine, Ira N. (1991). Quantum Chemistry (4th ed.). Prentice Hall. p. 52. ISBN 0-205-12770-3.CS1 maint: uses authors parameter (link)
3. ^ a b Messiah, Albert (1967). Quantum mechanics (3rd ed.). Amsterdam, NLD: North-Holland. pp. 98–106. ISBN 0471887021.CS1 maint: uses authors parameter (link)
Further readingEdit |
812b86303c603f53 | I'm trying to solve a full-vectorial wave equation for an arbitrarily shaped wave guide, by using NDSolve and perfectly matched layer (PML) conditions.
The PML conditions can be stated as a coordinate transformation of the form $\text{$\partial $/$\partial $x $\to $ $\alpha $x(x) $\partial $/$\partial $x}$ and so on. As the functionality of the Mathematica built-in Curl doesn't extend so far I set up a function that acts as intended:
coordTransfVector[[2]] D[applicationVector[[3]],coordNamesVector[[2]]] -
coordTransfVector[[3]] D[applicationVector[[2]],coordNamesVector[[3]]],
coordTransfVector[[3]] D[applicationVector[[1]],coordNamesVector[[3]]] -
coordTransfVector[[1]] D[applicationVector[[3]],coordNamesVector[[1]]],
coordTransfVector[[1]] D[applicationVector[[2]],coordNamesVector[[1]]] -
coordTransfVector[[2]] D[applicationVector[[1]],coordNamesVector[[2]]]
If the coordinate transformation is set to unity $\text{$\{$1,1,1$\}$}$ the function is identical to the normal Curl, as the comparison yields True:
generalizedCurl3D[{1, 1, 1}, {ψx[x, y, z], ψy[x, y, z], ψz[x, y, z]},
{x, y, z}] == {-Derivative[0, 0, 1][ψy][x, y, z] +
Derivative[0, 1, 0][ψz][x, y, z], Derivative[0, 0, 1][ψx][x, y, z] -
Derivative[1, 0, 0][ψz][x, y, z], -Derivative[0, 1, 0][ψx][x, y, z] +
Derivative[1, 0, 0][ψy][x, y, z]}
Now I derive the wave equation I need to solve. In the following I will consider only the magnetic field $\text{$\psi $[x,y,z]}$ because it is continuous at the boundary of the wave guide. The refractive index profile $n$ depends on $x$ and $y$ ($n=n[x,y]$) and is constant along the $z$ direction. The functions for PML will be set up later, but at this moment it is only important that the coordinate transformation functions $\text{$\alpha $x}$ and $\text{$\alpha $y}$ depend only on $x$ and $y$ ($\text{$\alpha $x[x]}$ and $\text{$\alpha $y[y]}$) while $\text{$\alpha $z}=1$.
I start with the Helmholtz equation, assuming that the time separation ansatz works. This way I only have to assume that the field will have fast oscillations in z direction with the effective refractive index $n0$ and the wave number $k$ ($e^{-i \cdot k \cdot \text{n0}\cdot z}$). The rest should be only slow oscillations of the field components in $x$ and $y$ direction. I also tend to set the $\text{$\psi $z}$ component of the $\psi$-vector to zero, because probably the cross-talk of $x$,$y$ components to the $z$ field component should be negligible.
Edit: In case you try to check my calculation, please make sure that the 3rd component of $\psi $ is defined to be zero. To keep it general for 3dimensional analysis (with easy "switching on or off") I define it nevertheless, but multiply it with 0.
cTV = {αx[x], αy[y], 1};
cNV = {x, y, z};
ψ = {ψx[x, y, z] E^(-I k n0 z), ψy[x, y, z] E^(-I k n0 z), 0 ψz[x, y, z] E^(-I k n0 z)};
Plugging now the still analytical expressions into the Helmholtz equation
($\text{$\nabla \times \nabla \times \psi $== - }\mu \text{$\epsilon $ }\partial ^2\left/\partial t^2\right.\text{ $\psi $ = + }k^2\text{ $\psi $}$)
and dividing all by the fast changing $z$-term ($e^{-i \cdot k \cdot \text{n0}\cdot z}$) gives me the following :
eqs = FullSimplify[(generalizedCurl3D[cTV, 1/n[x, y]^2 generalizedCurl3D[cTV, ψ, cNV], cNV]
- k^2 ψ ) 1/E^(-I k n0 z)]
Now I replace the analytical expressions by numerical values and functions and define the size of the boundary and the PML-layer (all in SI units). (Note that $n0$ is effectively a propagation constant that has to be found by trial and error: if it is chosen wrong, then the field should spread outside the waveguide.)
Edit: in the meanwhile I found out that I should use numerical quantities close to 1 instead of SI units for micrometer length scales, because obviously Mathematica performs numerical integration with machine precision and if the numbers are too close to machine precision the numerical "signal-to-noise-ratio" can cause singularities in the solution which causes in turn NDSolve to get stuck or to blow up the required memory.
So now I am using the following values (instead of 10^-6):
xBound = 12; yBound = 12; zBound = 5;
nVak = 1.; nMat = 1.5; λ = 0.8; k = (2 π)/λ;
n0 = 1.48 (*this is the "arbitrarily chosen" because yet unknown propagation constant*);
waveGuideR = 1; (*wave guide radius*)
n[x_, y_] := If[x^2 + y^2 <= waveGuideR^2, nMat, nVak];
(*let's have a circular waveguide with refractive index of 1.5*)
theoReflCoeff = 10.^-2;(*1/theoReflCoeff is the theoretical damping
coefficient of the PML layer*)
pmlWidth = 1;(*size of the PML-layer*)
αx[x_] := Piecewise[{{1 - I 3 λ (x + (xBound - pmlWidth))^2/(4 π nVak pmlWidth^3)
Log[1/theoReflCoeff], x < -(xBound - pmlWidth)}, {nVak, -(xBound - pmlWidth)
<= x <= (xBound - pmlWidth)}, {1 - I 3 λ (x - (xBound - pmlWidth))^2/(4 π nVak
pmlWidth^3) Log[1/theoReflCoeff], x > (xBound - pmlWidth)}}];
αy[y_] := Piecewise[{{1 - I 3 λ (y + (yBound - pmlWidth))^2/(4 π nVak pmlWidth^3)
Log[1/theoReflCoeff], y < -(yBound - pmlWidth)}, {nVak, -(yBound - pmlWidth)
<= y <= (yBound - pmlWidth)}, {1 - I 3 λ (y - (yBound - pmlWidth))^2/(4 π nVak
pmlWidth^3) Log[1/theoReflCoeff], y > (yBound - pmlWidth)}}];
The PML - coordinate transformation is formed such that outside the PML layer the derivatives are multiplied by one and inside the PML are multiplied by a complex value (so that they should cause damping on the wave). For clarity the imaginary part and the absolute part of the PML layer is shown in the following:
Structure of the perfectly matched layer
Of course the Helmholtz equation in input line 6 is actually a vector in 3 dimensions. However, I am interested only in the evolution of the field in $x$ and $y$ direction. That is why I will neglect the calculation for the field in $z$-dimension. (Here I am not totally sure that what I am doing is physically correct, since I chose for $\text{$\psi $z}$ to be zero in the initial calculation, but due to the Curl-operation I still get the contribution in the z-dimension. Anyway, if plug eqs[[3]] into NDSolve Mathematica tells me that the system is overdetermined).
The 2 coupled differential equations that have to be solved are then:
diffEq1 = eqs[[1]] == 0;
diffEq2 = eqs[[2]] == 0;
Now I define at the edges (after the PML layers) periodic boundary conditions for $x$ and $y$ directions and fields.
boundDirectionXfieldX = ψx[-xBound, y, z] == ψx[xBound, y, z];
boundDirectionYfieldX = ψx[x, -yBound, z] == ψx[x, yBound, z];
boundDirectionXfieldY = ψy[-xBound, y, z] == ψy[xBound, y, z];
boundDirectionYfieldY = ψy[x, -yBound, z] == ψy[x, yBound, z];
Furthermore I need to define the starting field that is launched into the waveguide. In this case I assume the y-component of the field to be zero, because due to the coupling of the equations also higher and more complicated transversal field modes should develop as the they indeed do in real experiments. Since I am looking only for solutions that will be guided it is unimportant what kind of initial field I chose. (Even if it is the wrong field distribution for the correct $n0$ a stable distribution in the waveguide should evolve.)
startCondDirectionZfieldX = ψx[x, y, 0] == E^(-((x^2 + y^2)/waveGuideR^2));
startCondDirectionZfieldY = ψy[x, y, 0] == 0;
Because there is a second derivative of $\psi$ in $z$ - direction, I need additionally a Neumann - boundary condition for the field launched into the waveguide at $z=0$. I think this is physically correct, because if the waveguide is infinite in z-direction and I manage somehow to generate a field with the above shape along an extended length of the waveguide then all the physical effects should still occur after this. (Correct me if I'm wrong here)
neumannCondFieldX = Derivative[0, 0, 1][ψx][x, y, 0] == 0;
neumannCondFieldY = Derivative[0, 0, 1][ψy][x, y, 0] == 0;
With this I can set up a equation system for NDSolve to solve. Additionally I used EvaluationMonitor to have at least an inkling where NDSolve currently is as well as a MemoryConstrained evaluation limited to 13 GB of RAM (but I don't it works the way I implemented it)
Monitor[Fkt = {ψx, ψy} /. First@NDSolve[{diffEq1, diffEq2,
boundDirectionXfieldX, boundDirectionYfieldX,
boundDirectionXfieldY, boundDirectionYfieldY,
startCondDirectionZfieldX, startCondDirectionZfieldY,
neumannCondFieldX, neumannCondFieldY}, {ψx, ψy}, {z,
0., zBound}, {x, -xBound, xBound}, {y, -yBound, yBound},
"DifferenceOrder" -> "Pseudospectral"}},
EvaluationMonitor :> (stepz = z)], AngularGauge[Dynamic[stepz/zBound], {0, 1},
GaugeLabels -> Automatic]], 13 2^30]
The most puzzling part during is the error message:
NDSolve::mxsst: Using maximum number of grid points 100 allowed by the MaxPoints or MinStepSize options for independent variable y.
This is where the strange things happen and I would like to know how to arrive at a proper well-behaved solution:
• Mathematica gives me an answer in spite of the above error message. The solution is not well behaved because it changes, depending on how large I chose the size of the boundaries. Even more troubling is the fact that for some combinations of the boundary sizes Mathematica Kernel hangs itself up as it uses almost the full available memory.
• The obvious thing to do would be to change the number of grid points. However if I chose smaller or larger number of "MaxPoints" than the default 100, NDSolve runs always into a memory limit. I wonder how this can be if I chose a smaller size? I would have thought that in such case the solution would be just less precise?
• The setting "DifferenceOrder"->"Pseudospectral" seems to be paramount. Anything else runs into the above memory problems. Only thanks to the postings of Complex valued 2+1D PDE Schrödinger equation, numerical method for `NDSolve`? I was able to get any result at all.
Here is the result calculated and displayed:
zSlices = Table[Plot3D[Abs[Fkt[[1]][x, y, z]], {x, -xBound, xBound},
{y, -yBound, yBound}, PlotRange -> {0, All}, PlotPoints -> 200],
{z, 0, zBound, zBound/6.}];
Export["UnstableSolution.gif", zSlices, AnimationRepetitions -> Infinity,
"DisplayDurations" -> .4]
enter image description here
Edit: Another culprit in my calculations seems to be the PML itself. If I discard the PML and use the normal Helmholtz-equation without any coordinate transformations and just pure periodic boundary condition I get a physically plausible solution:
enter image description here
But my joy is slightly marred by the fact that since I want to find stable propagating solutions in the waveguide, I must somehow get rid off the "reflected noisy waves". I would greatly appreciate if somebody could help me with the perfectly matched layer conditions. Thank you!
Edit: you can probably skip the next part, because as long as my university doesn't give me access to Mathematica 10.4 I will be stuck on the Finite Element Method. But it would be still cool to know if the Finite Element method makes better work of the PML-conditions than the pseudospectral decomposition I'm forced to use above.. ;-)
Ok, the next thing I thought to use were the new capabilities of Mathematica considering Finite Elements for calculation, because probably the observed issues are caused by the discontinuities on the waveguide or the PML transformation of cartesian coordinates. In such a case the Method in NDSolve must be changed as in the following, if I understand Mathematica help correctly. Although I had to change the boundary conditions to be zero instead of periodic and starting condition to be 0 at the boundaries to be consistent everywhere...
ψx[-xBound, y, z] == ψx[xBound, y, z] == 0,
ψx[x, -yBound, z] == ψx[x, yBound, z] == 0,
ψy[-xBound, y, z] == ψy[xBound, y, z] == 0,
ψy[x, -yBound, z] == ψy[x, yBound, z] == 0,
ψx[x, y, 0] == If[x^2 + y^2 <= waveGuideR^2, 1, 0],
ψy[x, y, 0] == 0,
{ψx, ψy}, {z, 0., 20 zBound}, {x, -xBound, xBound}, {y, -yBound, yBound},
Method -> {"PDEDiscretization" -> {"MethodOfLines",
"SpatialDiscretization" -> "FiniteElement"}}]
Now I get two new errors:
CompiledFunction::cfex: Could not complete external evaluation at instruction 10; proceeding with uncompiled evaluation.
NDSolve::femdpop: The FEMStiffnessElements operator failed.
where one of them has already been asked about in Error message for FEMStiffnessElements. @ilian mentioned that this problem has been resolved in Mathematica 10.4.
The problem is that the newest Mathematica version my university is able to provide is 10.3 (I don't know how long it will take for the data processing center to distribute the 10.4 version).
So I currently I have no opportunity to see whether the 10.4 can solve my problem so that I just wait a little bit or if I have to find a different way, e.g. to use COMSOL, or something similar, with which I'm not really familiar...
Thanks for bearing with me and my rantings up to the end!
• $\begingroup$ Your equations still have variables inside them, nMat and nVak` for example. Write a function that gets all parameters as arguments and that returns your equation with them filled in. Try a simpler example first and add more stuff later. The FEM message means that no all PDE coefficients could be compiled and for the complex value case you'd need 10.4. With the tensor product gird you might get away with an earlier version. $\endgroup$ – user21 Mar 14 '16 at 22:02
• $\begingroup$ @user21: But I assign a numeric value to $nMat$ and $nVak$ before I run NDSolve. Do you mean that the problem comes from the SetDelayed (:=) definition of my refractive index function? $\endgroup$ – Quit007 Mar 15 '16 at 9:42
• 1
$\begingroup$ In 10.4 I do not get the error message NDSolve::mxsst: that you report here. For the FEM code, is this a stationary problem - then why MethodOfLines? If it's transient which is the time variable? For the FEM code, those warnings will disappear if you remove the constants from your definitions, as those, when compiled will need to call back. So that for example ?n just depends on x and y. $\endgroup$ – user21 Mar 31 '16 at 11:40
• 1
$\begingroup$ Er… as to the PML for Helmholtz equation, what material/paper/book are you referring to? Actually I know a little about PML for Maxwell equation, whose form seems to be quite different from the one you're using. $\endgroup$ – xzczd Nov 23 '16 at 16:53
• 1
$\begingroup$ You need to add @xzczd in every comment of yours or I won't get the reminder… I think I found a possible trouble source. Your Helmholtz equation needs to be simplified further to something similar to this form based on Gauss's law. It has been discovered in the discussion under this post (Read those comments carefully) that, if Gauss's law isn't utilized, the equation will be hard to solve. $\endgroup$ – xzczd Dec 17 '16 at 4:40
I'm not that familiar with electromagnetism, either, but I think there're at least 4 issues in your solving process:
1. There's no need to "consider only the magnetic field", because electric field is also continuous at the boundary of the wave guide in your case.
2. Your definition for curl in stretched coordinate is wrong. When the scale factor is $(s_x(x), s_y(y), s_z(z))$, the generalized curl should be $(\frac{\psi _z{}^{(0,1,0)}(x,y,z)}{s_{\text{y}}(y)}-\frac{\psi _y{}^{(0,0,1)}(x,y,z)}{s_{\text{z}}(z)},\frac{\psi _x{}^{(0,0,1)}(x,y,z)}{s_{\text{z}}(z)}-\frac{\psi _z{}^{(1,0,0)}(x,y,z)}{s_{\text{x}}(x)},\frac{\psi _y{}^{(1,0,0)}(x,y,z)}{s_{\text{x}}(x)}-\frac{\psi _x{}^{(0,1,0)}(x,y,z)}{s_{\text{y}}(y)})$.
3. It's not clear to me if you're dealing with a 2D problem or 3D problem, but whatever it is, the deduction for Helmholtz equation seems to be wrong. For 2D case, there should be no derivative of z; for 3D case, "fast oscillations in z direction" should be introduced as a nonhomogeneous term of the equation.
4. In the final step, you're trying to solve initial-boundary value problem of Helmholtz equation, but it's a ill-posed problem. There exists techniques for dealing with this problem of course, but the standard approach is to set up a boundary value problem.
…Well, I'm not sure if the material you're referring to is improper, but, to have a better understanding for PML, you can refer to e.g. this, this or this. To help you understand PML better, in the rest part of this answer, I'll show you my implementation for SC-PML in 2D case. ($\text{TE}^\text{z}$ mode. )
First, we need the equation. The specific formulas can be found in numerous materials, but here I'll deduce the governing equation
$$\nabla _s\times {\mu^{-1} }{\nabla _s\times E}- \omega ^2 \epsilon E =-i \omega J$$
with Mathematica to make the code more instructive and elegant. The key point is implementing the generalized curl $\nabla _s\times$. There're many possible solutions, for example:
Cross[{d@x/sx, d@y/sy, d@z/sz}, {f, g, h}[x, y, z] // Through] /. d[v_] h_ :> D[h, v]
Or make use of DChange:
DChange[Curl[{f[x, y, z], g[x, y, z], h[x, y, z]}, {x, y, z}], {x == xx s["x"],
y == yy s@"y", z == zz s@"z"}, {x, y, z}, {xx, yy, zz}, {f[x, y, z], g[x, y, z],
h[x, y, z]}]
But I think the simplest approach is the one mentioned in tutorial/VectorAnalysis:
inde = {x, y, z};
sf = s[ToString@#]@# & /@ inde
vf = Times @@ sf
curlS = (sf Curl[sf #, {x, y, z}]/vf) &
The remaining part is straightforward:
Εlst = Ε[ToString@#] @@ Most@inde & /@ inde /. Ε["z"] -> (0 &)
jlst = j[ToString@#] @@ Most@inde & /@ inde /. j["z"] -> (0 &)
eqnS = Simplify[
curlS[curlS@Εlst/mu0] - omega^2 e0 Εlst == -I omega jlst // Thread // Most,
{mu0 > 0, s[ToString@#]@# != 0 & /@ inde} // Flatten]
Next, define the formula for scale factor $s$. Here I simply follow the one in the paper linked above, as far as I can tell, this is also the most popular way to define $s$:
sigmamax[thick_] = -(((m + 1) Log@R)/(2 eta0 thick))
sigma[l_, thick_] = sigmamax[thick] (l/thick)^m
sgenerator[x_, {lb_, rb_}, thick_] =
Piecewise[{{1 - I sigma[x - (rb - thick), thick]/(omega e0) ,
x > rb - thick}, {1 - I sigma[(lb + thick) - x, thick]/(omega e0) , x < lb + thick}},
{sfunc["x"][x_], sfunc["y"][y_]} =
MapThread[sgenerator, {{x, y}, {{lb@#, rb@#}, {lb@#2, rb@#2}}, {th@#, th@#2}} &["x",
"y"]] // Simplify
Then substitute specific values into the equation:
lam = 532 10^-9;
domain = {-2 lam, 2 lam};
thickness = 4 lam/10;
pdeS = Block[{e0 = 8854/10^3*10^-12, mu0 = 1257/10^3*10^-6, conduct = 10^7},
Block[{omega = (2 Pi)/(lam Sqrt[mu0 e0]), m = 4, R = E^-16, eta0 = Sqrt[mu0/e0], j,
lb, rb, th}, ({{lb@#, rb@#}, {lb@#2, rb@#2}, {th@#, th@#2}, {j[#][x, y],
j[#2][x, y]}} = {domain,
domain, {thickness, thickness}, {conduct Exp[10 (-(x/lam)^2 - (y/lam)^2)],
0}}) &["x", "y"]; eqnS /. s -> sfunc]];
bc = Function[{x,
y}, {Ε["x"][x, y] == 0, Ε["y"][x, y] ==
0}] @@@ {{domain[[1]], y}, {domain[[2]], y}, {x, domain[[1]]}, {x, domain[[2]]}} //
Last step is to solve the equation set. We can solve {pdeS, bc} with NDSolve directly if $VersionNumber >= 11.1:
<< NDSolve`FEM`
mesh = ToElementMesh[FullRegion[2], {domain, domain}, "MaxCellMeasure" -> lam/10^9];
bcfem = DirichletCondition[{Ε["x"][x, y] == 0, Ε["y"][x, y] == 0}, True];
sol = NDSolveValue[{pdeS, bcfem}, {Ε["x"], Ε["y"]},
Element[{x, y}, mesh]]
With[{domain = Sequence @@ (domain + {thickness, -thickness})},
Outer[Plot3D[#1[First[#2][x, y]], Evaluate[{x, domain}],
Evaluate[{y, domain}], PlotRange -> All,
PlotLabel -> #1[Last[#2]]] & , {Re, Im},
{{sol[[1]], "Ex"}, {sol[[2]], "Ey"}}, 1]],
ImageSize -> Large]
Mathematica graphics
If you're still in or before v9 (where "FiniteElement" isn't introduced yet), or between v10.0 and v11.0 (where a bug isn't fixed yet), finite difference method (FDM) can be used for solving the problem. I'll use pdetoae for discretizing:
points = 50;
grid = Array[# &, points, domain];
difforder = 2;
(*Definition of pdetoae isn't included in this code piece,
please find it in the link above.*)
ptoa = pdetoae[{Ε["x"], Ε["y"]}[x, y], {grid, grid}, difforder];
del = Most@Rest@# &;
ae = del /@ del@# & /@ ptoa@Simplify`PWToUnitStep@pdeS;
aebc = MapAt[del, ptoa@bc, List /@ Range@4];
{b, mat} = CoefficientArrays[{ae, aebc} // Flatten,
Outer[#[#2, #3] &, {Ε["x"], Ε["y"]}, grid, grid] // Flatten];
sollst = LinearSolve[-mat, N@b];
(* Alternatively, if you're confused about del: *)
fullae = ptoa@Simplify`PWToUnitStep@pdeS;
fullaebc = ptoa@bc;
{b, mat} = CoefficientArrays[{fullae, fullaebc} // Flatten,
sollst = LeastSquares[-mat, N@b, Method -> Direct]; // AbsoluteTiming
solmat = ArrayReshape[sollst, {2, points, points}];
{solΕx, solΕy} = ListInterpolation[#, {grid, grid}] & /@ solmat;
Outer[DensityPlot[#1[First[#2][x, y]], Evaluate[{x, domain}], Evaluate[{y, domain}],
PlotRange -> All, PlotLabel -> #1[Last[#2]], ColorFunction -> "AvocadoColors",
PlotPoints -> 50] &, {Re,
Im}, {{solΕx,
"\!\(\*SubscriptBox[\(Ε\), \(x\)]\)"}, {solΕy,
"\!\(\*SubscriptBox[\(Ε\), \(y\)]\)"}}, 1]], ImageSize -> Large]
Mathematica graphics
• 1
$\begingroup$ Something does not look quite right in the FEM solution, as both the Re and Im part do not show the requested 0 DirichletConditions. I think even for a noisy solution the DirichletCondition should be respected. But I don't immediately see what the problem is. $\endgroup$ – user21 Jan 9 '17 at 11:07
• $\begingroup$ @user21 Yeah, to be precise, it seems that only the DirichletConditions for Ε["y"] is ignored. Very strange. $\endgroup$ – xzczd Jan 9 '17 at 11:56
• $\begingroup$ @xzczd Thanks a lot for starting me in the right direction in the with the 4 issues I had with understanding of my problem and for finding out that as of now the finite element approach is not yet working. $\endgroup$ – Quit007 Feb 14 '17 at 9:26
• 2
$\begingroup$ @xzczd I just executed your code on Mathematica v11.0 and v11.1. With v11.1 Re(Ey) and Im(Ey) are calculated correctly by NDSolve. With v11.0 I get noise in Re(Ey) and Im(Ey), too. $\endgroup$ – Matthias Bernien Feb 21 '18 at 10:01
• $\begingroup$ @MatthiasBernien Oh, happy to know the bug is fixed! $\endgroup$ – xzczd Feb 21 '18 at 10:26
Your Answer
|
9f580b18fc0281f3 | The atomic emission spectrum for sodium ($\ce{Na}$) is completely dominated by a line in the range of yellow, about $590~\mathrm{nm}$ (to be more precise, it's a doublet). Here is how it looks like:
a) shows the emission spectrum and b) the absorption
This line is due to the transition $3\mathrm{p} \to 3\mathrm{s}$. I know the Rydberg equation which can be used to predict the transitions in hydrogen. I would like to know if there is a way to predict the transitions for other elements, such as the alkali metal $\ce{Na}$.
• $\begingroup$ When the Schrödinger equation is able to be solved for multi-electron systems, then it should be possible to predict the transition energies. $\endgroup$ – LDC3 May 3 '15 at 16:13
The origin of the Rydberg equation is the Bohr model's equation for the energy of a hydrogenic state: $E_\text{B} = -\dfrac{R_{H}hc}{n^2}$
This is accurate for other hydrogenic atoms, with only one electron too (bearing in mind that $R_{X}$ is proportional to the atomic number squared).
When you get to non-hydrogenic atoms you need to model inter-electron repulsion as well. This can be done by introducing an empirical parameter $\delta$, the quantum defect: $E_\text{B} = -\dfrac{R_{X}hc}{(n-\delta_X)^2}$. For sodium the quantum defect is 1.37, allowing good approximation of the transition energies between n-states in the electronic spectrum.
I don't think there is a similar back-of-the-envelope method to calculate the differences in energies between multi-electron atom l-energy levels (like the 3s-3p). The theory of angular momentum tends to mathematical complexity. There are likely good computationally derived approximations though.
Your Answer
|
469a60689101fa5a | Skip to main content
Chemistry LibreTexts
4: Postulates and Principles of Quantum Mechanics
• Page ID
• 4.1: The Wavefunction Specifies the State of a System
Postulate 1: Every physically-realizable state of the system is described in quantum mechanics by a state function that contains all accessible physical information about the system in that state.
• 4.2: Quantum Operators Represent Classical Variables
Every observable in quantum mechanics is represented by an operator which is used to obtain physical information about the observable from the state function. For an observable that is represented in classical physics by a function \(Q(x,p)\), the corresponding operator is \(Q(\hat{x},\hat{p})\).
• 4.3: Observable Quantities Must Be Eigenvalues of Quantum Mechanical Operators
• 4.4: The Time-Dependent Schrödinger Equation
While the time-dependent Schrödinger equation predicts that wavefunctions can form standing waves (i.e., stationary states), that if classified and understood, becomes easier to solve the time-dependent Schrödinger equation for any state. Stationary states can also be described by the time-independent Schrödinger equation (used when the Hamiltonian is not explicitly time dependent). The solutions to the time-independent Schrödinger equation still have a time dependency.
• 4.5: The Eigenfunctions of Operators are Orthogonal
The eigenvalues of operators associated with experimental measurements are all real; this is because the eigenfunctions of the Hamiltonian operator are orthogonal, and we also saw that the position and momentum of the particle could not be determined exactly. We now examine the generality of these insights by stating and proving some fundamental theorems. These theorems use the Hermitian property of quantum mechanical operators, which is described first.
• 4.6: Heisenburg Uncertainy Principle III - Commuting Operators
If two operators commute then both quantities can be measured at the same time with infinite precision, if not then there is a tradeoff in the accuracy in the measurement for one quantity vs. the other. This is the mathematical representation of the Heisenberg Uncertainty principle.
• 4.E: Postulates and Principles of Quantum Mechanics (Exercises)
These are homework exercises to accompany Chapter 4 of McQuarrie and Simon's "Physical Chemistry: A Molecular Approach" Textmap. |
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Probability Seminar
The probability seminar takes place on Wednesdays at 4:00 pm in room B3.02.
Organisers: Wei Wu, Stefan Adams, Stefan Grosskinsky
Term 3 2018-19
April 24: Giuseppe Cannizzaro (Warwick)
Title: A new Universality Class for random interfaces in (1+1)-dimensions: the Brownian Castle
Abstract: In the context of randomly fluctuating interfaces in (1+1)-dimensions two Universality Classes have generally been considered, the Kardar-Parisi-Zhang (KPZ) and the Edwards-Wilkinson (EW). Models within these classes exhibit universal fluctuations under 1:2:3 and 1:2:4 scaling respectively. Starting from a modification of the classical Ballistic Deposition model we will show that this picture is not exhaustive and another Universality Class, whose scaling exponents are 1:1:2, has to be taken into account. We will describe how it arises, briefly discuss its connections to KPZ and EW and introduce a new stochastic process, the Brownian Castle, deeply connected to the Brownian Web, which should capture the large-scale behaviour of models within this Class. This is joint work with M. Hairer.
May 1: Jhih-Huang Li (Academia Sinica)
Title: Universality of the Random-Cluster Model
Abstract: The random-cluster model is a generalization of Bernoulli percolation, Ising model and Potts model. Most of the results we knew were only valid for the square lattice. In this talk, we explain how to use star-triangle transformations to transport a connection property of the model from the square lattice onto an isoradial graph, thus getting a universality result.
May 8: Erik Slivken (Paris)
Title: Large random pattern-avoiding permutations
Abstract: A pattern in a permutation is a subsequence with a specific relative order. What can we say about a typical large random permutation that avoids a particular pattern? We use a variety of approaches. For certain classes we give a geometric description that relates these classes to other types of well-studied concepts like random walks or random trees. Using the right geometric description we can find the the distribution of certain statistics like the number and location of fixed points. This is based on joint work with Christopher Hoffman and Douglas Rizzolo.
May 15: Remi Rhodes (Marseilles)
Title: Exploring the Liouville Conformal Field Theory
Abstract: I will review the construction of the Liouville conformal field theory (LCFT), which was introduced in the eighties by Polyakov in the context of string theory. Nowadays it has become a topic of interest in probability theory as a prototype of random Riemannian geometry in 2D and as the conjectural scaling limit of random planar maps. Then I will review recent progress related to exact formulae for correlation functions and discuss how the conformal bootstrap can be implemented mathematically.
Based on joint works with F. David, A. Kupiainen et V. Vargas.
May 22: Nina Holden (Zurich)
Title: Cardy embedding of uniform triangulations
Abstract: A uniformly sampled triangulation is a canonical model for a discrete random surface. The Cardy embedding is a discrete conformal embedding of triangulations which is based on percolation observables. We present a series of works where we prove convergence of Cardy embedded uniform triangulations to the continuum random surface known as Liouville quantum gravity. The project is a collaboration with Xin Sun, and is also based on our joint works with Bernardi, Garban, Gwynne, Lawler, Li, and Sepulveda
May 29: Lisa Hartung (Mainz)
Title: The Ginibre characteristic polynomial and Gaussian Multiplicative Chaos
Abstract: It was proven by Rider and Virag that the logarithm of the characteristic polynomial of the Ginibre ensemble converges to a logarithmically correlated random field. In this talk we will see how this connection can be established on the level if powers of the characteristic polynomial by proving convergence to Gaussian multiplicative chaos. We consider the range of powers in the whole so-called subcritical phase. (Joint work in progress with Paul Bourgade and Guillaume Dubach).
June 5: Giambattista Giacomin (Paris Diderot)
Title: Polymer pinning models: localization structures and effect of disorder.
Abstract. Localization/delocalization transitions transitions appear in several polymer models. One example is the DNA denaturation, i.e. separation of the strands of the molecule at high temperature. But other transitions may happen in the localized state, which correspond to different geometrical binding configurations of the two strands. This has been understood in homogeneous models, typically via exact solution. But more faithful modeling demands dealing with the inhomogeneous character of the strands (that are sequences of different monomers). The aim of the talk is to present these configuration transitions and explain that they can be seen as «condensation» transitions. We will then tackle the issue of the effect of disorder, seen as a way of taking into account the inhomogeneous character of the chains.
June 12: Maite Wilke Berenguer (Bochum)
Title: Simultaneous migration in the seed bank coalescent
Abstract: The geometric seed bank model was introduced to describe the evolution of a population with active and dormant forms (`seeds') on a structure Markovian in both directions of time, whose limiting objects posses the advantageous property of being moment duals of each other: The (biallelic) Fisher-Wright diffusion with seed bank component describing the frequency of a given type of alleles forward in time and a new coalescent structure named the seed bank coalescent describing the genealogy backwards in time.
In this talk more recent results on extensions of this model will be discussed, focusing on the seed bank model with simultaneous migration: in addition to the spontaneous migration modeled before, where individuals decided to migrate independently of each other, correlated migration where several individuals become dormant (or awake) simultaneously is included. In particular, we will discuss the effect of the correlation on the property of coming down from infinity.
This is joint work with J. Blath (TU Berlin), A. Gonzalez Casanova (UNAM), and N. Kurt (TU Berlin).
East Midlands Stochastic Analysis Seminar
June 19: Huaizhong Zhao (Loughborough)
Title: Random Periodicity: Theory and Modelling
Abstract: Random periodicity is ubiquitous in the real world. In this talk, I will provide the concepts of random periodic paths and periodic measures to mathematically describe random periodicity. It is proved that these two different notions are “equivalent”. Existence and uniqueness of random periodic paths and periodic measures for certain stochastic differential equations are proved. An ergodic theory is established. It is proved that for a Markovian random dynamical system, in the random periodic case, the infinitesimal generator of its Markovian semigroup has infinite number of equally placed simple eigenvalues including $0$ on the imaginary axis. This is in contrast to the mixing stationary case in which the Koopman-von Neumann Theorem says there is only one eigenvalue $0$, which is simple, on the imaginary axis. Geometric ergodicity for some stochastic systems is obtained. Possible applications e.g. in stochastic resonance will be discussed.
Term 2 2018-19
January 9: Benoit Laslier (Paris VII)
Title: Logarithmic variance for uniform homomorphisms on Z^2
Abstract: We study random functions from Z^2 to Z that change by exactly 1 between neighboring vertices and show that the variance in the center of a box grows logarithmically with the size of the box, together with various RSW type property for level lines of such function. This model is interesting both as a natural discrete version of taking a continuous function from R^2 to R at random, and also because it is an instance of the 6-vertex model (more precisely the square-ice point) which connects combinatorially (for different values of it's parameters) many well known models such as all FK model, UST or ASEP. The approach does not really on any exact solvability of the model, instead we use a new FKG inequality to adopt the renormalization approach that was developed for the continuity of phase transition for FK. This is joint work with Hugo Duminil-Copin, Matan Harel, Gourab Ray and Aran Raoufi.
January 16: TALK MOVED TO MS.05
Sander Dommers (Hull)
Title: Metastability in the reversible inclusion process
Abstract: In the reversible inclusion process with N particles on a finite graph each particle at a site x jumps to site y at rate (d+η_y)r(x,y), where d is a diffusion parameter, η_y is the number of particles on site y and r(x,y) is the jump rate from x to y of an underlying reversible random walk. When the diffusion d tends to 0 as the number of particles tends to infinity, the particles cluster together to form a condensate. It turns out that these condensates only form on the sites where the underlying random walk spends the most time.
Once such a condensate is formed the particles stick together and the condensate performs a random walk itself on much longer timescales, which can be seen as metastable (or tunnelling) behaviour. We study the rates at which the condensate jumps and show that in the reversible case there are several time scales on which these jumps occur depending on how far (in graph distance) the sites are from each other. This generalises work by Grosskinsky, Redig and Vafayi who study the symmetric case where only one timescale is present. Our analysis is based on the martingale approach by Beltrán and Landim.
This is joint work with Alessandra Bianchi and Cristian Giardinà.
January 23: Thomas Bothner (King's College)
Title: When J. Ginibre met E. Schrödinger
Abstract: The real Ginibre ensemble consists of square real matrices whose entries are i.i.d. standard normal random variables. In sharp contrast to the complex and quaternion Ginibre ensemble, real eigenvalues in the real Ginibre ensemble attain positive likelihood. In turn, the spectral radius of a real Ginibe matrix follows a different limiting law for purely real eigenvalues than for non-real ones. Building on previous work by Rider, Sinclair and Poplavskyi, Tribe, Zaboronski, we will show that the limiting distribution of the largest real eigenvalue admits a closed form expression in terms of a distinguished solution to an inverse scattering problem for the Zakharov-Shabat system. This system is directly related to several of the most interesting nonlinear evolution equations in 1+1 dimensions which are solvable by the inverse scattering method, for instance the nonlinear Schrödinger equation. The results of this talk are based on the recent preprint arXiv:1808.02419, joint with Jinho Baik.
January 30: Cyril Labbé (Ceremade)
Title: Localisation of the continuous Anderson hamiltonian in 1d
Abstract: Consider the so-called Anderson hamiltonian obtained by perturbing the Laplacian with a white noise on a segment of size L. This operator is intimately connected to random matrix models and plays an important role for the study of the parabolic Anderson model. I will present a complete description of the bottom of the spectrum of this operator when L goes to infinity. Joint work with Laure Dumaz (Paris Dauphine).
February 6: Ewain Gwynne (Cambridge)
Title: The fractal dimension of Liouville quantum gravity: monotonicity, universality, and bounds
Abstract: It is an open problem to construct a metric on $\gamma$-Liouville quantum gravity (LQG) for $\gamma \in (0,2)$, except in the special case $\gamma=\sqrt{8/3}$. Nevertheless, the Hausdorff dimension $d_\gamma$ of the conjectural LQG metric is well-defined in the following sense. For a large class of approximations of $\gamma$-LQG distances --- including random planar maps, Liouville first passage percolation, Liouville graph distance, and the Liouville heat kernel --- there is a notion of dimension (in terms of a certain exponent associated with the model) and these exponents all agree with one another.
I will give an overview of some recent progress on understanding $d_\gamma$. In particular, I will discuss the relationships between different exponents, the proof the $\gamma\mapsto d_\gamma$ is strictly increasing, and new upper and lower bounds for $d_\gamma$. These bounds are consistent with (and numerically quite close to) the Watabiki prediction for the value of $d_\gamma$ for $\gamma \in (0,2)$. However, in an extended regime corresponding Liouville first passage percolation with parameter $\xi >2/d_2$, or equivalently LQG with central charge greater than 1, the bounds are inconsistent with the analytic continuation of Watabiki's prediction for certain parameter values.
Based on joint works with Jian Ding, Nina Holden, Tom Hutchcroft, Jason Miller, Josh Pfeffer, and Xin Sun.
February 13: Nicos Georgiou (Sussex)
Title: Last passage times in discontinuous environments
Abstract: We are studying a last passage percolation model on the two dimensional lattice, where the environment is a field of independent random exponential weights with different parameters. Each variable is associated with a lattice vertex and its parameter is selected according to a discretization of lower semi-continuous parameter function that may admit discontinuities on a set of curves.
We prove a law of large numbers for the sequence of last passage times, defined as the maximum sum of weights which a directed path can collect from (0, 0) to a target point (Nx, Ny) as N tends to infinity and the mesh of the discretisation of the parameter function tends to 0 as 1/N. The LLN is cast in the form of a variational formula, optimised over a given set of macroscopic paths. Properties of maximizers to the variational formula above are investigated in two models where the parameter function allows for analytical tractability. This is joint work with Federico Ciech.
February 20: George Deligiannidis (Oxford)
Title: Boundary of the range of a random walk and the Folner property
Abstract: In this work we deal with the question of whether the range of a random walk is almost surely a Folner sequence and show the following results: (a) The size of the inner boundary of the range of recurrent, aperiodic random walks with finite second moment on the two-dimensional integer lattice and of aperiodic, integer-valued random walks in the standard domain of attraction of the symmetric Cauchy distribution, is almost surely of order $n\log^2 (n)$. (b) We establish a formula for the Folner asymptotic of transient co-cycles over an ergodic probability preserving transformation and use it to show that for transient random walk on groups which are not virtually cyclic, for almost every path, the range is not a F ̈olner sequence. (c) For aperiodic random walks in the domain of attraction of symmetric alpha- stable distributions with 1< α≤2, we prove a sharp polynomial upper bound for the decay at infinity of |∂Rn|/|Rn|. This last result shows that the range process of these random walks is almost surely a Folner sequence.
Joint work with Z.Kosloff and S. Gouezel.
February 27: Pierre-Francois Rodriguez (IHES)
Title: Sign cluster geometry of the Gaussian free field
Abstract: We consider the Gaussian free field on a class of transient weighted graphs G, and show that its sign clusters fall into a regime of strong supercriticality, in which two infinite sign clusters dominate (one for each sign), and finite sign clusters are necessarily tiny, with overwhelming probability. Examples of graphs G belonging to this class include cases in which the random walk on G exhibits anomalous diffusive behavior. Our findings also imply the existence of a nontrivial percolating regime for the vacant set of random interlacements on G. Based on joint work with A. Prévost and A. Drewitz.
March 6: Jere Koskera (Warwick)
Title: Asymptotic genealogies of interacting particle systems
Abstract: We consider time-evolving, weighted particle systems of fixed size in which a time step consists of a selection stage, during which each particle has a random number of offspring proportional to its weight, and a mutation stage, during which offspring locations are randomly perturbed, and the resulting particles are reweighted based on their new locations. Such interacting particle systems form a rich class of processes with applications in areas including, but not limited to, computational statistics and population genetics. The genealogical tree embedded into the particle system by the selection stages is a key analytical tool, as well as an object of interest in its own right. It is well known that in the neutral case, where particles always have equal weights, the genealogical tree of a fixed number of particles converges to the Kingman coalescent in the infinite system size limit. I will review this classical result, and show that it can be extended to non-neutral models under practically verifiable conditions. This is joint work with Paul Jenkins, Adam Johansen, and Dario Spano.
March 13: Erik Slivken (Paris) MOVED TO MAY 8
Title: Large random pattern-avoiding permutations
Term 1 2018-19
October 3: no seminar
October 10: Milton Jara (IMPA)
Title: Entropy methods in Markov chains
Abstract: Building upon Yau's relative entropy method, we derive a new strategy to obtain scaling properties of Markov chains with a large number of components. As an application, we obtain very precise estimates on the mixing properties of a mean-field spin system. Time permitting, we will also discuss the derivation of the speed of convergence of the hydrodynamic limit and non-equilibrium fluctuations of interacting particle systems.
October 17: Christian Webb (Aalto)
Title: On the statistical behavior of the Riemann zeta function
Abstract: A notoriously difficult problem of analytic number theory is to describe the behavior of the Riemann zeta function on the critical line. After reviewing some basic facts about the zeta function, I will discuss what can be said if the problem is relaxed by considering the behavior of the zeta function in the vicinity of a random point on the critical line.
Time permitting, I will also discuss how this problem is related to various models
of probability theory and mathematical physics. The talk is based on joint work with Eero Saksman.
October 24: no seminar
October 31: Paul Dario (ENS Paris)
Title: Quantitative result on the gradient field model through homogenization.
Abstract: Consider the standard uniformly elliptic gradient field model. It was proved by Funaki and Spohn in 1997, by a subadditivity argument, that the finite volume surface tension of this model converges to a limit called the surface tension. The goal of this talk is to show how one can use the tools developed in recent the theory of stochastic homogenization to obtain an algebraic rate of convergence of the surface tension. The analysis relies on the study of dual subadditive quantities, useful in stochastic homogenization, as well as a variational formulation of the partition function and the notion of displacement convexity from the theory of optimal transport.
November 7: Alexandros Eskenazis (Princeton)
Title: Nonpositive curvature is not coarsely universal
Abstract: A complete geodesic metric space of global nonpositive curvature in the sense of Alexandrov is called a Hadamard space. In this talk we will show that there exist metric spaces which do not admit a coarse embedding into any Hadamard space, thus answering a question of Gromov (1993). The main technical contribution of this work lies in the use of metric space valued martingales to derive the metric cotype 2 inequality with sharp scaling parameter for Hadamard spaces. The talk is based on joint work with M. Mendel and A. Naor.
November 14: David Dereudre (Lille)
Title: DLR equations and rigidity for the Sine-beta process
Abstract: We investigate Sine β, the universal point process arising as the thermodynamic limit of the microscopic scale behavior in the bulk of
one-dimensional log-gases, or β- ensembles, at inverse temperature β > 0. We adopt a statistical physics perspective, and give a description of
Sineβ using the Dobrushin-Landford-Ruelle (DLR) formalism by proving that it satisfies the DLR equations: the restriction of Sine β to a compact set,
conditionally to the exterior configuration, reads as a Gibbs measure given by a finite log-gas in a potential generated by the exterior configuration.
Moreover, we show that Sineβ is number-rigid and tolerant in the sense of Ghosh-Peres, i.e. the number, but not the position, of particles lying
inside a compact set is a deterministic function of the exterior configuration. Our proof of the rigidity differs from the usual strategy and is robust enough
to include more general long range interactions in arbitrary dimension. (joint work with A. Hardy, M. Maïda and T. Leblé)
November 21: Nadia Sidorova (UCL)
Title: Localisation and delocalisation in the parabolic Anderson model
Abstract: The parabolic Anderson problem is the Cauchy problem for the heat equation on the integer lattice with random potential. It describes the mean-field behaviour of a continuous-time branching random walk. It is well-known that, unlike the standard heat equation, the solution of the parabolic Anderson model exhibits strong localisation. In particular, for a wide class of iid potentials it is localised at just one point. However, in a partially symmetric parabolic Anderson model, the one-point localisation breaks down for heavy-tailed potentials and remains unchanged for light-tailed potentials, exhibiting a range of phase transitions.
November 28: Steffen Dereich (Münster)
Abstract: The concept of quasi-process stems from probabilistic potential theory. Although the notion may not be that familiar nowadays, it is connected to various current developments in probability. For instance, interlacements, as introduced by Sznitman, are Poisson point processes with the intensity measure being a quasi-process. Furthermore, extensions of Markov families as recently derived for certain self-similar Markov processes are closely related to quasi-processes and entrance boundaries.In this talk, we start with a basic introduction of parts of the classical potential theory and then focus on branching Markov chains. The main result will be a spine construction of a branching quasi-process.
The talk is based on joint work with Martin Maiwald (WWU Münster).
December 5: Weijun Xu (Oxford)
Title: Weakuniversality of KPZ
Abstract: We establish weak universality of the KPZ equation for a class of continuous interface fluctuation models initiated by Hairer-Quastel, but now with general nonlinearities beyond polynomials. Joint work with Martin Hairer. |
f19f5ce4cbfca8f9 | Is electric charge truly conserved for bosonic matter? | PhysicsOverflow
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Is electric charge truly conserved for bosonic matter?
+ 6 like - 0 dislike
Even before quantization, charged bosonic fields exhibit a certain "self-interaction". The body of this post demonstrates this fact, and the last paragraph asks the question.
Notation/ Lagrangians
Let me first provide the respective Lagrangians and elucidate the notation.
I am talking about complex scalar QED with the Lagrangian $$\mathcal{L} = \frac{1}{2} D_\mu \phi^* D^\mu \phi - \frac{1}{2} m^2 \phi^* \phi - \frac{1}{4} F^{\mu \nu} F_{\mu \nu}$$ Where $D_\mu \phi = (\partial_\mu + ie A_\mu) \phi$, $D_\mu \phi^* = (\partial_\mu - ie A_\mu) \phi^*$ and $F^{\mu \nu} = \partial^\mu A^\nu - \partial^\nu A^\mu$. I am also mentioning usual QED with the Lagrangian $$\mathcal{L} = \bar{\psi}(iD_\mu \gamma^\mu-m) \psi - \frac{1}{4} F^{\mu \nu} F_{\mu \nu}$$ and "vector QED" (U(1) coupling to the Proca field) $$\mathcal{L} = - \frac{1}{4} (D^\mu B^{* \nu} - D^\nu B^{* \mu})(D_\mu B_\nu-D_\nu B_\mu) + \frac{1}{2} m^2 B^{* \nu}B_\nu - \frac{1}{4} F^{\mu \nu} F_{\mu \nu}$$
The four-currents are obtained from Noether's theorem. Natural units $c=\hbar=1$ are used. $\Im$ means imaginary part.
Noether currents of particles
Consider the Noether current of the complex scalar $\phi$ $$j^\mu = \frac{e}{m} \Im(\phi^* \partial^\mu\phi)$$ Introducing local $U(1)$ gauge we have $\partial_\mu \to D_\mu=\partial_\mu + ie A_\mu$ (with $-ieA_\mu$ for the complex conjugate). The new Noether current is $$\mathcal{J}^\mu = \frac{e}{m} \Im(\phi^* D^\mu\phi) = \frac{e}{m} \Im(\phi^* \partial^\mu\phi) + \frac{e^2}{m} |\phi|^2 A^\mu$$ Similarly for a Proca field $B^\mu$ (massive spin 1 boson) we have $$j^\mu = \frac{e}{m} \Im(B^*_\mu(\partial^\mu B^\nu-\partial^\nu B^\mu))$$ Which by the same procedure leads to $$\mathcal{J}^\mu = \frac{e}{m} \Im(B^*_\mu(\partial^\mu B^\nu-\partial^\nu B^\mu))+ \frac{e^2}{m} |B|^2 A^\mu$$
Similar $e^2$ terms also appear in the Lagrangian itself as $e^2 A^2 |\phi|^2$. On the other hand, for a bispinor $\psi$ (spin 1/2 massive fermion) we have the current $$j^\mu = \mathcal{J}^\mu = e \bar{\psi} \gamma^\mu \psi$$ Since it does not have any $\partial_\mu$ included.
Now consider very slowly moving or even static particles, we have $\partial_0 \phi, \partial_0 B \to \pm im\phi, \pm im B$ and the current is essentially $(\rho,0,0,0)$. For $\phi$ we have thus approximately $$\rho = e (|\phi^+|^2-|\phi^-|^2) + \frac{e^2}{m} (|\phi^+|^2 + |\phi^-|^2) \Phi$$ Where $A^0 = \Phi$ is the electrostatic potential and $\phi^\pm$ are the "positive and negative frequency parts" of $\phi$ defined by $\partial_0 \phi^\pm = \pm im \phi^\pm$. A similar term appears for the Proca field.
For the interpretation let us pass back to SI units, in this case we only get a $1/c^2$ factor. The "extra density" is $$\Delta \rho = e\cdot \frac{e \Phi}{mc^2}\cdot |\phi|^2$$ That is, there is an extra density proportional to the ratio of the energy of the electrostatic field $e \Phi$ and the rest mass of the particle $mc^2$. The sign of this extra density is dependent only on the sign of the electrostatic potential and both frequency parts contribute with the same sign (which is superweird). This would mean that classicaly, the "bare" charge of bosons in strong electromagnetic fields is not conserved, only this generalized charge is.
After all, it seems a bad convention to call $\mathcal{J}^\mu$ the electric charge current. By multiplying it by $m(c^2)/e$ it becomes a matter density current with the extra term corresponding to mass gained by electrostatic energy. However, that does not change the fact that the "bare charge density" $j^0$ seems not to be conserved for bosons.
Now to the questions:
• On a theoretical level, is charge conservation at least temporarily or virtually violated for bosons in strong electromagnetic fields? (Charge conservation will quite obviously not be violated in the final S-matrix, and as an $\mathcal{O}(e^2)$ effect it will probably not be reflected in first order processes.) Is there an intuitive physical reason why such a violation is not true for fermions even on a classical level?
• Charged bosons do not have a high abundance in fundamental theories, but they do often appear in effective field theories. Is this "bare charge" non-conservation anyhow reflected in them and does it have associated experimental phenomena?
• Extra clarifying question: Say we have $10^{23}$ bosons with charge $e$ so that their charge is $e 10^{23}$. Now let us bring these bosons from very far away to very close to each other. As a consequence, they will be in a much stronger field $\Phi$. Does their measured charge change from $e 10^{23}$? If not, how do the bosons compensate in terms of $\phi, B, e, m$? If this is different for bosons rather than fermions, is there an intuitive argument why?
This post imported from StackExchange Physics at 2015-06-09 14:50 (UTC), posted by SE-user Void
asked Sep 24, 2014 in Theoretical Physics by Void (1,620 points) [ revision history ]
edited Jun 9, 2015 by Void
Most voted comments show all comments
By Noether's theorem, Noether currents are conserved since they are derived from an infinitesimal symmetry; they are observable iff they are gauge invariant. Are you missing something in the answer by Qmechanic?
@ArnoldNeumaier I added an extra clarifying question to what bugs me. I am well aware about the conservation and observability, I mainly wanted to inquire about the deeper physical explanation of these facts.
The charge doesn't change, as it is an integral over the whole space - only the charge density develops a very localized peak. What should need compensation?
Note that bare stuff doesn't matter; it is irrelevant scaffolding removed by renormalization.
Just a dumb idea: Maybe this is somehow related to the fact in the SM, introducing mass-terms for the bosons simply as $\frac{1}{2}m\phi^{*}\phi$ without a higgs field or mechanism breaks the gauge symmetry, and therefore is no conserved current corresponding to the by the mass term broken symmetry?
@Dilaton: Yes, there seems to be something funky about massive or charged elementary bosons. I was just hoping there is an established argument what exactly is the crux of this funkiness -- perhaps through such things as charged pions and their relation to $U(1)$.
Most recent comments show all comments
@drake I just meant that for example Proca mass terms such as $\frac{1}{2}m^2 B^{*\nu} B_{\nu}$ break gauge symmetries such as $U(1)$, and could therefore spoil charge conservation.
@Dilaton I don't get your point... Here the gauge field is $A$, which doesn't have any mass term. In the SM one wants to give mass to the gauge fields. I think you are wrong.
4 Answers
+ 3 like - 0 dislike
Comments to the question (v3):
1. In contrast to QED with fermionic matter, in QED with bosonic matter, the full Noether current ${\cal J}^{\mu}$ (for global gauge transformations) tends to depend explicitly on the gauge potential $A^{\mu}$, see e.g. Refs. 1-2 and this Phys.SE post.
2. The reason for this difference is because the QED Lagrangian for fermionic (bosonic) matter typically contains one (two) spacetime derivative(s) $\partial_{\mu}$, which after minimal coupling $\partial_{\mu}\to D_{\mu}$ leads to e.g. no (a) quartic matter-matter-photon-photon coupling term, respectively.
3. The full Noether current ${\cal J}^{\mu}$ is a gauge-invariant and conserved quantity, $d_{\mu }{\cal J}^{\mu} \approx 0$. [Here $d_{\mu}\equiv\frac{d}{dx^{\mu}}$ means a total spacetime derivative, and the $\approx$ symbol means equality modulo eom.] The electric charge $Q=\int \! d^3x ~{\cal J}^{0}$ is a conserved quantity.
4. The only physical observables in a gauge theory are gauge-invariant quantities. The quantity $j^{\mu}$, which OP calls the "bare current", is not gauge-invariant, and hence not a consistent physical observable to consider.
5. As Trimok mentions in a comment, the situation for non-Abelian (as opposed to Abelian) Yang-Mills is radically different. The full Noether current ${\cal J}^{\mu a}$ (for global gauge transformations) is a conserved $d_{\mu }{\cal J}^{\mu a} \approx 0$, but ${\cal J}^{\mu a}$ is not gauge-invariant (or even gauge covariant), and hence not a consistent physical observable to consider. There is not a well-defined observable for color charge that one can measure. This follows also from Weinberg-Witten theorem (for spin 1): A theory with a global non-Abelian symmetry under which massless spin-1 particles are charged does not admit a gauge- and Lorentz-invariant conserved current, cf. Ref. 3.
1. M. Srednicki, QFT, Chapter 61.
2. M.D. Schwartz, QFT and the Standard Model, Section 8.3 and Chapter 9.
3. M.D. Schwartz, QFT and the Standard Model, Section 25.3.
answered Sep 24, 2014 by Qmechanic (2,860 points) [ no revision ]
Yes, some of these are the observations which lead me to this question. But say we have a macroscopic material with bosonic charged particles, object it to a very strong electrostatic field and measure it's charge. Would we have to be measuring $\mathcal{J}^0$ under all conditions? I guess 3. implies yes, and that means we would measure the object to have a charge different from the zero field situation. The extra "non-bare" charge obviously comes from the field, but this is a very different notion from the usual intuition of "charge".
${\cal J}^{\mu}$ is a covariant quantity, then it should verify $D_\mu {\cal J}^{\mu}=0$, but a conserved quantity corresponds to $\partial_\mu {\cal J}^{\mu}=0$. So, here, are covariant and conserved current compatible notions ? (for instance, this is not the case in Yang-Mills theories).
I updated the answer.
+ 1 like - 0 dislike
I have actually taken the time to compute the equations of motion and the situation is more complicate than I previously thought. The Lagrangian in the static situation $\vec{A} = 0, \partial_t \to 0$ reads
$$\mathcal{L} = -\frac{1}{2} |\nabla \phi|^2 - \frac{1}{2} m^2 |\phi|^2 + e^2 |\Phi|^2 |\phi|^2 + \frac{1}{2} |\nabla \Phi|^2 $$
which leads to EOM:
$$(\Delta - m^2 + 2 e^2 |\Phi|^2) \phi = 0$$
$$ (\Delta - 2 e^2 |\phi|^2) \Phi = 0 $$
Amongst other things, this implies that minimally coupled bosons do not act as a usual source of the electromagnetic field at all. As it stands (a more detailed analysis of the non-stationary equations might show otherwise), the bosons actually "easen" their motion (effectively loose mass) in the presence of the electromagnetic field at the cost of weakening (rendering massive and short-range) the electromagnetic field.
The coupling constant $e$ really does not have any reasonable interpretation in terms of a usual charge. For instance, the sign of $e$ is irrelevant and the particles and antiparticles of quantized $\phi$ have the same effect on $\Phi$. The $U(1)$ charge is just a conserved quantity with no intuitive interpretation in terms of the usual charge. Hence, the original form of the question does not have a proper meaning; $U(1)$ coupling for bosons simply means something totally different than for fermions.
(If you have any more observations or a different view, please contribute, I am interested.)
answered Jun 10, 2015 by Void (1,620 points) [ no revision ]
Are you allowed to simply put $A=0$? It changes the dynamics.
@ArnoldNeumaier: If we still hold $\partial_t \to 0$ a nonzero $\vec{A}$ would only make $|\Phi|^2 \to |\Phi|^2 - |A|^2$ and an extra $\vec{A}$ equation coupled to $\phi$ similarly as in the $\Phi$ case.
+ 0 like - 0 dislike
Dear mods, I am sorry this answer is not graduate-upward level, but I have not been able to come up with a more sophisticated one.
1) Yes, the charge is truly conserved but the respective current depends on the 4-potential A. What it is confusing you, I think, is that the current for a scalar field depends on 4-potential $A$, whereas that of a spin-1/2 does not. This is obviously related to the number of derivatives in the Lagrangian kinetic term and, likewise, to the number of derivatives in the current. It can help you understand what it's going on to adopt the canonical formalism (also known as the language of gentlemen), in which in both cases the density (and the charge too) involves the product of the canonical momentum and the field, as it could not be otherwise because the charge is nothing else but the infinitesimal generator of $U(1)$ transformations for both the field and the canonical momentum.
2) What you call the "bare charge", which probably is not a good name since this term is reserved for something else, lacks of physical content before fixing a gauge, as it is not a gauge invariant quantity. Note however that one can always choose one's favorite gauge. And if one picks the temporal gauge (\(A_0 = 0\)), the charge does not depend on the 4-potential and the form is the same as your "bare charge", which is conserved in this gauge.
3) The only difference in the movement of spin-one-half particles and spin zero-particles in an electromagnetic field is a term proportional to \[\sigma_{\mu\nu}\, F^{\mu\nu}\]
in the equation for spin-1/2 particles. This term gives rise to the term
\[\bf{S}\cdot \bf{B} \]
in the non-relativistic limit, that is, the interaction between the spin of the particle and the magnetic field.
4) It can help you to get the equation in your answer to first think of the equation of motion in the non-relativist limit, which is the Schrödinger equation in an electromagnetic field, that is, the Schrödinger equation replacing partial derivatives with gauge-covariant ones (for scalar particles, for spin-1/2 there is the additional term I wrote above).
answered Jun 12, 2015 by drake (875 points) [ revision history ]
edited Jun 12, 2015 by drake
+ 0 like - 4 dislike
The charge $e$ introduced into your Lagrangians/equations is a constant in time by definition, no Noether theorem is necessary to "conserve" it: $\frac{de}{dt}=0$.
Another thing is your equations/theory or "charge definition" via equations/solutions (as an integral bla-bla-bla). Here everything depends on your equations. Do not think that equations for bosons are already well established and finalized. For one formulation you get one result, for another you do another. So, there is no 'truly" thing, keep it firmly in your mind!
answered Jun 9, 2015 by Vladimir Kalitvianski (112 points) [ revision history ]
edited Jun 9, 2015 by Vladimir Kalitvianski
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594318db973802e8 |
Margaret Mead
Review article
The Awareness of the Fascial System
Fascia is a cacophony of functions and information, a completely adaptable entropy complex. The fascial system has a solid and a liquid component, acting in a perfect symbiotic synchrony. Each cell communicates with the other cells by sending and receiving signals; this concept is a part of quantum physics and it is known as quantum entanglement: a physical system cannot be described individually, but only as a juxtaposition of multiple systems, where the measurement of a quantity determines the value for other systems. Fascial continuum serves as a target for different manual approaches, such as physiotherapy, osteopathy and chiropractic. Cellular behaviour and the inclusion of quantum physics background are hardly being considered to find out what happens between the operator and the patient during a manual physical contact. The article examines these topics. According to the authors' knowledge, this is the first scientific text to offer manual operators’ new perspectives to understand what happens during palpatory contact. A fascial cell has not only memory but also the awareness of the mechanometabolic information it feels, and it has the anticipatory predisposition in preparing itself for alteration of its natural environment.
Introduction & Background
The fascial system supports the human body in its vital functions: it ensures the maintenance of posture and motor expression and helps achieve a salutogenic homeostasis [1-5]. Fascia also influences the emotional sphere [6]. In a previous study, we showed that the fascia not only functions in support and communication, protection and sustenance but also provides protection to the entire body through the epidermis that is an inherent part of the fascia [7]. We provide a new definition of the fascia: “The fascia is any tissue that contains features capable of responding to mechanical stimuli. The fascial continuum is the result of the evolution of the perfect synergy among different tissues, capable of supporting, dividing, penetrating and connecting all the districts of the body, from the epidermis to the bone, involving all the functions and organic structures. The continuum constantly transmits and receives mechanometabolic information that can influence the shape and function of the entire body. These afferent/efferent impulses come from the fascia and the tissues that are not considered as part of the fascia in a bi-univocal mode [7].” Fascia consists partly of solid matter (bones, fat, muscles, ligaments and reciprocal tension membrane) and partly of liquid fascia (blood and lymph), combined in a single functional continuum [8]. Based on the idea of liquid and solid fascia, we have recently presented a new theoretical model, with the aim of explaining the importance of liquids (pressure, direction and velocity) in a final and functional expression of the fascial system: Rapid Adaptability of Internal Network (RAIN) [8]. The fascial system cannot be divided into layers because of its entropic nature based on the highest capacity to adapt itself to different stress scenarios; fascia has the freedom to respond to any stimuli (internal/external), thanks to the lack of predefined structural and fluidic patterns or negentropic behaviour [9]. The nervous system does not regulate the morphological features of the fascial system. The latter is a holobiont, an asymptotic behaviour between the mechanical environment inside and outside the cell and the modification of the environment itself. A non-movement syntropic is based on a heuristic basis: the maximum configuration of the order and at the same time maximum differentiation with the aim to have access to all information [10-12]. Tissues use a stigmergic communication through a stochastic process to achieve optimal adaptation strategies; tissues change their characteristics and the means of transmission of external information inward. It is not only about a tissue, but it is, in fact, an awareness [10,12-15]. The article discusses the fascial cellular response modality to mechanical stimuli and the possible influence on the fascial tissue by a manual palpation during a manual treatment, in terms of quantum physics and physiology.
Palpation is a mechanical induction (with perpendicular or tangential pressure) towards a static (solid) and hydrostatic (liquid) tissue, within a specific period. Palpation is an important part of the physical examination, a manual exploration of the tactile perception [16]. The tactile perception system gathers information about the environment using mechanoreceptors and thermoreceptors residing in the skin, as well as from the deeper mechanoreceptors located in the myofascial and articular system [16]. The palm of the hand has specific receptors, which permit to determine the size of the palpated tissues (Meissner corpuscle and Merkel cell complex) and understand the tissues' ability to deform under a rapid or continuous touch (Ruffini and Pacinian corpuscles). Touch can discriminate a solid feature for about 200 microns [16]. Furthermore, thermoreceptors are capable of detecting temperature variations (myelinated type Aδ fibres and non-myelinated type C fibres) [16-17]. According to the Bayesian perceptive interference, mechanoreceptors and thermoreceptors can detect wetness of the palpated tissues through a multimodal integration [17]. Inspection and palpation activate the superior and inferior parietal lobules in the operator's cortex [16]. Palpation is part of a personal experiential memory bank useful to find tissue anomalies, and it is a manual art [18]. As with all arts, the result is not always reproducible in the same way. In literature, there seems to be some disagreement in the palpatory results between different operators using the same patients; even the palpatory practical experience does not seem to make the difference in deciphering patients' tissue abnormalities [19]. The first tissue touched by hands is the epidermis. If the pressure increases, the soft tissues perceive the tension created, for example, the muscles and the visceral fascia tissues that connect or cover all organs in the body. Fascia is an interrelation of liquids and solids [8]. In physics, the “state of rest” is the macroscopic condition of a body that is not subject to motion [20]. The static (solid objects) and the hydrostatics (fluids objects) study bodies in a state of rest. In case of a fluid at rest, the individual constituent particles (atoms and molecules) move because of the phenomenon of thermal agitation (absent at absolute zero temperature); hence, macroscopically, the fluid is at rest, but microscopically, the individual particles keep moving. Similarly, in a solid body, the individual constituent particles (atoms, ions and molecules) are constantly in motion; however, this movement is less evident than in fluid bodies. While particles in fluids move freely within a volume containing them, this motion of particles in a solid is more like a vibrational motion, i.e., in the case of an ionic crystalline solid, ions undergo minimum translations around their reticular position [20-22]. The pressure resulting from palpation and received by the tissues creates numerous vectors that are dispersed in multiple directions, above ground and in depth [23]. Different models try to explain what happens to a tissue deformed by mechanical stress, but none with a satisfactory solution [24]. What we know for sure is that palpation and manual approaches to tissues can alter the cellular behaviour of the fascial system [25].
Changes in Cell Behavior in Case of Mechanical Stress
Fascial cells derive from tissues developed from the mesoderm and partly from the neural crests of the ectoderm (neck and face): skin, fat, blood and lymph, connective tissues and muscles, other tissues that cover and sustain the nervous, blood and lymphatic system, as well as bone tissues. A seamless web of connective tissues that covers, supports and penetrates the viscera is part of the fascial system [2,7]. Although the difference between the cells in different tissues is evident, their behaviour in case of mechanical stress is very similar. Cellular deformation makes the cell aware of what happens inside it and in the environment in which it lives, resulting in behaviours that can anticipate the deformation [26]. The shape of a cell is a stochastic process based on a perfect relationship between entropy and syntropy, and the second law of thermodynamics is not violated. Every living system cannot die of thermal death; it is in response to the entropy that the system has an opposite syntrophic process of natural restoration of order. An example is the metabolism of living organisms in which anabolism is present in response to catabolism. In living beings, syntropy would act as a tendency to project itself into the future, being a feature coming from the future towards the past [26]. This means that the exploration of the surrounding environment allows us to improve and not repeat the same mistakes: to understand what will happen, thanks to the accumulated experiences. A reasonable attempt to craft a logical explanation for that feature is a part of various studies on pre-stimulus response found in human beings and animals. They discovered that both heart rate and skin conductance and other biological parameters vary before emotional phenomena appear. This would demonstrate the man's instinctive tendency to prevent future acts according to the principle of syntropy [26]. Mechanical events suffered by the fascial holobiont are actively maintained in its memory, with the aim of being already predisposed to a new action of the same stressors [27]. A fascial cell has not only memory, but also has the awareness of the mechanometabolic information it feels, and it has the anticipatory predisposition in preparing itself for alteration of its natural intra- and extracellular milieu. A cellular genome can monitor itself in response to mechano-metabolic stimuli, obtaining information not only from the extracellular matrix (ECM) but also from other cells and tissues. Ribonucleic acid (RNA) is not only a carrier of proteins but also has a great influence on epigenetics. Micro RNAs and other non-coding RNAs (ncRNA, endo-siRNAs, piRNAs, antisense and long ncRNAs) determine cell-to-cell gene activation and expression. Particularly, RNA interference (RNAi) is involved in learning, self-propagation and amplification of information, involving different tissues [28]. Deoxyribonucleic acid (DNA) is involved in transmitting and transporting information outside the cells in different and distant sites from its original site of replication. Circulating ring-like non-genomic DNA seems to be involved in other cells specialization and the production of specific proteins such as titin in cardiac muscles [29]. A cell communicates with the other cells by sending and receiving signals; this concept is part of quantum physics and it is known as quantum entanglement: a physical system cannot be described individually, but only as a juxtaposition of multiple systems, and the measurement of a quantity determines the value for other systems [30]. According to the quantum theory, each element has a non-hierarchical form of organization, and it only responds when necessary (mechanical and metabolic stimulation) [31].
Cell membrane
Nearly all the cell membranes have an action potential, that is, an electric potential difference (voltage) between the cytosol, the interior of the cell that has a negative voltage and the extracellular space that has positive charges. The resting membrane potential is determined by the uneven distribution of the phospholipids and because of this potential difference across the cell membrane, the membrane is said to be polarized. A typical voltage across a cell membrane is between -60 mV and -70 mV [32-34]. This time-varying electric field becomes the source of magnetic and electromagnetic fields. The study of electromagnetic fields in living cells is placed under the aegis of magnetobiology [35]. Any change in the electromagnetic field can deform a cell for a very short time; this deformation improves and accelerates adaptation processes and cellular functions, such as the synthesis of adenosine triphosphate (ATP) or controlling the enzymatic processes of DNA [35]. Cell deformation and the electromagnetic field are influenced and facilitated by the anisotropic rotation of the phospholipids and water molecules sited in the membrane [36-37]. The ions exchanged during an action potential make a change in the cell volume, causing a cell deformation and inducing a transient electromagnetic field. This mechanism creates microwaves, which radiate to other cell membranes influencing the rotation or the orientation of electrons and affecting other electromagnetic fields [35]. We could compare this phenomenon, already decoded for the central nervous system, to what happens to neurons. Whenever an electrical impulse runs along a neuron, a small electric field surrounds that cell. The sum of all the electric fields created by the neural activity modifies the activation of the single neurons, increasing the synchronism of the neural activity. The effect of an increased neural coordination is defined as an ephaptic effect or ephaptic coupling [38]. All cells present in the liquid and solid fascial system have electromagnetic fields stimulated by membrane deformation; the greater the synchronicity of this event among multiple cells, the higher the quantum coherence and the functional cellular effectiveness. This quantum synchronicity in physics is called the Larmor precession [39]. The electromagnetic fields can travel faster than the electrical signal, and they can cross the whole body almost instantly [39]. The cells that make up the tissues show a greater capacity for awareness if they act together [5,12]. The cell membrane has unstable areas, which allow mechanical information to spread widely and to be different. These fewer complex areas or system of pervasive information fields (PIFs) are the cellular entropic border, where PIFs represent the cellular negentropy (also called syntropy), in a perfect balance to keep the homeostasis maintained [26]. Negentropy allows information to be less fragmented and reach the internal part of a cell more effectively, optimizing the entropic actions of the different cellular components for the survival and evolution of the cell. This mechanism is under the aegis of the Schrödinger equation that describes the changes over time of a physical system (the cell) [26]. The remaining membrane is semi-permeable, permitting an oxidative phosphorylation that is the process in which ATP is formed or chemiosmosis: entropy and chemiosmosis are complementary, and they can be considered fractal reiteration within the evolution [40]. Palpation deforms tissues and cells: we can assume that the palpatory act is the beginning of a therapeutic act on patients, where the palpated tissues gather information about the operator (awareness of the fascial system). In biological systems, it is the ability to maintain cellular homeostasis through the evaluation of information and the energy transfer resulting from a cell-cell interaction [40]. Cell-cell junctions are specialized membrane areas consisting of multiprotein complexes that provide contact between the neighbouring cells or between a cell and the extracellular matrix. These intercellular contacts may be transient, permitting the passage of information and mesenchymal cells, or they can create stable bonds to form a barrier [41]. Types of specific transmembrane junctions are defined depending on the specific receptor, which mediates the transmission of information between the membrane and the cytoskeleton; furthermore, receptors are regulated by membrane trafficking, the existence of membrane lipids and the membrane shape when a mechanical stress occurs. These junctions rapidly communicate membrane deformation to other cells, irrespective of the liquids present around such as the extracellular matrix [41]. Cellular aggregation through intercellular junctions forms a moving entity as a viscous fluid with irregular behaviour (entropic), with the aim to minimize friction across cell surfaces [42]. Cell aggregates are always present in the extracellular matrix, interstitial fluids and the bloodstream and lymphatic flow; living cells behave like fluid-filled sponges. Taking a "liquid" behaviour, cellular movement within these liquids creates waves or vibrations, which results in another way of communication between cells and tissues: vibration frequency is measurable with a spatial and temporal scale [43]. This fascial "wet network" strengthens our RAIN theoretical model [8].
The cytoskeleton is a complex network of interlinking of microtubules and cytoplasmic filaments. The cytoskeleton is a structure that helps cells maintain their shape and internal organization and provides specific characteristics to them (such as stiffness, flexibility and motility). Many forces acting on a junction between cells come from within, especially when a junction is combined with the actomyosin complex that forms within the cytoskeleton (intrinsic force), via the cadherin-catenin complex [41]. This connection allows the cell to communicate faster, especially if the extracellular matrix (biopolymers in a three-dimensional context) is in small quantities; the same transfer of information (mechanometabolic) using the same modalities could spend hours to arrive at other tissues; this behavioural and temporal context of the cell is not completely understood [41]. Cell deformation by intrinsic forces could give rise to very fast or extremely slow messages. We can assume that palpation can create mechanical stresses that continue over time. Cells are deformed following vectors, as the shape of the man's footprinted in the sand [41]. Cellular morphology affects the extracellular matrix shape, influencing how the mechanometabolic resulting message will be transported: slow, fast or conditioning its direction [44]. Probably this kind of "mirror" behaviour would allow the cell to better respond to stress solicitation, improving its adaptation [45]. The cytoskeleton plays an important role for cell conformational memory, thanks to a metabolic regulator as the target of rapamycin (TOR), which acts on polymerization of actin, determining its cytoskeletal conformation [45]. A fundamental role played by the actomyosin complex is collecting information outside the cell and at the same time reinforcing the cell. The actin forms a network able to branch out within the cell by G-actin monomers exchange; these monomers place themselves in the terminal part of the neighbouring filament [11]. The growth of the actin filament pushes against the inner cell membrane, creating small curvatures on the outer surface of the plasma membrane (lamellipodium) or longer ramifications (filopodium): this phenomenon is called treadmilling of actin [11]. The growth of these ramifications is interrupted when capping proteins will be associated at the terminal part of the actin filament (F-actin); instead, their disassembly is due to the action of depolymerizing factors (actin depolymerizing factor - ADF) [11]. The treadmilling of the actin phenomenon is entropic. This behaviour allows the cell to be aware of its surroundings; the cell changes its morphology or implements specific mechanisms of mechanotransduction by the resistance encountered. The myosin acts as a stabilizing protein, located ventrally and dorsally each branch; it produces a contractile force equal to the tension produced by the actin filament. Myosin counterbalances and simultaneously sends back the mechanical information into the cell [46]. These two opposing forces make the membrane stiffer: the actin that pushes out and the myosin that pulls inward. This is not a negative result, tight as a guitar string; the cell becomes more sensitive to the changes in tension and improves the accuracy of mechanotransduction [46]. The information received readily crosses the cell nucleus (microseconds) and produces an instantaneous response of the genes [47].
The cell must have an entropic organization because it is aware that the knowledge is asymptotic, which means that the stimuli received come from an ambiguous environment and the cell can change its morphology in real time because of this absence of syntropy. The fact that it is impossible to know the cellular external environment is in accordance with Heisenberg's uncertainty principle in quantum mechanics: it asserts a fundamental limit to the precision with which certain physical properties of a particle, such as position and momentum, can be measured (at the same time or in subsequent occasions) and known [48]. Microtubules (MTs) form a part of the cytoskeleton that provides cells with structure and shape, or more specifically, microtubule-associated proteins (MAPs) [5]. MTs are tubular polymers of tubulin, formed by the polymerization of a dimer of two tubulin proteins that might have a different length. Each tubulin is defined as a dimer (two monomers: alpha and beta), with a dipole (positive and negative electrical charge); the latter gives the electromagnetic property to the tubulin structure and the MAPs complex [5]. Motor proteins (dynein and kinesin) are found in all MAPs, and they can move rapidly along the MPs, transporting different molecules [5]. The same motor proteins can help MPs in contraction, to improve cell adaptation resulting from a mechanical information received; MPs can produce forces measurable in pN (piconewton) [47]. MAPs are a more stable network than the actomyosin filaments, keeping the memory of mechanometabolic events longer and encoding them faster. According to Sherrington, MAPs can be compared to a cell of the nervous system put inside another cell [5]. MAPs carry electromagnetic information and vibrations as a rapid communication tool, in response to a cellular morphological change, towards the inside of the cell (DNA) and towards the neighbouring cells. This mechanism can be compared to a cellular awareness [5]. Electromagnetism is associated with another law of quantum physics, the "non-local entanglement": when two cells or molecules are in contact, this connection creates an unbroken microbiological link; in this way, every cell is aware of what happens to another cell, no matter how far away [5]. During a morphological change, the energy released by MAPs and by other cellular structures is minimal or quantum. It is possible to summarize this quantum energy by the formula: E = hv ("E" represents a particle's energy, "v" is the oscillation, "h" is the Planck's constant) [5]. Another example of the close relationship between biology and quantum physics is represented by biophotons or quantum particles. These are contained and then emitted by the entire fascial system (ultraweak photon emission or UPE), both liquid and solid, as already discussed in another article [49]. DNA carries and produces electromagnetic fields and UPE; its filaments carry electrons, produced internally or in other cellular structures, or even from other distant cells. The DNA adapts itself to cellular morphological changes, increasing the transcription of genes activated by specific regions of DNA which are sensitive to the flow of electromagnetic energy: electromagnetic response elements or EMRE [39]. The deformation of the cellular structures also activates the transcription of other genes, which are specific to a mechanical stimulus [50]. Palpation is an instrument useful for interacting with cells, not only locally but also in a long-distance, thanks to the different means of communication provided by the tissue on the surface and below. Further scientific and experimental research will be needed to better understand how the principles of quantum physics work within biology. We quote a sentence that summarizes our intent to associate these two themes, the fascial system and the palpation: “quantum physics and electro-dynamics shape all molecules and thus determine molecular recognition, the workings of proteins, and DNA…all this is quantum physics and a natural basis for life and everything we see [14].”
The fascial system supports, protects, evolves and connects the human body. It can be divided into solid and liquid fascia, closely inter-linked, without interruption between the different components, making the subdivision of the fascia into layers unnecessary. Healthcare professionals, such as medical doctors and physiotherapists, have different clinical tools for patient assessment, including palpation. The touch meets the skin as the first fascial tissue, but the resulting cell deformation can get deeper and it can reach the DNA of different cell tissues. The morphological deformation of the cellular components starts numerous mechanometabolic and electromagnetic messages; this information will affect the entire body structure, as the palpated area and the remaining non-palpated tissues. The mechanisms that allow cells to communicate with each other are based on the principles of physiology and quantum physics. The article reviewed these scientific concepts to understand the importance of palpation in the clinical setting and the complexity of cellular behaviour, not completely understood. Further research and studies are needed to implement our knowledge of two fundamental sciences: Biology and Physics.
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Review article
The Awareness of the Fascial System
Author Information
Bruno Bordoni Corresponding Author
Cardiology, Foundation Don Carlo Gnocchi / (IRCCS) Institute of Hospitalization and Care, Milano, ITA
Marta Simonelli
Osteopathy, (SOFI) School of French-Italian Osteopathy, Pisa, ITA
Ethics Statement and Conflict of Interest Disclosures
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6d05b10ca4ae698c | Mathematical model
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A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, biology, earth science, chemistry) and engineering disciplines (such as computer science, electrical engineering), as well as in non-physical systems such as the social sciences (such as economics, psychology, sociology, political science). Mathematical models are also used in music,[1] linguistics[2] and philosophy (for example, intensively in analytic philosophy).
A model may help to explain a system and to study the effects of different components, and to make predictions about behavior.
Elements of a mathematical model[edit]
Mathematical models can take many forms, including dynamical systems, statistical models, differential equations, or game theoretic models. These and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include logical models. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed.
In the physical sciences, a traditional mathematical model contains most of the following elements:
1. Governing equations
2. Supplementary sub-models
1. Defining equations
2. Constitutive equations
3. Assumptions and constraints
1. Initial and boundary conditions
2. Classical constraints and kinematic equations
Mathematical models are usually composed of relationships and variables. Relationships can be described by operators, such as algebraic operators, functions, differential operators, etc. Variables are abstractions of system parameters of interest, that can be quantified. Several classification criteria can be used for mathematical models according to their structure:
• Linear vs. nonlinear: If all the operators in a mathematical model exhibit linearity, the resulting mathematical model is defined as linear. A model is considered to be nonlinear otherwise. The definition of linearity and nonlinearity is dependent on context, and linear models may have nonlinear expressions in them. For example, in a statistical linear model, it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables. Similarly, a differential equation is said to be linear if it can be written with linear differential operators, but it can still have nonlinear expressions in it. In a mathematical programming model, if the objective functions and constraints are represented entirely by linear equations, then the model is regarded as a linear model. If one or more of the objective functions or constraints are represented with a nonlinear equation, then the model is known as a nonlinear model.
Linear structure implies that a problem can be decomposed into simpler parts that can be treated independently and/or analyzed at a different scale and the results obtained will remain valid for the initial problem when recomposed and rescaled.
Nonlinearity, even in fairly simple systems, is often associated with phenomena such as chaos and irreversibility. Although there are exceptions, nonlinear systems and models tend to be more difficult to study than linear ones. A common approach to nonlinear problems is linearization, but this can be problematic if one is trying to study aspects such as irreversibility, which are strongly tied to nonlinearity.
• Static vs. dynamic: A dynamic model accounts for time-dependent changes in the state of the system, while a static (or steady-state) model calculates the system in equilibrium, and thus is time-invariant. Dynamic models typically are represented by differential equations or difference equations.
• Explicit vs. implicit: If all of the input parameters of the overall model are known, and the output parameters can be calculated by a finite series of computations, the model is said to be explicit. But sometimes it is the output parameters which are known, and the corresponding inputs must be solved for by an iterative procedure, such as Newton's method or Broyden's method. In such a case the model is said to be implicit. For example, a jet engine's physical properties such as turbine and nozzle throat areas can be explicitly calculated given a design thermodynamic cycle (air and fuel flow rates, pressures, and temperatures) at a specific flight condition and power setting, but the engine's operating cycles at other flight conditions and power settings cannot be explicitly calculated from the constant physical properties.
• Discrete vs. continuous: A discrete model treats objects as discrete, such as the particles in a molecular model or the states in a statistical model; while a continuous model represents the objects in a continuous manner, such as the velocity field of fluid in pipe flows, temperatures and stresses in a solid, and electric field that applies continuously over the entire model due to a point charge.
• Deterministic vs. probabilistic (stochastic): A deterministic model is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables; therefore, a deterministic model always performs the same way for a given set of initial conditions. Conversely, in a stochastic model—usually called a "statistical model"—randomness is present, and variable states are not described by unique values, but rather by probability distributions.
• Deductive, inductive, or floating: A deductive model is a logical structure based on a theory. An inductive model arises from empirical findings and generalization from them. The floating model rests on neither theory nor observation, but is merely the invocation of expected structure. Application of mathematics in social sciences outside of economics has been criticized for unfounded models.[3] Application of catastrophe theory in science has been characterized as a floating model.[4]
• Strategic vs non-strategic Models used in game theory are different in a sense that they model agents with incompatible incentives, such as competing species or bidders in an auction. Strategic models assume that players are autonomous decision makers who rationally choose actions that maximize their objective function. A key challenge of using strategic models is defining and computing solution concepts such as Nash equilibrium. An interesting property of strategic models is that they separate reasoning about rules of the game from reasoning about behavior of the players.[5]
In business and engineering, mathematical models may be used to maximize a certain output. The system under consideration will require certain inputs. The system relating inputs to outputs depends on other variables too: decision variables, state variables, exogenous variables, and random variables.
Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as parameters or constants. The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system (represented by the state variables).
Objectives and constraints of the system and its users can be represented as functions of the output variables or state variables. The objective functions will depend on the perspective of the model's user. Depending on the context, an objective function is also known as an index of performance, as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved (computationally) as the number increases.
For example, economists often apply linear algebra when using input-output models. Complicated mathematical models that have many variables may be consolidated by use of vectors where one symbol represents several variables.
A priori information[edit]
Mathematical modeling problems are often classified into black box or white box models, according to how much a priori information on the system is available. A black-box model is a system of which there is no a priori information available. A white-box model (also called glass box or clear box) is a system where all necessary information is available. Practically all systems are somewhere between the black-box and white-box models, so this concept is useful only as an intuitive guide for deciding which approach to take.
Usually it is preferable to use as much a priori information as possible to make the model more accurate. Therefore, the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly. Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in a human system, we know that usually the amount of medicine in the blood is an exponentially decaying function. But we are still left with several unknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine in blood? This example is therefore not a completely white-box model. These parameters have to be estimated through some means before one can use the model.
In black-box models one tries to estimate both the functional form of relations between variables and the numerical parameters in those functions. Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately. If there is no a priori information we would try to use functions as general as possible to cover all different models. An often used approach for black-box models are neural networks which usually do not make assumptions about incoming data. Alternatively the NARMAX (Nonlinear AutoRegressive Moving Average model with eXogenous inputs) algorithms which were developed as part of nonlinear system identification[6] can be used to select the model terms, determine the model structure, and estimate the unknown parameters in the presence of correlated and nonlinear noise. The advantage of NARMAX models compared to neural networks is that NARMAX produces models that can be written down and related to the underlying process, whereas neural networks produce an approximation that is opaque.
Subjective information[edit]
Sometimes it is useful to incorporate subjective information into a mathematical model. This can be done based on intuition, experience, or expert opinion, or based on convenience of mathematical form. Bayesian statistics provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: we specify a prior probability distribution (which can be subjective), and then update this distribution based on empirical data.
An example of when such approach would be necessary is a situation in which an experimenter bends a coin slightly and tosses it once, recording whether it comes up heads, and is then given the task of predicting the probability that the next flip comes up heads. After bending the coin, the true probability that the coin will come up heads is unknown; so the experimenter would need to make a decision (perhaps by looking at the shape of the coin) about what prior distribution to use. Incorporation of such subjective information might be important to get an accurate estimate of the probability.
In general, model complexity involves a trade-off between simplicity and accuracy of the model. Occam's razor is a principle particularly relevant to modeling, its essential idea being that among models with roughly equal predictive power, the simplest one is the most desirable. While added complexity usually improves the realism of a model, it can make the model difficult to understand and analyze, and can also pose computational problems, including numerical instability. Thomas Kuhn argues that as science progresses, explanations tend to become more complex before a paradigm shift offers radical simplification.[7]
For example, when modeling the flight of an aircraft, we could embed each mechanical part of the aircraft into our model and would thus acquire an almost white-box model of the system. However, the computational cost of adding such a huge amount of detail would effectively inhibit the usage of such a model. Additionally, the uncertainty would increase due to an overly complex system, because each separate part induces some amount of variance into the model. It is therefore usually appropriate to make some approximations to reduce the model to a sensible size. Engineers often can accept some approximations in order to get a more robust and simple model. For example, Newton's classical mechanics is an approximated model of the real world. Still, Newton's model is quite sufficient for most ordinary-life situations, that is, as long as particle speeds are well below the speed of light, and we study macro-particles only.
Note that better accuracy does not necessarily mean a better model. Statistical models are prone to overfitting which means that a model is fitted to data too much and it has lost its ability to generalize to new events that were not observed before.
Training and tuning[edit]
Any model which is not pure white-box contains some parameters that can be used to fit the model to the system it is intended to describe. If the modeling is done by an artificial neural network or other machine learning, the optimization of parameters is called training, while the optimization of model hyperparameters is called tuning and often uses cross-validation.[8] In more conventional modeling through explicitly given mathematical functions, parameters are often determined by curve fitting[citation needed].
Model evaluation[edit]
A crucial part of the modeling process is the evaluation of whether or not a given mathematical model describes a system accurately. This question can be difficult to answer as it involves several different types of evaluation.
Fit to empirical data[edit]
Usually, the easiest part of model evaluation is checking whether a model fits experimental measurements or other empirical data. In models with parameters, a common approach to test this fit is to split the data into two disjoint subsets: training data and verification data. The training data are used to estimate the model parameters. An accurate model will closely match the verification data even though these data were not used to set the model's parameters. This practice is referred to as cross-validation in statistics.
Defining a metric to measure distances between observed and predicted data is a useful tool for assessing model fit. In statistics, decision theory, and some economic models, a loss function plays a similar role.
While it is rather straightforward to test the appropriateness of parameters, it can be more difficult to test the validity of the general mathematical form of a model. In general, more mathematical tools have been developed to test the fit of statistical models than models involving differential equations. Tools from nonparametric statistics can sometimes be used to evaluate how well the data fit a known distribution or to come up with a general model that makes only minimal assumptions about the model's mathematical form.
Scope of the model[edit]
Assessing the scope of a model, that is, determining what situations the model is applicable to, can be less straightforward. If the model was constructed based on a set of data, one must determine for which systems or situations the known data is a "typical" set of data.
The question of whether the model describes well the properties of the system between data points is called interpolation, and the same question for events or data points outside the observed data is called extrapolation.
As an example of the typical limitations of the scope of a model, in evaluating Newtonian classical mechanics, we can note that Newton made his measurements without advanced equipment, so he could not measure properties of particles travelling at speeds close to the speed of light. Likewise, he did not measure the movements of molecules and other small particles, but macro particles only. It is then not surprising that his model does not extrapolate well into these domains, even though his model is quite sufficient for ordinary life physics.
Philosophical considerations[edit]
Many types of modeling implicitly involve claims about causality. This is usually (but not always) true of models involving differential equations. As the purpose of modeling is to increase our understanding of the world, the validity of a model rests not only on its fit to empirical observations, but also on its ability to extrapolate to situations or data beyond those originally described in the model. One can think of this as the differentiation between qualitative and quantitative predictions. One can also argue that a model is worthless unless it provides some insight which goes beyond what is already known from direct investigation of the phenomenon being studied.
An example of such criticism is the argument that the mathematical models of optimal foraging theory do not offer insight that goes beyond the common-sense conclusions of evolution and other basic principles of ecology.[9]
Significance in the natural sciences[edit]
Mathematical models are of great importance in the natural sciences, particularly in physics. Physical theories are almost invariably expressed using mathematical models.
Throughout history, more and more accurate mathematical models have been developed. Newton's laws accurately describe many everyday phenomena, but at certain limits theory of relativity and quantum mechanics must be used.
It is common to use idealized models in physics to simplify things. Massless ropes, point particles, ideal gases and the particle in a box are among the many simplified models used in physics. The laws of physics are represented with simple equations such as Newton's laws, Maxwell's equations and the Schrödinger equation. These laws are a basis for making mathematical models of real situations. Many real situations are very complex and thus modeled approximate on a computer, a model that is computationally feasible to compute is made from the basic laws or from approximate models made from the basic laws. For example, molecules can be modeled by molecular orbital models that are approximate solutions to the Schrödinger equation. In engineering, physics models are often made by mathematical methods such as finite element analysis.
Different mathematical models use different geometries that are not necessarily accurate descriptions of the geometry of the universe. Euclidean geometry is much used in classical physics, while special relativity and general relativity are examples of theories that use geometries which are not Euclidean.
Some applications[edit]
Often when engineers analyze a system to be controlled or optimized, they use a mathematical model. In analysis, engineers can build a descriptive model of the system as a hypothesis of how the system could work, or try to estimate how an unforeseeable event could affect the system. Similarly, in control of a system, engineers can try out different control approaches in simulations.
A mathematical model usually describes a system by a set of variables and a set of equations that establish relationships between the variables. Variables may be of many types; real or integer numbers, boolean values or strings, for example. The variables represent some properties of the system, for example, measured system outputs often in the form of signals, timing data, counters, and event occurrence (yes/no). The actual model is the set of functions that describe the relations between the different variables.
• One of the popular examples in computer science is the mathematical models of various machines, an example is the deterministic finite automaton (DFA) which is defined as an abstract mathematical concept, but due to the deterministic nature of a DFA, it is implementable in hardware and software for solving various specific problems. For example, the following is a DFA M with a binary alphabet, which requires that the input contains an even number of 0s.
The state diagram for M
M = (Q, Σ, δ, q0, F) where
S1 S2 S1
S2 S1 S2
The state S1 represents that there has been an even number of 0s in the input so far, while S2 signifies an odd number. A 1 in the input does not change the state of the automaton. When the input ends, the state will show whether the input contained an even number of 0s or not. If the input did contain an even number of 0s, M will finish in state S1, an accepting state, so the input string will be accepted.
The language recognized by M is the regular language given by the regular expression 1*( 0 (1*) 0 (1*) )*, where "*" is the Kleene star, e.g., 1* denotes any non-negative number (possibly zero) of symbols "1".
• Many everyday activities carried out without a thought are uses of mathematical models. A geographical map projection of a region of the earth onto a small, plane surface is a model which can be used for many purposes such as planning travel.[10]
• Another simple activity is predicting the position of a vehicle from its initial position, direction and speed of travel, using the equation that distance traveled is the product of time and speed. This is known as dead reckoning when used more formally. Mathematical modeling in this way does not necessarily require formal mathematics; animals have been shown to use dead reckoning.[11][12]
• Population Growth. A simple (though approximate) model of population growth is the Malthusian growth model. A slightly more realistic and largely used population growth model is the logistic function, and its extensions.
• Model of a particle in a potential-field. In this model we consider a particle as being a point of mass which describes a trajectory in space which is modeled by a function giving its coordinates in space as a function of time. The potential field is given by a function and the trajectory, that is a function , is the solution of the differential equation:
that can be written also as:
Note this model assumes the particle is a point mass, which is certainly known to be false in many cases in which we use this model; for example, as a model of planetary motion.
• Model of rational behavior for a consumer. In this model we assume a consumer faces a choice of n commodities labeled 1,2,...,n each with a market price p1, p2,..., pn. The consumer is assumed to have an ordinal utility function U (ordinal in the sense that only the sign of the differences between two utilities, and not the level of each utility, is meaningful), depending on the amounts of commodities x1, x2,..., xn consumed. The model further assumes that the consumer has a budget M which is used to purchase a vector x1, x2,..., xn in such a way as to maximize U(x1, x2,..., xn). The problem of rational behavior in this model then becomes a mathematical optimization problem, that is:
subject to:
This model has been used in a wide variety of economic contexts, such as in general equilibrium theory to show existence and Pareto efficiency of economic equilibria.
See also[edit]
1. ^ D. Tymoczko, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice (Oxford Studies in Music Theory), Oxford University Press; Illustrated Edition (March 21, 2011), ISBN 978-0195336672
2. ^ Andras Kornai, Mathematical Linguistics (Advanced Information and Knowledge Processing),Springer, ISBN 978-1849966948
3. ^ Andreski, Stanislav (1972). Social Sciences as Sorcery. St. Martin’s Press. ISBN 0-14-021816-5.
4. ^ Truesdell, Clifford (1984). An Idiot's Fugitive Essays on Science. Springer. pp. 121–7. ISBN 3-540-90703-3.
5. ^ Li, C., Xing, Y., He, F., & Cheng, D. (2018). A Strategic Learning Algorithm for State-based Games. ArXiv.
6. ^ Billings S.A. (2013), Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains, Wiley.
7. ^ "Thomas Kuhn". Stanford Encyclopedia of Philosophy. 13 August 2004. Retrieved 15 January 2019.
8. ^ Thornton, Chris. "Machine Learning Lecture". Retrieved 2019-02-06.
9. ^ Pyke, G. H. (1984). "Optimal Foraging Theory: A Critical Review". Annual Review of Ecology and Systematics. 15: 523–575. doi:10.1146/
10. ^ "GIS Definitions of Terminology M-P". LAND INFO Worldwide Mapping. Retrieved January 27, 2020.
11. ^ Gallistel (1990). The Organization of Learning. Cambridge: The MIT Press. ISBN 0-262-07113-4.
12. ^ Whishaw, I. Q.; Hines, D. J.; Wallace, D. G. (2001). "Dead reckoning (path integration) requires the hippocampal formation: Evidence from spontaneous exploration and spatial learning tasks in light (allothetic) and dark (idiothetic) tests". Behavioural Brain Research. 127 (1–2): 49–69. doi:10.1016/S0166-4328(01)00359-X. PMID 11718884. S2CID 7897256.
Further reading[edit]
Specific applications[edit]
External links[edit]
General reference |
778722c086ae805f | PSICAN - Paranormal Studies and Inquiry Canada
Written by PSICAN Editorial Staff
Non Local SETI
Discussing this issue five years ago with some colleagues about SETI I extensively answered as follows.
<< Concerning quantum entanglement, it is an ascertained reality when two particles have first interacted together and then are separated even at the longest distance: in addition to Dr. Aspect’s experiment in 1982 and previous EPR (Einstein, Podolsky, Rosen) “gedanken” experiment in the fifties, what fully proves the reality of entanglement is just experiments on quantum teleportation of simple particles such as photons or electrons. Up to here everyone agrees.
But several new hypotheses postulate that this mechanism might be active not only in the micro world but also in the mesoscopic (intermediate scale) and even macroscopic domains.
Not only this. Someone (including Prof. John Wheeler with his hypothesis on so called retrocausation) is convinced that, considering just particles, some hidden link may exist between all particles in the Universe. Why? Because at the zero time (Big Bang start) particles were all connected and strictly interacting. It is not yet known which particle parameters are affected here, in addition to the spin and polarization (maybe the quark color too?), but if this hypothesis is true then, at a certain level, everything should be non-locally linked inside this universe, and possibly also between multiverses and possible other dimensions. In the sense that maybe a sort of “fossil link” might be still present now. The framework of this idea stands upon the so called “implicate order” elaborated many years ago by quantum physicist David Bohm, who best than others (including nobel prize Wolfgang Pauli) created the mathematical apparatus that describes what happens in the entanglement process, by expanding the Schrödinger equation (most important equation of quantum mechanics) with an additional parameter called “quantum potential” whose character is non-local (namely instantaneous). According to Bohm’s physics reality is constituted of two interconnected domains: a local and a non-local one. The first obeys to Newton/Einstein physics (finite light speed, etc., on which no one argues), the second obeys to another law of which quantum theory is only the tip of an iceberg. Someone (mostly philosophers) think that the second realm is just “consciousness” while the first is matter/energy.
In reality, if particles all over the universe maintain some hidden link together, this means that even the cells of our body are affected. In particular: neurons. And now I come to Dr. Thaheld hypothesis on “non-local astrobiology”. The hypothesis is that neurons (being constituted themselves by particles) are able to receive non-locally some kind of sentient information, which is then explicated to brainwaves (alpha, delta, theta). From this point on all the investigation becomes absolutely conventional, because whatever is the method for sending information, then that information is deposited inside neurons whose electric activity produce brainwaves. So you just have to look into them, first using an EEG apparatus (of very high-resolution kind, in this specific case) and then using a specific algorithm (Fourier, Karhunen-Loewe, multi-scale computational procedures, or even a simple time-series analysis) which is able to extract a signal from the background noise inside brainwaves (in fact what is of interest here is not the way in which brainwave goes but if something is deposited inside). There might be a very structured message that can be decoded by such an algorithm, so that the analysis becomes exactly identical to the one used in standard SETI. The only difference between NLSETI and SETI is that in the first case information is assumed to be received instantaneously through the quantum entanglement mechanism, and in the second case through radio (or optical) photons whose intensity decreases with the inverse of the square of distance.
Of course many detractors of this hypothesis will say that the entanglement mechanism is not able to transfer information because we acknowledge a quantum entanglement state only when we observe one of the two particles and at that moment we make the wave function linking them collapse, which is true per se, of course. But they persist not to understand that when a non-local link like quantum entanglement is established that can be used to transfer information from a quantum to a classical state (such as the neurons in the brain) in form of information in the brainwave, which we can measure indeed.
It is obvious that standard SETI is totally limited by the distance factor, while the probability to find an intelligent signal increases just with the source’s distance, but at the same time the signal amplitude diminishes exponentially as well: here is the trap. We have tried to increase radiotelescope aperture or acquisition mode (such as the recent Square Kilometer Array technique, for instance), receiver sensitivity, amplifier power, number of channels that can be detected simultaneously (up to one billion, nowadays) through a multi-channel spectrum analyzer, power of the algorithm of analysis, etc. After 50 years, according to the SETI Institute’s protocol (see Note * at the end about SETI PROTOCOL), the result is just discomforting.
Therefore trying the NLSETI way is not that bad and not even so expensive. Of course it has nothing to do with telepathy, because the quantitative analysis is intended to be done directly doing measures on neurons through brainwave. If something true were found (after checking all possible sources of systematic error or interference) we should have two results: a) the entanglement mechanism is extended everywhere in the universe; b) an informative sentient message could be quantitatively decoded. And, apart from the hypothesis per se, I am interested only in the quantitative/mathematical aspect of the test.
Where does the “message” come from? We can just hypothesize. Assuming that all the sources of background noise can be eliminated, we have two possibilities: a) someone particularly intelligent has sent the message through non-local means using maybe “quantum repeaters” placed somewhere in the universe (in order to avoid decoherence); b) the test subject himself has been able to connect non-locally to a sort of “server” that is placed not in the cyberspace but rather in the quantum void, where a sort of “big informative library” has been deposited since eons. Maybe everyone everywhere uploads spontaneously this kind of information there all the time without even knowing to do it. If a completely new information is found this means that the test subject has downloaded something from there and then transferred it to the neural electrical activity which then manifests the info and we reconstruct it technically.
This is, grossly speaking, the assumption of NLSETI. As you see it is of double importance: for fundamental physics and for SETI. An attempt does not hurt.
Some other considerations:
1. It is potentially possible to send an answer in quasi-real time by irradiating neural cells using a nanopulsed (with a modulated structure) Laser and/or a magnetic field. Some experiments have been already done in medical labs concerning the entanglement between two test tubes containing neural cells that have been linked previously together through a chemical substance such as an anesthetic.
2. Independently from all of this a quantum theory of the brain already exists due to mathematical physicist Roger Penrose and to neurophysiologist Stuart Hameroff. In brief this theory says that microtubules inside each neuron work in a so called “orchestrated entanglement”. They are all together (the entire ensemble of them) described by a wave function (typical equation of quantum mechanics). Normally this wave function collapses when a quantum system is observed, before which all possibilities coexist being overlapped all together. Differently from normal quantum systems, in the brain the wave function collapses spontaneously more or less every 1/40 sec: this collapse is a physical (geometrodynamic in terms of spacetime) collapse at the Planck scale level (quantum void: 10^-33 cm) where both relativity and quantum theory (due to the micro-scale involved here) are required. What then in simple terms the wave function collapse consist of? It is a “consciousness moment”. We experience it normally one million of times or so every day. Therefore the so called “consciousness” in order to manifest itself needs a neural correlate: the brain. Otherwise the wave function remains suspended and it doesn’t collapse. This means in few words that consciousness and physical matter cannot exist the one without the other (thus contradicting almost everything of religions of any kind). Prof. Hameroff found that just microtubules are the ideal physical vectors to permit entanglement inside the brain, because they are well insulated by any kind of interaction that might destroy quantum information. The quantity of consciousness depends on how much energy is inside the brain, namely how much mass of active elements (microtubules) are present able to trigger full consciousness, whose “power” is inversely proportional to the velocity of the process in terms of time taken to make so that the wave function (uniting all microtubules in the brain in a quantum coherent domain) collapses. So: according to this hypothesis, if it will be demonstrated to be true, the brain is a purely quantum system. From this (even if Penrose & Hameroff are maybe unaware of Dr. Thaheld hypothesis on NLSETI) it is not difficult to deduce that: a) if all brains are quantum systems based on the entanglement mechanism within its components they are ideal communication centers; b) whatever comes from outside affects also consciousness. But we can measure only neurons and not what a person “feels”. This doesn’t exclude that at a consciousness (and not neural) level a person can potentially acquire suddenly ideas: namely, able to connect to a universal “server” or to receive “non-local emails” directly from someone (by the way: what is exactly a genius? And how exactly one becomes a genius?). It is exactly the same mechanism of Internet: the only difference is that the mechanism here is non-local. Therefore NLSETI, being experimental and not speculative, can permit to quantitatively prove or disprove the hypothesis of connection between intelligent beings in the universe through quantum entanglement.
Concerning the "quantum mind" theory by Penrose & Hameroff, in spite of Max Tegmark's rebuttal (and other's) it is based on the fact that microtubules (inside neurons) are highly isolated by a specific gel, therefore there is sufficient time to transfer information before the overall wave function collapses due to thermal effects.
I am sure that who is largely more advanced than us did two things: a) used and manipulated the quantum vacuum (playing with virtual particles using them as the elements of a quantum computer) in the same way in which we do with a silicon chip in order to order a library of universal information of every kind; b) send deliberately information everywhere hoping that someone catch it. Of course some persons catch the message unconsciously but not scientifically: “they” know it and so they decided to leave a track in the brainwave too in order to permit us to demonstrate the mechanism scientifically. Our duty is to verify scientifically this and, if present, to decode the information. Doing many trials and using many test subjects.
Fortunately this is not science fiction. Simply I think it is time to turn the page, if we really want to attempt a real communication with alien intelligence. I have the impression that if we really want to know more on true alien intelligence we have to understand more what exactly “reality” is. But I will never tell this at my public conferences (that I have not any more time to do for now) because then the idiot of the moment would immediately tell: “Oh yeessss. We live inside a “Matrix” ”. New Agers are truly a big problem here, as if they were created on purpose by someone in order to block research in its depth (more or less like CSICOP on the opposite side, from where I just quit recently much to my pleasure).
Scientists are alone when it’s time to lift the black curtain. But they are never alone when true science is replaced by accountancy.
* SETI PROTOCOL - They say that a SETI signal is considered as such only if it is persistent in time, namely that it comes always from the same alpha-delta coordinates (which of course must be detected by many observers everywhere in the world). Correct, of course. But at the same time highly limitative. In this truly bureaucratic way of the above said protocol, everything else is excluded, not only internal/external noise or interference (as it often happens), but also possible high-proper motion sources, namely: possible sources transiting inside the solar system (in substance they want to throw away dirty water together with the baby inside). Of course it is not so difficult to span the antenna inside an error circle that is slightly bigger and bigger until we find again the same signal at a slightly different coordinate position so that we can reconstruct the orbit and track it like a comet (with full happiness of Dr. Freeman Dyson). But this is not done. Why? Because the SETV branch of SETI is not politically correct. But this is not a scientific aptitude, this is religion, or even politics. Of course I still support (but not do it any more) standard SETI: sooner or later we’ll find something. But that something will be the result of a pure selection effect: just like to find some kind of aliens of the monkey type using black glasses or a smoked filter. There is more out there, methink. >> |
8aa3220cba8447d8 | Archive for the ‘Math’ Category
New approaches to cancer therapy using mathematics
Reporter: Irina Robu, PhD
Our bodies are made up of trillions of cells grouped together to form tissues and organs such as muscles, bones, the lungs and the liver. Genes inside each cell tell it when to grow, work, divide and die. Usually, our cells follow these commands and we stay healthy. Nevertheless, occasionally the instructions get mixed up, triggers our cells to grow and divide out of control or not die when they should. As more and more of these abnormal cells grow and divide, they can form a lump in the body called a tumor.
Cancer therapy thrives in shrinking tumors and frequently fails in the long run; however, a small number of cancer cells are resistant to treatment. The cancer cells expand to fill the space left by the cells that were destroyed. Using mathematical analysis and numerical simulations, Dr. Noble and Dr. Viossat, a mathematician at Université Paris-Dauphine proposed new approach to validate the concept of using a combination of biological, computational and mathematical models and they show how spatial constraints within tumors can be exploited to suppress resistance to targeted therapy.
Lately, mathematical oncologists have designed a new method to tackling this problem based on evolutionary principles. Known as adaptive therapy, this as-yet unproven strategy aims to stop or delay the failure of cancer treatment by manipulating competition between drug-sensitive and resistant cells. It uses relatively low doses and has the additional potential benefits of reducing side effects and enhance quality of life.
As a way to solve the problem, Dr. Noble and Dr. Viossat organized a workshop for mathematical modelers to determine the state of art of adaptive therapy, discuss future directions and foster collaborations. The virtual event was attended by one hundred persons who participated in more than twenty talks, interacting via the Sococo virtual meeting platform.
Dr. Noble plans to continue developing mathematical models to improve cancer treatment. His long-term objective is to project optimal treatment regimens for each tumor type and each patient.
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Reporter: Stephen J. Williams, Ph.D.
Other 2019 Conference Announcement Posts on this Open Access Journal Include:
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Reporter and Curator: Dr. Sudipta Saha, Ph.D.
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Reporter and Curator: Dr. Sudipta Saha, Ph.D.
Babies born at or before 25 weeks have quite low survival outcomes, and in the US it is the leading cause of infant mortality and morbidity. Just a few weeks of extra ‘growing time’ can be the difference between severe health problems and a relatively healthy baby.
Researchers from The Children’s Hospital of Philadelphia (USA) Research Institute have shown it’s possible to nurture and protect a mammal in late stages of gestation inside an artificial womb; technology which could become a lifesaver for many premature human babies in just a few years.
The researchers took eight lambs between 105 to 120 days gestation (the physiological equivalent of 23 to 24 weeks in humans) and placed them inside the artificial womb. The artificial womb is a sealed and sterile bag filled with an electrolyte solution which acts like amniotic fluid in the uterus. The lamb’s own heart pumps the blood through the umbilical cord into a gas exchange machine outside the bag.
The artificial womb worked in this study and after just four weeks the lambs’ brains and lungs had matured like normal. They had also grown wool and could wiggle, open their eyes, and swallow. Although this study is looking incredibly promising but getting the research up to scratch for human babies still requires a big leap.
Nevertheless, if all goes well, the researchers hope to test the device on premature humans within three to five years. Potential therapeutic applications of this invention may include treatment of fetal growth retardation related to placental insufficiency or the salvage of preterm infants threatening to deliver after fetal intervention or fetal surgery.
The technology may also provide the opportunity to deliver infants affected by congenital malformations of the heart, lung and diaphragm for early correction or therapy before the institution of gas ventilation. Numerous applications related to fetal pharmacologic, stem cell or gene therapy could be facilitated by removing the possibility for maternal exposure and enabling direct delivery of therapeutic agents to the isolated fetus.
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Curator: Larry H. Bernstein, MD, FCAP
Frankenstein Proteins Stitched Together by Scientists
Histone Mutation Deranges DNA Methylation to Cause Cancer
Histone H3K36 mutations promote sarcomagenesis through altered histone methylation landscape
An oncohistone deranges inhibitory chromatin
Science, this issue p. 844
Mitochondria? We Don’t Need No Stinking Mitochondria!
Mysterious Eukaryote Missing Mitochondria
By Anna Azvolinsky | May 12, 2016
organellesmitochondriagenetics & genomics and evolution
A Eukaryote without a Mitochondrial Organelle
• Monocercomonoides sp. is a eukaryotic microorganism with no mitochondria
HIV Particles Used to Trap Intact Mammalian Protein Complexes
Trapping mammalian protein complexes in viral particles
Sven Eyckerman, Kevin Titeca, …Kris GevaertJan Tavernier
Concept of the Virotrap system
Figure 1: Schematic representation of the Virotrap strategy.
Virotrap for the detection of binary interactions
Virotrap for unbiased discovery of novel interactions
Figure 3: Use of Virotrap for unbiased interactome analysis
New Autism Blood Biomarker Identified
A Search for Blood Biomarkers for Autism: Peptoids
Association between peptoid binding and ADOS and ADI-R subdomains
Computational Model Finds New Protein-Protein Interactions
Schizophrenia interactome with 504 novel protein–protein interactions
MK GanapathirajuM Thahir,…, CE LoscherEM Bauer & S Chaparala
Figure 1
SZ interactome
Webserver of SZ interactome
Functional and pathway enrichment in SZ interactome
Massimo Stefani · Christopher M. Dobson
Amyloid formation is a generic property of polypeptide chains ….
Precursors of amyloid fibrils can be toxic to cells
Structural basis and molecular features of amyloid toxicity
Concluding remarks
Shared Genetic Risk Factors for Late-Life Depression and Alzheimer’s Disease
Ye, Qing | Bai, Feng* | Zhang, Zhijun
Notes from Kurzweill
This vitamin stops the aging process in organs, say Swiss researchers
A potential breakthrough for regenerative medicine, pending further studies
Mitochondria —> stem cells —> organs
How to revitalize stem cells
Sean WhalenRebecca M Truty & Katherine S Pollard
Nature Genetics 2016; 48:488–496
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Beyond Moore’s Law
Beyond Moore’s Law
Larry H. Bernstein, MD, FCAP, Curator
Experiments show magnetic chips could dramatically increase computing’s energy efficiency
Beyond Moore’s law: the challenge in computing today is reducing chips’ energy consumption, not increasing packing density
Magnetic microscope image of three nanomagnetic computer bits. Each bit is a tiny bar magnet only 90 nanometers long. The image hows a bright spot at the “North” end and a dark spot at the “South” end of the magnet. The “H” arrow shows the direction of magnetic field applied to switch the direction of the magnets. (credit: Jeongmin Hong et al./Science Advances) http://www.kurzweilai.net/images/Nanomagnetic-Bit.jpg
The findings were published Mar. 11 an open-access paper in the peer-reviewed journal Science Advances.
This is critical at two ends of the size scale: for mobile devices, which demand powerful processors that can run for a day or more on small, lightweight batteries; and on an industrial scale, as computing increasingly moves into “the cloud,” where the electricity demands of the giant cloud data centers are multiplying, collectively taking an increasing share of the country’s — and world’s — electrical grid.
“The biggest challenge in designing computers and, in fact, all our electronics today is reducing their energy consumption,” aid senior author Jeffrey Bokor, a UC Berkeley professor of electrical engineering and computer sciences and a faculty scientist at the Lawrence Berkeley National Laboratory.
Lowering energy use is a relatively recent shift in focus in chip manufacturing after decades of emphasis on packing greater numbers of increasingly tiny and faster transistors onto chips to keep up with Moore’s law.
“Making transistors go faster was requiring too much energy,” said Bokor, who is also the deputy director the Center for Energy Efficient Electronics Science, a Science and Technology Center at UC Berkeley funded by the National Science Foundation. “The chips were getting so hot they’d just melt.”
So researchers have been turning to alternatives to conventional transistors, which currently rely upon the movement of electrons to switch between 0s and 1s. Partly because of electrical resistance, it takes a fair amount of energy to ensure that the signal between the two 0 and 1 states is clear and reliably distinguishable, and this results in excess heat.
Nanomagnetic computing: how low can you get?
The UC Berkeley team used an innovative technique to measure the tiny amount of energy dissipation that resulted when they flipped a nanomagnetic bit. The researchers used a laser probe to carefully follow the direction that the magnet was pointing as an external magnetic field was used to rotate the magnet from “up” to “down” or vice versa.
They determined that it only took 15 millielectron volts of energy — the equivalent of 3 zeptojoules — to flip a magnetic bit at room temperature, effectively demonstrating the Landauer limit (the lowest limit of energy required for a computer operation). *
This is the first time that a practical memory bit could be manipulated and observed under conditions that would allow the Landauer limit to be reached, the authors said. Bokor and his team published a paper in 2011 that said this could theoretically be done, but it had not been demonstrated until now.
While this paper is a proof of principle, he noted that putting such chips into practical production will take more time. But the authors noted in the paper that “the significance of this result is that today’s computers are far from the fundamental limit and that future dramatic reductions in power consumption are possible.”
The National Science Foundation and the U.S. Department of Energy supported this research.
* The Landauer limit was named after IBM Research Lab’s Rolf Landauer, who in 1961 found that in any computer, each single bit operation must expend an absolute minimum amount of energy. Landauer’s discovery is based on the second law of thermodynamics, which states that as any physical system is transformed, going from a state of higher concentration to lower concentration, it gets increasingly disordered. That loss of order is called entropy, and it comes off as waste heat. Landauer developed a formula to calculate this lowest limit of energy required for a computer operation. The result depends on the temperature of the computer; at room temperature, the limit amounts to about 3 zeptojoules, or one-hundredth the energy given up by a single atom when it emits one photon of light.
Abstract of Experimental test of Landauer’s principle in single-bit operations on nanomagnetic memory bits
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Wave Theory Updated
Larry H. Bernstein, MD, FCAP, Curator
Mathematical Advance in Describing Waves Captures Essence of Modulational Instabilitywave theory
2/25/2016 – Charlotte Hsu, University at Buffalo
Researchers have shown, mathematically, that many different kinds of disturbances evolve to produce wave forms belonging to a single class, denoted by their identical asymptotic state.
BUFFALO, NY — One of the great joys in mathematics is the ability to use it to describe phenomena seen in the physical world, says University at Buffalo mathematician Gino Biondini.
With UB postdoctoral researcher Dionyssios Mantzavinos, Biondini has published a new paper that advances the art — or shall we say, the math — of describing a wave. The findings, published January 27, 2016, in Physical Review Letters, are thought to apply to wave forms ranging from light waves in optical fibers to water waves in the sea.
The study explores what happens when a regular wave pattern has small irregularities, a question that scientists have been trying to answer for the last 50 years.
Researchers have long known that, in many cases, such minor imperfections grow and eventually completely distort the original wave as it travels over long distances, a phenomenon known as “modulational instability.” But the UB team has added to this story by showing, mathematically, that many different kinds of disturbances evolve to produce wave forms belonging to a single class, denoted by their identical asymptotic state.
“Ever since Isaac Newton used math to describe gravity, applied mathematicians have been inventing new mathematics or using existing forms to describe natural phenomena,” says Biondini, a professor of mathematics in the UB College of Arts and Sciences and an adjunct faculty member in the UB physics department. “Our research is, in a way, an extension of all the work that’s come before.”
He says the first great success in using math to represent waves came in the 1700s. The so-called wave equation, used to describe the propagation of waves such as light, sound and water waves, was discovered by Jean le Rond d’Alembert in the middle of that century. But the model has limitations.
“The wave equation is a great first approximation, but it breaks down when the waves are very large — or, in technical parlance — ‘nonlinear,’” Biondini said. “So, for example, in optical fibers, the wave equation is great for moderate distances, but if you send a laser pulse (which is an electromagnetic wave) through an optical fiber across the ocean or the continental U.S., the wave equation is not a good approximation anymore. “Similarly, when a water wave whitecaps and overturns, the wave equation is not a good description of the physics anymore.”
Over the next 250 years, scientists and mathematicians continued to develop new and better ways to describe waves. One of the models that researchers derived in the middle of the 20th century is the nonlinear Schrödinger equation, which helps to characterize wave trains in a variety of physical contexts, including in nonlinear optics and in deep water.
But many questions remained unanswered, including what happens when a wave has small imperfections at its origin.
This is the topic of Biondini and Mantzavinos’ new paper.
“Modulational instability has been known since the 1960s. When you have small perturbations at the input, you’ll have big changes at the output. But is there a way to describe precisely what happens?” Biondini said. “After laying out the foundations in two earlier papers, it took us a year of work to obtain a mathematical description of the solutions. We then used computers to test whether our math was correct, and the simulation results were pretty good — it appears that we have captured the essence of the phenomenon.”
The next step, Biondini said, is to partner with experimental researchers to see if the theoretical findings hold when applied to tangible, physical waves. He has started to collaborate with research groups in optics as well as water waves, and he hopes that it will soon be possible to test the theoretical predictions with real experiments.
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The Music of the Elements
Larry H. Bernstein, MD, FCAP, Curator
A Scientist is Creating Music from the Periodic Table
Mon, 02/29/2016 – Suzanne Tracy, Editor-in-Chief, Scientific Computing and HPC Source
When researchers ran computer simulations to study vibrations, they noticed that some of the polymers didn’t behave as expected. If they tweaked the starting parameters, the system evolved normally up to a point, but then it diverged into a patterned series of vibrations that were not random. The simulated polymer becomes thermally superconductive — that is, capable of transporting heat with no resistance, much like the existing class of superconducting materials that conduct electricity without resistance.
A researcher at Georgia Institute of Technology has applied for a National Science Foundation grant to create an educational app that would catalog a unique musical signature for each element in the periodic table so that scientists would have a new tool to use in identifying the differences between the molecular structures of solids, liquids and gases. Asegun Henry, Director of the Atomistic Simulation & Energy (ASE) Research Group, and an Assistant Professor in the George W. Woodruff School of Mechanical Engineering, is also in the process of setting all of the elements in the table to music.
“My hope is that it will be an interesting tool to teach the periodic table, but also to give people some notion about the idea that the entire universe is moving around and making noise,” Henry told Gizmodo. “You just can’t hear it.”
As Gizmodo’s Jennifer Ouellette explains, it’s more than just a fun exercise. “Henry and his graduate student, Wei Lv, were interested in a peculiar feature of polymers, long chains of molecules all strung together, with thousands upon thousands of different modes of vibration that interact with each other. Polymers are much more complicated than the simple toy models, so it’s harder to describe their interactions mathematically. Scientists must rely on computer simulations to study the vibrations.”
“How the energy of the interaction changes with respect to the distance between the molecules dictates a lot of the physics,” says Henry. “We have to slow down the vibrations of the atoms so you can hear them, because they’re too fast, and at too high frequencies. But you’ll be able to hear the difference between something low on the periodic table and something like carbon that’s very high. One will sound high-pitched, and one will sound low.”
However, when Henry and Lv ran their computer simulations, they noticed that some of the polymers they were modeling didn’t behave as expected, Ouellette reports. If they tweaked the starting parameters a bit, the system evolved normally up to a point, but then it diverged into a patterned series of vibrations that were not random. The simulated polymer becomes thermally superconductive — that is, capable of transporting heat with no resistance, much like the existing class of superconducting materials that conduct electricity without resistance (albeit at very low temperatures).
“Toy models are fictitious and designed to be really simple and plain so that you can analyze them easily,” said Henry. “We did this with a real system, and the [effect] actually persisted.”
Henry and Lv successfully identified three vibrational modes out of several thousand responsible for the phenomenon. However, traditional analysis techniques — like plotting the amplitudes of the modes over time in a visual graph — didn’t reveal anything significant. It wasn’t until the researchers decided to sonify the data that they pinpointed what was going on. This involved mapping pitch, timbre and amplitude onto the data to translate it into a kind of molecular music. The three modes faded in and out over time and eventually synchronized, creating a kind of sonic feedback loop until the simulated material became thermally superconductive.
“As soon as you play it, your ears pick up on it immediately,” said Henry. So, it’s solid proof-of-principle of sonification as an analytical tool for materials science.
Henry is attempting to identify the underlying mechanism behind the phenomenon in order to understand why it manifests in some polymer systems, but not others. This information could help to actually construct physical thermal superconducting materials. “It would change the world,” said Henry. “Conceptually you’d be able to run a thermal superconducting pipe from the Sahara desert and provide heat to the rest of the world.”
Phonon transport at interfaces: Determining the correct modes of vibration
Kiarash Gordiz1 and Asegun Henry1,2,a)
J. Appl. Phys. 119, 015101 (2016); http://dx.doi.org/10.1063/1.4939207
For many decades, phonon transport at interfaces has been interpreted in terms of phonons impinging on an interface and subsequently transmitting a certain fraction of their energy into the other material. It has also been largely assumed that when one joins two bulk materials,interfacialphonon transport can be described in terms of the modes that exist in each material separately. However, a new formalism for calculating the modal contributions to thermal interface conductance with full inclusion of anharmonicity has been recently developed, which now offers a means for checking the validity of this assumption. Here, we examine the assumption of using the bulk materials’ modes to describe the interfacial transport. The results indicate that when two materials are joined, a new set of vibrational modes are required to correctly describe the transport. As the modes are analyzed, certain classifications emerge and some of the most important modes are localized at the interface and can exhibit large conductance contributions that cannot be explained by the current physical picture based on transmission probability.
Article outline:
A. EMD simulations
B. Wave-packet simulations
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Discovery of Pi
Larry H. Bernstein, MD, FCAP, Curator
How a Farm Boy from Wales Gave the World Pi
3/14/2016 – Gareth Ffowc Roberts, Bangor University http://www.scientificcomputing.com/articles/2016/03/how-farm-boy-wales-gave-world-pi
Maths pi-oneer. William Hogarth/National Portrait Gallery
Maths pi-oneer. William Hogarth/National Portrait Gallery
One of the most important numbers in maths might today be named after the Greek letter π or “pi,” but the convention of representing it this way actually doesn’t come from Greece at all. It comes from the pen of an 18th century farmer’s son and largely self-taught mathematician from the small island of Anglesey in Wales. The Welsh Government has even renamed Pi Day (on March 14 or 3/14, which matches the first three digits of pi, 3.14) as “Pi Day Cymru.”
The importance of the number we now call pi has been known about since ancient Egyptian times. It allows you to calculate the circumference and area of a circle from its diameter (and vice versa). But it’s also a number that crops up across all scientific disciplines from cosmology to thermodynamics. Yet even after mathematicians worked out how to calculate pi accurately to over 100 decimal places at the start of the 18th century, we didn’t have an agreed symbol for the number.
From accountant to maths pioneer
This all changed thanks to William Jones who was born in 1674 in the parish of Llanfihangel Tre’r Beirdd. After attending a charity school, Jones landed a job as a merchant’s accountant and then as a maths teacher on a warship, before publishing A New Compendium of the Whole Art of Navigation, his first book in 1702 on the mathematics of navigation. On his return to Britain he began to teach maths in London, possibly starting by holding classes in coffee shops for a small fee.
Shortly afterwards he published Synopsis palmariorum matheseos, a summary of the current state of the art developments in mathematics which reflected his own particular interests. In it is the first recorded use of the symbol π as the number that gives the ratio of a circle’s circumference to its diameter.
We typically think of this number as being about 3.14, but Jones rightly suspected that the digits after its decimal point were infinite and non-repeating. This meant it could never be “expressed in numbers,” as he put it. That was why he recognised the number needed its own symbol. It is commonly thought that he chose pi either because it is the first letter of the word for periphery (περιφέρεια) or because it is the first letter of the word for perimeter (περίμετρος), or both.
Finding pi _Synopsis palmariorum matheseos_
In the pages of his Synopsis, Jones also showed his familiarity with the notion of an infinite series and how it could help calculate pi far more accurately than was possible just by drawing and measuring circles. An infinite series is the total of all the numbers in a sequence that goes on forever, for example ½ + ¼ + ⅛ + and so on. Adding an infinite sequence of ever-smaller fractions like this can bring you closer and closer to a number with an infinite number of digits after the decimal point — just like pi. So by defining the right sequence, mathematicians were able to calculate pi to an increasing number of decimal places.
Infinite series also assist our understanding of rational numbers, more commonly referred to as fractions. Irrational numbers are the ones, like pi, that can’t be written as a fraction, which is why Jones decided it needed its own symbol. What he wasn’t able to do was prove with maths that the digits of pi definitely were infinite and non-repeating and so that the number was truly irrational. This would eventually be achieved in 1768 by the French mathematician Johann Heinrich Lambert. Jones dipped his toes into the subject and showed an intuitive grasp of the complexity of pi but lacked the analytical tools to enable him to develop his ideas further.
Scientific success
Despite this — and his obscure background — Jones’s book was a success and led him to become an important and influential member of the scientific establishment. He was noticed and befriended by two of Britain’s foremost mathematicians — Edmund Halley and Sir Isaac Newton — and was elected a fellow of the Royal Society in 1711. He later became the editor and publisher of many of Newton’s manuscripts and built up an extraordinary library that was one of the greatest collections of books on science and mathematics ever known, and only recently fully dispersed.
Despite this success, the use of the symbol π spread slowly at first. It was popularised in 1737 by the Swiss mathematician Leonhard Euler (1707–83), one of the most eminent mathematicians of the 18th century, who likely came across Jones’ work while studying Newton at the University of Basel. His endorsement of the symbol in his own work ensured that it received wide publicity, yet even then the symbol wasn’t adopted universally until as late as 1934. Today π is instantly recognised worldwide but few know that its history can be traced back to a small village in the heart of Anglesey.
Gareth Ffowc Roberts, Emeritus Professor of Education, Bangor University. This article was originally published on The Conversation. Read the original article.
The Search for the Value of Pi
Tue, 03/15/2016 – 4:44pm
This “pi plate” shows some of the progress toward finding all the digits of pi. In 1946, ENIAC, the first electronic general-purpose computer, calculated 2,037 digits of pi in 70 hours. The most recent calculation found more than 13 trillion digits of pi in 208 days. Piledhigheranddeeper, CC BY-SA
The number represented by pi (π) is used in calculations whenever something round (or nearly so) is involved, such as for circles, spheres, cylinders, cones and ellipses. Its value is necessary to compute many important quantities about these shapes, such as understanding the relationship between a circle’s radius and its circumference and area (circumference=2πr; area=πr2). Pi also appears in the calculations to determine the area of an ellipse and in finding the radius, surface area and volume of a sphere.
Our world contains many round and near-round objects; finding the exact value of pi helps us build, manufacture and work with them more accurately. Historically, people had only very coarse estimations of pi (such as 3, or 3.12, or 3.16), and while they knew these were estimates, they had no idea how far off they might be. The search for the accurate value of pi led not only to more accuracy, but also to the development of new concepts and techniques, such as limits and iterative algorithms, which then became fundamental to new areas of mathematics.
Finding the actual value of pi
Archimedes. André Thévet (1584)
Between 3,000 and 4,000 years ago, people used trial-and-error approximations of pi, without doing any math or considering potential errors. The earliest written approximations of pi are 3.125 in Babylon (1900-1600 B.C.) and 3.1605 in ancient Egypt (1650 B.C.). Both approximations start with 3.1 — pretty close to the actual value, but still relatively far off.
Archimedes’ method of calculating pi involved polygons with more and more sides. Leszek Krupinski, CC BY-SA
The first rigorous approach to finding the true value of pi was based on geometrical approximations. Around 250 B.C., the Greek mathematician Archimedes drew polygons both around the outside and within the interior of circles. Measuring the perimeters of those gave upper and lower bounds of the range containing pi. He started with hexagons; by using polygons with more and more sides, he ultimately calculated three accurate digits of pi: 3.14. Around A.D. 150, Greek-Roman scientist Ptolemy used this method to calculate a value of 3.1416.
Liu Hui’s method of calculating pi also used polygons, but in a slightly different way. Gisling and Pbroks13, CC BY-SA
Independently, around A.D. 265, Chinese mathematician Liu Hui created another simple polygon-based iterative algorithm. He proposed a very fast and efficient approximation method, which gave four accurate digits. Later, around A.D. 480, Zu Chongzhi adopted Liu Hui’s method and achieved seven digits of accuracy. This record held for another 800 years.
In 1630, Austrian astronomer Christoph Grienberger arrived at 38 digits, which is the most accurate approximation manually achieved using polygonal algorithms.
Moving beyond polygons
The development of infinite series techniques in the 16th and 17th centuries greatly enhanced people’s ability to approximate pi more efficiently. An infinite series is the sum (or much less commonly, product) of the terms of an infinite sequence, such as ½, ¼, 1/8, 1/16, … 1/(2n). The first written description of an infinite series that could be used to compute pi was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji around 1500 A.D., the proof of which was presented around 1530 A.D.
Sir Isaac Newton Wellcome Trust, CC BY
In 1665, English mathematician and physicist Isaac Newton used infinite series to compute pi to 15 digits using calculus he and German mathematician Gottfried Wilhelm Leibniz discovered. After that, the record kept being broken. It reached 71 digits in 1699, 100 digits in 1706, and 620 digits in 1956 — the best approximation achieved without the aid of a calculator or computer.
Carl Louis Ferdinand von Lindemann
In tandem with these calculations, mathematicians were researching other characteristics of pi. Swiss mathematician Johann Heinrich Lambert (1728-1777) first proved that pi is an irrational number — it has an infinite number of digits that never enter a repeating pattern. In 1882, German mathematician Ferdinand von Lindemann proved that pi cannot be expressed in a rational algebraic equation (such as pi²=10 or 9pi4 – 240pi2 + 1492 = 0).
Toward even more digits of pi
Bursts of calculations of even more digits of pi followed the adoption of iterative algorithms, which repeatedly build an updated value by using a calculation performed on the previous value. A simple example of an iterative algorithm allows you to approximate the square root of 2 as follows, using the formula (x+2/x)/2:
• (2+2/2)/2 = 1.5
• (1.5+2/1.5)/2 = 1.4167
• (1.4167+2/1.4167)/2 = 1.4142, which is a very close approximation already.
Advances toward more digits of pi came with the use of a Machin-like algorithm (a generalization of English mathematician John Machin’s formula developed in 1706) and the Gauss-Legendre algorithm (late 18th century) in electronic computers (invented mid-20th century). In 1946, ENIAC, the first electronic general-purpose computer, calculated 2,037 digits of pi in 70 hours. The most recent calculation found more than 13 trillion digits of pi in 208 days!
It has been widely accepted that for most numerical calculations involving pi, a dozen digits provides sufficient precision. According to mathematicians Jörg Arndt and Christoph Haenel, 39 digits are sufficient to perform most cosmological calculations, because that’s the accuracy necessary to calculate the circumference of the observable universe to within one atom’s diameter. Thereafter, more digits of pi are not of practical use in calculations; rather, today’s pursuit of more digits of pi is about testing supercomputers and numerical analysis algorithms.
Calculating pi by yourself
There are also fun and simple methods for estimating the value of pi. One of the best-known is a method called “Monte Carlo.”
A square with inscribed circle. Deweirdifier
The method is fairly simple. To try it at home, draw a circle and a square around it (as at left) on a piece of paper. Imagine the square’s sides are of length 2, so its area is 4; the circle’s diameter is therefore 2, and its area is pi. The ratio between their areas is pi/4, or about 0.7854.
Now pick up a pen, close your eyes and put dots on the square at random. If you do this enough times, and your efforts are truly random, eventually the percentage of times your dot landed inside the circle will approach 78.54 percent — or 0.7854.
Now you’ve joined the ranks of mathematicians who have calculated pi through the ages.
Xiaojing Ye, Assistant Professor of Mathematics and Statistics, Georgia State University. This article was originally published on The Conversation. Read the original article.
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History of Quantum Mechanics
Curator: Larry H. Bernstein, MD, FCAP
A history of Quantum Mechanics
It is hard to realise that the electron was only discovered a little over 100 years ago in 1897. That it was not expected is illustrated by a remark made by J J Thomson, the discoverer of the electron. He said
I was told long afterwards by a distinguished physicist who had been present at my lecture that he thought I had been pulling their leg.
The neutron was not discovered until 1932 so it is against this background that we trace the beginnings of quantum theory back to 1859.
In 1859 Gustav Kirchhoff proved a theorem about blackbody radiation. A blackbody is an object that absorbs all the energy that falls upon it and, because it reflects no light, it would appear black to an observer. A blackbody is also a perfect emitter and Kirchhoff proved that the energy emitted E depends only on the temperature T and the frequency v of the emitted energy, i.e.
E = J(T,v).
He challenged physicists to find the function J.
In 1879 Josef Stefan proposed, on experimental grounds, that the total energy emitted by a hot body was proportional to the fourth power of the temperature. In the generality stated by Stefan this is false. The same conclusion was reached in 1884 by Ludwig Boltzmann for blackbody radiation, this time from theoretical considerations using thermodynamics and Maxwell‘s electromagnetic theory. The result, now known as the StefanBoltzmann law, does not fully answer Kirchhoff‘s challenge since it does not answer the question for specific wavelengths.
In 1896 Wilhelm Wien proposed a solution to the Kirchhoff challenge. However although his solution matches experimental observations closely for small values of the wavelength, it was shown to break down in the far infrared by Rubens and Kurlbaum.
Kirchhoff, who had been at Heidelberg, moved to Berlin. Boltzmann was offered his chair in Heidelberg but turned it down. The chair was then offered to Hertz who also declined the offer, so it was offered again, this time to Planck and he accepted.
Rubens visited Planck in October 1900 and explained his results to him. Within a few hours of Rubens leaving Planck‘s house Planck had guessed the correct formula for Kirchhoff‘s J function. This guess fitted experimental evidence at all wavelengths very well but Planck was not satisfied with this and tried to give a theoretical derivation of the formula. To do this he made the unprecedented step of assuming that the total energy is made up of indistinguishable energy elements – quanta of energy. He wrote
Experience will prove whether this hypothesis is realised in nature
Planck himself gave credit to Boltzmann for his statistical method but Planck‘s approach was fundamentally different. However theory had now deviated from experiment and was based on a hypothesis with no experimental basis. Planck won the 1918 Nobel Prize for Physics for this work.
In 1901 Ricci and Levi-Civita published Absolute differential calculus. It had been Christoffel‘s discovery of ‘covariant differentiation’ in 1869 which let Ricci extend the theory of tensor analysis to Riemannian space of n dimensions. The Ricci and Levi-Civita definitions were thought to give the most general formulation of a tensor. This work was not done with quantum theory in mind but, as so often happens, the mathematics necessary to embody a physical theory had appeared at precisely the right moment.
In 1905 Einstein examined the photoelectric effect. The photoelectric effect is the release of electrons from certain metals or semiconductors by the action of light. The electromagnetic theory of light gives results at odds with experimental evidence. Einstein proposed a quantum theory of light to solve the difficulty and then he realised that Planck‘s theory made implicit use of the light quantum hypothesis. By 1906 Einstein had correctly guessed that energy changes occur in a quantum material oscillator in changes in jumps which are multiples of v where is Planck‘s reduced constant and v is the frequency. Einstein received the 1921 Nobel Prize for Physics, in 1922, for this work on the photoelectric effect.
In 1913 Niels Bohr wrote a revolutionary paper on the hydrogen atom. He discovered the major laws of the spectral lines. This work earned Bohr the 1922 Nobel Prize for Physics. Arthur Compton derived relativistic kinematics for the scattering of a photon (a light quantum) off an electron at rest in 1923.
However there were concepts in the new quantum theory which gave major worries to many leading physicists. Einstein, in particular, worried about the element of ‘chance’ which had entered physics. In fact Rutherford had introduced spontaneous effect when discussing radio-active decay in 1900. In 1924 Einstein wrote:-
There are therefore now two theories of light, both indispensable, and – as one must admit today despite twenty years of tremendous effort on the part of theoretical physicists – without any logical connection.
In the same year, 1924, Bohr, Kramers and Slater made important theoretical proposals regarding the interaction of light and matter which rejected the photon. Although the proposals were the wrong way forward they stimulated important experimental work. Bohr addressed certain paradoxes in his work.
(i) How can energy be conserved when some energy changes are continuous and some are discontinuous, i.e. change by quantum amounts.
(ii) How does the electron know when to emit radiation.
Einstein had been puzzled by paradox (ii) and Pauli quickly told Bohr that he did not believe his theory. Further experimental work soon ended any resistance to belief in the electron. Other ways had to be found to resolve the paradoxes.
Up to this stage quantum theory was set up in Euclidean space and used Cartesian tensors of linear and angular momentum. However quantum theory was about to enter a new era.
The year 1924 saw the publication of another fundamental paper. It was written by Satyendra Nath Bose and rejected by a referee for publication. Bose then sent the manuscript to Einstein who immediately saw the importance of Bose‘s work and arranged for its publication. Bose proposed different states for the photon. He also proposed that there is no conservation of the number of photons. Instead of statistical independence of particles, Bose put particles into cells and talked about statistical independence of cells. Time has shown that Bose was right on all these points.
Work was going on at almost the same time as Bose‘s which was also of fundamental importance. The doctoral thesis of Louis de Broglie was presented which extended the particle-wave duality for light to all particles, in particular to electrons. Schrödinger in 1926 published a paper giving his equation for the hydrogen atom and heralded the birth of wave mechanics. Schrödingerintroduced operators associated with each dynamical variable.
The year 1926 saw the complete solution of the derivation of Planck‘s law after 26 years. It was solved by Dirac. Also in 1926 Born abandoned the causality of traditional physics. Speaking of collisions Born wrote
One does not get an answer to the question, What is the state after collision? but only to the question, How probable is a given effect of the collision? From the standpoint of our quantum mechanics, there is no quantity which causally fixes the effect of a collision in an individual event.
Heisenberg wrote his first paper on quantum mechanics in 1925 and 2 years later stated his uncertainty principle. It states that the process of measuring the position x of a particle disturbs the particle’s momentum p, so that
Dx Dp ≥ = h/2π
where Dx is the uncertainty of the position and Dp is the uncertainty of the momentum. Here h is Planck‘s constant and is usually called the ‘reduced Planck‘s constant’. Heisenberg states that
the nonvalidity of rigorous causality is necessary and not just consistently possible.
Heisenberg‘s work used matrix methods made possible by the work of Cayley on matrices 50 years earlier. In fact ‘rival’ matrix mechanics deriving from Heisenberg‘s work and wave mechanics resulting from Schrödinger‘s work now entered the arena. These were not properly shown to be equivalent until the necessary mathematics was developed by Riesz about 25 years later.
Also in 1927 Bohr stated that space-time coordinates and causality are complementary. Pauli realised that spin, one of the states proposed by Bose, corresponded to a new kind of tensor, one not covered by the Ricci and Levi-Civita work of 1901. However the mathematics of this had been anticipated by Eli Cartan who introduced a ‘spinor’ as part of a much more general investigation in 1913.
Dirac, in 1928, gave the first solution of the problem of expressing quantum theory in a form which was invariant under the Lorentz group of transformations of special relativity. He expressedd’Alembert‘s wave equation in terms of operator algebra.
The uncertainty principle was not accepted by everyone. Its most outspoken opponent was Einstein. He devised a challenge to Niels Bohr which he made at a conference which they both attended in 1930. Einstein suggested a box filled with radiation with a clock fitted in one side. The clock is designed to open a shutter and allow one photon to escape. Weigh the box again some time later and the photon energy and its time of escape can both be measured with arbitrary accuracy. Of course this is not meant to be an actual experiment, only a ‘thought experiment’.
Niels Bohr is reported to have spent an unhappy evening, and Einstein a happy one, after this challenge by Einstein to the uncertainty principle. However Niels Bohr had the final triumph, for the next day he had the solution. The mass is measured by hanging a compensation weight under the box. This is turn imparts a momentum to the box and there is an error in measuring the position. Time, according to relativity, is not absolute and the error in the position of the box translates into an error in measuring the time.
Although Einstein was never happy with the uncertainty principle, he was forced, rather grudgingly, to accept it after Bohr‘s explanation.
In 1932 von Neumann put quantum theory on a firm theoretical basis. Some of the earlier work had lacked mathematical rigour, but von Neumann put the whole theory into the setting of operator algebra.
References (33 books/articles)
Article by: J J O’Connor and E F Robertson
A Brief History of Quantum Mechanics
Appendix A of
The Strange World of Quantum Mechanics
written by Dan StyerOberlin College Physics Department;
copyright © Daniel F. Styer 1999
One must understand not only the cleanest and most direct experimental evidence supporting our current theories (like the evidence presented in this book), but must understand also how those theories came to be accepted through a tightly interconnected web of many experiments, no one of which was completely convincing but which taken together presented an overwhelming argument
Thus a full history of quantum mechanics would have to discuss Schrödinger’s many mistresses, Ehrenfest’s suicide, and Heisenberg’s involvement with Nazism. It would have to treat the First World War’s effect on the development of science. It would need to mention “the Thomson model” of the atom, which was once the major competing theory to quantum mechanics. It would have to give appropriate weight to both theoretical and experimental developments.
Much of the work of science is done through informal conversations, and the resulting written record is often sanitized to avoid offending competing scientists. The invaluable oral record is passed down from professor to student repeatedly before anyone ever records it on paper. There is a tendency for the exciting stories to be repeated and the dull ones to be forgotten.
The fact is that scientific history, like the stock market and like everyday life, does not proceed in an orderly, coherent pattern. The story of quantum mechanics is a story full of serendipity, personal squabbles, opportunities missed and taken, and of luck both good and bad.
Status of physics: January 1900
In January 1900 the atomic hypothesis was widely but not universally accepted. Atoms were considered point particles, and it wasn’t clear how atoms of different elements differed. The electron had just been discovered (1897) and it wasn’t clear where (or even whether) electrons were located within atoms. One important outstanding problem concerned the colors emitted by atoms in a discharge tube (familiar today as the light from a fluorescent tube or from a neon sign). No one could understand why different gas atoms glowed in different colors. Another outstanding problem concerned the amount of heat required to change the temperature of a diatomic gas such as oxygen: the measured amounts were well below the value predicted by theory. Because quantum mechanics is important when applied to atomic phenomena, you might guess that investigations into questions like these would give rise to the discovery of quantum mechanics. Instead it came from a study of heat radiation.
Heat radiation
You know that the coals of a campfire, or the coils of an electric stove, glow red. You probably don’t know that even hotter objects glow white, but this fact is well known to blacksmiths. When objects are hotter still they glow blue. (This is why a gas stove should be adjusted to make a blue flame.) Indeed, objects at room temperature also glow (radiate), but the radiation they emit is infrared, which is not detectable by the eye. (The military has developed — for use in night warfare — special eye sets that convert infrared radiation to optical radiation.)
In the year 1900 several scientists were trying to turn these observations into a detailed explanation of and a quantitatively accurate formula for the color of heat radiation as a function of temperature. On 19 October 1900 the Berliner Max Planck (age 42) announced a formula that fit the experimental results perfectly, yet he had no explanation for the formula — it just happened to fit. He worked to find an explanation through the late fall and finally was able to derive his formula by assuming that the atomic jigglers could not take on any possible energy, but only certain special “allowed” values. He announced this result on 14 December 1900. We know this because the assumption of allowed energy values raises certain obvious questions. If a jiggling atom can only assume certain allowed values of energy, then there must also be restrictions on the positions and speeds that the atom can have. What are they?
Planck wrote (31 years after his discovery):
I had already fought for six years (since 1894) with the problem of equilibrium between radiation and matter without arriving at any successful result. I was aware that this problem was of fundamental importance in physics, and I knew the formula describing the energy distribution . . .
Here is another wonderful story, this one related by Werner Heisenberg:
In a period of most intensive work during the summer of 1900 [Planck] finally convinced himself that there was no way of escaping from this conclusion [of “allowed” energies]. It was told by Planck’s son that his father spoke to him about his new ideas on a long walk through the Grunewald, the wood in the suburbs of Berlin. On this walk he explained that he felt he had possibly made a discovery of the first rank, comparable perhaps only to the discoveries of Newton.
(the son would probably remember the nasty cold he caught better than any remarks his father made.)
The old quantum theory
Classical mechanics was assumed to hold, but with the additional assumption that only certain values of a physical quantity (the energy, say, or the projection of a magnetic arrow) were allowed. Any such quantity was said to be “quantized”. The trick seemed to be to guess the right quantization rules for the situation under study, or to find a general set of quantization rules that would work for all situations.
For example, in 1905 Albert Einstein (age 26) postulated that the total energy of a beam of light is quantized. Just one year later he used quantization ideas to explain the heat/temperature puzzle for diatomic gases. Five years after that, in 1911, Arnold Sommerfeld (age 43) at Munich began working on the implications of energy quantization for position and speed.
In the same year Ernest Rutherford (age 40), a New Zealander doing experiments in Manchester, England, discovered the atomic nucleus — only at this relatively late stage in the development of quantum mechanics did physicists have even a qualitatively correct picture of the atom! In 1913, Niels Bohr (age 28), a Dane who had recently worked in Rutherford’s laboratory, introduced quantization ideas for the hydrogen atom. His theory was remarkably successful in explaining the colors emitted by hydrogen glowing in a discharge tube, and it sparked enormous interest in developing and extending the old quantum theory.
During the WWI (in 1915) William Wilson (age 40, a native of Cumberland, England, working at King’s College in London) made progress on the implications of energy quantization for position and speed, and Sommerfeld also continued his work in that direction.
With the coming of the armistice in 1918, work in quantum mechanics expanded rapidly. Many theories were suggested and many experiments performed. To cite just one example, in 1922 Otto Stern and his graduate student Walther Gerlach (ages 34 and 23) performed their important experiment that is so essential to the way this book presents quantum mechanics. Jagdish Mehra and Helmut Rechenberg, in their monumental history of quantum mechanics, describe the situation at this juncture well:
At the turn of the year from 1922 to 1923, the physicists looked forward with enormous enthusiasm towards detailed solutions of the outstanding problems, such as the helium problem and the problem of the anomalous Zeeman effects. However, within less than a year, the investigation of these problems revealed an almost complete failure of Bohr’s atomic theory.
The matrix formulation of quantum mechanics
As more and more situations were encountered, more and more recipes for allowed values were required. This development took place mostly at Niels Bohr’s Institute for Theoretical Physics in Copenhagen, and at the University of Göttingen in northern Germany. The most important actors at Göttingen were Max Born (age 43, an established professor) and Werner Heisenberg (age 23, a freshly minted Ph.D. from Sommerfeld in Munich). According to Born “At Göttingen we also took part in the attempts to distill the unknown mechanics of the atom out of the experimental results. . . . The art of guessing correct formulas . . . was brought to considerable perfection.”
Heisenberg particularly was interested in general methods for making guesses. He began to develop systematic tables of allowed physical quantities, be they energies, or positions, or speeds. Born looked at these tables and saw that they could be interpreted as mathematical matrices. Fifty years later matrix mathematics would be taught even in high schools. But in 1925 it was an advanced and abstract technique, and Heisenberg struggled with it. His work was cut short in June 1925.
It was late spring in Göttingen, and Heisenberg suffered from an allergy attack so severe that he could hardly work. He asked his research director, Max Born, for a vacation, and spent it on the rocky North Sea island of Helgoland. At first he was so ill that could only stay in his rented room and admire the view of the sea. As his condition improved he began to take walks and to swim. With further improvement he began also to read Goethe and to work on physics. With nothing to distract him, he concentrated intensely on the problems that had faced him in Göttingen.
Heisenberg reproduced his earlier work, cleaning up the mathematics and simplifying the formulation. He worried that the mathematical scheme he invented might prove to be inconsistent, and in particular that it might violate the principle of the conservation of energy. In Heisenberg’s own words:
By the end of the summer Heisenberg, Born, and Pascual Jordan (age 22) had developed a complete and consistent theory of quantum mechanics. (Jordan had entered the collaboration when he overheard Born discussing quantum mechanics with a colleague on a train.)
This theory, called “matrix mechanics” or “the matrix formulation of quantum mechanics”, is not the theory I have presented in this book. It is extremely and intrinsically mathematical, and even for master mathematicians it was difficult to work with. Although we now know it to be complete and consistent, this wasn’t clear until much later. Heisenberg had been keeping Wolfgang Pauli apprised of his progress. (Pauli, age 25, was Heisenberg’s friend from graduate student days, when they studied together under Sommerfeld.) Pauli found the work too mathematical for his tastes, and called it “Göttingen’s deluge of formal learning”. On 12 October 1925 Heisenberg could stand Pauli’s biting criticism no longer. He wrote to Pauli:
With respect to both of your last letters I must preach you a sermon, and beg your pardon… When you reproach us that we are such big donkeys that we have never produced anything new in physics, it may well be true. But then, you are also an equally big jackass because you have not accomplished it either . . . . . . (The dots denote a curse of about two-minute duration!) Do not think badly of me and many greetings.
The wavefunction formulation of quantum mechanics
While this work was going on at Göttingen and Helgoland, others were busy as well. In 1923 Louis de Broglie (age 31), associated an “internal periodic phenomenon” — a wave — with a particle. He was never very precise about just what that meant. (De Broglie is sometimes called “Prince de Broglie” because his family descended from the French nobility. To be strictly correct, however, only his eldest brother could claim the title.)
It fell to Erwin Schroedinger, an Austrian working in Zürich, to build this vague idea into a theory of wave mechanics. He did so during the Christmas season of 1925 (at age 38), at the alpine resort of Arosa, Switzerland, in the company of “an old girlfriend [from] Vienna”, while his wife stayed home in Zürich.
In short, just twenty-five years after Planck glimpsed the first sight of a new physics, there was not one, but two competing versions of that new physics!
The two versions seemed utterly different and there was an acrimonious debate over which one was correct. In a footnote to a 1926 paper Schrödinger claimed to be “discouraged, if not repelled” by matrix mechanics. Meanwhile, Heisenberg wrote to Pauli (8 June 1926) that
Fortunately the debate was soon stilled: in 1926 Schrödinger and, independently, Carl Eckert (age 24) of Caltech proved that the two new mechanics, although very different in superficial appearance, were equivalent to each other. [Very much as the process of adding arabic numerals is quite different from the process of adding roman numerals, but the two processes nevertheless always give the same result.] (Pauli also proved this, but never published the result.)
With not just one, but two complete formulations of quantum mechanics in hand, the quantum theory grew explosively. It was applied to atoms, molecules, and solids. It solved with ease the problem of helium that had defeated the old quantum theory. It resolved questions concerning the structure of stars, the nature of superconductors, and the properties of magnets. One particularly important contributor was P.A.M. Dirac, who in 1926 (at age 22) extended the theory to relativistic and field-theoretic situations. Another was Linus Pauling, who in 1931 (at age 30) developed quantum mechanical ideas to explain chemical bonding, which previously had been understood only on empirical grounds. Even today quantum mechanics is being applied to new problems and new situations. It would be impossible to mention all of them. All I can say is that quantum mechanics, strange though it may be, has been tremendously successful.
The Bohr-Einstein debate
The extraordinary success of quantum mechanics in applications did not overwhelm everyone. A number of scientists, including Schrödinger, de Broglie, and — most prominently — Einstein, remained unhappy with the standard probabilistic interpretation of quantum mechanics. In a letter to Max Born (4 December 1926), Einstein made his famous statement that
In concrete terms, Einstein’s “inner voice” led him, until his death, to issue occasional detailed critiques of quantum mechanics and its probabilistic interpretation. Niels Bohr undertook to reply to these critiques, and the resulting exchange is now called the “Bohr-Einstein debate”. At one memorable stage of the debate (Fifth Solvay Congress, 1927), Einstein made an objection similar to the one quoted above and Bohr
replied by pointing out the great caution, already called for by ancient thinkers, in ascribing attributes to Providence in every-day language.
These two statements are often paraphrased as, Einstein to Bohr: “God does not play dice with the universe.” Bohr to Einstein: “Stop telling God how to behave!” While the actual exchange was not quite so dramatic and quick as the paraphrase would have it, there was nevertheless a wonderful rejoinder from what must have been a severely exasperated Bohr.
The Bohr-Einstein debate had the benefit of forcing the creators of quantum mechanics to sharpen their reasoning and face the consequences of their theory in its most starkly non-intuitive situations. It also had (in my opinion) one disastrous consequence: because Einstein phrased his objections in purely classical terms, Bohr was compelled to reply in nearly classical terms, giving the impression that in quantum mechanics, an electron is “really classical” but that somehow nature puts limits on how well we can determine those classical properties. … this is a misconception: the reason we cannot measure simultaneously the exact position and speed of an electron is because an electron does not have simultaneously an exact position and speed — an electron is not just a smaller, harder edition of a marble. This misconception — this picture of a classical world underlying the quantum world … avoid it.
On the other hand, the Bohr-Einstein debate also had at least one salutary product. In 1935 Einstein, in collaboration with Boris Podolsky and Nathan Rosen, invented a situation in which the results of quantum mechanics seemed completely at odds with common sense, a situation in which the measurement of a particle at one location could reveal instantly information about a second particle far away. The three scientists published a paper which claimed that “No reasonable definition of reality could be expected to permit this.” Bohr produced a recondite response and the issue was forgotten by most physicists, who were justifiably busy with the applications of rather than the foundations of quantum mechanics. But the ideas did not vanish entirely, and they eventually raised the interest of John Bell. In 1964 Bell used the Einstein-Podolsky-Rosen situation to produce a theorem about the results from certain distant measurements for any deterministic scheme, not just classical mechanics. In 1982 Alain Aspect and his collaborators put Bell’s theorem to the test and found that nature did indeed behave in the manner that Einstein (and others!) found so counterintuitive.
The amplitude formulation of quantum mechanics
The version of quantum mechanics presented in this book is neither matrix nor wave mechanics. It is yet another formulation, different in approach and outlook, but fundamentally equivalent to the two formulations already mentioned. It is called amplitude mechanics (or “the sum over histories technique”, or “the many paths approach”, or “the path integral formulation”, or “the Lagrangian approach”, or “the method of least action”), and it was developed by Richard Feynman in 1941 while he was a graduate student (age 23) at Princeton. Its discovery is well described by Feynman himself in his Nobel lecture:
I went to a beer party in the Nassau Tavern in Princeton. There was a gentleman, newly arrived from Europe (Herbert Jehle) who came and sat next to me. Europeans are much more serious than we are in America because they think a good place to discuss intellectual matters is a beer party. So he sat by me and asked, “What are you doing” and so on, and I said, “I’m drinking beer.” Then I realized that he wanted to know what work I was doing and I told him I was struggling with this problem, and I simply turned to him and said “Listen, do you know any way of doing quantum mechanics starting with action — where the action integral comes into the quantum mechanics?” “No,” he said, “but Dirac has a paper in which the Lagrangian, at least, comes into quantum mechanics. I will show it to you tomorrow.”
Next day we went to the Princeton Library (they have little rooms on the side to discuss things) and he showed me this paper.
Dirac’s short paper in the Physikalische Zeitschrift der Sowjetunion claimed that a mathematical tool which governs the time development of a quantal system was “analogous” to the classical Lagrangian.
Professor Jehle showed me this; I read it; he explained it to me, and I said, “What does he mean, they are analogous; what does that mean,analogous? What is the use of that?” He said, “You Americans! You always want to find a use for everything!” I said that I thought that Dirac must mean that they were equal. “No,” he explained, “he doesn’t mean they are equal.” “Well,” I said, “let’s see what happens if we make them equal.”
So, I simply put them equal, taking the simplest example . . . but soon found that I had to put a constant of proportionality A in, suitably adjusted. When I substituted . . . and just calculated things out by Taylor-series expansion, out came the Schrödinger equation. So I turned to Professor Jehle, not really understanding, and said, “Well you see Professor Dirac meant that they were proportional.” Professor Jehle’s eyes were bugging out — he had taken out a little notebook and was rapidly copying it down from the blackboard and said, “No, no, this is an important discovery.”
Feynman’s thesis advisor, John Archibald Wheeler (age 30), was equally impressed. He believed that the amplitude formulation of quantum mechanics — although mathematically equivalent to the matrix and wave formulations — was so much more natural than the previous formulations that it had a chance of convincing quantum mechanics’s most determined critic. Wheeler writes:
Visiting Einstein one day, I could not resist telling him about Feynman’s new way to express quantum theory. “Feynman has found a beautiful picture to understand the probability amplitude for a dynamical system to go from one specified configuration at one time to another specified configuration at a later time. He treats on a footing of absolute equality every conceivable history that leads from the initial state to the final one, no matter how crazy the motion in between. The contributions of these histories differ not at all in amplitude, only in phase. . . . This prescription reproduces all of standard quantum theory. How could one ever want a simpler way to see what quantum theory is all about! Doesn’t this marvelous discovery make you willing to accept the quantum theory, Professor Einstein?” He replied in a serious voice, “I still cannot believe that God plays dice. But maybe”, he smiled, “I have earned the right to make my mistakes.”
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c093a142ab0cdd6c | Self-Consciousness Explained #2 Orthogonal Complementarity and the Transcendental Philosophical Foundation of the Unity of Physical and Psychological Concepts
Marcus Schmieke, Kränzlin, 17 July 2018
The four circle model of self-consciousness
In the transcendental philosophy of the Kant successors Fichte, Schelling and Hegel, the I is constituted by self-reflection as a pure subject. The sociophilosopher Johannes Heinrichs, who stands in the tradition of the philosophy of reflection, regards this transcendental consummation of consciousness as the subject of a phenomenological model of mind, matter and I and You consisting of four sense elements, which in its ontological interpretation becomes the triad of mind, matter and psyche[i]. In his social theory, the four-circle model of mind, matter, I and you emerge from this. [1]
The model proposed in this work is also in the thinking tradition of German idealism, which sees in consciousness as self-reflection the transcendental reason of the objectified description of reality. From the necessary distinction between reflection in itself and other, or between self-reflection and external reflection, the separation of the experience of reality into an objective external reality and an internal subjective experience is derived. The Cartesian dualism of an objective res extensa and a subjective res cogitans is joined by the transcendental subject, whereby the res cogitans in the reflection light of the subject’s execution becomes the objective content of consciousness of the psyche. The psyche experiences reality in a complementarity of material and spiritual contents, whereby in the two complementary limit values the material concept of substance (mass) and purely spiritual contents of knowledge, such as mathematical laws, stand in opposition as extremes. All concrete mental contents have complementary material and spiritual qualities, which justifies the concept of complementarity in this context. Complementarity is understood here in analogy to the term coined by Niels Bohr. He describes pairs of terms or characteristics which represent mutually exclusive perspectives on a system, but which are necessary for a complete description. They are characterized by maximum possible incompatibility in the respective context[ii]. In this work, complementarity is used both in the strictly scientific quantum-theoretical and in this analogous sense, since a key to the connection of physical and psychological knowledge is presumed in this term[iii].
In this context, the spiritual, as in Johannes Heinrichs, is understood as a medium of meaning, an a priori of the communication community, since it organizes material as well as psychological things in a meaningful way and relates them to each other. [iv] It can neither be reduced to the material nor to the psychological, nor can it be regarded as dependent on these two categories.
The theoretical physicist, cosmologist and mathematician Roger Penrose bases his scientific understanding on an analogous three-world model that supplements a platonic-mental and a physical-material world with a mental world that can know the spiritual contents, which in turn are the arrangements of physical processes[v]. According to Penrose, the physical processes in turn form the basis of the empirical consciousness of the psyche. Empirical consciousness is dependent on representation in mentally permeated material spaces but is ultimately transcendental in self-reflection and therefore independent of a concrete physical embodiment.
Three worlds after Roger Penrose
The repeated self-reflection is the motor of the interaction of material and spiritual contents in the psychological consciousness and appears there as empirical time. Empirical time is reflected in material-spiritual processes as well as in human experience, which focuses on the present. In the classical scientific models of Newton´schen mechanics, Maxwell´schen electrodynamics and the Schrödinger equation of quantum physics, however, the now is not found as an excellent element of the per se linear understanding of time. It was Carl Friedrich von Weizsäcker who, in his justification of quantum theory, first pointed out the fundamental significance of time as a present experience and placed this term at the beginning of his derivation of the structure of physics. [vi] The concept of the present is derived from the distinction between the factual of the past and the possible of the future and its dynamic transformation into one another in the present execution of the present.
Quantum Theoretical Conditions of Empirical Consciousness
Quantum theory has established itself for about 100 years as the basis of almost all scientific theories. It requires the division of reality into the observer to be described in factual terms by classical physics and the observed system, which is described with the use of the Schrödinger equation as a superposition of possibilities by the wave function. The interaction between the observer and the observed system takes place only in the now corresponding observation, whereby both change into an entangled state, which must be described by a common wave function. This is a superposition of possibilities which then, at the moment of observation, merge into a single definite factual state (reduction of the wave function) which can then be described by the terms of classical physics. The division of one reality into an observer and an observed system is called a Heisenberg section. This section reflects the Descarte´schen section as well as the relation between the transcendental subject and the empirical consciousness.[vii]
Only the factual result of an observation is suitable as the content of consciousness and thus becomes the conscious content of the psyche. The superposition of possibilities of the wave function and its dynamic temporal development according to the Schrödinger equation do not represent conscious contents, since they have no unambiguity and clarity.
Current theories of quantum neurobiology see the reduction of complex quantum physical fields in the brain to concrete factual molecular neuronal structures and activities as the physiological correspondence of consciousness processes. Here the classical result of a quantum physical measurement of the electromagnetic fields and states of the brain is mapped as a measurement result in consciousness. In this way, conscious contents of the psyche can be assigned to the factual content of quantum observations.
The model developed by Penrose and Hameroff for the orchestrated reduction of coherent neuronal quantum fields sees consciousness as a regular consequence of such reductions with a frequency between 40 and 80 Hz.[viii] The empirically experienced continuity of consciousness would thus result from a high-frequency overlap of conscious moments, each resulting from a factual reduction of quantum-physical possibility fields.
Within the framework of this theory, the wave function between its reductions develops into the factual according to the Schrödinger equation. For about 25 ms, a holographic quantum field extending over several centimetres is created in which an infinite number of possible states overlap. This superposition of possibilities generally does not correspond to conscious psychic contents due to a lack of precision but could be assigned to the unconscious psychic processes that lie between conscious thoughts or feelings and form the unconscious background of conscious events. This classification is useful because definiteness and clarity are essential properties of conscious events, while the unconscious psyche may include blurred contours and an overlapping of mutually exclusive thoughts and feelings. Later we will argue that in certain circumstances such extended quantum superpositions may appear as special extraordinary states of consciousness.
Just as in quantum physics, where a reality must be described by terms of factual and possible models, the description of psychological processes requires the coexistence of conscious and unconscious elements. In analogy to the quantum-theoretical concept of complementarity, the psychological opposition can be consciously-unconsciously regarded as a complementary system, as the following quote from C.G. Jung expresses[2]:
“Thus, we come to the paradoxical conclusion that there is no content of consciousness that is not unconscious in any other respect. Perhaps there is also no unconscious psychic that is not conscious at the same time, with the explicit exception of the unconscious and only soul-like.” [ix]
According to these considerations, in both physics and psychology it is necessary to describe both the objective and the subjective side of reality with complementary concepts, whereby the pure transcendental subject owed to the dualism of objectification remains in the background as an excluded third. In both disciplines, the objective side is represented by the complementarity of mind and matter, while the subjective side is characterized by the pairs of terms factually-possible and consciously-unconscious.
In both physics and psychology, the dynamical empirical process is based on repeated self-reflection as the pure subject. In physics, empirical time is expressed in repeated quantum observation, while in psychology it corresponds to conscious experience itself.
Orthogonal Complementarity
An orthogonal complementarity consists of two complementary pairs of terminologies. Since the involved complementarities already represent products of reflexive relations, a double complementarity is a double reflection, which is analogous to the self-reflection of the transcendental subject. Such an orthogonal complementarity could form the basis of the structural unity of physics, psychology and philosophy. The property of orthogonality indicates that the one complementary pair is already complete in the sense of a bivalent logic and the tertium non datur that is implied therein, and contains the other complementarity as its absolute negation, i.e. reflection.[x] The material-spirit duality thus needs to be supplemented by the perspective of the conscious unconscious and vice versa in order to describe the underlying introscendent origin of self-reflection. In quantum physics, the property of complementarity of non-interchangeable observables such as location and momentum can be related to the need to represent the wave function in the complex number space. The property of complementarity corresponds in some respects to the representation by complex numbers, since only the special calculus properties of complex numbers enable the common definition of complementary property spaces.[xi] The representation of the dynamics of quantum states in complex number spaces also leads to the fact that in the interior of quantum states, i.e. in their subjective being, an imaginary time can run cyclically, which does not appear in the outer empirical time. A connection to psychological phenomena and to the distinction between conscious and unconscious perception of time could be investigated against this background.
It is not surprising that physics Nobel Prize winner and co-founder of quantum physics Wolfgang Pauli came to a similar conclusion, as he conducted an intensive dialogue with C.G. Jung for more than twenty years, focusing on the unification of the physical with the psychological point of view. In “Modern Examples of Background Physics” Pauli wrote in an article not intended for publication:
“The complementarity of physics has … a profound analogy to the terms “consciousness” and “unconscious” in psychology.”[xii]
Further Pauli writes in the same article:
“According to the view held here, quaternity would not be valid within physics, but a quaternity would probably be assigned to the wholeness consisting of physics and psychology, insofar as the complementary pair of opposites of physics is reflected again in the psychic. It would be conceivable, and it even seems plausible to me, that there could be phenomena where the whole fourness plays an essential role, not only the physical and the psychological pair of opposites alone. In such phenomena, conceptual distinctions such as “physical” and “psychological” would no longer be meaningful.”
He sees the complementary pair of opposites of physics mentioned here by Pauli in analogy to the terms “conscious” and “unconscious” as the observer and the observed, whereby he sees consciousness as the subjective observer and the unconscious as the objective observed.[3] Pauli takes this view in a letter to C.G. Jung from 1954, which Jung quotes for the first time in “The Spirit of Psychology”:
“The physicist will indeed expect a correspondence in psychology at this point, because the epistemological situation concerning the terms “consciousness” and “unconscious” seems to show a far-reaching analogy to the situation of “complementarity” in physics outlined below. On the one hand the unconscious can only be opened indirectly through its (ordering) effects on contents of consciousness, on the other hand every “observation of the unconscious”, i.e. every making conscious of unconscious contents, has an initially uncontrollable retroactive effect on these unconscious contents themselves (which, as is well known, excludes in principle an “exhaustion” of the unconscious through “making conscious”). Physics will therefore conclude per analogia that precisely this uncontrollable reaction of the observing subject to the unconscious limits the objective character of its reality and at the same time lends it a subjectivity.
Furthermore, although the position of the “cut” between consciousness and unconscious (at least to some extent) is left to the free choice of the “psychological experimenter”, the existence of this “cut” remains an inevitable necessity. The “observed system” from the point of view of psychology would therefore not only consist of physical objects, but would also include the unconscious, while consciousness would play the role of the ” medium of observation “. It is unmistakable that the development of “microphysics” has brought the nature of the description of nature in this science closer to recent psychology: While the former, due to the fundamental situation referred to as “complementarity”, is confronted with the impossibility of eliminating the effects of the observer through deterministic corrections, and must therefore in principle dispense the objective recording of all physical phenomena, while the latter could fundamentally supplement the only subjective psychology of consciousness through the postulate of the existence of an unconscious of objective reality to a large extent.”[xiii]
The quaternity to be formed from Pauli’s suggestion would thus consist of the poles observer – observed – consciousness – unconscious. This shows convincingly how Pauli saw the quaternity as a reflection of the complementarities of physics and psychology in the other discipline, while the quaternity on which this article is based consists of two complementary complementarities which can be found in each of the disciplines. Pauli’s above-mentioned assumption that a common quaternity of physics and psychology is possible in certain phenomena seems to be realized in this approach, since matter and mind are concepts of both disciplines, while the subjective axis can be formulated in psycho-physical terms. Human experience as a whole thus, seems to be such a phenomenon to be described by a psycho-physical quaternity.
The Four-Circle Model of the Human Experience Space
The double complementarity can now be represented in the form of four circles which penetrate and overlap each other in such a way that the intersection of all four circles, four triple intersections around them, flanked by four double intersections and the four simple residual circle segments are created in the centre. This image allows the representation of the mutual overlapping of two complementarities in relation to different reflection levels, whereby the number of overlapping circles characterizes the reflection level. The possible assignment to basic concepts and elements of physics and psychology, in particular quantum physics and depth psychology, suggest that the diagram and the assumption of the underlying orthogonal complementarity of physics and psychology represent common structures of reality. In this way, this structure could help to find a common language for the psychological and physical realms of reality.
The simple residual circle segments correspond to the first depth of reflection[4] and thus to the immediate consciousness of matter, mind, conscious/factual and unconscious/possible.[xiv] These sensory elements in themselves are not capable of consciousness separately from each other. The unconscious without reference to the spiritual or to matter is not conscious, neither is the purely spiritual without reference to the conscious. In this context, special consideration should be given to the spiritual, which also includes the concept of information in the physical and psychological context. At the level of the first depth of reflection it is necessarily meaningless information which cannot yet be assigned to any psychic, material or living object or process. Thomas Görnitz, the quantum physicist and long-time collaborator of Carl Friedrich von Weizsäckers, has introduced such meaningless information in contrast to the classical Shannon concept of information in order to enable a derivation of the basic physical concepts without having to refer to the psyche.[xv]
The four basic psychic functions according to C.G. Jung
The second depth of reflection of the double circle elements corresponds in the psychological domain to the four basic functions of consciousness, which in the representation of C.G. Jung result in an equally orthogonal system of two complementary axes.[xvi] Here thinking and feeling are just as complementary as feeling and intuition. The latter pair of concepts forms the axis of perception, while feeling and thinking form the axis of judgement. Perception and judgement are also in complementary relation to each other.[5]
A perception should be as non-judgmental as possible, but at the same time requires the ability to recognize, which cannot be non-judgmental. At the same time the judgement changes the perception and is at the same time a perception of the same from a different perspective. Pure perception and pure judgment do not exist without the other. Likewise, the two opposite poles on the two axes are complementary to each other. The sensation perceives the being now of an object, while the intuition recognizes its possible becoming. In feeling and thinking, the overall judgement again depends on the sequence. First thinking and then feeling probably results in a different result than first feeling and then thinking about it. In this respect thinking and feeling behave like the quantum physical measurement of the location and the impulse of a particle.
In physical terms, these reflection elements of the second order of depth form the basic concepts of physical description such as mass (feeling), the phenomenology of classical observation (feeling), its measurement results (thinking) and the quantum theoretical wave function (intuition).
In the following, the four basic functions of consciousness are presented individually and set in relation to their physical counterparts. These correspondences form a core thesis of this model, since they reveal the structural and content-related similarity of analytical psychology according to C.G. Jung and quantum physics and represent an offer for further interdisciplinary theory formation:
Thinking: In the interface between the mind and the conscious pole of the psyche, thinking emerges as self-reflection with simultaneous reflection on the object of the spiritual. In this way, the human being recognises mental connections and carries out theory formation, with the help of which he can arrange the sensory perceptions gained empirically through perception. This includes, for example, the recognition and comprehension of mathematical laws, which in turn can serve as a quantitative arrangement of the results of physical measurements. The physical correspondence of thoughts is thus the quantitative measurement result, which is a transformation of sensory perception by thinking. In the language of quantum physics, this is the current information obtained by a measurement within the framework of the given theoretical model. It characterizes a classical state of the measuring device, on which the common system of measuring device and object of observation was mapped by the measurement or observation. Thinking produces systems of logic, mathematics and philosophy, which can be described as conscious figures of the spiritual. The measurement results in their arrangement follow the laws of classical logic and the formulas of mathematical physics. Philosophy relates them from the perspective of thought to its transcendental reason, self-reflection.
Intuition: While thinking appears as a more active conscious reflection of the spiritual, the reflection of the mind on the unconscious aspect of the psyche results in the more passive function of intuition ascending from the unconscious. C.G. Jung describes intuition as the function that recognizes what is possible in the objects of perception and, so to speak, takes a look around the corner into the future. This corresponds to the aspect of spiritual information, which does not clearly exist, but provides information about possible future developments. In quantum theory, this is called potential information and is represented with the wave function. It is a mathematical function that describes the temporal development of all possible states of a system in a complete superposition. While the factual measurement results which correspond to the thinking always refer to the past, the wave function which corresponds to the intuition enables a probability view into the future of the possible. The wave function develops in time strictly causally determined by the mathematical formalism of the Schrödinger equation. However, this does not result in a causality for the relationship of the measurement results to each other or for the relationship of a state of the wave function and a possible measurement result, since the transition from superposition to the factual uniqueness of the measurement results occurs through the acausal process of reduction of the wave function within the framework of a quantum observation, which is located in the quadrivalent central field of the four-circle model. This spontaneous process maps the repeated self-reflection of the transcendental subject to the empirical objectified level. It is, so to speak, the clutch at which the shaft transmits the torque of the engine to the gearbox of the four-circuit model.
Sensation: Sensation is the conscious reflection of the material and its arrangement in the outer physical space. It consists of concrete impressions, which arrange sensory impressions such as colours, forms, smells, sounds and touches spatially next to each other and chronologically behind each other. From these directly gained sensory impressions, the quantification in measurement results, which are subject to thinking on the opposite mentally-conscious side, only becomes possible in extended theory formation by comparison with collectively defined scales. In physics, the conscious reflection of matter corresponds to observation itself as physiologically performed sensory perception with direct reading of the pointer position of the measuring instrument and the sensual evaluation of the actual material processes. Just as in thinking the potential abstract information of the wave function is updated, in perception in observation the abstract matter is realized as an unconscious expression of the extended being in subjective conscious perception.
Feeling: In feelings, the unconscious aspect of the psyche and the material overlap, leading to a passive ascent of psychological impulses that occur as a physically unconscious reaction to sensations or intuitions in the consciousness. Feeling refers to the inner side of the physical, just as feeling refers more to its outer side. Although there are also sensations purely related to the inside of the body, they are more conscious and externalized than the feelings ascending from the unconscious psyche, which can bring out the depths of the material just as thinking can lead the depths or vastness of spiritual connections to consciousness. In the physical context, the matter reflected in the unconscious corresponds to the concept of mass, which appears as the idea of pure substance detached from externally visible qualities such as movement, energy or information. From physics we know today that mass in the sense of rest mass can be converted into energy and converted. The dynamics and interaction associated with this, however, is implicit and hidden in the concept of mass inside and thus unconscious. Also, in the field of unconscious mirrored matter the spiritual aspect of information is hidden as entropy. An old insight of mystical experience and tradition can be found in this analogy: The mass-aspect of the material is the spatially externally visible expression of the unfeeling. Or formulated pragmatically: In the outer you encounter as material what you are not prepared to feel in the inner.
Two complementary modes of consciousness
Between the bivalent fields of mass and potential information, in the intersection of matter, spirit, and the unconscious aspect of the psyche, there is a dynamic penetration of spiritual and material contents in the unconscious psyche which is shifted toward the possible on the temporal axis of the psyche. While the middle tetravalent intersection of all four circles corresponds to the dynamic process of the reduction of the wave function in the present and thus to the creative actualization of the inner and outer reality in conscious self-reflection, the two tetravalent intersections above and below the middle, which are each shifted in the direction of the unconscious and conscious pole, must be interpreted as components of this present process of time pointing into the future or into the past. The possible material-spiritual forms are still shortly before the complete reduction of the wave function, while its actual correspondences represent the traces of these events in material events. In quantum theory there are two types of reduction of the wave function which are called strong and weak.[xvii] According to the quantum neurobiological models already described, processes in the brain that manifest themselves in molecular changes or actions of the neuronal network become particularly conscious. These complete reductions of large-scale coherent quantum superpositions are referred to as strong reduction and leave factual traces in the memory and thus generate clear and definite contents of consciousness. They could be assigned to the trivalent intersection of matter, mind and consciousness.[xviii]
The so-called weak quantum observation, on the other hand, does not lead to a complete reduction of the wave function to a definite state, but only to its deformation.[xix] Through this, certain possibilities become more probable and others lose probability, but there is no clear selection of a certain state. Some models of the quantum theory of consciousness assume that such quantum processes can also lead to conscious perceptions. These can be predictions of future events or contents of so-called expanded states of consciousness, as they can occur in dreams, near-death experiences or induced by psychedelic substances. These events could be assigned to the intersection of mind, matter and unconscious. The experience consciousness of the now in the central tetravalent intersection corresponds to the periodically repeated actualization of the wave function. In this context, the unconscious trivalent field can be assigned to the quantum fields of global coherent states in the nervous system and their states of consciousness modified by weak quantum observations and entanglements, while the conscious trivalent field corresponds to the classically actualized brain and consciousness processes emergent from the actualization of the wave function of the global coherent quantum fields of the brain.[xx]
In depth-psychological language, C.G. Jung expresses this connection in the following quote:
“But since the existence of highly complex, consciousness-like processes in the unconscious is at least made immensely probable by the experience of psychopathology and dream psychology, we are forced to conclude that the state of unconscious content is not more consciously equal to that of the brain, but somehow similar. Under these circumstances, there is probably nothing left to do but to assume a middle ground between the concept of an unconscious and a conscious state, namely an approximate consciousness.”[xxi]
The approximate consciousness is an approximation or approach to the superposition of its two described factual and potential varieties and does not occur as a pure form, as is characteristic for complementary conceptual systems. Carl Friedrich von Weizsäcker describes the existence of separate individual objects as a classical approximation as well as the individual conscious subjects, caused by Heisenberg’s cut.[xxii] From the point of view of quantum theory, empirical consciousness is thus an approximation of the transcendental subject and thus also an approximative one.
Material and mental poles of psychic dynamics
The central tetravalent field of self-consciousness is flanked on the spirit-matter axis by two further trivalent fields, each shifted either toward the spiritual or material pole. This represents a shift on the spatial axis which could result in different spatial manifestations of consciousness. In a similar way as shifting on the temporal axis leads to more future or past related forms of consciousness.
A shift of the overlapping of conscious and unconscious aspects of the psyche towards matter in the matter-psycho-field could rather produce organic disordered spatial structures of consciousness as found in the vegetative nervous system of the human abdominal brain. However, if the center of consciousness is shifted toward the spiritual as in the mind-psych field, more hierarchically structured forms such as those of the central nervous system could emerge. In the terms of the Swiss psychotherapist Remo Roth, the matter-psyche field would be assigned to Eros consciousness, which is carried by the vegetative abdominal brain, which spreads in a network-like manner in the body, while the mind-psyche field corresponds to the Logos consciousness of the hierarchically structured central nervous system.[xxiii] Thinking and intuition thus lead into Logos-consciousness and out of the body, while sensation and feeling lead introvertedly connected into the body. Only in their complementary combination do these two forms of the human psyche lead to a complete consciousness.
The Space and Time Axis
Space and time form a superordinate complementarity in relation to consciousness, since they cannot be experienced and described separately in empirical consciousness. However, the two orthogonal axes can be assigned to the terms space and time, since the conscious and unconscious pole of the psyche as the factual and possible correspond to the two modes of time, while matter and spirit constitute the classical res extensa of extended substance. While our consciousness reflects the past and thus its own history, the psyche in the unconscious prepares the near and distant future. While I am typing these words, in the unconscious aspect of my psyche, the words at the end of this sentence are already ready, even though my consciousness does not yet carry them within itself. At the same time, however, I am already aware of the meaning of the entire sentence at the beginning.
The spatial axis between matter and mind can be occupied again with the complementary properties concrete-abstract, while in relation to the psyche the conscious side can be described as external and the unconscious as internal. This results in the square concrete-abstract-external-internal, which can also be translated into the four functions of consciousness. The concrete-external consciousness expresses itself in sensation, while the opposite abstract-internal consciousness expresses itself in intuition. The external-abstract consciousness is thinking, while the internal-concrete can be assigned to feeling. The physicist Harald Atmanspacher comes to this assignment in his work Raum, Zeit und psychische Funktionen[xxiv] (Space, Time and Psychic Functions), but in relation to the concepts of space and time in Emmanuel Kant’s Kritik der reinen Vernunft (Critique of Pure Reason). However, he assigns sensation and intuition to space and thinking as well as feeling time. In the scheme shown here, the four functions of consciousness are not clearly assigned to space and time, but always connect both forms of perception with each other.
The structural diagram of two orthogonal complementary axes of human knowledge presented here can be understood from a transcendental philosophical point of view as an objectified representation of self-consciousness as repeated reflection in oneself and others. From an epistemological perspective, it thus combines psychological and physical concepts and concretizes them into neurobiological structures. The resulting analogy and clarity of the structural and sense references indicates that the orthogonal complementarity of objective (material/concrete-spirit/abstract) and subjective (conscious/external-unconscious/internal) represents fundamental reflexive structures of empirical consciousness. The focus is on the physical quantum observation process, the spontaneous psychic experience and the dynamic interplay of global coherent non-local neurobiological quantum fields, which, as extended quantum presences, reach some way into the past and future, as well as into mind and matter, thus forming the present as the fundamental mystery of human life.
Special thanks go to Prof. Johannes Heinrichs, whose work provided the basis for many of the thoughts behind this work and who helped to sharpen some concepts and con
The German philosopher and logician Gotthard Günther also sets up a space for reflection with four ontological components: being, nothing, I and you, whereby nothing is to be understood as a reflection of being, the I as a reflection on the not-being negation and the you as the thematic inversion of the I. The German philosopher and logician Gotthard Günther also sets up a space for reflection with four ontological components: being, nothing, I and you. The juxtaposition of mind and matter cannot be understood in the sense of a classical reflection, but as a thematic inversion, just like the relation between ego and you. While in the classical reflection according to Günther being and non-being face each other in the sense of a negation, the thematic inversion always represents the transition from the determining to the determining motif of reflection. Thus, thinking is the thematic inversion of self-consciousness, you the thematic inversion of the I, and spirit the thematic inversion of matter. In the latter relation, matter is no longer understood as pure being in the sense of classical logic, but already as a process reflected in itself, as is compellingly apparent from quantum physics. In the context of quantum theory, matter can only be regarded as an interplay of abstract dynamics of probability functions and empirical observation as a reduction of the probability function, which in itself represents a reflection process. Since matter itself is already reflexive, it cannot simply be represented by a classical negation in thought but requires a thematic inversion into the spiritual. Matter and spirit appear here as the objective and subjective side of an inversion relationship in which the inside is depicted in the outside and the abstract in the concrete in one another. Therefore, the matter appears objective and the spirit subjective, although each of these sense elements carries in itself the other pole as its essence. The thematic inversion has a strong correspondence to the concept of complementarity of quantum theory and to the relationship between the psychological concepts of the conscious and the unconscious. The conscious is the sense of the unconscious, just as the unconscious is the sense of the conscious. Both concepts need each other for mutual determination. The thematic inversion, so to speak, only turns its interior into the exterior and vice versa. The physical impulse is only defined by the temporal change of place, while the place as a spatial property is classically derived from movement. In the place the movement is contained and in the movement the place. Therefore, both terms are complementary as observables in quantum physics, as are the associated images of particles and waves. The wave is defined as the possibility of local interactions in the form of particles, while the particle appears as an update of a spatially extended wave process. The relation of the thematic inversion seems to me to be closely related to the concept of complementarity.
Der quantenphysikalische und hier vorgeschlagene psychologische Komplementaritäts-Begriff kann, wie in einer vorigen Fußnote bereits dargelegt, als thematische Inversion im Kontext einer auf den Sinn bezogenen Reflexions- oder Sinnlogik verstanden werden. Während in der klassischen Logik das Sein als mit sich selbst identisch nur widerspruchsfrei gedacht werden kann, definiert die Sinnlogik den Sinn als durch einen geschlossenen Reflexionskreis, der seine eigene Negation durchläuft. Sinn ist keine Identität, sondern ein Gegenverhältnis zweier Bewusstseinsmotive, die sich darin gegenseitig bestimmen wie z.B. Wahrheit und Irrtum oder das Endliche und das Unendliche.
This concept of the unconscious is therefore a summarizing objectification of many unconscious functions in actu (remark by Johannes Heinrichs).
Gotthard Günther’s non-classical logic contains four levels of depth of reflection on the third and final level of reflection, the first corresponding to immediate consciousness, the second to simply reflected consciousness, the third to infinitely iterable consciousness and the fourth to self-consciousness.
Judgement, strictly speaking, could also appear as a reflection on perception, which in my opinion does not fully do justice to the functions of consciousness it contains, since each of these is to be understood fundamentally at the same level as a reflection process of self-consciousness. Feeling could therefore also be understood as perception of a sensation, while intuition could also be seen as judgment of a thought.
Heinrichs, Johannes; Öko – Logik: Geistige Wege aus der Klima- und Umweltkatastrophe; Steno 2007
Atmanspacher, Harald. (1996). Metaphysics taken literally. In Honor of Kalervo Laurikainen’s
80th Birthday. In U. Ketvel (Ed.), Festschrift in Honor of K.V. Laurikainen’s 80th Birthday. (pp. 49-59). Helsinki: University of Helsinki Press.
Walach, Harald; Generalisierte Quantentheorie (Weak Quantum Theory): Eine theoretische Basis zum Verständnis transpersonaler Phänomene
Heinrichs, Johannes; Kritik der integralen Vernunft Bd. 1 und 2; 2018
Penrose, Roger; Das Große, das Kleine und der menschliche Geist, Spektrum Akademischer Verlag, 1997
Weizsäcker, Carl Friedrich von; Aufbau der Physik, DTV 1988
Lucadou, Walter von; Leiblichkeit – L’homme machine – Mensch-Maschinen-Interaktion: Erweiterung oder Konstriktion des Weltbezuges. Dynamische Psychiatrie 2016, Vol. 49, S. 208-234.
Stuart Hameroff, Quantum computation in brain microtubules? The Penrose-Hameroff Orch OR model of consciousness, Departments of Anesthesiology and Psychology, The University of Arizona, Tucson, AZ 85724, USA
Jung, Carl Gustav; Theoretische Überlegungen zum Wesen des Unbewussten, GW Band 8 § 385
Gotthard Günther, Metaphysik, Logik und die Theorie der Reflexion, Beiträge zur Grundlegung einer operationsfähigen Dialektik, Band 1, Felix Meiner Verlag, Hamburg 1991
Sautter , Ulrich; Komplexität und Korrelation: eine logische Propädeutik der Quantenmechanik, Tectum Verlag, Marburg 1999
Pauli, Wolfgang, Moderne Beispiele der Hintergrundsphysik, 1948
Jung, Carl Gustav; Theoretische Überlegungen zum Wesen des Psychischen, GW Band 8 S. 262, Anm.
Gotthard Günther, Metaphysik, Logik und die Theorie der Reflexion, Beiträge zur Grundlegung einer operationsfähigen Dialektik, Band 1, Felix Meiner Verlag, Hamburg 1991, S. 28
Thomas Görnitz/ Brigitte Görnitz; Die Evolution des Geistigen, Vandenhoeck & Ruprecht, Göttingen 2009
von Franz, Marie-Louise, Hillman, James: Zur Typologie C. G. Jungs. Die inferiore und die Fühlfunktion. Bonz Adolf, 1992
Aharonov, Y; Albert, D.; Vaidman, L.; How the Result of a Measurement of a Component of the Spin of a Spin 1/2 Particle Can Turn Out to be 100; PHYSICAL REVIEW LETTERS 1988; VOLUME 60, NUMBER 14
King, Chris; Space, Time and Consciousness; Cosciousness Became the Universe, Science Publisher 2017
Jung, Carl Gustav; Theoretische Überlegungen zum Wesen des Unbewussten, GW Band 8 § 387
Thomas Görnitz, Carl Friedrich v. Weizsäcker, Physiker, Philosoph, Visionär. Verlag der C.F.W. Stiftung, Enger 2012;
Remo F. Roth; Return of the World Soul: Wolfgang Pauli, C.G. Jung and the Challenge of
Psychophysical Reality; Part 1 and 2; Pari Publishing 2011, 2012
Atmanspacher, Harald; Raum, Zeit und psychische Funktion, Der Pauli-Jung-Dialog und seine Bedeutung für die moderne Wissenschaft, Springer Verlag 1995
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9cdc3d74d7f99c55 | Fachbereich Physik
Navigation und Suche der Universität Osnabrück
We develop and apply materials and methods for the investigation of electron spin coherence in nanoscopic systems and devices. The spin of the electron adds another dimension to electronics that can be useful for applications but also for device analysis. Our research programme consists of three branches:
Coherent Spin Systems
Spins in condensed matter are often paired since unpaired electrons are chemically very reactive. Two classes of spin systems with long coherence times are trapped atoms that retain their open-shell configuration when encapsulated in a molecular cage like the N@C60 endohedral fullerene, and defects in semiconductors like the NV centre in diamond.
Endohedral Fullerenes
Fullerenes such as the soccer ball-shaped C60 provide an ideal molecular cage for the encapsulation of paramagnetic atoms: the hollow space is just large enough to fit the spin S = 3/2 atoms nitrogen or phosphorus by repulsive van der Waals forces (shown on the right). There are thus no chemical bonds between the encapsulated atom and its molecular cage, and hence no transfer of spin density. This leads to relatively long electron spin coherence at room temperature (T2 = 50 µs), which is essentially limited by vibrations of the molecular cage (i.e. by T1 = 100 µs). At low temperatures (T < 100 K) where these vibrations are not excited, spin coherence can persist up to milliseconds.
In our group, we fabricate mostly N@C60 by N+ ion bombardment of bare C60 molecules in our implantation setup. The filling ratio is initially low and needs to be enhanced by repeated fractionation using high-performance liquid chromatography. This process is controlled by monitoring the electron spin resonance signal unique to N@C60 (see figure right). High-purity (> 99 %) N@C60 material can be obtained using this method, thus providing a true source of ready-made qubits that are stable at room temperature.
NV-Centres in Diamond
The NV (nitrogen-vacancy) centre is one of the many optically active defects in diamond. Its outstanding properties of high optical stability, relatively high fluorescence yield, combined with its paramagnetic ground state, have turned it into a highly popular qubit candidate. We are interested in very shallow NV centres that are sufficiently close to the surface so that they can couple to external spins. These can be prepared by N+ion bombardment with similar ion energies as those used in the production of endohedral fullerene.
Ultra-Sensitive Spin Detection
Traditional Electron Spin Resonance (ESR) uses microwave detection, which is inherently limited in sensitivity due to the small energy of microwave photons. It is however often the method of choice for systems with a paramagnetic ground state. Selection rules governing the lifetimes and transport of elementary excitations, on the other hand, can be quite strict even at room temperature, especially in organic materials. The well-known weakness of inter-system crossing is one of these rules, which may lead to spin-dependent fluorescence. A similar process governs charge carrier recombination in semiconductors and leads to spin-dependent currents. Since electrical charges or currents and also luminescence photons can be detected with very high quantum efficiency, electron spin resonance sensitivity can be increased hugely.
Spin-Dependent Currents
When a charge carrier (electron or hole) gets trapped and localized, it may form a pair state with other nearby carriers. If the total spin of such a pair is not zero, carrier recombination is suppressed and the pair can dissociate again and thus contribute to the total current. Using resonant microwaves, we can manipulate the spin pair state and detect the resulting changes in the (photo-) current.
We could detect the motion of only 3·103 spins in a C60 thin-film device at room temperature (W. Harneit et al., Phys. Rev. Lett. 98 (2007) 216601), demonstrating a 106- fold increase in sensitivity over traditional ESR. We obtained even better sensitivity for zinc phthalocyanine (ZnPc) thin films and could show that the interfaces between ZnPc and both, the metal contact and the C60 layer, are of primordial importance for the efficiency and stability in ZnPc-C60 bilayer organic solar cells.
More on this topic can be found in the applications section, see Organic Electronic Devices.
Spin-Dependent Fluorescence
The NV centre in diamond can be detected at the single defect level. This is possible due to the higher quantum efficiency of fluorescence detection (optical photons as opposed to microwave photons), confocal microscopy (spatial resolution of 300x300x600 nm3), and optical polarization of the paramagnetic ground state of the NV centre. We have developed our setup for NV spin detection with the special purpose of coupling experiments in mind. The following picture illustrates this setup.
More on this topic can be found in the applications section, see Scalable Quantum Register.
Our applied research is dedicated to areas where spin coherence is important. We focus mainly on two fields:
Organic Electronic Devices
Organic electronic devices are interesting for many applications, but they often suffer from limited efficiency, or limited stability, or both. A big advantage they offer is that molecular materials can be designed with far more versatility than crystalline semiconductors so that, for example, opto-electronic devices can be made much thinner than silicon devices. A well-known mass product is the AMOLED display used in certain smartphones.
The small thickness (typically 2 - 200 nm) of the active layer in an organic device can lead to an analysis problem. Defects in the active layer, or imperfections in its morphology, may occur in very small amounts that escape observation using bulk-sensitive methods. Ultra-sensitive methods like scanning probe or electron microscopy, on the other hand, often require special sample preparation and cannot be performed on complete devices. Trial-and-error is therefore still often used in device development.
We develop the technique of electrically detected electron spin resonance (EDESR) as a new method to characterize complete electronic devices at room temperature.
In organic semiconductors, spin selection rules have a detectable influence on the recombination of photo-generated electrons and holes. This means that we can monitor changes in the photo-current of a solar cell (its most important device characteristic) while manipulating the spin states with microwaves. Only long-lived spin states will be susceptible to this manipulation. But those are the spin states that indicate unwanted processes in the solar cell, i.e., electrons and holes that have become trapped at defects or at interfaces. We are thus very sensitive to such imperfections. Furthermore, the spectroscopic properties of the signal can tell us more about the nature and the location of the imperfection.
In a new project funded by the DFG in the context of the priority programme SPP 1601 “New Frontiers in Sensitivity for EPR Spectroscopy: From Biological Cells to Nano Materials”, we investigate the sensitivity limits of this method by going to true nano-devices like molecular wires.
Scalable Quantum Register
In a quantum computer, qubits are the basic unit of information. In principle, any two (or few)-level quantum system can be used to encode a logical 0 and 1. In the pure state, which acts very similarly to a classical bit (or state), the quantum system will be measured with 100 % probability as occupying one of these two states. The extra quantum property is to allow arbitrary superposition states ∣Ψ＞ = c0∣0＞ + c1∣1＞ where c0, c1 ∈ ℂ are two complex amplitudes that obey the normalization condition ∣c02 + ∣c12 = 1. Upon measurement, the square of these amplitudes describes the probability of the measurement result.
The power of quantum computing comes about when these amplitudes are deterministically manipulated using external parameters such as optical or microwave fields, or electrical gates. This permits very dense information coding and heavily parallel information processing. However, the superposition states are usually not energy eigenstates and thus not very robust. Any interaction with the environment can lead to changes of the wave function, i.e., to qubit errors. The key figure to consider is therefore the coherence time, which describes the amount of time that the system follows the coherent (and thus deterministic) evolution prescribed by a Schrödinger equation.
In earlier years, we have shown that N@C60 and P@C60 can be used as qubits in this sense and that they have long coherence times even at room temperature.
A quantum register can be considered as the next larger unit of information. The key idea we pursue is to use several of our N@C60 qubits in a device configuration as depicted below.
First, we put them into carbon nanotubes, which nicely align them in a linear chain. This chain is then places near an NV centre in diamond, which acts as a spin-state read-out for several of the N@C60 qubits. Our goals are to demonstrate the read-out mechanism for this configuration and to explore the scalability limits.
We have recently started this project in collaboration with FZJ and generous funding provided by the VolkswagenStiftung. First results are described in these papers. |
4c4816e4b0dbd856 | Saturday, May 15, 2021
Quantum Computing: Top Players 2021
[This is a transcript of the video embedded below.]
Quantum computing is currently one of the most exciting emergent technologies, and it’s almost certainly a topic that will continue to make headlines in the coming years. But there are now so many companies working on quantum computing, that it’s become really confusing. Who is working on what? What are the benefits and disadvantages of each technology? And who are the newcomers to watch out for? That’s what we will talk about today.
Quantum computers use units that are called “quantum-bits” or qubits for short. In contrast to normal bits, which can take on two values, like 0 and 1, a qubit can take on an arbitrary combination of two values. The magic of quantum computing happens when you entangle qubits.
Entanglement is a type of correlation, so it ties qubits together, but it’s a correlation that has no equivalent in the non-quantum world. There are a huge number of ways qubits can be entangled and that creates a computational advantage - if you want to solve certain mathematical problems.
Quantum computer can help for example to solve the Schrödinger equation for complicated molecules. One could use that to find out what properties a material has without having to synthetically produce it. Quantum computers can also solve certain logistic problems of optimize financial systems. So there is a real potential for application.
But quantum computing does not help for *all types of calculations, they are special purpose machines. They also don’t operate all by themselves, but the quantum parts have to be controlled and read out by a conventional computer. You could say that quantum computers are for problem solving what wormholes are for space-travel. They might not bring you everywhere you want to go, but *if they can bring you somewhere, you’ll get there really fast.
What makes quantum computing special is also what makes it challenging. To use quantum computers, you have to maintain the entanglement between the qubits long enough to actually do the calculation. And quantum effects are really, really sensitive even to smallest disturbances. To be reliable, quantum computer therefore need to operate with several copies of the information, together with an error correction protocol. And to do this error correction, you need more qubits. Estimates say that the number of qubits we need to reach for a quantum computer to do reliable and useful calculations that a conventional computer can’t do is about a million.
The exact number depends on the type of problem you are trying to solve, the algorithm, and the quality of the qubits and so on, but as a rule of thumb, a million is a good benchmark to keep in mind. Below that, quantum computers are mainly of academic interest.
Having said that, let’s now look at what different types of qubits there are, and how far we are on the way to that million.
1. Superconducting Qubits
Superconducting qubits are by far the most widely used, and most advanced type of qubits. They are basically small currents on a chip. The two states of the qubit can be physically realized either by the distribution of the charge, or by the flux of the current.
The big advantage of superconducting qubits is that they can be produced by the same techniques that the electronics industry has used for the past 5 decades. These qubits are basically microchips, except, here it comes, they have to be cooled to extremely low temperatures, about 10-20 milli Kelvin. One needs these low temperatures to make the circuits superconducting, otherwise you can’t keep them in these neat two qubit states.
Despite the low temperatures, quantum effects in superconducting qubits disappear extremely quickly. This disappearance of quantum effects is measured in the “decoherence time”, which for the superconducting qubits is currently a few 10s of micro-seconds.
Superconducting qubits are the technology which is used by Google and IBM and also by a number of smaller companies. In 2019, Google was first to demonstrate “quantum supremacy”, which means they performed a task that a conventional computer could not have done in a reasonable amount of time. The processor they used for this had 53 qubits. I made a video about this topic specifically, so check this out for more. Google’s supremacy claim was later debated by IBM. IBM argued that actually the calculation could have been performed within reasonable time on a conventional super-computer, so Google’s claim was somewhat premature. Maybe it was. Or maybe IBM was just annoyed they weren’t first.
IBM’s quantum computers also use superconducting qubits. Their biggest one currently has 65 qubits and they recently put out a roadmap that projects 1000 qubits by 2023. IBMs smaller quantum computers, the ones with 5 and 16 qubits, are free to access in the cloud.
The biggest problem for superconducting qubits is the cooling. Beyond a few thousand or so, it’ll become difficult to put all qubits into one cooling system, so that’s where it’ll become challenging.
2. Photonic quantum computing
In photonic quantum computing the qubits are properties related to photons. That may be the presence of a photon itself, or the uncertainty in a particular state of the photon. This approach is pursued for example by the company Xanadu in Toronto. It is also the approach that was used a few months ago by a group of Chinese researchers, which demonstrated quantum supremacy for photonic quantum computing.
The biggest advantage of using photons is that they can be operated at room temperature, and the quantum effects last much longer than for superconducting qubits, typically some milliseconds but it can go up to some hours in ideal cases. This makes photonic quantum computers much cheaper and easier to handle. The big disadvantage is that the systems become really large really quickly because of the laser guides and optical components. For example, the photonic system of the Chinese group covers a whole tabletop, whereas superconducting circuits are just tiny chips.
The company PsiQuantum however claims they have solved the problem and have found an approach to photonic quantum computing that can be scaled up to a million qubits. Exactly how they want to do that, no one knows, but that’s definitely a development to have an eye on.
3. Ion traps
In ion traps, the qubits are atoms that are missing some electrons and therefore have a net positive charge. You can then trap these ions in electromagnetic fields, and use lasers to move them around and entangle them. Such ion traps are comparable in size to the qubit chips. They also need to be cooled but not quite as much, “only” to temperatures of a few Kelvin.
The biggest player in trapped ion quantum computing is Honeywell, but the start-up IonQ uses the same approach. The advantages of trapped ion computing are longer coherence times than superconducting qubits – up to a few minutes. The other advantage is that trapped ions can interact with more neighbors than superconducting qubits.
But ion traps also have disadvantages. Notably, they are slower to react than superconducting qubits, and it’s more difficult to put many traps onto a single chip. However, they’ve kept up with superconducting qubits well.
Honeywell claims to have the best quantum computer in the world by quantum volume. What the heck is quantum volume? It’s a metric, originally introduced by IBM, that combines many different factors like errors, crosstalk and connectivity. Honeywell reports a quantum volume of 64, and according to their website, they too are moving to the cloud next year. IonQ’s latest model contains 32 trapped ions sitting in a chain. They also have a roadmap according to which they expect quantum supremacy by 2025 and be able to solve interesting problems by 2028.
4. D-Wave
Now what about D-Wave? D-wave is so far the only company that sells commercially available quantum computers, and they also use superconducting qubits. Their 2020 model has a stunning 5600 qubits.
However, the D-wave computers can’t be compared to the approaches pursued by Google and IBM because D-wave uses a completely different computation strategy. D-wave computers can be used for solving certain optimization problems that are defined by the design of the machine, whereas the technology developed by Google and IBM is good to create a programmable computer that can be applied to all kinds of different problems. Both are interesting, but it’s comparing apples and oranges.
5. Topological quantum computing
Topological quantum computing is the wild card. There isn’t currently any workable machine that uses the technique. But the idea is great: In topological quantum computers, information would be stored in conserved properties of “quasi-particles”, that are collective motions of particles. The great thing about this is that this information would be very robust to decoherence.
According to Microsoft “the upside is enormous and there is practically no downside.” In 2018, their director of quantum computing business development, told the BBC Microsoft would have a “commercially relevant quantum computer within five years.” However, Microsoft had a big setback in February when they had to retract a paper that demonstrated the existence of the quasi-particles they hoped to use. So much about “no downside”.
6. The far field
These were the biggest players, but there are two newcomers that are worth having an eye on.
The first is semi-conducting qubits. They are very similar to the superconducting qubits, but here the qubits are either the spin or charge of single electrons. The advantage is that the temperature doesn’t need to be quite as low. Instead of 10 mK, one “only” has to reach a few Kelvin. This approach is presently pursued by researchers at TU Delft in the Netherlands, supported by Intel.
The second are Nitrogen Vacancy systems where the qubits are places in the structure of a carbon crystal where a carbon atom is replaced with nitrogen. The great advantage of those is that they’re both small and can be operated at room temperatures. This approach is pursued by The Hanson lab at Qutech, some people at MIT, and a startup in Australia called Quantum Brilliance.
So far there hasn’t been any demonstration of quantum computation for these two approaches, but they could become very promising.
So, that’s the status of quantum computing in early 2021, and I hope this video will help you to make sense of the next quantum computing headlines, which are certain to come.
I want to thank Tanuj Kumar for help with this video.
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2. It is not my speciality but, like you, I think it is an important new discipline in modern sciences. Presumably we will have to make enorm progresses in controlling the fabrication of superconducting devices at room temperature or and of cooling systems. Once again the technological progresses are meaningful only if they bring advantages, for example in chemistry and-as consequence- in biology and then in medicine (bioengeneery). Otherwise, if the long range purpose is only a military suppremacy... We are actually facing more important and immediate problems, I think !
1. There is a challenge that involves China on this. It is interesting how China has become the bogie-man these days. Their government has been playing dishonestly for many years, but American companies were making profits anyway, so nobody paid attention. It took t'Rump to raise the alarm, but his way of doing this was hopelessly wrong. With all his stuff about Kung-Flu and China-virus we now have a sickening east-Asian hate thing going on.
China has its sights on gaining a monopoly on as many technological areas as possible. They have a lot of world outside their sphere of influence to surpass. Whether they succeed and if the US and EU (UK if there is such a thing before long) rise to the challenge is to be seen.
Russia is also a bit of a challenge, but their military developments are being built on a weak national and economic basis. From the white Tsars, to the red Tsars and now the blue-grey Tsars this pattern has happened repeatedly.
China wants to master a global quantum internet. EPR and other entanglements W, GHZ etc will probably be implemented on fiber optic and U-verse. They may succeed, and the logo image of Huawei looks a bit like a sliced up apple.
It is too bad in a way that this all feeds into power games and militarization.
2. If I am not mistaking Roger Penrose in the Emperors New Mind supposes that the Human Brain might be another type of quantum computer, quite advanced one indeed.
3. While Penrose has written extensively on this and related topics, it's all philosophic speculation with not one whit of actual evidence behind it.
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5. Penrose's idea of humans performing quantum computations is not likely right. For one thing, bounded quantum polynomial space is a subset of PSPACE, which is the set of algorithms that satisfy the Church-Turing thesis --- well modulo oracle inputs. This most likely means the human brain does not perform self-referential loop calculations that skirt the limits set by Godel and Turing.
3. The D-wave computer is an annealing machine. This is a sort of quantum version of a neural network. It is an optimizing system.
Quantum computers are based on linear algebra on a complex field. As such quantum computers only really solve linear algebra problems. The Shor algorithm is a Fourier transform method. A lot of mathematics, and by extension physics, involves linear algebra. Many mathematical theorems are solved by transforming the problem into linear algebra, where the methods are well known.
The quantum computer will creep into the computing world slowly at first and in time will assume some level of importance. There are other architectures that will also assume more importance, artificial neural nets, spintronics, and others. As computing most probably must conform to the Church-Turing thesis it is likely though these systems will be supplementary to a standard von-Neumann computer, such as what we have.
1. Hi Lawrence,
Perhaps neural nets and other AI computing will be among the specialist applications that quantum computing will be used for, with hybrid systems made to increase efficiency.
And WRT China, I think where they lack advantage is in collaboration and information exchange internationally; same with Russia. I personally think the greatest leaps will come with cross-pollination of methods and ideas.
4. If you have some spare time and enjoy a quality Sci-Fi you may find recent "DEVS" series quite enjoying..Long story short a billioner builds a working quantum computer capable of emulating physical Reality; the story is smart, intriguing and the ending is quite unexpected.
5. ion traps - shouldn't be positive charge (instead of negative
1. Yes, sorry, I have fixed that in the text. Can't fix it in the video. It's in the info.
6. Interesting video. So how long do you think before Shor's Algorithm will be run for non trivial cases?
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8. Is Shor's Algorithm the fastest algorithm, or is the mindset on what is logically fastest wrong? I think fixing that issue would be necessary for understanding if quantum or regular computers will be the fastest. Only example I have that can produce evidence of screwing up the notion of run time is Simple algorithm I figured out years ago. Take N odd, and instead of trying to find PQ as a rectangle like kids would do with unit blocks, you find a unit block trapazoid it converges to and then that shape can be broken into a rectangle. While it can be simplified in computer speak, geometry of it is just first try to make a right triangle with the square blocks. If the blocks make a right triangle then break the triangle if its even height in half and flip around and you got a rectangle, which gives two factors. If its odd break off the nose and flip it up and you got a rectangle, which gives factors. But likely there remainder blocks at bottom of triangle, so you take rows off the top and throw them on the bottom till there is no remainder, you converge to a block trapezoid. The run time for this is bizaree. If one takes N odd, and trying to find p or q, but say takes 3N, it can actually often does converge faster.. it can pop out p or q or 3p or 3q or pq. This is because of the geometry, because it will tend to converge to the largest factor first below the initial triangle height. That why increasing the size it can converge faster. Shor's algorithm fits this nice notion of Log N convergence... not this chaos algorithm convergence time of periodic unknown. Its runtime is like rolling a dice and sometimes it will be instant, not matter how large the key size of something like RSA. That is why I think its important for the discussion of quantum computers verses regular computers that the computer science notion of Run Time itself be challenged with evidence like the chaos algorithm above has produced. If we live in an evidence based system, and there out of the blue pops up evidence that run time itself may need core logic retuning, then that needs to be done. I really should publish the algorithm and the data charts for how it defies the notion of runtime. But I can say with evidence that Shor's algorithm while nice and pretty, these screwy unexplored functions at times can beat it. And if a set of these chaos algorithms can be put together and ran at the same time, there possible a high density of constant run time convergence where N size is relatively unimportant. Entire class of algorithms that are unexplored. The beauty of math has no boundaries.
1. Shor's algorithm is the fastest polynomial time quantum algorithm. Non-quantum integer factorization algorithms are exponential so incomparably slower. That's why RSA and majority of all our cryptosystems are in danger if a ~1M qubits size working quantum computer goes online...
2. FWIW, Wikipedia reports that there are multiple cryptographic systems that are not breakable by the known quantum computer algorithms, and that several of these date back to the last century*. It'll be a minor irritation to have to switch to a different system, but my understanding is that software that implements quantum-safe cryptography has already been written and is ready to use.
*: At least two (from the 1970s) predated Shor's Algorithm, and a couple more were invented after that.
9. I am surprised that when a quantum computer needs about a million qubit, Google already declared quantum supremacy with only 53 qubit. If this claim is correct then, with a million qubit, the quantum computer will be really fantastic. Do you agree?
1. FWIW, the "problem" that Google "solved" in achieving "quantum supremacy" was simulating quantum gates. It's not particularly surprising (to me, anyway) that quantum gates are good at simulating quantum gates, but, whatever. The main spokesperson for quantum computing has been very specific in stating that this "problem" and it's "solution" are unrelated to anything anyone would ever want or need to do with a computer, but he insists that (a) it's true that quantum supremacy has been achieved, and (b) it's really good that someone found something to actually do with current quantum computers.
As I said, "whatever".
I will admit, though, that as a 1970s/1980s generation Comp. Sci. type, I'm surprised how few things other than Shor's algorithm have been found that can be done with quantum computers. According to a recent blog post at a Comp. Sci. blog, there's really only one other algorithm. And it's been 25 years.
10. The algorithms which supposedly demonstrate "quantum speedup" tend to have caveats, for example the quantum Fourier transform part of Shor's algorithm would scale well but the exponentiation part which is required to load your number (and test integer) into the QFT is much heavier. There are algorithms which supposedly demonstrate that an oracle can be interrogated once in order to obtain all the information about it, but the oracle is necessarily part of your quantum circuit so you already knew how to program it. They tend to rely on a conditional-NOT gate kicking its phase back to the control qubit if your control and target qubits are not in pure |0> or |1>.
(Generally, even the problem of how to make the most efficient transpilation of a quantum circuit onto real hardware isn't even solved.)
But "entanglement" isn't always necessary, but rather superposition.
(I make material for semiconductor qubits by the way, but I'm not affiliated with TU Delft).
1. For oracle problems, if you put your oracle on one half of the computer chip and the algorithm circuit on the other half, I really don't see why this wouldn't demonstrate quantum speedup.
11. Quantum computing is a scam to pump up stock, the base physics of it is flood/wrong, they will use specialied hardware (ex: Cuda Cores) with AI to get certain calculations done then will call it quantum computer.
Let's look at other things like Light Computing and see what goes on there.
1. Lee, I agree entirely. The 'ideal' they still aim for remains an infinite distance away because the theory behind it is wrong. The Swiss banks had the good sense to turn down Anton Zeilingers proposal for quantum cryptographic security because is was founded on Poppers 'mud'. Those throwing millions into trying to develop true quantum computers may also one day see through the hype. Uncertainty has a real fundamental physical cause (SpringerNature paper imminent) and I suggest it can't be overcome.
2. Hi Lee and Peter,
so you both think Dr. Hossenfelder is mistaken about the current developments, or what?
3. C Thompson,
Those are guys who don't understand how quantum computing works and who also haven't noticed, apparently, that as a matter of fact it does work, and that we know -- again, for a fact -- that the theory behind it is correct (in the parameter range tested, etc etc).
World's full with people who have strong opinions on things they know very little about.
4. Dr. Hossenfelder,
Indeed. I wondered what they thought made them better-informed about quantum computing that you supposedly missed with your well-researched and comprehensive summary, and why they saw fit to comment thusly on this blog, with no evidence to back up their claims, especially Lee's comment.
5. C Thompson,
Yes, you are asking very good questions...
12. 1000000 qubits? Still a lot of questions.
I think I can show even this would live up to the hype with the following thought experiment:
For the sake of discussion, lets imagine we want to break an encryption which uses 100 digit prime numbers for keys. So you would have factor a 200 digit number into a couple of 100 digit primes to break the code.
Now this is going to be pretty tough. Consider how many 100 digit prime numbers there are, and that defines the space you have search with your quantum code breaker in order to find your answer.
Now remember that getting "close" with some kind of refinement process isn't going to hack it. The nature of beast is such that you either find the answer or you don't. What is more, you are constrained by your search because you have to deal with the whole search space directly as a whole, because the information held by your superimposition is only held holistically. You can't look at just a part of the space and hope to find your answer.
So just how big is our search space? A reasonable approximation for discussion purposes is the set of all 100 digit integers. So lets get a handle on just how large this search space is on an intuitive level.
Lets look for our needle in a haystack by getting an idea how large the haystack is. Assume a 0.5 mm x 0.5 mm x 3 cm as our needle size. Now what is the size of the haystack?
The volume of our needle would be 7.5 mm^3 and the volume of our haystack would be 7.5 x 10^100 mm^3
Converting 1 light-year to millimeters we have:
(1 lightyear)x365.35x24x60x60x186000x5280x12x2.54x10 = 9.45x10^18 mm
or (1 lightyear) / 9.45x10^18 = 1 mm
Which means 1 cubic lightyear / (8.43x10^56) = 1 cubic millimeter
So our haystack is (7.5x10^100 / 8.43x10^56) cubic lightyears
or 8.897x10^43 cubic lightyears
Assuming the diameter of the Universe is a cube 93 billion lightyears on a side, then the volume of the Universe is about 8.04x10^32 cubic lightyears.
So 8.897x10^43 / 8.04x10^32 = a haystack about 111 billion times as big as the whole of the known Universe.
In other words we need to find and isolate 1 needle in a haystack over 100 billion times as big as our whole Universe.
Somehow I doubt we will ever achieve such a feat even with a million qubit quantum computer because our error rate would have to be low enough to distinguish that one prime number "needle" from all other 100 digit prime numbers without error.
13. Dear John, I think your estimation is incorrect here. 1000000 qubit would roughly size up to 2^1000000 which is incomparably bigger to whatever size you mention above.
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09b2552c65e8449b | Editor's Note: This story was originally printed in the December 2007 issue of Scientific American and is being reposted from our archive in light of a new documentary on PBS, Parallel Worlds, Parallel Lives.
Hugh Everett III was a brilliant mathematician, an iconoclastic quantum theorist and, later, a successful defense contractor with access to the nation’s most sensitive military secrets. He introduced a new conception of reality to physics and influenced the course of world history at a time when nuclear Armageddon loomed large. To science-fiction aficionados, he remains a folk hero: the man who invented a quantum theory of multiple universes. To his children, he was someone else again: an emotionally unavailable father; “a lump of furniture sitting at the dining room table,” cigarette in hand. He was also a chain-smoking alcoholic who died prematurely.
At least that is how his history played out in our fork of the universe. If the many-worlds theory that Everett developed when he was a student at Princeton University in the mid-1950s is correct, his life took many other turns in an unfathomable number of branching universes.
Everett’s revolutionary analysis broke apart a theoretical logjam in interpreting the how of quantum mechanics. Although the many-worlds idea is by no means universally accepted even today, his methods in devising the theory presaged the concept of quantum decoherence— a modern explanation of why the probabilistic weirdness of quantum mechanics resolves itself into the concrete world of our experience.
Everett’s work is well known in physics and philosophical circles, but the tale of its discovery and of the rest of his life is known by relatively few. Archival research by Russian historian Eugene Shikhovtsev, myself and others and interviews I conducted with the late scientist’s colleagues and friends, as well as with his rock-musician son, unveil the story of a radiant intelligence extinguished all too soon by personal demons.
Ridiculous Things
Everett’s scientific journey began one night in 1954, he recounted two decades later, “after a slosh or two of sherry.” He and his Princeton classmate Charles Misner and a visitor named Aage Petersen (then an assistant to Niels Bohr) were thinking up “ridiculous things about the implications of quantum mechanics.” During this session Everett had the basic idea behind the many-worlds theory, and in the weeks that followed he began developing it into a dissertation.
The core of the idea was to interpret what the equations of quantum mechanics represent in the real world by having the mathematics of the theory itself show the way instead of by appending interpretational hypotheses to the math. In this way, the young man challenged the physics establishment of the day to reconsider its foundational notion of what constitutes physical reality.
In pursuing this endeavor, Everett boldly tackled the notorious measurement problem in quantum mechanics, which had bedeviled physicists since the 1920s. In a nutshell, the problem arises from a contradiction between how elementary particles (such as electrons and photons) interact at the microscopic, quantum level of reality and what happens when the particles are measured from the macroscopic, classical level. In the quantum world, an elementary particle, or a collection of such particles, can exist in a superposition of two or more possible states of being. An electron, for example, can be in a superposition of different locations, velocities and orientations of its spin. Yet anytime scientists measure one of these properties with precision, they see a definite result—just one of the elements of the superposition, not a combination of them. Nor do we ever see macroscopic objects in superpositions. The measurement problem boils down to this question: How and why does the unique world of our experience emerge from the multiplicities of alternatives available in the superposed quantum world?
Physicists use mathematical entities called wave functions to represent quantum states. A wave function can be thought of as a list of all the possible configurations of a superposed quantum system, along with numbers that give the probability of each configuration’s being the one, seemingly selected at random, that we will detect if we measure the system. The wave function treats each element of the superposition as equally real, if not necessarily equally probable from our point of view.
The Schrödinger equation delineates how a quantum system’s wave function will change through time, an evolution that it predicts will be smooth and deterministic (that is, with no randomness). But that elegant mathematics seems to contradict what happens when humans observe a quantum system, such as an electron, with a scientific instrument (which itself may be regarded as a quantum-mechanical system). For at the moment of measurement, the wave function describing the superposition of alternatives appears to collapse into one member of the superposition, thereby interrupting the smooth evolution of the wave function and introducing discontinuity. A single measurement outcome emerges, banishing all the other possibilities from classically described reality. Which alternative is produced at the moment of measurement appears to be arbitrary; its selection does not evolve logically from the information- packed wave function of the electron before measurement. Nor does the mathematics of collapse emerge from the seamless flow of the Schrödinger equation. In fact, collapse has to be added as a postulate, as an additional process that seems to violate the equation.
Universal Wave Function
In stark contrast, Everett addressed the measurement problem by merging the microscopic and macroscopic worlds. He made the observer an integral part of the system observed, introducing a universal wave function that links observers and objects as parts of a single quantum system. He described the macroscopic world quantum mechanically and thought of large objects as existing in quantum superpositions as well. Breaking with Bohr and Heisenberg, he dispensed with the need for the discontinuity of a wave-function collapse.
Everett’s radical new idea was to ask, What if the continuous evolution of a wave function is not interrupted by acts of measurement? What if the Schrödinger equation always applies and applies to everything—objects and observers alike? What if no elements of superpositions are ever banished from reality? What would such a world appear like to us?
Everett saw that under those assumptions, the wave function of an observer would, in effect, bifurcate at each interaction of the observer with a superposed object. The universal wave function would contain branches for every alternative making up the object’s superposition. Each branch has its own copy of the observer, a copy that perceived one of those alternatives as the outcome. According to a fundamental mathematical property of the Schrödinger equation, once formed, the branches do not influence one another. Thus, each branch embarks on a different future, independently of the others.
Consider a person measuring a particle that is in a superposition of two states, such as an electron in a superposition of location A and location B. In one branch, the person perceives that the electron is at A. In a nearly identical branch, a copy of the person perceives that the same electron is at B. Each copy of the person perceives herself or himself as being one of a kind and sees chance as cooking up one reality from a menu of physical possibilities, even though, in the full reality, every alternative on the menu happens.
Explaining how we would perceive such a universe requires putting an observer into the picture. But the branching process happens regardless of whether a human being is present. In general, at each interaction between physical systems the total wave function of the combined systems would tend to bifurcate in this way. Today’s understanding of how the branches become independent and each turn out looking like the classical reality we are accustomed to is known as decoherence theory. It is an accepted part of standard modern quantum theory, although not everyone agrees with the Everettian interpretation that all the branches represent realities that exist.
Everett was not the first physicist to criticize the Copenhagen collapse postulate as inadequate. But he broke new ground by deriving a mathematically consistent theory of a universal wave function from the equations of quantum mechanics itself. The existence of multiple universes emerged as a consequence of his theory, not a predicate. In a footnote in his thesis, Everett wrote: “From the viewpoint of the theory, all elements of a superposition (all ‘branches’) are ‘actual,’ none any more ‘real’ than the rest.”
The draft containing all these ideas provoked a remarkable behind-the-scenes struggle, uncovered about five years ago in archival research by Olival Freire, Jr., a historian of science at the Federal University of Bahia in Brazil. In the spring of 1956 Everett’s academic adviser at Princeton, John Archibald Wheeler, took the draft dissertation to Copenhagen to convince the Royal Danish Academy of Sciences and Letters to publish it. He wrote to Everett that he had “three long and strong discussions about it” with Bohr and Petersen. Wheeler also shared his student’s work with several other physicists at Bohr’s Institute for Theoretical Physics, including Alexander W. Stern.
Wheeler’s letter to Everett reported: “Your beautiful wave function formalism of course remains unshaken; but all of us feel that the real issue is the words that are to be attached to the quantities of the formalism.” For one thing, Wheeler was troubled by Everett’s use of “splitting” humans and cannonballs as scientific metaphors. His letter revealed the Copenhagen-ists’ discomfort over the meaning of Everett’s work. Stern dismissed Everett’s theory as “theology,” and Wheeler himself was reluctant to challenge Bohr. In a long, politic letter to Stern, he explicated and excused Everett’s theory as an extension, not a refutation, of the prevailing interpretation of quantum mechanics:
I think I may say that this very fine and able and independently thinking young man has gradually come to accept the present approach to the measurement problem as correct and self-consistent, despite a few traces that remain in the present thesis draft of a past dubious attitude. So, to avoid any possible misunderstanding, let me say that Everett’s thesis is not meant to question the present approach to the measurement problem, but to accept it and generalize it. [Emphasis in original.]
Everett would have completely disagreed with Wheeler’s description of his opinion of the Copenhagen interpretation. For example, a year later, when responding to criticisms from Bryce S. DeWitt, editor of the journal Reviews of Modern Physics, he wrote:
The Copenhagen Interpretation is hopelessly incomplete because of its a priori reliance on classical physics ... as well as a philosophic monstrosity with a “reality” concept for the macroscopic world and denial of the same for the microcosm.
While Wheeler was off in Europe arguing his case, Everett was in danger of losing his student draft deferment. To avoid going to boot camp, he decided to take a research job at the Pentagon. He moved to the Washington, D.C., area and never came back to theoretical physics.
During the next year, however, he communicated long-distance with Wheeler as he reluctantly whittled down his thesis to a quarter of its original length. In April 1957 Everett’s thesis committee accepted the abridged version—without the “splits.” Three months later Reviews of Modern Physics published the shortened version, entitled “‘Relative State’ Formulation of Quantum Mechanics.” In the same issue, a companion paper by Wheeler lauded his student’s discovery.
When the paper appeared in print, it slipped into instant obscurity. Wheeler gradually distanced himself from association with Everett’s theory, but he kept in touch with the theorist, encouraging him, in vain, to do more work in quantum mechanics. In an interview last year, Wheeler, then 95, commented that “[Everett] was disappointed, perhaps bitter, at the nonreaction to his theory. How I wish that I had kept up the sessions with Everett. The questions that he brought up were important.”
Nuclear Military Strategies
Princeton awarded Everett his doctorate nearly a year after he had begun his first project for the Pentagon: calculating potential mortality rates from radioactive fallout in a nuclear war. He soon headed the mathematics division in the Pentagon’s nearly invisible but extremely influential Weapons Systems Evaluation Group (WSEG). Everett advised high-level officials in the Eisenhower and Kennedy administrations on the best methods for selecting hydrogen bomb targets and structuring the nuclear triad of bombers, submarines and missiles for optimal punch in a nuclear strike.
In 1960 he helped write WSEG No. 50, a catalytic report that remains classified to this day. According to Everett’s friend and WSEG colleague George E. Pugh, as well as historians, WSEG No. 50 rationalized and promoted military strategies that were operative for decades, including the concept of Mutually Assured Destruction. WSEG provided nuclear warfare policymakers with enough scary information about the global effects of radioactive fallout that many became convinced of the merit of waging a perpetual standoff—as opposed to, as some powerful people were advocating, launching preemptive first strikes on the Soviet Union, China and other communist countries.
One final chapter in the struggle over Everett’s theory also played out in this period. In the spring of 1959 Bohr granted Everett an interview in Copenhagen. They met several times during a six-week period but to little effect: Bohr did not shift his position, and Everett did not reenter quantum physics research. The excursion was not a complete failure, though. One afternoon, while drinking beer at the Hotel Østerport, Everett wrote out on hotel stationery an important refinement of the other mathematical tour de force for which he is renowned, the generalized Lagrange multiplier method, also known as the Everett algorithm. The method simplifies searches for optimum solutions to complex logistical problems—ranging from the deployment of nuclear weapons to just-in-time industrial production schedules to the routing of buses for maximizing the desegregation of school districts.
In 1964 Everett, Pugh and several other WSEG colleagues founded a private defense company, Lambda Corporation. Among other activities, it designed mathematical models of anti-ballistic missile systems and computerized nuclear war games that, according to Pugh, were used by the military for years. Everett became enamored of inventing applications for Bayes’ theorem, a mathematical method of correlating the probabilities of future events with past experience. In 1971 Everett built a prototype Bayesian machine, a computer program that learns from experience and simplifies decision making by deducing probable outcomes, much like the human faculty of common sense. Under contract to the Pentagon, Lambda used the Bayesian method to invent techniques for tracking trajectories of incoming ballistic missiles.
In 1973 Everett left Lambda and started a data-processing company, DBS, with Lambda colleague Donald Reisler. DBS researched weapons applications but specialized in analyzing the socioeconomic effects of government affirmative action programs. When they first met, Reis-ler recalls, Everett “sheepishly” asked whether he had ever read his 1957 paper. “I thought for an instant and replied, ‘Oh, my God, you are that Everett, the crazy one who wrote that insane paper,’” Reisler says. “I had read it in graduate school and chuckled, rejected it out of hand.” The two became close friends but agreed not to talk about multiple universes again.
Three-Martini Lunches
Despite all these successes, Everett’s life was blighted in many ways. He had a reputation for drinking, and friends say the problem seemed only to grow with time. According to Reisler, his partner usually enjoyed a three-martini lunch, sleeping it off in his office—although he still managed to be productive.
Yet his hedonism did not reflect a relaxed, playful attitude toward life. “He was not a sympathetic person,” Reisler says. “He brought a cold, brutal logic to the study of things. Civil-rights entitlements made no sense to him.”
John Y. Barry, a former colleague of Everett’s at WSEG, also questioned his ethics. In the mid-1970s Barry convinced his employers at J. P. Morgan to hire Everett to develop a Bayesian method of predicting movement in the stock market. By several accounts, Everett succeeded— and then refused to turn the product over to J. P. Morgan. “He used us,” Barry recalls. “[He was] a brilliant, innovative, slippery, untrustworthy, probably alcoholic individual.”
Everett was egocentric. “Hugh liked to espouse a form of extreme solipsism,” says Elaine Tsiang, a former employee at DBS. “Although he took pains to distance his [many-worlds] theory from any theory of mind or consciousness, obviously we all owed our existence relative to the world he had brought into being.”
And he barely knew his children, Elizabeth and Mark.
As Everett pursued his entrepreneurial career, the world of physics was starting to take a hard look at his once ignored theory. DeWitt swung around 180 degrees and became its most devoted champion. In 1967 he wrote an article presenting the Wheeler-DeWitt equation: a universal wave function that a theory of quantum gravity should satisfy. He credited Everett for having demonstrated the need for such an approach. DeWitt and his graduate student Neill Graham then edited a book of physics papers, The Many-Worlds Interpretation of Quantum Mechanics, which featured the unamputated version of Everett’s dissertation. The epigram “many worlds” stuck fast, popularized in the science-fiction magazine Analog in 1976.
Not everybody agrees, however, that the Copenhagen interpretation needs to give way. Cornell University physicist N. David Mermin maintains that the Everett interpretation treats the wave function as part of the objectively real world, whereas he sees it as merely a mathematical tool. “A wave function is a human construction,” Mer-min says. “Its purpose is to enable us to make sense of our macroscopic observations. My point of view is exactly the opposite of the many-worlds interpretation. Quantum mechanics is a device for enabling us to make our observations coherent, and to say that we are inside of quantum mechanics and that quantum mechanics must apply to our perceptions is inconsistent.”
But many working physicists say that Everett’s theory should be taken seriously.
“When I heard about Everett’s interpretation in the late 1970s,” says Stephen Shenker, a theoretical physicist at Stanford University, “I thought it was kind of crazy. Now most of the people I know that think about string theory and quantum cosmology think about something along an Everett-style interpretation. And because of recent developments in quantum computation, these questions are no longer academic.”
One of the pioneers of decoherence, Wojciech H. Zurek, a fellow at Los Alamos National Laboratory, comments that “Everett’s accomplishment was to insist that quantum theory should be universal, that there should not be a division of the universe into something which is a priori classical and something which is a priori quantum. He gave us all a ticket to use quantum theory the way we use it now to describe measurement as a whole.”
String theorist Juan Maldacena of the Institute for Advanced Study in Princeton, N.J., reflects a common attitude among his colleagues: “When I think about the Everett theory quantum mechanically, it is the most reasonable thing to believe. In everyday life, I do not believe it.”
In 1977 DeWitt and Wheeler invited Everett, who hated public speaking, to make a presentation on his interpretation at the University of Texas at Austin. He wore a rumpled black suit and chain-smoked throughout the seminar. David Deutsch, now at the University of Oxford and a founder of the field of quantum computation (itself inspired by Everett’s theory), was there. “Everett was before his time,” Deutsch says in summing up Everett’s contribution. “He represents the refusal to relinquish objective explanation. A great deal of harm was done to progress in both physics and philosophy by the abdication of the original purpose of those fields: to explain the world. We got irretrievably bogged down in formalisms, and things were regarded as progress which are not explanatory, and the vacuum was filled by mysticism and religion and every kind of rubbish. Everett is important because he stood out against it.”
After the Texas visit, Wheeler tried to hook Everett up with the Institute for Theoretical Physics in Santa Barbara, Calif. Everett reportedly was interested, but nothing came of the plan.
Totality of Experience
Everett died in bed on July 19, 1982. He was just 51. His son, Mark, then a teenager, remembers finding his father’s lifeless body that morning. Feeling the cold body, Mark realized he had no memory of ever touching his dad before. “I did not know how to feel about the fact that my father just died,” he told me. “I didn’t really have any relationship with him.”
Not long afterward, Mark moved to Los Angeles. He became a successful songwriter and the lead singer for a popular rock band, Eels. Many of his songs express the sadness he experienced as the son of a depressed, alcoholic, emotionally detached man. It was not until years after his father’s death that Mark learned of Everett’s career and accomplishments.
Mark’s sister, Elizabeth, made the first of many suicide attempts in June 1982, only a month before Everett died. Mark discovered her unconscious on the bathroom floor and got her to the hospital just in time. When he returned home later that night, he recalled, his father “looked up from his newspaper and said, ‘I didn’t know she was that sad.’” In 1996 Elizabeth killed herself with an overdose of sleeping pills, leaving a note in her purse saying she was going to join her father in another universe.
In a 2005 song, “Things the Grandchildren Should Know,” Mark wrote: “I never really understood/ what it must have been like for him/living inside his head.” His solipsistically inclined father would have understood that dilemma. “Once we have granted that any physical theory is essentially only a model for the world of experience,” Everett concluded in the unedited version of his dissertation, “we must renounce all hope of finding anything like the correct theory ... simply because the totality of experience is never accessible to us.” |
98a1734131a6c183 | 2019 Impact factor 0.630
Applied Physics
EPJ A Highlight - Paving the way for effective field theories
Cover picture: image courtesy of Germain Caminade (http://germaincaminade.com/)
A detailed analysis of theories which approximate the underlying properties of physical systems could lead to new advances in studies of low-energy nuclear processes
Over the past century, a wide variety of models have emerged to explain the complex behaviours which unfold within atomic nuclei at low energies. However, these theories bring up deep philosophical questions regarding their scientific value. Indeed, traditional epistemological tools have been rather elaborated to account for a unified and stabilised theory rather than to apprehend a plurality of models. Ideally, a theory is meant to be reductionist, unifying and fundamentalist. In view of the intrinsic limited precision of their prediction and of the difficulty in assessing a priori their range of applicability, as well as of their specific and disconnected character, traditional nuclear models are necessarily deficient when analysed by means of standard epistemological interpretative frameworks.
EPJ A Highlight - Automated symmetry adaption in nuclear many-body theory
Symmetry reduction process of a prototypical many-body expression leading to an equivalent symmetry-reduced form. Recoupling coefficients arising from the AMC program are shown in red.
The extreme cost of solving the A-nucleon Schrödinger equation can be minimized by leveraging rotational symmetry and, thus, enable the computation of observables in heavy nuclei and/or with high precision.
The associated reduction process, which amounts to re-expressing the working equations in terms of rotationally-invariant objects, requires lengthy symbolic manipulations of elaborate algebraic identities.
For the first time, this involved process is automated by a powerful graph-theory-based tool, the AMC code, which condenses months of error-prone derivations into a simple computational task performed within seconds.
The AMC program tightens the gap for a full automation of the many-body workflow, thereby lowering the time required to build and test novel quantum many-body formalisms.
EPJ A Highlight - Emergence of nuclear rotation from elementary interactions between the nucleons
Rotational bands in an ab initio calculations for the nuclear excitation spectrum of 11Be.
Nuclei are quantum many-body systems which exhibit emergent degrees of freedom, from shell structure and clustering to collective rotations and vibrations. Such emergent phenomena are traditionally the domain of phenomenological models, yet their description can now be placed on a more fundamental footing in terms of microscopic theory. The nature and emergence of rotational bands are presently investigated in light nuclei through ab initio nuclear many-body calculations. Beyond simply analyzing spectroscopic signatures, the structural insight are investigated in terms of angular momentum coupling schemes and group theoretical correlations as underpinnings for the rotational structure.
EPJ A Highlight - Advancing AGATA – Future Science with The Advanced Gamma Tracking Array
Artist's view of the 4p AGATA spectrometer showing the mechanical holding frame (yellow) and cryostat dewars (blue) of the Ge detectors.
AGATA – the Advanced Gamma Tracking Array is a multi-national European project for the ultimate high-resolution gamma-ray spectrometer for nuclear physics capable of measuring γ rays from a few tens of keV to beyond 10 MeV, with unprecedented efficiency, excellent position resolution for individual γ-ray interactions and correspondingly unparalleled angular resolution, and very high count-rate capability. AGATA will be a flag ship spectrometer and have an enormous impact on nuclear structure studies at the extremes of isospin, mass, angular momentum, excitation energy and temperature. It will enable us to uncover and understand hitherto hidden secrets of the atomic nucleus.
EPJ A Highlight - Towards the solution of the “hyperon puzzle”
Neutron star’s mass-radius relation with and without hyperons. Masses of the pulsars PSR J0348+0432 and PSR J0740+6620 are shown with their observation uncertainties.
The possible presence of strange matter in the core of neutron stars has given rise to the so-called hyperon puzzle: hyperonic degrees of freedom are energetically allowed in the extreme density conditions believed to exist in the core of Neutron Stars, but hyperons reduce the internal pressure of the star, which then cannot compensate the gravitational field to sustain the most massive compact stars observed.
This work reports on the effect of three-body interactions when including a Lambda hyperon on the properties of hyper-nuclei and Neutron Stars. State-of-the-art three-body chiral effective interactions are introduced in a microscopic Brueckner-Hartree-Fock calculation.
EPJ A Highlight - Confirming the validity of the Silver-Blaze property for QCD at finite chemical potential
Sketch of the QCD phase diagram in the temperature and baryon chemical potential plane.
The properties of the theory of strong interactions, QCD, at finite chemical potential are of great interest for at least two reasons: (i) model studies suggest a potentially rich landscape of different phases with highly interesting analogies to those found in solid state physics; (ii) the resulting thermodynamic properties have far reaching consequences for the physics of neutron stars and neutron star mergers.
EPJ A Highlight - A Liquid-Lithium Target for Nuclear Physics
The free-surface LiLiT flow, photographed while bombarded by a ~ 3 kW continuous-wave proton beam from the SARAF linac. The liquid lithium jet, ~1.5 mm thick, forced-flown at a velocity of 2.5 m/s at ~ 195 °C and supported by a 0.5 mm thick stainless steel backing wall, serves both as a neutron producing target and the power beam dump. The target chamber pressure connected to the accelerator beam line is 1×10-6 mbar.
A liquid-lithium target (LiLiT) bombarded by a 1.5 mA, 1.92 MeV proton beam from the SARAF superconducting linac acts as a ~30 keV quasi-Maxwellian neutron source via the 7Li(p,n) reaction with the highest intensity (5×1010 neutrons/s) available todate. We activate samples relevant to stellar nucleosynthesis by slow neutron capture (s-process). Activation products are detected by α, β or γ spectrometry or by direct atom counting (accelerator mass spectrometry, atom-trap trace analysis). The neutron capture cross sections, corrected for systematic effects using detailed simulations of neutron production and transport, lead to experimental astrophysical Maxwellian averaged cross sections (MACS). A parallel effort to develop a LiLiT-based neutron source for cancer therapy is ongoing, taking advantage of the neutron spectrum suitability for Boron Neutron Capture Therapy (BNCT) and the high neutron yield available.
EPJ A Highlight - Shape stability of pasta phases: Lasagna case
Exotic non-spherical shapes of nuclear matter, so called pasta phases, are possible because of the competition between the short-ranged nuclear attraction and the long-ranged Coulomb repulsion, leading to the phenomenon of Coulomb frustration, well known in statistical mechanics. Such complex phases are expected in the inner crust of neutron stars, as well as in core-collapse supernova cores.
The authors of the EPJ A (2018) 54:215 paper examine for the first time the stability of the «lasagna» phase, consisting of periodically placed slabs, by means of exact geometrical methods. Calculations are done in the framework of the compressible liquid drop model but obtained results are universal and do not depend on model parameters like surface tension and charge density. The stability analysis is done with respect to the different types of deformations corresponding to the eigenvalues of the deformation matrix.
EPJ A Highlight - Lattice Improvement in Lattice Effective Field Theory
The dimer-boson inverse scattering length $1/a_{3}$ versus lattice spacing at LO, NLO, and N2LO. The vertical lines give the upper limits of the fit range
Lattice calculations using the framework of effective field theory have been applied to a wide range of few-body and many-body systems. One of the challenges of these calculations is to remove systematic errors arising from the nonzero lattice spacing. While the lattice improvement program pioneered by Symanzik provides a formalism for doing this and has already been utilized in lattice effective field theory calculations, the effectiveness of the improvement program has not been systematically benchmarked.
In this work lattice improvement is used to remove lattice errors for a one-dimensional system of bosons with zero-range interactions. To this aim the improved lattice action up to next-to-next-to-leading order is constructed and it is verified that the remaining errors scale as the fourth power of the lattice spacing for observables involving as many as five particles. These results provide a guide for increasing the accuracy of future calculations in lattice effective field theory with improved lattice actions.
EPJ A Highlight - The P2-Experiment - A future high-precision measurement of the weak mixing angle at low momentum transfer
The experimental setup of the P2-experiment to measure the weak mixing angle at the new electron accelerator MESA in Mainz.
The P2-experiment at the new electron accelerator MESA in Mainz aims at a high-precision determination of the weak mixing angle at the permille level at low Q2. This accuracy is comparable to existing measurements at the Z-pole but allows for sensitive tests of the Standard Model up to a mass scale of 50 TeV. The weak mixing angle will be extracted from a measurement of the parity violating asymmetry in elastic electron-proton scattering. The asymmetry measured at P2 is smaller than any asymmetry measured so far in electron scattering, with an unprecedented accuracy. This review just published in EPJ A describes the underlying physics and the innovative experimental techniques, such as the Cherenkov detector, beam control, polarimetry, and the construction of a novel liquid hydrogen high-power target. The physics program of the MESA facility comprises indirect, high-precision search for physics beyond the Standard Model, measurement of the neutron distribution in nuclei, transverse single-spin asymmetries, and a possible future extension to the measurement of hadronic parity violation.
S. Giorgio and D. Jacob
ISSN (Print Edition): 1286-0042
ISSN (Electronic Edition): 1286-0050
© EDP Sciences |
58c4f94c4ddee533 | TY - JOUR AB - In the recent years important experimental advances in resonant electro-optic modulators as high-efficiency sources for coherent frequency combs and as devices for quantum information transfer have been realized, where strong optical and microwave mode coupling were achieved. These features suggest electro-optic-based devices as candidates for entangled optical frequency comb sources. In the present work, I study the generation of entangled optical frequency combs in millimeter-sized resonant electro-optic modulators. These devices profit from the experimentally proven advantages such as nearly constant optical free spectral ranges over several gigahertz, and high optical and microwave quality factors. The generation of frequency multiplexed quantum channels with spectral bandwidth in the MHz range for conservative parameter values paves the way towards novel uses in long-distance hybrid quantum networks, quantum key distribution, enhanced optical metrology, and quantum computing. AU - Rueda Sanchez, Alfredo R ID - 9242 IS - 2 JF - Physical Review A SN - 2469-9926 TI - Frequency-multiplexed hybrid optical entangled source based on the Pockels effect VL - 103 ER - TY - JOUR AB - Microelectromechanical systems and integrated photonics provide the basis for many reliable and compact circuit elements in modern communication systems. Electro-opto-mechanical devices are currently one of the leading approaches to realize ultra-sensitive, low-loss transducers for an emerging quantum information technology. Here we present an on-chip microwave frequency converter based on a planar aluminum on silicon nitride platform that is compatible with slot-mode coupled photonic crystal cavities. We show efficient frequency conversion between two propagating microwave modes mediated by the radiation pressure interaction with a metalized dielectric nanobeam oscillator. We achieve bidirectional coherent conversion with a total device efficiency of up to ~60%, a dynamic range of 2 × 10^9 photons/s and an instantaneous bandwidth of up to 1.7 kHz. A high fidelity quantum state transfer would be possible if the drive dependent output noise of currently ~14 photons s^−1 Hz^−1 is further reduced. Such a silicon nitride based transducer is in situ reconfigurable and could be used for on-chip classical and quantum signal routing and filtering, both for microwave and hybrid microwave-optical applications. AU - Fink, Johannes M AU - Kalaee, M. AU - Norte, R. AU - Pitanti, A. AU - Painter, O. ID - 8038 IS - 3 JF - Quantum Science and Technology TI - Efficient microwave frequency conversion mediated by a photonics compatible silicon nitride nanobeam oscillator VL - 5 ER - TY - JOUR AB - Practical quantum networks require low-loss and noise-resilient optical interconnects as well as non-Gaussian resources for entanglement distillation and distributed quantum computation. The latter could be provided by superconducting circuits but existing solutions to interface the microwave and optical domains lack either scalability or efficiency, and in most cases the conversion noise is not known. In this work we utilize the unique opportunities of silicon photonics, cavity optomechanics and superconducting circuits to demonstrate a fully integrated, coherent transducer interfacing the microwave X and the telecom S bands with a total (internal) bidirectional transduction efficiency of 1.2% (135%) at millikelvin temperatures. The coupling relies solely on the radiation pressure interaction mediated by the femtometer-scale motion of two silicon nanobeams reaching a Vπ as low as 16 μV for sub-nanowatt pump powers. Without the associated optomechanical gain, we achieve a total (internal) pure conversion efficiency of up to 0.019% (1.6%), relevant for future noise-free operation on this qubit-compatible platform. AU - Arnold, Georg M AU - Wulf, Matthias AU - Barzanjeh, Shabir AU - Redchenko, Elena AU - Rueda Sanchez, Alfredo R AU - Hease, William J AU - Hassani, Farid AU - Fink, Johannes M ID - 8529 JF - Nature Communications KW - General Biochemistry KW - Genetics and Molecular Biology KW - General Physics and Astronomy KW - General Chemistry SN - 2041-1723 TI - Converting microwave and telecom photons with a silicon photonic nanomechanical interface VL - 11 ER - TY - JOUR AB - Superconductor insulator transition in transverse magnetic field is studied in the highly disordered MoC film with the product of the Fermi momentum and the mean free path kF*l close to unity. Surprisingly, the Zeeman paramagnetic effects dominate over orbital coupling on both sides of the transition. In superconducting state it is evidenced by a high upper critical magnetic field 𝐵𝑐2, by its square root dependence on temperature, as well as by the Zeeman splitting of the quasiparticle density of states (DOS) measured by scanning tunneling microscopy. At 𝐵𝑐2 a logarithmic anomaly in DOS is observed. This anomaly is further enhanced in increasing magnetic field, which is explained by the Zeeman splitting of the Altshuler-Aronov DOS driving the system into a more insulating or resistive state. Spin dependent Altshuler-Aronov correction is also needed to explain the transport behavior above 𝐵𝑐2. AU - Zemlicka, Martin AU - Kopčík, M. AU - Szabó, P. AU - Samuely, T. AU - Kačmarčík, J. AU - Neilinger, P. AU - Grajcar, M. AU - Samuely, P. ID - 8944 IS - 18 JF - Physical Review B SN - 24699950 TI - Zeeman-driven superconductor-insulator transition in strongly disordered MoC films: Scanning tunneling microscopy and transport studies in a transverse magnetic field VL - 102 ER - TY - JOUR AB - Microwave photonics lends the advantages of fiber optics to electronic sensing and communication systems. In contrast to nonlinear optics, electro-optic devices so far require classical modulation fields whose variance is dominated by electronic or thermal noise rather than quantum fluctuations. Here we demonstrate bidirectional single-sideband conversion of X band microwave to C band telecom light with a microwave mode occupancy as low as 0.025 ± 0.005 and an added output noise of less than or equal to 0.074 photons. This is facilitated by radiative cooling and a triply resonant ultra-low-loss transducer operating at millikelvin temperatures. The high bandwidth of 10.7 MHz and total (internal) photon conversion efficiency of 0.03% (0.67%) combined with the extremely slow heating rate of 1.1 added output noise photons per second for the highest available pump power of 1.48 mW puts near-unity efficiency pulsed quantum transduction within reach. Together with the non-Gaussian resources of superconducting qubits this might provide the practical foundation to extend the range and scope of current quantum networks in analogy to electrical repeaters in classical fiber optic communication. AU - Hease, William J AU - Rueda Sanchez, Alfredo R AU - Sahu, Rishabh AU - Wulf, Matthias AU - Arnold, Georg M AU - Schwefel, Harald G.L. AU - Fink, Johannes M ID - 9114 IS - 2 JF - PRX Quantum SN - 2691-3399 TI - Bidirectional electro-optic wavelength conversion in the quantum ground state VL - 1 ER - TY - JOUR AB - Quantum transduction, the process of converting quantum signals from one form of energy to another, is an important area of quantum science and technology. The present perspective article reviews quantum transduction between microwave and optical photons, an area that has recently seen a lot of activity and progress because of its relevance for connecting superconducting quantum processors over long distances, among other applications. Our review covers the leading approaches to achieving such transduction, with an emphasis on those based on atomic ensembles, opto-electro-mechanics, and electro-optics. We briefly discuss relevant metrics from the point of view of different applications, as well as challenges for the future. AU - Lauk, Nikolai AU - Sinclair, Neil AU - Barzanjeh, Shabir AU - Covey, Jacob P AU - Saffman, Mark AU - Spiropulu, Maria AU - Simon, Christoph ID - 9194 IS - 2 JF - Quantum Science and Technology SN - 2058-9565 TI - Perspectives on quantum transduction VL - 5 ER - TY - JOUR AB - Quantum information technology based on solid state qubits has created much interest in converting quantum states from the microwave to the optical domain. Optical photons, unlike microwave photons, can be transmitted by fiber, making them suitable for long distance quantum communication. Moreover, the optical domain offers access to a large set of very well‐developed quantum optical tools, such as highly efficient single‐photon detectors and long‐lived quantum memories. For a high fidelity microwave to optical transducer, efficient conversion at single photon level and low added noise is needed. Currently, the most promising approaches to build such systems are based on second‐order nonlinear phenomena such as optomechanical and electro‐optic interactions. Alternative approaches, although not yet as efficient, include magneto‐optical coupling and schemes based on isolated quantum systems like atoms, ions, or quantum dots. Herein, the necessary theoretical foundations for the most important microwave‐to‐optical conversion experiments are provided, their implementations are described, and the current limitations and future prospects are discussed. AU - Lambert, Nicholas J. AU - Rueda Sanchez, Alfredo R AU - Sedlmeir, Florian AU - Schwefel, Harald G. L. ID - 9195 IS - 1 JF - Advanced Quantum Technologies SN - 2511-9044 TI - Coherent conversion between microwave and optical photons - An overview of physical implementations VL - 3 ER - TY - JOUR AB - The superconducting circuit community has recently discovered the promising potential of superinductors. These circuit elements have a characteristic impedance exceeding the resistance quantum RQ ≈ 6.45 kΩ which leads to a suppression of ground state charge fluctuations. Applications include the realization of hardware protected qubits for fault tolerant quantum computing, improved coupling to small dipole moment objects and defining a new quantum metrology standard for the ampere. In this work we refute the widespread notion that superinductors can only be implemented based on kinetic inductance, i.e. using disordered superconductors or Josephson junction arrays. We present modeling, fabrication and characterization of 104 planar aluminum coil resonators with a characteristic impedance up to 30.9 kΩ at 5.6 GHz and a capacitance down to ≤ 1 fF, with lowloss and a power handling reaching 108 intra-cavity photons. Geometric superinductors are free of uncontrolled tunneling events and offer high reproducibility, linearity and the ability to couple magnetically - properties that significantly broaden the scope of future quantum circuits. AU - Peruzzo, Matilda AU - Trioni, Andrea AU - Hassani, Farid AU - Zemlicka, Martin AU - Fink, Johannes M ID - 8755 IS - 4 JF - Physical Review Applied TI - Surpassing the resistance quantum with a geometric superinductor VL - 14 ER - TY - JOUR AB - Quantum illumination uses entangled signal-idler photon pairs to boost the detection efficiency of low-reflectivity objects in environments with bright thermal noise. Its advantage is particularly evident at low signal powers, a promising feature for applications such as noninvasive biomedical scanning or low-power short-range radar. Here, we experimentally investigate the concept of quantum illumination at microwave frequencies. We generate entangled fields to illuminate a room-temperature object at a distance of 1 m in a free-space detection setup. We implement a digital phase-conjugate receiver based on linear quadrature measurements that outperforms a symmetric classical noise radar in the same conditions, despite the entanglement-breaking signal path. Starting from experimental data, we also simulate the case of perfect idler photon number detection, which results in a quantum advantage compared with the relative classical benchmark. Our results highlight the opportunities and challenges in the way toward a first room-temperature application of microwave quantum circuits. AU - Barzanjeh, Shabir AU - Pirandola, S. AU - Vitali, D AU - Fink, Johannes M ID - 7910 IS - 19 JF - Science Advances TI - Microwave quantum illumination using a digital receiver VL - 6 ER - TY - CONF AB - Quantum illumination is a sensing technique that employs entangled signal-idler beams to improve the detection efficiency of low-reflectivity objects in environments with large thermal noise. The advantage over classical strategies is evident at low signal brightness, a feature which could make the protocol an ideal prototype for non-invasive scanning or low-power short-range radar. Here we experimentally investigate the concept of quantum illumination at microwave frequencies, by generating entangled fields using a Josephson parametric converter which are then amplified to illuminate a room-temperature object at a distance of 1 meter. Starting from experimental data, we simulate the case of perfect idler photon number detection, which results in a quantum advantage compared to the relative classical benchmark. Our results highlight the opportunities and challenges on the way towards a first room-temperature application of microwave quantum circuits. AU - Barzanjeh, Shabir AU - Pirandola, Stefano AU - Vitali, David AU - Fink, Johannes M ID - 9001 IS - 9 SN - 1097-5659 T2 - IEEE National Radar Conference - Proceedings TI - Microwave quantum illumination with a digital phase-conjugated receiver VL - 2020 ER - TY - JOUR AB - We propose an efficient microwave-photonic modulator as a resource for stationary entangled microwave-optical fields and develop the theory for deterministic entanglement generation and quantum state transfer in multi-resonant electro-optic systems. The device is based on a single crystal whispering gallery mode resonator integrated into a 3D-microwave cavity. The specific design relies on a new combination of thin-film technology and conventional machining that is optimized for the lowest dissipation rates in the microwave, optical, and mechanical domains. We extract important device properties from finite-element simulations and predict continuous variable entanglement generation rates on the order of a Mebit/s for optical pump powers of only a few tens of microwatts. We compare the quantum state transfer fidelities of coherent, squeezed, and non-Gaussian cat states for both teleportation and direct conversion protocols under realistic conditions. Combining the unique capabilities of circuit quantum electrodynamics with the resilience of fiber optic communication could facilitate long-distance solid-state qubit networks, new methods for quantum signal synthesis, quantum key distribution, and quantum enhanced detection, as well as more power-efficient classical sensing and modulation. AU - Rueda Sanchez, Alfredo R AU - Hease, William J AU - Barzanjeh, Shabir AU - Fink, Johannes M ID - 7156 JF - npj Quantum Information SN - 2056-6387 TI - Electro-optic entanglement source for microwave to telecom quantum state transfer VL - 5 ER - TY - CONF AB - We demonstrate electro-optic frequency comb generation using a doubly resonant system comprising a whispering gallery mode disk resonator made of lithium niobate mounted inside a three dimensional copper cavity. We observe 180 sidebands centred at 1550 nm. AU - Rueda Sanchez, Alfredo R AU - Sedlmeir, Florian AU - Leuchs, Gerd AU - Kumari, Madhuri AU - Schwefel, Harald G.L. ID - 7233 SN - 9781557528209 T2 - Nonlinear Optics, OSA Technical Digest TI - Resonant electro-optic frequency comb generation in lithium niobate disk resonator inside a microwave cavity ER - TY - JOUR AB - We prove that the observable telegraph signal accompanying the bistability in the photon-blockade-breakdown regime of the driven and lossy Jaynes–Cummings model is the finite-size precursor of what in the thermodynamic limit is a genuine first-order phase transition. We construct a finite-size scaling of the system parameters to a well-defined thermodynamic limit, in which the system remains the same microscopic system, but the telegraph signal becomes macroscopic both in its timescale and intensity. The existence of such a finite-size scaling completes and justifies the classification of the photon-blockade-breakdown effect as a first-order dissipative quantum phase transition. AU - Vukics, A. AU - Dombi, A. AU - Fink, Johannes M AU - Domokos, P. ID - 7451 JF - Quantum SN - 2521-327X TI - Finite-size scaling of the photon-blockade breakdown dissipative quantum phase transition VL - 3 ER - TY - JOUR AB - Recent technical developments in the fields of quantum electromechanics and optomechanics have spawned nanoscale mechanical transducers with the sensitivity to measure mechanical displacements at the femtometre scale and the ability to convert electromagnetic signals at the single photon level. A key challenge in this field is obtaining strong coupling between motion and electromagnetic fields without adding additional decoherence. Here we present an electromechanical transducer that integrates a high-frequency (0.42 GHz) hypersonic phononic crystal with a superconducting microwave circuit. The use of a phononic bandgap crystal enables quantum-level transduction of hypersonic mechanical motion and concurrently eliminates decoherence caused by acoustic radiation. Devices with hypersonic mechanical frequencies provide a natural pathway for integration with Josephson junction quantum circuits, a leading quantum computing technology, and nanophotonic systems capable of optical networking and distributing quantum information. AU - Kalaee, Mahmoud AU - Mirhosseini, Mohammad AU - Dieterle, Paul B. AU - Peruzzo, Matilda AU - Fink, Johannes M AU - Painter, Oskar ID - 6053 IS - 4 JF - Nature Nanotechnology SN - 1748-3387 TI - Quantum electromechanics of a hypersonic crystal VL - 14 ER - TY - JOUR AB - Light is a union of electric and magnetic fields, and nowhere is the complex relationship between these fields more evident than in the near fields of nanophotonic structures. There, complicated electric and magnetic fields varying over subwavelength scales are generally present, which results in photonic phenomena such as extraordinary optical momentum, superchiral fields, and a complex spatial evolution of optical singularities. An understanding of such phenomena requires nanoscale measurements of the complete optical field vector. Although the sensitivity of near- field scanning optical microscopy to the complete electromagnetic field was recently demonstrated, a separation of different components required a priori knowledge of the sample. Here, we introduce a robust algorithm that can disentangle all six electric and magnetic field components from a single near-field measurement without any numerical modeling of the structure. As examples, we unravel the fields of two prototypical nanophotonic structures: a photonic crystal waveguide and a plasmonic nanowire. These results pave the way for new studies of complex photonic phenomena at the nanoscale and for the design of structures that optimize their optical behavior. AU - Le Feber, B. AU - Sipe, J. E. AU - Wulf, Matthias AU - Kuipers, L. AU - Rotenberg, N. ID - 6102 IS - 1 JF - Light: Science and Applications SN - 20955545 TI - A full vectorial mapping of nanophotonic light fields VL - 8 ER - TY - JOUR AB - High-speed optical telecommunication is enabled by wavelength-division multiplexing, whereby hundreds of individually stabilized lasers encode information within a single-mode optical fibre. Higher bandwidths require higher total optical power, but the power sent into the fibre is limited by optical nonlinearities within the fibre, and energy consumption by the light sources starts to become a substantial cost factor1. Optical frequency combs have been suggested to remedy this problem by generating numerous discrete, equidistant laser lines within a monolithic device; however, at present their stability and coherence allow them to operate only within small parameter ranges2,3,4. Here we show that a broadband frequency comb realized through the electro-optic effect within a high-quality whispering-gallery-mode resonator can operate at low microwave and optical powers. Unlike the usual third-order Kerr nonlinear optical frequency combs, our combs rely on the second-order nonlinear effect, which is much more efficient. Our result uses a fixed microwave signal that is mixed with an optical-pump signal to generate a coherent frequency comb with a precisely determined carrier separation. The resonant enhancement enables us to work with microwave powers that are three orders of magnitude lower than those in commercially available devices. We emphasize the practical relevance of our results to high rates of data communication. To circumvent the limitations imposed by nonlinear effects in optical communication fibres, one has to solve two problems: to provide a compact and fully integrated, yet high-quality and coherent, frequency comb generator; and to calculate nonlinear signal propagation in real time5. We report a solution to the first problem. AU - Rueda Sanchez, Alfredo R AU - Sedlmeir, Florian AU - Kumari, Madhuri AU - Leuchs, Gerd AU - Schwefel, Harald G.L. ID - 6348 IS - 7752 JF - Nature SN - 00280836 TI - Resonant electro-optic frequency comb VL - 568 ER - TY - JOUR AB - Mechanical systems facilitate the development of a hybrid quantum technology comprising electrical, optical, atomic and acoustic degrees of freedom1, and entanglement is essential to realize quantum-enabled devices. Continuous-variable entangled fields—known as Einstein–Podolsky–Rosen (EPR) states—are spatially separated two-mode squeezed states that can be used for quantum teleportation and quantum communication2. In the optical domain, EPR states are typically generated using nondegenerate optical amplifiers3, and at microwave frequencies Josephson circuits can serve as a nonlinear medium4,5,6. An outstanding goal is to deterministically generate and distribute entangled states with a mechanical oscillator, which requires a carefully arranged balance between excitation, cooling and dissipation in an ultralow noise environment. Here we observe stationary emission of path-entangled microwave radiation from a parametrically driven 30-micrometre-long silicon nanostring oscillator, squeezing the joint field operators of two thermal modes by 3.40 decibels below the vacuum level. The motion of this micromechanical system correlates up to 50 photons per second per hertz, giving rise to a quantum discord that is robust with respect to microwave noise7. Such generalized quantum correlations of separable states are important for quantum-enhanced detection8 and provide direct evidence of the non-classical nature of the mechanical oscillator without directly measuring its state9. This noninvasive measurement scheme allows to infer information about otherwise inaccessible objects, with potential implications for sensing, open-system dynamics and fundamental tests of quantum gravity. In the future, similar on-chip devices could be used to entangle subsystems on very different energy scales, such as microwave and optical photons. AU - Barzanjeh, Shabir AU - Redchenko, Elena AU - Peruzzo, Matilda AU - Wulf, Matthias AU - Lewis, Dylan AU - Arnold, Georg M AU - Fink, Johannes M ID - 6609 JF - Nature TI - Stationary entangled radiation from micromechanical motion VL - 570 ER - TY - CONF AB - Optical frequency combs (OFCs) are light sources whose spectra consists of equally spaced frequency lines in the optical domain [1]. They have great potential for improving high-capacity data transfer, all-optical atomic clocks, spectroscopy, and high-precision measurements [2]. AU - Rueda Sanchez, Alfredo R AU - Sedlmeir, Florian AU - Leuchs, Gerd AU - Kuamri, Madhuri AU - Schwefel, Harald G. L. ID - 7032 SN - 9781728104690 T2 - 2019 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference TI - Electro-optic frequency comb generation in lithium niobate whispering gallery mode resonators ER - TY - JOUR AB - In this paper, we discuss biological effects of electromagnetic (EM) fields in the context of cancer biology. In particular, we review the nanomechanical properties of microtubules (MTs), the latter being one of the most successful targets for cancer therapy. We propose an investigation on the coupling of electromagnetic radiation to mechanical vibrations of MTs as an important basis for biological and medical applications. In our opinion, optomechanical methods can accurately monitor and control the mechanical properties of isolated MTs in a liquid environment. Consequently, studying nanomechanical properties of MTs may give useful information for future applications to diagnostic and therapeutic technologies involving non-invasive externally applied physical fields. For example, electromagnetic fields or high intensity ultrasound can be used therapeutically avoiding harmful side effects of chemotherapeutic agents or classical radiation therapy. AU - Salari, Vahid AU - Barzanjeh, Shabir AU - Cifra, Michal AU - Simon, Christoph AU - Scholkmann, Felix AU - Alirezaei, Zahra AU - Tuszynski, Jack ID - 287 IS - 8 JF - Frontiers in Bioscience - Landmark TI - Electromagnetic fields and optomechanics In cancer diagnostics and treatment VL - 23 ER - TY - JOUR AB - Conventional ultra-high sensitivity detectors in the millimeter-wave range are usually cooled as their own thermal noise at room temperature would mask the weak received radiation. The need for cryogenic systems increases the cost and complexity of the instruments, hindering the development of, among others, airborne and space applications. In this work, the nonlinear parametric upconversion of millimeter-wave radiation to the optical domain inside high-quality (Q) lithium niobate whispering-gallery mode (WGM) resonators is proposed for ultra-low noise detection. We experimentally demonstrate coherent upconversion of millimeter-wave signals to a 1550 nm telecom carrier, with a photon conversion efficiency surpassing the state-of-the-art by 2 orders of magnitude. Moreover, a theoretical model shows that the thermal equilibrium of counterpropagating WGMs is broken by overcoupling the millimeter-wave WGM, effectively cooling the upconverted mode and allowing ultra-low noise detection. By theoretically estimating the sensitivity of a correlation radiometer based on the presented scheme, it is found that room-temperature radiometers with better sensitivity than state-of-the-art high-electron-mobility transistor (HEMT)-based radiometers can be designed. This detection paradigm can be used to develop room-temperature instrumentation for radio astronomy, earth observation, planetary missions, and imaging systems. AU - Botello, Gabriel AU - Sedlmeir, Florian AU - Rueda Sanchez, Alfredo R AU - Abdalmalak, Kerlos AU - Brown, Elliott AU - Leuchs, Gerd AU - Preu, Sascha AU - Segovia Vargas, Daniel AU - Strekalov, Dmitry AU - Munoz, Luis AU - Schwefel, Harald ID - 22 IS - 10 JF - Optica SN - 23342536 TI - Sensitivity limits of millimeter-wave photonic radiometers based on efficient electro-optic upconverters VL - 5 ER - TY - CONF AB - There is currently significant interest in operating devices in the quantum regime, where their behaviour cannot be explained through classical mechanics. Quantum states, including entangled states, are fragile and easily disturbed by excessive thermal noise. Here we address the question of whether it is possible to create non-reciprocal devices that encourage the flow of thermal noise towards or away from a particular quantum device in a network. Our work makes use of the cascaded systems formalism to answer this question in the affirmative, showing how a three-port device can be used as an effective thermal transistor, and illustrates how this formalism maps onto an experimentally-realisable optomechanical system. Our results pave the way to more resilient quantum devices and to the use of thermal noise as a resource. AU - Xuereb, André AU - Aquilina, Matteo AU - Barzanjeh, Shabir ED - Andrews, D L ED - Ostendorf, A ED - Bain, A J ED - Nunzi, J M ID - 155 TI - Routing thermal noise through quantum networks VL - 10672 ER - TY - JOUR AB - There has been significant interest recently in using complex quantum systems to create effective nonreciprocal dynamics. Proposals have been put forward for the realization of artificial magnetic fields for photons and phonons; experimental progress is fast making these proposals a reality. Much work has concentrated on the use of such systems for controlling the flow of signals, e.g., to create isolators or directional amplifiers for optical signals. In this Letter, we build on this work but move in a different direction. We develop the theory of and discuss a potential realization for the controllable flow of thermal noise in quantum systems. We demonstrate theoretically that the unidirectional flow of thermal noise is possible within quantum cascaded systems. Viewing an optomechanical platform as a cascaded system we show here that one can ultimately control the direction of the flow of thermal noise. By appropriately engineering the mechanical resonator, which acts as an artificial reservoir, the flow of thermal noise can be constrained to a desired direction, yielding a thermal rectifier. The proposed quantum thermal noise rectifier could potentially be used to develop devices such as a thermal modulator, a thermal router, and a thermal amplifier for nanoelectronic devices and superconducting circuits. AU - Barzanjeh, Shabir AU - Aquilina, Matteo AU - Xuereb, André ID - 436 IS - 6 JF - Physical Review Letters TI - Manipulating the flow of thermal noise in quantum devices VL - 120 ER - TY - JOUR AB - Spontaneous emission spectra of two initially excited closely spaced identical atoms are very sensitive to the strength and the direction of the applied magnetic field. We consider the relevant schemes that ensure the determination of the mutual spatial orientation of the atoms and the distance between them by entirely optical means. A corresponding theoretical description is given accounting for the dipole-dipole interaction between the two atoms in the presence of a magnetic field and for polarizations of the quantum field interacting with magnetic sublevels of the two-atom system. AU - Redchenko, Elena AU - Makarov, Alexander AU - Yudson, Vladimir ID - 307 IS - 4 JF - Physical Review A - Atomic, Molecular, and Optical Physics TI - Nanoscopy of pairs of atoms by fluorescence in a magnetic field VL - 97 ER - TY - JOUR AB - We present the fabrication and characterization of an aluminum transmon qubit on a silicon-on-insulator substrate. Key to the qubit fabrication is the use of an anhydrous hydrofluoric vapor process which selectively removes the lossy silicon oxide buried underneath the silicon device layer. For a 5.6 GHz qubit measured dispersively by a 7.1 GHz resonator, we find T1 = 3.5 μs and T∗2 = 2.2 μs. This process in principle permits the co-fabrication of silicon photonic and mechanical elements, providing a route towards chip-scale integration of electro-opto-mechanical transducers for quantum networking of superconducting microwave quantum circuits. The additional processing steps are compatible with established fabrication techniques for aluminum transmon qubits on silicon. AU - Keller, Andrew J AU - Dieterle, Paul AU - Fang, Michael AU - Berger, Brett AU - Fink, Johannes M AU - Painter, Oskar ID - 796 IS - 4 JF - Applied Physics Letters SN - 00036951 TI - Al transmon qubits on silicon on insulator for quantum device integration VL - 111 ER - TY - JOUR AB - Phasenübergänge helfen beim Verständnis von Vielteilchensystemen in der Festkörperphysik und Fluiddynamik bis hin zur Teilchenphysik. Unserer internationalen Kollaboration ist es gelungen, einen neuartigen Phasenübergang in einem Quantensystem zu beobachten [1]. In einem Mikrowellenresonator konnte erstmals die spontane Zustandsänderung von undurchsichtig zu transparent nachgewiesen werden. AU - Fink, Johannes M ID - 797 IS - 3 JF - Physik in unserer Zeit TI - Photonenblockade aufgelöst VL - 48 ER - TY - JOUR AB - Nonreciprocal circuit elements form an integral part of modern measurement and communication systems. Mathematically they require breaking of time-reversal symmetry, typically achieved using magnetic materials and more recently using the quantum Hall effect, parametric permittivity modulation or Josephson nonlinearities. Here we demonstrate an on-chip magnetic-free circulator based on reservoir-engineered electromechanic interactions. Directional circulation is achieved with controlled phase-sensitive interference of six distinct electro-mechanical signal conversion paths. The presented circulator is compact, its silicon-on-insulator platform is compatible with both superconducting qubits and silicon photonics, and its noise performance is close to the quantum limit. With a high dynamic range, a tunable bandwidth of up to 30 MHz and an in situ reconfigurability as beam splitter or wavelength converter, it could pave the way for superconducting qubit processors with multiplexed on-chip signal processing and readout. AU - Barzanjeh, Shabir AU - Wulf, Matthias AU - Peruzzo, Matilda AU - Kalaee, Mahmoud AU - Dieterle, Paul AU - Painter, Oskar AU - Fink, Johannes M ID - 798 IS - 1 JF - Nature Communications SN - 20411723 TI - Mechanical on chip microwave circulator VL - 8 ER - TY - CONF AB - We present results on nonlinear electro-optical conversion of microwave radiation into the optical telecommunication band with more than 0.1% photon number conversion efficiency with MHz bandwidth, in a crystalline whispering gallery mode resonator AU - Rueda Sanchez, Alfredo R AU - Sedlmeir, Florian AU - Collodo, Michele AU - Vogl, Ulrich AU - Stiller, Birgit AU - Schunk, Gerhard AU - Strekalov, Dmitry AU - Marquardt, Christoph AU - Fink, Johannes M AU - Painter, Oskar AU - Leuchs, Gerd AU - Schwefel, Harald ID - 485 SN - 978-155752820-9 TI - Single sideband microwave to optical photon conversion-an-electro-optic-realization VL - F54 ER - TY - JOUR AB - Microtubules provide the mechanical force required for chromosome separation during mitosis. However, little is known about the dynamic (high-frequency) mechanical properties of microtubules. Here, we theoretically propose to control the vibrations of a doubly clamped microtubule by tip electrodes and to detect its motion via the optomechanical coupling between the vibrational modes of the microtubule and an optical cavity. In the presence of a red-detuned strong pump laser, this coupling leads to optomechanical-induced transparency of an optical probe field, which can be detected with state-of-the art technology. The center frequency and line width of the transparency peak give the resonance frequency and damping rate of the microtubule, respectively, while the height of the peak reveals information about the microtubule-cavity field coupling. Our method opens the new possibilities to gain information about the physical properties of microtubules, which will enhance our capability to design physical cancer treatment protocols as alternatives to chemotherapeutic drugs. AU - Barzanjeh, Shabir AU - Salari, Vahid AU - Tuszynski, Jack AU - Cifra, Michal AU - Simon, Christoph ID - 700 IS - 1 JF - Physical Review E Statistical Nonlinear and Soft Matter Physics SN - 24700045 TI - Optomechanical proposal for monitoring microtubule mechanical vibrations VL - 96 ER - TY - JOUR AB - From microwave ovens to satellite television to the GPS and data services on our mobile phones, microwave technology is everywhere today. But one technology that has so far failed to prove its worth in this wavelength regime is quantum communication that uses the states of single photons as information carriers. This is because single microwave photons, as opposed to classical microwave signals, are extremely vulnerable to noise from thermal excitations in the channels through which they travel. Two new independent studies, one by Ze-Liang Xiang at Technische Universität Wien (Vienna), Austria, and colleagues [1] and another by Benoît Vermersch at the University of Innsbruck, also in Austria, and colleagues [2] now describe a theoretical protocol for microwave quantum communication that is resilient to thermal and other types of noise. Their approach could become a powerful technique to establish fast links between superconducting data processors in a future all-microwave quantum network. AU - Fink, Johannes M ID - 1013 IS - 32 JF - Physics TI - Viewpoint: Microwave quantum states beat the heat VL - 10 ER - TY - JOUR AB - Cellulose is the most abundant biopolymer on Earth. Cellulose fibers, such as the one extracted form cotton or woodpulp, have been used by humankind for hundreds of years to make textiles and paper. Here we show how, by engineering light-matter interaction, we can optimize light scattering using exclusively cellulose nanocrystals. The produced material is sustainable, biocompatible, and when compared to ordinary microfiber-based paper, it shows enhanced scattering strength (×4), yielding a transport mean free path as low as 3.5 μm in the visible light range. The experimental results are in a good agreement with the theoretical predictions obtained with a diffusive model for light propagation. AU - Caixeiro, Soraya AU - Peruzzo, Matilda AU - Onelli, Olimpia AU - Vignolini, Silvia AU - Sapienza, Riccardo ID - 1020 IS - 9 JF - ACS Applied Materials and Interfaces SN - 19448244 TI - Disordered cellulose based nanostructures for enhanced light scattering VL - 9 ER - TY - JOUR AB - Nonequilibrium phase transitions exist in damped-driven open quantum systems when the continuous tuning of an external parameter leads to a transition between two robust steady states. In second-order transitions this change is abrupt at a critical point, whereas in first-order transitions the two phases can coexist in a critical hysteresis domain. Here, we report the observation of a first-order dissipative quantum phase transition in a driven circuit quantum electrodynamics system. It takes place when the photon blockade of the driven cavity-atom system is broken by increasing the drive power. The observed experimental signature is a bimodal phase space distribution with varying weights controlled by the drive strength. Our measurements show an improved stabilization of the classical attractors up to the millisecond range when the size of the quantum system is increased from one to three artificial atoms. The formation of such robust pointer states could be used for new quantum measurement schemes or to investigate multiphoton phases of finite-size, nonlinear, open quantum systems. AU - Fink, Johannes M AU - Dombi, András AU - Vukics, András AU - Wallraff, Andreas AU - Domokos, Peter ID - 1114 IS - 1 JF - Physical Review X SN - 21603308 TI - Observation of the photon blockade breakdown phase transition VL - 7 ER - TY - CONF AB - Nonlinear electro-optical conversion of microwave radiation into the optical telecommunication band is achieved within a crystalline whispering gallery mode resonator, reaching 0.1% photon number conversion efficiency with MHz bandwidth. AU - Rueda, Alfredo AU - Sedlmeir, Florian AU - Collodo, Michele AU - Vogl, Ulrich AU - Stiller, Birgit AU - Schunk, Gerhard AU - Strekalov, Dmitry AU - Marquardt, Christoph AU - Fink, Johannes M AU - Painter, Oskar AU - Leuchs, Gerd AU - Schwefel, Harald ID - 482 TI - Nonlinear single sideband microwave to optical conversion using an electro-optic WGM-resonator ER - TY - JOUR AB - We present a microelectromechanical system, in which a silicon beam is attached to a comb-drive actuator, which is used to tune the tension in the silicon beam and thus its resonance frequency. By measuring the resonance frequencies of the system, we show that the comb-drive actuator and the silicon beam behave as two strongly coupled resonators. Interestingly, the effective coupling rate (1.5 MHz) is tunable with the comb-drive actuator (10%) as well as with a side-gate (10%) placed close to the silicon beam. In contrast, the effective spring constant of the system is insensitive to either of them and changes only by 60.5%. Finally, we show that the comb-drive actuator can be used to switch between different coupling rates with a frequency of at least 10 kHz. AU - Verbiest, Gerard AU - Xu, Duo AU - Goldsche, Matthias AU - Khodkov, Timofiy AU - Barzanjeh, Shabir AU - Von Den Driesch, Nils AU - Buca, Dan AU - Stampfer, Christoph ID - 1339 JF - Applied Physics Letter TI - Tunable mechanical coupling between driven microelectromechanical resonators VL - 109 ER - TY - JOUR AB - Fabrication processes involving anhydrous hydrofluoric vapor etching are developed to create high-Q aluminum superconducting microwave resonators on free-standing silicon membranes formed from a silicon-on-insulator wafer. Using this fabrication process, a high-impedance 8.9-GHz coil resonator is coupled capacitively with a large participation ratio to a 9.7-MHz micromechanical resonator. Two-tone microwave spectroscopy and radiation pressure backaction are used to characterize the coupled system in a dilution refrigerator down to temperatures of Tf=11 mK, yielding a measured electromechanical vacuum coupling rate of g0/2π=24.6 Hz and a mechanical resonator Q factor of Qm=1.7×107. Microwave backaction cooling of the mechanical resonator is also studied, with a minimum phonon occupancy of nm≈16 phonons being realized at an elevated fridge temperature of Tf=211 mK. AU - Dieterle, Paul AU - Kalaee, Mahmoud AU - Fink, Johannes M AU - Painter, Oskar ID - 1354 IS - 1 JF - Physical Review Applied TI - Superconducting cavity electromechanics on a silicon-on-insulator platform VL - 6 ER - TY - JOUR AB - Radiation pressure has recently been used to effectively couple the quantum motion of mechanical elements to the fields of optical or microwave light. Integration of all three degrees of freedom—mechanical, optical and microwave—would enable a quantum interconnect between microwave and optical quantum systems. We present a platform based on silicon nitride nanomembranes for integrating superconducting microwave circuits with planar acoustic and optical devices such as phononic and photonic crystals. Using planar capacitors with vacuum gaps of 60 nm and spiral inductor coils of micron pitch we realize microwave resonant circuits with large electromechanical coupling to planar acoustic structures of nanoscale dimensions and femtoFarad motional capacitance. Using this enhanced coupling, we demonstrate microwave backaction cooling of the 4.48 MHz mechanical resonance of a nanobeam to an occupancy as low as 0.32. These results indicate the viability of silicon nitride nanomembranes as an all-in-one substrate for quantum electro-opto-mechanical experiments. AU - Fink, Johannes M AU - Kalaee, Mahmoud AU - Pitanti, Alessandro AU - Norte, Richard AU - Heinzle, Lukas AU - Davanço, Marcelo AU - Srinivasan, Kartik AU - Painter, Oskar ID - 1355 JF - Nature Communications TI - Quantum electromechanics on silicon nitride nanomembranes VL - 7 ER - TY - JOUR AB - We study coherent phonon oscillations and tunneling between two coupled nonlinear nanomechanical resonators. We show that the coupling between two nanomechanical resonators creates an effective phonon Josephson junction, which exhibits two different dynamical behaviors: Josephson oscillation (phonon-Rabi oscillation) and macroscopic self-trapping (phonon blockade). Self-trapping originates from mechanical nonlinearities, meaning that when the nonlinearity exceeds its critical value, the energy exchange between the two resonators is suppressed, and phonon Josephson oscillations between them are completely blocked. An effective classical Hamiltonian for the phonon Josephson junction is derived and its mean-field dynamics is studied in phase space. Finally, we study the phonon-phonon coherence quantified by the mean fringe visibility, and show that the interaction between the two resonators may lead to the loss of coherence in the phononic junction. AU - Barzanjeh, Shabir AU - Vitali, David ID - 1370 IS - 3 JF - Physical Review A - Atomic, Molecular, and Optical Physics TI - Phonon Josephson junction with nanomechanical resonators VL - 93 ER - TY - JOUR AB - Solitons are localized waves formed by a balance of focusing and defocusing effects. These nonlinear waves exist in diverse forms of matter yet exhibit similar properties including stability, periodic recurrence and particle-like trajectories. One important property is soliton fission, a process by which an energetic higher-order soliton breaks apart due to dispersive or nonlinear perturbations. Here we demonstrate through both experiment and theory that nonlinear photocarrier generation can induce soliton fission. Using near-field measurements, we directly observe the nonlinear spatial and temporal evolution of optical pulses in situ in a nanophotonic semiconductor waveguide. We develop an analytic formalism describing the free-carrier dispersion (FCD) perturbation and show the experiment exceeds the minimum threshold by an order of magnitude. We confirm these observations with a numerical nonlinear Schrödinger equation model. These results provide a fundamental explanation and physical scaling of optical pulse evolution in free-carrier media and could enable improved supercontinuum sources in gas based and integrated semiconductor waveguides. AU - Husko, Chad AU - Wulf, Matthias AU - Lefrançois, Simon AU - Combrié, Sylvain AU - Lehoucq, Gaëlle AU - De Rossi, Alfredo AU - Eggleton, Benjamin AU - Kuipers, Laurens ID - 1429 JF - Nature Communications TI - Free-carrier-induced soliton fission unveiled by in situ measurements in nanophotonic waveguides VL - 7 ER - TY - CONF AB - We present a coherent microwave to telecom signal converter based on the electro-optical effect using a crystalline WGM-resonator coupled to a 3D microwave cavity, achieving high photon conversion efficiency of 0.1% with MHz bandwidth. AU - Rueda, Alfredo AU - Sedlmeir, Florian AU - Collodo, Michele AU - Vogl, Ulrich AU - Stiller, Birgit AU - Schunk, Georg AU - Strekalov, Dimitry AU - Marquardt, Christoph AU - Fink, Johannes M AU - Painter, Oskar AU - Leuchs, Gerd AU - Schwefel, Harald ID - 1115 TI - Efficient single sideband microwave to optical conversion using a LiNbO inf 3 inf WGM-resonator ER - TY - JOUR AB - We study a polar molecule immersed in a superfluid environment, such as a helium nanodroplet or a Bose–Einstein condensate, in the presence of a strong electrostatic field. We show that coupling of the molecular pendular motion, induced by the field, to the fluctuating bath leads to formation of pendulons—spherical harmonic librators dressed by a field of many-particle excitations. We study the behavior of the pendulon in a broad range of molecule–bath and molecule–field interaction strengths, and reveal that its spectrum features a series of instabilities which are absent in the field-free case of the angulon quasiparticle. Furthermore, we show that an external field allows to fine-tune the positions of these instabilities in the molecular rotational spectrum. This opens the door to detailed experimental studies of redistribution of orbital angular momentum in many-particle systems. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim AU - Redchenko, Elena AU - Lemeshko, Mikhail ID - 1206 IS - 22 JF - ChemPhysChem TI - Libration of strongly oriented polar molecules inside a superfluid VL - 17 ER - TY - JOUR AB - Near-field imaging is a powerful tool to investigate the complex structure of light at the nanoscale. Recent advances in near-field imaging have indicated the possibility for the complete reconstruction of both electric and magnetic components of the evanescent field. Here we study the electro-magnetic field structure of surface plasmon polariton waves propagating along subwavelength gold nanowires by performing phase- and polarization-resolved near-field microscopy in collection mode. By applying the optical reciprocity theorem, we describe the signal collected by the probe as an overlap integral of the nanowire's evanescent field and the probe's response function. As a result, we find that the probe's sensitivity to the magnetic field is approximately equal to its sensitivity to the electric field. Through rigorous modeling of the nanowire mode as well as the aperture probe response function, we obtain a good agreement between experimentally measured signals and a numerical model. Our findings provide a better understanding of aperture-based near-field imaging of the nanoscopic plasmonic and photonic structures and are helpful for the interpretation of future near-field experiments. AU - Kabakova, Irina AU - De Hoogh, Anouk AU - Van Der Wel, Ruben AU - Wulf, Matthias AU - Le Feber, Boris AU - Kuipers, Laurens ID - 1246 JF - Scientific Reports TI - Imaging of electric and magnetic fields near plasmonic nanowires VL - 6 ER - TY - JOUR AB - Linking classical microwave electrical circuits to the optical telecommunication band is at the core of modern communication. Future quantum information networks will require coherent microwave-to-optical conversion to link electronic quantum processors and memories via low-loss optical telecommunication networks. Efficient conversion can be achieved with electro-optical modulators operating at the single microwave photon level. In the standard electro-optic modulation scheme, this is impossible because both up- and down-converted sidebands are necessarily present. Here, we demonstrate true single-sideband up- or down-conversion in a triply resonant whispering gallery mode resonator by explicitly addressing modes with asymmetric free spectral range. Compared to previous experiments, we show a 3 orders of magnitude improvement of the electro-optical conversion efficiency, reaching 0.1% photon number conversion for a 10 GHz microwave tone at 0.42 mW of optical pump power. The presented scheme is fully compatible with existing superconducting 3D circuit quantum electrodynamics technology and can be used for nonclassical state conversion and communication. Our conversion bandwidth is larger than 1 MHz and is not fundamentally limited. AU - Rueda, Alfredo AU - Sedlmeir, Florian AU - Collodo, Michele AU - Vogl, Ulrich AU - Stiller, Birgit AU - Schunk, Gerhard AU - Strekalov, Dmitry AU - Marquardt, Christoph AU - Fink, Johannes M AU - Painter, Oskar AU - Leuchs, Gerd AU - Schwefel, Harald ID - 1263 IS - 6 JF - Optica TI - Efficient microwave to optical photon conversion: An electro-optical realization VL - 3 ER - |
b2d856f19da2635f | The LAPW basis for thin film systems
Calculations on thin films can be performed in different ways. A popular approach to this task is to define a bulk unit cell in which the film is provided along the xy-plane and the width of the film in z direction only covers a small fraction of the unit cell's length in that direction. Such a setup defines periodic repetitions of the film in z direction and one has to make sure that the vacuum in between the films is large enough to decouple them.
On the basis of the FLAPW method a more elegant treatment of thin film systems is possible. As sketched in figure 1 one can set up unit cells with semi-infinite vacuum regions above and below the film.
The unit cell partitioning for thin films.
An LAPW basis function for such a setup is defined as
where the extension in the vacuum regions consists of the two functions and are solutions and energy derivatives to the Schrödinger equation in the respective vacuum region
at the energy parameter . The coefficients and are determined by enforcing continuity of value and slope of the basis function at the vacuum boundary defined by the parameter . The z components of the plane waves in the interstitial region are constructed such that they feature a periodicty in agreement to the parameter . Note that is larger than to avoid a kink of the implicitely periodic interstitial representation of the wave functions at the interstitial-vacuum boundaries.
For film setups the parameters and are specified in cell/filmLattice/@dVac and cell/filmLattice/@dTilda. For each vacuum the energy parameters are set in the respective cell/filmLattice/vacuumEnergyParameters element for the spin-up (cell/filmLattice/vacuumEnergyParameters/@spinUp) and spin-down (cell/filmLattice/vacuumEnergyParameters/@spinDown) channel separately, relative to the vacuum potential at an infinite distance from the film. In the case of a nonmagnetic calculation the value for the spin-up electrons is used for the calculation of the energy parameter. If both vacua are equivalent only a single cell/filmLattice/vacuumEnergyParameters element for vacuum 1 is present. It is then also used for vacuum 2. |
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Editor-In-Chief: C. Michael Gibson, M.S., M.D. [3]
Template:Infobox hydrogen Hydrogen (Template:PronEng), is the chemical element represented by the symbol H and an atomic number of 1. At standard temperature and pressure it is a colourless, odorless, nonmetallic, tasteless, highly flammable diatomic gas (H2). With an atomic mass of 1.00794 amu, hydrogen is the lightest element.
Hydrogen is the most abundant of the chemical elements, constituting roughly 75% of the universe's elemental mass.[1] Stars in the main sequence are mainly composed of hydrogen in its plasma state. Elemental hydrogen is relatively rare on Earth, and is industrially produced from hydrocarbons such as methane, after which most elemental hydrogen is used "captively" (meaning locally at the production site), with the largest markets about equally divided between fossil fuel upgrading (e.g., hydrocracking) and ammonia production (mostly for the fertilizer market). Hydrogen may be produced from water using the process of electrolysis, but this process is presently significantly more expensive commercially than hydrogen production from natural gas[2].
The most common naturally occurring isotope of hydrogen, known as protium, has a single proton and no neutrons. In ionic compounds it can take on either a positive charge (becoming a cation composed of a bare proton) or a negative charge (becoming an anion known as a hydride). Hydrogen can form compounds with most elements and is present in water and most organic compounds. It plays a particularly important role in acid-base chemistry, in which many reactions involve the exchange of protons between soluble molecules. As the only neutral atom for which the Schrödinger equation can be solved analytically, study of the energetics and bonding of the hydrogen atom has played a key role in the development of quantum mechanics.
The solubility and characteristics of hydrogen with various metals are very important in metallurgy (as many metals can suffer hydrogen embrittlement) and in developing safe ways to store it for use as a fuel. Hydrogen is highly soluble in many compounds composed of rare earth metals and transition metals[3] and can be dissolved in both crystalline and amorphous metals.[4] Hydrogen solubility in metals is influenced by local distortions or impurities in the metal crystal lattice.[5]
Hydrogen is highly combustible in air. It burned rapidly in the Hindenburg disaster on May 6 1937
Hydrogen gas is highly flammable and will burn at concentrations as low as 4% H2 in air. The enthalpy of combustion for hydrogen is – 286 kJ/mol; it burns according to the following balanced equation.
2 H2(g) + O2(g) → 2 H2O(l) + 572 kJ/mol
When mixed with oxygen across a wide range of proportions, hydrogen explodes upon ignition. Hydrogen burns violently in air. It ignites automatically at a temperature of 560 C [4] Pure hydrogen-oxygen flames burn in the ultraviolet color range and are nearly invisible to the naked eye, as illustrated by the faintness of flame from the main space shuttle engines (as opposed to the easily visible flames from the shuttle boosters). Thus it is difficult to visually detect if a hydrogen leak is burning. The explosion of the Hindenburg airship was an infamous case of hydrogen combustion (pictured), although the tragedy was due mainly to combustible materials in the skin, which were also responsible for the coloring of the flames.[6] Another characteristic of hydrogen fires is that the flames tend to ascend rapidly with the gas in air, as illustrated by the Hindenberg flames, causing less damage than hydrocarbon fires. Two-thirds of the Hindenburg passengers survived the fire, and many of the deaths which occurred were from falling or from diesel fuel burns.[7]
Electron energy levels
File:Hydrogen atom.svg
The ground state energy level of the electron in a hydrogen atom is -13.6 eV, which is equivalent to an ultraviolet photon of roughly 92 nm.
The energy levels of hydrogen can be calculated fairly accurately using the Bohr model of the atom, which conceptualizes the electron as "orbiting" the proton in analogy to the Earth's orbit of the sun. However, the electromagnetic force attracts electrons and protons to one another, while planets and celestial objects are attracted to each other by gravity. Because of the discretization of angular momentum postulated in early quantum mechanics by Bohr, the electron in the Bohr model can only occupy certain allowed distances from the proton, and therefore only certain allowed energies. A more accurate description of the hydrogen atom comes from a purely quantum mechanical treatment that uses the Schrödinger equation or the equivalent Feynman path integral formulation to calculate the probability density of the electron around the proton. Treating the electron as a matter wave reproduces chemical results such as shape of the hydrogen atom more naturally than the particle-based Bohr model, although the energy and spectral results are the same. Modeling the system fully using the reduced mass of nucleus and electron (as one would do in the two-body problem in celestial mechanics) yields an even better formula for the hydrogen spectra, and also the correct spectral shifts for the isotopes deuterium and tritium. Very small adjustments in energy levels in the hydrogen atom, which correspond to actual spectral effects, may be determined by using a full quantum mechanical theory which corrects for the effects of special relativity (see Dirac equation), and by accounting for quantum effects arising from production of virtual particles in the vacuum and as a result of electric fields (see quantum electrodynamics).
In hydrogen liquid, the electronic ground state energy level is split into hyperfine structure levels because of magnetic effects of the quantum mechanical spin of the electron and proton. The energy of the atom when the proton and electron spins are aligned is higher than when they are not aligned. The transition between these two states can occur through emission of a photon through a magnetic dipole transition. Radio telescopes can detect the radiation produced in this process, which is used to map the distribution of hydrogen in the galaxy.
H2 reacts directly with other oxidizing elements. A violent and spontaneous reaction can occur at room temperature with chlorine and fluorine, forming the corresponding hydrogen halides: hydrogen chloride and hydrogen fluoride.
Elemental molecular forms
First tracks observed in liquid hydrogen bubble chamber at the Bevatron.
There are two different types of diatomic hydrogen molecules that differ by the relative spin of their nuclei.[8] In the orthohydrogen form, the spins of the two protons are parallel and form a triplet state; in the parahydrogen form the spins are antiparallel and form a singlet. At standard temperature and pressure, hydrogen gas contains about 25% of the para form and 75% of the ortho form, also known as the "normal form".[9] The equilibrium ratio of orthohydrogen to parahydrogen depends on temperature, but since the ortho form is an excited state and has a higher energy than the para form, it is unstable and cannot be purified. At very low temperatures, the equilibrium state is composed almost exclusively of the para form. The physical properties of pure parahydrogen differ slightly from those of the normal form.[10] The ortho/para distinction also occurs in other hydrogen-containing molecules or functional groups, such as water and methylene.
The uncatalyzed interconversion between para and ortho H2 increases with increasing temperature; thus rapidly condensed H2 contains large quantities of the high-energy ortho form that convert to the para form very slowly.[11] The ortho/para ratio in condensed H2 is an important consideration in the preparation and storage of liquid hydrogen: the conversion from ortho to para is exothermic and produces enough heat to evaporate the hydrogen liquid, leading to loss of the liquefied material. Catalysts for the ortho-para interconversion, such as iron compounds, are used during hydrogen cooling.[12]
A molecular form called protonated molecular hydrogen, or H3+, is found in the interstellar medium (ISM), where it is generated by ionization of molecular hydrogen from cosmic rays. It has also been observed in the upper atmosphere of the planet Jupiter. This molecule is relatively stable in the environment of outer space due to the low temperature and density. H3+ is one of the most abundant ions in the Universe, and it plays a notable role in the chemistry of the interstellar medium.[13]
Covalent and organic compounds
While H2 is not very reactive under standard conditions, it does form compounds with most elements. Millions of hydrocarbons are known, but they are not formed by the direct reaction of elementary hydrogen and carbon (although synthesis gas production followed by the Fischer-Tropsch process to make hydrocarbons comes close to being an exception, as this begins with coal and the elemental hydrogen is generated in situ). Hydrogen can form compounds with elements that are more electronegative, such as halogens (e.g., F, Cl, Br, I) and chalcogens (O, S, Se); in these compounds hydrogen takes on a partial positive charge. When bonded to fluorine, oxygen, or nitrogen, hydrogen can participate in a form of strong noncovalent bonding called hydrogen bonding, which is critical to the stability of many biological molecules. Hydrogen also forms compounds with less electronegative elements, such as the metals and metalloids, in which it takes on a partial negative charge. These compounds are often known as hydrides.
Hydrogen forms a vast array of compounds with carbon. Because of their general association with living things, these compounds came to be called organic compounds; the study of their properties is known as organic chemistry and their study in the context of living organisms is known as biochemistry. By some definitions, "organic" compounds are only required to contain carbon (as a classic historical example, urea). However, most of them also contain hydrogen, and since it is the carbon-hydrogen bond which gives this class of compounds most of its particular chemical characteristics, carbon-hydrogen bonds are required in some definitions of the word "organic" in chemistry. (This latter definition is not perfect, however, as in this definition urea would not be included as an organic compound).
In inorganic chemistry, hydrides can also serve as bridging ligands that link two metal centers in a coordination complex. This function is particularly common in group 13 elements, especially in boranes (boron hydrides) and aluminum complexes, as well as in clustered carboranes.[14]
Compounds of hydrogen are often called hydrides, a term that is used fairly loosely. To chemists, the term "hydride" usually implies that the H atom has acquired a negative or anionic character, denoted H. The existence of the hydride anion, suggested by G.N. Lewis in 1916 for group I and II salt-like hydrides, was demonstrated by Moers in 1920 with the electrolysis of molten lithium hydride (LiH), that produced a stoichiometric quantity of hydrogen at the anode.[15] For hydrides other than group I and II metals, the term is quite misleading, considering the low electronegativity of hydrogen. An exception in group II hydrides is BeH2, which is polymeric. In lithium aluminum hydride, the AlH4 anion carries hydridic centers firmly attached to the Al(III). Although hydrides can be formed with almost all main-group elements, the number and combination of possible compounds varies widely; for example, there are over 100 binary borane hydrides known, but only one binary aluminum hydride.[16] Binary indium hydride has not yet been identified, although larger complexes exist.[17]
"Protons" and acids
Oxidation of H2 formally gives the proton, H+. This species is central to discussion of acids, though the term proton is used loosely to refer to positively charged or cationic hydrogen, denoted H+. A bare proton H+ cannot exist in solution because of its strong tendency to attach itself to atoms or molecules with electrons. To avoid the convenient fiction of the naked "solvated proton" in solution, acidic aqueous solutions are sometimes considered to contain the hydronium ion (H3O+) organized into clusters to form H9O4+.[18] Other oxonium ions are found when water is in solution with other solvents.[19]
Although exotic on earth, one of the most common ions in the universe is the H3+ ion, known as protonated molecular hydrogen or the triatomic hydrogen cation.[20]
Hydrogen has three naturally occurring isotopes, denoted 1H, ²H, and ³H. Other, highly unstable nuclei (4H to 7H) have been synthesized in the laboratory but not observed in nature.[21][22]
• ²H, the other stable hydrogen isotope, is known as deuterium and contains one proton and one neutron in its nucleus. Deuterium comprises 0.0026 – 0.0184% (by mole-fraction or atom-fraction) of hydrogen samples on Earth, with the lower number tending to be found in samples of hydrogen gas and the higher enrichments (0.015% or 150 ppm) typical of ocean water. Deuterium is not radioactive, and does not represent a significant toxicity hazard. Water enriched in molecules that include deuterium instead of normal hydrogen is called heavy water. Deuterium and its compounds are used as a non-radioactive label in chemical experiments and in solvents for 1H-NMR spectroscopy. Heavy water is used as a neutron moderator and coolant for nuclear reactors. Deuterium is also a potential fuel for commercial nuclear fusion.
• ³H is known as tritium and contains one proton and two neutrons in its nucleus. It is radioactive, decaying into Helium-3 through beta decay with a half-life of 12.32 years.[14] Small amounts of tritium occur naturally because of the interaction of cosmic rays with atmospheric gases; tritium has also been released during nuclear weapons tests. It is used in nuclear fusion reactions, as a tracer in isotope geochemistry, and specialized in self-powered lighting devices. Tritium was once routinely used in chemical and biological labeling experiments as a radiolabel (this has become less common).
Hydrogen is the only element that has different names for its isotopes in common use today. (During the early study of radioactivity, various heavy radioactive isotopes were given names, but such names are no longer used). The symbols D and T (instead of ²H and ³H) are sometimes used for deuterium and tritium, but the corresponding symbol P is already in use for phosphorus and thus is not available for protium. IUPAC states that while this use is common it is not preferred.
Natural occurrence
NGC 604, a giant region of ionized hydrogen in the Triangulum Galaxy
Hydrogen is the most abundant element in the universe, making up 75% of normal matter by mass and over 90% by number of atoms.[23] This element is found in great abundance in stars and gas giant planets. Molecular clouds of H2 are associated with star formation. Hydrogen plays a vital role in powering stars through proton-proton reaction nuclear fusion.
Throughout the universe, hydrogen is mostly found in the atomic and plasma states whose properties are quite different from molecular hydrogen. As a plasma, hydrogen's electron and proton are not bound together, resulting in very high electrical conductivity and high emissivity (producing the light from the sun and other stars). The charged particles are highly influenced by magnetic and electric fields. For example, in the solar wind they interact with the Earth's magnetosphere giving rise to Birkeland currents and the aurora. Hydrogen is found in the neutral atomic state in the Interstellar medium. The large amount of neutral hydrogen found in the damped Lyman-alpha systems is thought to dominate the cosmological baryonic density of the Universe up to redshift z=4.[24]
Under ordinary conditions on Earth, elemental hydrogen exists as the diatomic gas, H2 (for data see table). However, hydrogen gas is very rare in the Earth's atmosphere (1 ppm by volume) because of its light weight, which enables it to escape from Earth's gravity more easily than heavier gases. Although H atoms and H2 molecules are abundant in interstellar space, they are difficult to generate, concentrate, and purify on Earth. Still, hydrogen is the third most abundant element on the Earth's surface.[25] Most of the Earth's hydrogen is in the form of chemical compounds such as hydrocarbons and water.[14] Hydrogen gas is produced by some bacteria and algae and is a natural component of flatus. Methane is a hydrogen source of increasing importance.
Discovery of H2
Hydrogen gas, H2, was first artificially produced and formally described by T. Von Hohenheim (also known as Paracelsus, 1493 – 1541) via the mixing of metals with strong acids. He was unaware that the flammable gas produced by this chemical reaction was a new chemical element. In 1671, Robert Boyle rediscovered and described the reaction between iron filings and dilute acids, which results in the production of hydrogen gas.[26] In 1766, Henry Cavendish was the first to recognize hydrogen gas as a discrete substance, by identifying the gas from a metal-acid reaction as "inflammable air" and further finding that the gas produces water when burned. Cavendish had stumbled on hydrogen when experimenting with acids and mercury. Although he wrongly assumed that hydrogen was a liberated component of the mercury rather than the acid, he was still able to accurately describe several key properties of hydrogen. He is usually given credit for its discovery as an element. In 1783, Antoine Lavoisier gave the element the name of hydrogen when he (with Laplace) reproduced Cavendish's finding that water is produced when hydrogen is burned. Lavoisier's name for the gas won out.
One of the first uses of H2 was for balloons, and later airships. The H2 was obtained by reacting sulfuric acid and metallic iron. Infamously, H2 was used in the Hindenburg airship that was destroyed in a midair fire. The highly flammable hydrogen (H2) was later replaced for airships and most balloons by the unreactive helium (He).
Role in history of quantum theory
Because of its relatively simple atomic structure, consisting only of a proton and an electron, the hydrogen atom, together with the spectrum of light produced from it or absorbed by it, has been central to the development of the theory of atomic structure. Furthermore, the corresponding simplicity of the hydrogen molecule and the corresponding cation H2+ allowed fuller understanding of the nature of the chemical bond, which followed shortly after the quantum mechanical treatment of the hydrogen atom had been developed in the mid-1920s.
Large quantities of H2 are needed in the petroleum and chemical industries. The largest application of H2 is for the processing ("upgrading") of fossil fuels, and in the production of ammonia. The key consumers of H2 in the petrochemical plant include hydrodealkylation, hydrodesulfurization, and hydrocracking.[28] H2 has several other important uses. H2 is used as a hydrogenating agent, particularly in increasing the level of saturation of unsaturated fats and oils (found in items such as margarine), and in the production of methanol. It is similarly the source of hydrogen in the manufacture of hydrochloric acid. H2 is also used as a reducing agent of metallic ores.
Apart from its use as a reactant, H2 has wide applications in physics and engineering. It is used as a shielding gas in welding methods such as atomic hydrogen welding. H2 is used as the rotor coolant in electrical generators at power stations, because it has the highest thermal conductivity of any gas. Liquid H2 is used in cryogenic research, including superconductivity studies. Since H2 is lighter than air, having a little more than 1/15th of the density of air, it was once widely used as a lifting agent in balloons and airships. However, this use was curtailed after the Hindenburg disaster erroneously convinced the public that the gas was too dangerous for this purpose. Hydrogen is still regularly used for the inflation of weather balloons.
In more recent application Hydrogen is used pure or mixed with Nitrogen (sometime called Forming Gas) as a tracer gas for minute leak detection. Applications can be found in automotive, aircraft, consumer goods, medical device and chemical industry. Hydrogen is an authorized food additive (E 949) that allows food package leak testing among other anti-oxidizing properties.[29]
Hydrogen's rarer isotopes also each have specific applications. Deuterium (hydrogen-2) is used in nuclear fission applications as a moderator to slow neutrons, and in nuclear fusion reactions. Deuterium compounds have applications in chemistry and biology in studies of reaction isotope effects. Tritium (hydrogen-3), produced in nuclear reactors, is used in the production of hydrogen bombs, as an isotopic label in the biosciences, and as a radiation source in luminous paints.
Energy carrier
Hydrogen is not an energy source, except in the hypothetical context of commercial nuclear fusion power plants using deuterium or tritium, a technology presently far from development. The sun's energy comes from nuclear fusion of hydrogen but this process is difficult to achieve on earth. Elemental hydrogen from solar, biological, or electrical sources costs more in energy to make than is obtained by burning it. Hydrogen may be obtained from fossil sources (such as methane) for less energy than required to make it, but these sources are unsustainable, and are also themselves direct energy sources (and are rightly regarded as the basic source of the energy in the hydrogen obtained from them).
Molecular hydrogen has been widely discussed in the context of energy, as a possible carrier of energy on an economy-wide scale. A theoretical advantage of using H2 as an energy carrier is the localization and concentration of environmentally unwelcome aspects of hydrogen manufacture from fossil fuel energy sources. For example, CO2 sequestration followed by carbon capture and storage could be conducted at the point of H2 production from methane. Hydrogen used in transportation would burn cleanly, without carbon emissions. However, the infrastructure costs associated with full conversion to a hydrogen economy would be substantial.[30] In addition, the energy density per unit volume of both liquid hydrogen and hydrogen gas at any practicable pressure is significantly less than that of traditional fuel sources. (Although the energy density per unit mass is higher)
Laboratory syntheses
In the laboratory, H2 is usually prepared by the reaction of acids on metals such as zinc.
Zn + 2 H+ → Zn2+ + H2
Aluminum produces H2 upon treatment with acids but also with base:
2 Al + 6 H2O → 2 Al(OH)3 + 3 H2
The electrolysis of water is a simple method of producing hydrogen, although the resulting hydrogen necessarily has less energy content than was required to produce it. A low voltage current is run through the water, and gaseous oxygen forms at the anode while gaseous hydrogen forms at the cathode. Typically the cathode is made from platinum or another inert metal when producing hydrogen for storage. If, however, the gas is to be burnt on site, oxygen is desirable to assist the combustion, and so both electrodes would be made from inert metals. (Iron, for instance, would oxidize, and thus decrease the amount of oxygen given off.) The theoretical maximum efficiency (electricity used vs. energetic value of hydrogen produced) is between 80 – 94%. Bellona Report on Hydrogen
2H2O(aq) → 2H2(g) + O2(g)
In 2007, it was discovered that an alloy of aluminium and gallium in pellet form added to water could be used to generate hydrogen.[31] The process creates also creates alumina, but the expensive gallium, which prevents to formation of an oxide skin on the pellets, can be re-used. This potentially has important implications for a hydrogen economy, since hydrogen can be produced on-site and does not need to be transported.
Industrial syntheses
Hydrogen can be prepared in several different ways but the economically most important processes involve removal of hydrogen from hydrocarbons. Commercial bulk hydrogen is usually produced by the steam reforming of natural gas.[32] At high temperatures (700 – 1100 °C; 1,300 – 2,000 °F), steam (water vapor) reacts with methane to yield carbon monoxide and H2.
CH4 + H2OCO + 3 H2
This reaction is favored at low pressures but is nonetheless conducted at high pressures (20 atm; 600 inHg) since high pressure H2 is the most marketable product. The product mixture is known as "synthesis gas" because it is often used directly for the production of methanol and related compounds. Hydrocarbons other than methane can be used to produce synthesis gas with varying product ratios. One of the many complications to this highly optimized technology is the formation of coke or carbon:
CH4 → C + 2 H2
Consequently, steam reforming typically employs an excess of H2O.
Additional hydrogen from steam reforming can be recovered from the carbon monoxide through the water gas shift reaction, especially with an iron oxide catalyst. This reaction is also a common industrial source of carbon dioxide:[32] :CO + H2OCO2 + H2
Other important methods for H2 production include partial oxidation of hydrocarbons:
CH4 + 0.5 O2CO + 2 H2
and the coal reaction, which can serve as a prelude to the shift reaction above:[32] :C + H2OCO + H2
Hydrogen is sometimes produced and consumed in the same industrial process, without being separated. In the Haber process for the production of ammonia (the world's fifth most produced industrial compound), hydrogen is generated from natural gas.
Hydrogen is also produced in usable quantities as a co-product of the major petrochemical processes of steam cracking and reforming. Electrolysis of brine to yield chlorine also produces hydrogen as a co-product.
Biological syntheses
Water splitting, in which water is decomposed into its component protons, electrons, and oxygen, occurs in the light reactions in all photosynthetic organisms. Some such organisms — including the alga Chlamydomonas reinhardtii and cyanobacteria — have evolved a second step in the dark reactions in which protons and electrons are reduced to form H2 gas by specialized hydrogenases in the chloroplast.[34] Efforts have been undertaken to genetically modify cyanobacterial hydrogenases to efficiently synthesize H2 gas even in the presence of oxygen.[35]
Other rarer but mechanistically interesting routes to H2 production also exist in nature. Nitrogenase produces approximately one equivalent of H2 for each equivalent of N2 reduced to ammonia. Some phosphatases reduce phosphite to H2.
Hydrogen, Template:Lang-la, is from Ancient Greek ὕδωρ (hydor): "water" and (genes): "forming". Ancient Greek γείνομαι (geinomai): "to beget or sire")[36]
The word "hydrogen" has several different meanings;
1. the name of an element.
2. an atom, sometimes called "H dot", that is abundant in space but essentially absent on Earth, because it dimerizes.
3. a diatomic molecule that occurs naturally in trace amounts in the Earth's atmosphere; chemists increasingly refer to H2 as dihydrogen,[37] or hydrogen molecule, to distinguish this molecule from atomic hydrogen and hydrogen found in other compounds.
4. the atomic constituent within all organic compounds, water, and many other chemical compounds.
The elemental forms of hydrogen should not be confused with hydrogen as it appears in chemical compounds.
See also
1. Hydrogen in the Universe, NASA Website. URL accessed on 2 June 2006.
2. Hydrogen Basics - Production Florida Solar Energy Center.
3. Takeshita T, Wallace WE, Craig RS. (1974). Hydrogen solubility in 1:5 compounds between yttrium or thorium and nickel or cobalt. Inorg Chem 13(9):2282.
4. Kirchheim R, Mutschele T, Kieninger W. (1988). Hydrogen in amorphous and nanocrystalline metals Mater. Sci. Eng. 99: 457–462.
5. Kirchheim R. (1988). Hydrogen solubility and diffusivity in defective and amorphous metals. Prog. Mater. Sci. 32(4):262–325.
6. Dziadecki, John (2005). "Hindenburg Hydrogen Fire". Retrieved 2007-01-16.
7. "The Hindenburg Disaster". Swiss Hydrogen Association. Retrieved 2007-01-16.
8. "Universal Industrial Gases, Inc. – Hydrogen (H2) Applications and Uses". Unknown parameter |accessyear= ignored (|access-date= suggested) (help); Unknown parameter |accessmonthday= ignored (help)
9. Tikhonov VI, Volkov AA. (2002). Separation of water into its ortho and para isomers. Science 296(5577):2363.
10. NASA Glenn Research Center Glenn Safety Manual. CH. 6 - Hydrogen. Document GRC-MQSA.001, March 2006. [1]
11. Milenko YY, Sibileva RM, Strzhemechny MA. (1997). Natural ortho-para conversion rate in liquid and gaseous hydrogen. J Low Temp Phys 107(1-2):77–92.
12. Svadlenak RE, Scott AB. (1957). The Conversion of Ortho-to Parahydrogen on Iron Oxide-Zinc Oxide Catalysts. J Am Chem Soc 79(20); 5385–5388.
13. "H3+ Resource Center". Universities of Illinois and Chicago. Retrieved 2007-02-09.
14. 14.0 14.1 14.2 Miessler GL, Tarr DA. (2004). Inorganic Chemistry 3rd ed. Pearson Prentice Hall: Upper Saddle River, NJ, USA
15. K. Moers, (1920). 2. Z. Anorg. Allgem. Chem., 113:191.
16. Downs AJ, Pulham CR. (1994). The hydrides of aluminium, gallium, indium, and thallium: a re-evaluation. Chem Soc Rev 23:175–83.
17. Hibbs DE, Jones C, Smithies NA. (1999). A remarkably stable indium trihydride complex: synthesis and characterization of [InH3{P(C6H11)3}]. Chem Commum 185–6.
18. Okumura M, Yeh LI, Myers JD, Lee YT. (1990). Infrared spectra of the solvated hydronium ion: vibrational predissociation spectroscopy of mass-selected H3O+•H2On•H2m.
19. Perdoncin G, Scorrano G. (1977). Protonation equilibria in water at several temperatures of alcohols, ethers, acetone, dimethyl sulfide, and dimethyl sulfoxide. 99(21); 6983–6986.
20. Carrington A, McNab IR. (1989). The infrared predissociation spectrum of triatomic hydrogen cation (H3+). Accounts of Chemical Research 22:218–22.
21. Gurov YB, Aleshkin DV, Berh MN, Lapushkin SV, Morokhov PV, Pechkurov VA, Poroshin NO, Sandukovsky VG, Tel'kushev MV, Chernyshev BA, Tschurenkova TD. (2004). Spectroscopy of superheavy hydrogen isotopes in stopped-pion absorption by nuclei. Physics of Atomic Nuclei 68(3):491–497.
22. Korsheninnikov AA. et al. (2003). Experimental Evidence for the Existence of 7H and for a Specific Structure of 8He. Phys Rev Lett 90, 082501.
23. "Jefferson Lab – Hydrogen". Unknown parameter |accessyear= ignored (|access-date= suggested) (help); Unknown parameter |accessmonthday= ignored (help)
24. "Surveys for z > 3 Damped Lyα Absorption Systems: The Evolution of Neutral Gas". Unknown parameter |accessyear= ignored (|access-date= suggested) (help); Unknown parameter |accessmonthday= ignored (help)
25. "Basic Research Needs for the Hydrogen Economy." Argonne National Laboratory, U.S. Department of Energy, Office of Science Laboratory. 15 May 2003. [2]
26. "Webelements – Hydrogen historical information". Unknown parameter |accessyear= ignored (|access-date= suggested) (help); Unknown parameter |accessmonthday= ignored (help)
27. Berman R, Cooke AH, Hill RW. Cryogenics, Ann. Rev. Phys. Chem. 7 (1956). 1–20.
28. "Los Alamos National Laboratory – Hydrogen". Unknown parameter |accessyear= ignored (|access-date= suggested) (help); Unknown parameter |accessmonthday= ignored (help)
29. additives
30. See Romm, Joseph (2004). The Hype about Hydrogen, Fact and Fiction in the Race to Save the Climate. New York: Island Press. ISBN 1-55963-704-8.
32. 32.0 32.1 32.2 Oxtoby DW, Gillis HP, Nachtrieb NH. (2002). Principles of Modern Chemistry 5th ed. Thomson Brooks/Cole
33. Cammack, R.; Frey, M.; Robson, R. Hydrogen as a Fuel: Learning from Nature; Taylor & Francis: London, 2001
34. Kruse O, Rupprecht J, Bader KP, Thomas-Hall S, Schenk PM, Finazzi G, Hankamer B. (2005). Improved photobiological H2 production in engineered green algal cells. J Biol Chem 280(40):34170–7.
35. United States Department of Energy FY2005 Progress Report. IV.E.6 Hydrogen from Water in a Novel Recombinant Oxygen-Tolerant Cyanobacteria System. HO Smith, Xu Q. http://www.hydrogen.energy.gov/pdfs/progress05/iv_e_6_smith.pdf Accessed 16 August 2006.
36. LSJ, "of the father to beget, rarely of the mother to give birth.
37. Kubas, G. J., Metal Dihydrogen and σ-Bond Complexes, Kluwer Academic/Plenum Publishers: New York, 2001
Further reading
• Template:Cite paper
• Ferreira-Aparicio, P (2005). "New Trends in Reforming Technologies: from Hydrogen Industrial Plants to Multifuel Microreformers". Catalysis Reviews. 47: 491–588. Unknown parameter |coauthors= ignored (help)
• Krebs, Robert E. (1998). The History and Use of Our Earth's Chemical Elements: A Reference Guide. Westport, Conn.: Greenwood Press. ISBN 0-313-30123-9.
• Newton, David E. (1994). The Chemical Elements. New York, NY: Franklin Watts. ISBN 0-531-12501-7.
• Rigden, John S. (2002). Hydrogen: The Essential Element. Cambridge, MA: Harvard University Press. ISBN 0-531-12501-7.
• Romm, Joseph, J. (2004). The Hype about Hydrogen, Fact and Fiction in the Race to Save the Climate. Island Press. ISBN 1-55963-703-X. Author interview at Global Public Media.
• Stwertka, Albert (2002). A Guide to the Elements. New York, NY: Oxford University Press. ISBN 0-19-515027-9.
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fe8d63ab09a49c3f | Hydrogen atom
A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen constitutes about of the baryonic mass of the universe. All the power of Jupyter kernels, inside your favorite text editor. Since we cannot say exactly where an electron is, the Bohr picture of the atom, with electrons in neat orbits, cannot be correct.
Quantum theory describes electron probability distributions: Quantum Mechanics and the hydrogen atom.
E_0$ and $a_0$ are defined in Eqs. Here, it is assumed that $E0$ , since we are only interested in bound-states of the hydrogen atom. The above differential equation transforms to . To determine the wave functions of the hydrogen -like atom , we use a Coulomb potential to describe the attractive interaction between the single electron and the nucleus, and a spherical reference frame centred on the centre of gravity of the two-body system.
The Schrödinger equation is solved by separation of variables to . A Hydrogen atom consists of a proton and an electron which are “bound” together – the proton (positive charge) and electron (negative charge) stay together and continually interact with each other. If the electron escapes, the Hydrogen atom (now a single proton) is positively ionized. Similarly, the Hydrogen atom can .
The hydrogen atom consists of a single negatively charged electron that moves about a positively charged proton (Figure ). The hydrogen atom is the simplest atom in nature an therefore, a good starting point to study atoms and atomic structure. Hej jeg har et meget akut spørgsmål vedr. Opgaven lyder således: a) Et hydrogenatom befinder sig i grundtilstanden. Beregn den energi, der skal tilføres for at blive exciteret til tilstand nr.
Hydrogen is the simplest kind of atom, and in the very earliest days after the Big Bang hydrogen was the only kind of atom in the new Universe. The nucleus of a hydrogen atom is made of just one proton. Around the nucleus, there is just one electron, which . For a complete description of the hydrogen atom we should describe the motions of both the proton and the electron. It is possible to do this in quantum mechanics in a way that is analogous to the classical idea of describing the motion of each particle relative to the center of gravity, but we will not do so. There are many good reasons to address the hydrogen atom beyond its historical significance.
Though hydrogen spectra motivated much of the early quantum theory, research involving the hydrogen remains at the cutting edge of science and technology. How did scientists figure out the structure of atoms without looking at them? Try out different models by shooting light at the atom. Check how the prediction of the model matches the experimental.
Thomson discovered the electron, a negatively charged particle more than two thousand times lighter than a hydrogen atom. Since atoms are neutral, the charge of these electrons must be balanced . |
1aaab2100ccee6ee | Sunday, August 9, 2015
A very brief introduction to the electron correlation energy
RHF is often not accurate enough for predicting the change in energies due to a chemical reaction, no matter how big a basis set we use. The reason is the error due to the molecular orbital approximation
$$ \Psi ({{\bf{r}}_1},{{\bf{r}}_2}, \ldots {{\bf{r}}_N}) \approx \left| {{\phi _1}(1){{\bar \phi }_1}(2) \ldots {\phi _{N/2}}(N - 1){{\bar \phi }_{N/2}}(N)} \right\rangle \equiv \Phi $$
and the energy difference due to this approximation is known as the correlation energy. Just like we improve the LCAO approximation by including more terms in an expansion, we can improve the orbital approximation by an expansion, in terms of Slater determinants
$$\Psi ({{\bf{r}}_1},{{\bf{r}}_2}, \ldots {{\bf{r}}_N}) \approx \sum\limits_{i = 1}^L {{C_i}{\Phi _i}({{\bf{r}}_1},{{\bf{r}}_2}, \ldots {{\bf{r}}_N})} $$
The “basis set” of Slater determinants $\{\Phi_i \}$ is generated by first computing an RHF wave function $\{\Phi_0 \}$ as usual, which also generates a lot of virtual orbitals, and then generating other determinants with these orbitals. For example, for an atom or molecule with two electrons the RHF wave function is $\left| {{\phi _1}{{\bar \phi }_1}} \right\rangle $ and we have $K-1$ virtual orbitals (${\phi _2}, \ldots ,{\phi _K}$ , where $K$ is the number of basis functions), which can be used to make other Slater determinants like $\Phi _1^2 = \left| {{\phi _1}{{\bar \phi }_2}} \right\rangle $ and $\Phi _{11}^{22} = \left| {{\phi _2}{{\bar \phi }_2}} \right\rangle $ (Figure 1).
Figure 1. Schematic representation of the electronic structure of some of the determinants used in Equation 3
Conceptually (in analogy to spectroscopy), an electron is excited from an occupied to a virtual orbital: $\left| {{\phi _1}{{\bar \phi }_2}} \right\rangle$ represents a single excitation and $\left| {{\phi _2}{{\bar \phi }_2}} \right\rangle $ a double excitation. For systems with more than two electrons higher excitations (like triple and quadruple excitations) are also possible. In general
$$\Psi \approx {C_0}{\Phi _0} + \sum\limits_a {\sum\limits_r {C_a^r\Phi _a^r} } + \sum\limits_a {\sum\limits_b {\sum\limits_r {\sum\limits_s {C_{ab}^{rs}\Phi _{ab}^{rs}} } } } + \ldots $$
The expansion coefficients can be found using the variational principle
$$\frac{{\partial E}}{{\partial {C_i}}} = 0 \ \textrm{for all} \ i$$
and this approach is called configuration interaction (CI). The more excitations we include (i.e. increase L in Eq 2.12.1) the more accurate the expansion and the resulting energy becomes. If the expansion includes all possible excitations (known as a full CI, FCI) then we have a numerically exact wave function for the particular basis set, and if we use a basis set where the HF limit is reached then we have a numerically exact solution to the electronic Schrödinger equation! That’s the good news …
The bad news is that the FCI “basis set of determinants” is much, much larger than the LCAO basis set (i.e. $L >> K$),
$$L = \frac{{K!}}{{N!(K - N)!}}$$
where $N$ is the number of electrons. Thus, an RHF/6-31G(d,p) calculation on water involves 24 basis functions and roughly $\tfrac{1}{8}K^4$ = 42,000 2-electron integrals but a corresponding FCI/6-31G(d) calculation involves nearly 2,000,000 Slater determinants.
Just like finding the LCAO coefficients involves the diagonalization of the Fock matrix, finding the CI coefficients (Ci) and the lowest energy also involves a matrix diagonalization.
$$\bf{E} = {{\bf{C}}^t}{\bf{HC}}$$
where $\bf{E}$ is a diagonal matrix whose smallest value ($E_0$) corresponds to the variational energy minimum. While the Fock matrix is a $K \times K$ matrix, the CI Hamiltonian ($\bf{H}$) is an $L \times L$ matrix. Just holding the 2 million by 2 million matrix for the water molecule using the 6-31G(d,p) basis set requires millions of gigabites! Clever programming and large computers actually makes a FCI/6-31G(d,p) calculation on $\ce{H2O}$ possible, but FCI is clearly not a routine molecular modeling tool.
Using, for example, only single excitations (called CI singles, CIS)
$${\Psi ^{CIS}} = {C_0}{\Phi _0} + \sum\limits_a {\sum\limits_r {C_a^r\Phi _a^r} } $$
is feasible, however is doesn’t result in any improvement. The CIS Hamiltonian has three kinds of contributions
& \langle \Phi _0\left| {\hat H} \right| \Phi_0 \rangle = E_{RHF}\\
\langle\Phi^{CIS}\left| {\hat H} \right| \Phi^{CIS} \rangle \rightarrow & \langle \Phi _0\left| {\hat H} \right| \Phi_a^r \rangle = F_{ar} = 0 \\
& \langle \Phi _a^r\left| {\hat H} \right| \Phi_a^r \rangle
which means that when this matrix is diagonalized $E_0=E_{RHF}$. Thus CIS does not give us any correlation energy. However, CIS is not completely useless. The second lowest value of $\bf{E}$, $E_1$, represents the energy of the first excited state, at roughly an RHF quality.
Thus, we need at least single and double excitations (CISD)
to get any correlation energy. However, in general including doubles already results in an $\bf{H}$ matrix that is impractically large for a matrix diagonalization. CI, i.e. finding the $C_i$ coefficients using the variational principle, is therefore rarely used to compute the correlation energy.
Perhaps the most popular means of finding the $C_i$’s is by perturbation theory, a standard mathematical technique in physics to compute corrections to a reference state (in this case RHF). Perturbation theory using this reference is called Møller-Plesset pertubation theory, and there are several successively more accurate and more expensive variants: MP2 (which includes some double excitations), MP3 (more double excitations than MP2), and MP4 (single, double, triple, and some quadruple excitations).
Another approach is called coupled cluster which has a similar hierarchy of methods, such as CCSD (singles and doubles) and CCSD(T) (CCSD plus an estimate of the triples contributions). In terms of accuracy vs expense, MP2 is the best choice of a cheap correlation method, followed by CCSD, and CCSD(T). For example, MP4 is not too much cheaper than CCSD(T), but the latter is much more accurate. In fact for many practical purposes it is rarely necessary to go beyond CCSD(T) in terms of accuracy, provided a triple-zeta or higher basis set it used. However, CCSD(T) is usually too computationally demanding for molecules with more than 10 non-hydrogen atoms. In general, the computational expense of these correlated methods scale much worse than RHF with respect to basis set size: MP2 ($K^5$), CCSD ($K^6$), and CCSD(T) ($K^7$). These methods also require a significant amount of computer memory, compared to RHF, which is often the practical limitation of these post-HF methods. Finally, it should be noted that all these calculations also imply an RHF calculation as the first step.
In conclusion we now have ways of systematically improving the wave function, and hence the energy, by increasing the number of basis functions ($K$) and the number of excitations ($L$) as shown in Figure 2.
Figure 2 Schematic representation of the increase in accuracy due to using better correlation methods and larger basis sets.
The most important implication of this is that in principle it is possible to check the accuracy of a given level of theory without comparison to experiment! If going to a better correlation method or a bigger basis set does not change the answer appreciably, then we have a genuine prediction with only the charges and masses of the particles involved as empirical input. These kinds of calculations are therefore known as ab initio or first principle calculations. In practice, different properties will converge at different rates, so it is better to monitor the convergence of the property you are actually interested in, than the total energy. For example, energy differences (e.g. between two conformers) converge earlier than the molecular energies. Furthermore, the molecular structure (bond lengths and angles) tends to converge faster than the energy difference. So it is common to optimize the geometry at a low level of theory [e.g. RHF/6-31G(d)] followed by an energy computation (a single point energy) at a higher level of theory [e.g. MP2/6-311+G(2d,p)]. This level of theory would be denoted MP2/6-311+G(2d,p)//RHF/6-31G(d).
Finally, the correlation energy is not just a fine-tuning of the RHF result but introduces an important intermolecular force called the dispersion energy. The dispersion energy (also known as the induced dipole-induced dipole interaction) is a result of the simultaneous excitation of at least two electrons and is not accounted for in the RHF energy. For example, the stacked orientation of base pairs in DNA is largely a result of dispersion interactions and cannot be predicted using RHF.
Monday, August 3, 2015
Computational Chemistry Highlights: July issue
The July issue of Computational Chemistry Highlights is out.
Simulations of Chemical Reactions with the Frozen Domain Formulation of the Fragment Molecular Orbital Method |
2afdf87ff8d62285 | Jones Calculus
Jones calculus
A quaternion valued wave equation \Psi_{tt} = D^2 \Psi can be solved as usual with a d’Alembert solution \Psi(t) = \cos(D t) \Psi(0) + \sin(D t) D^{-1} \Psi'(0). We can write this more generally as e^{\beta D t} (u(0) - \beta v(0)) where \beta is a unit space quaternion where \psi(0)=u(0) - \beta v(0) is the initial wave. Now, \exp(\beta x) = \cos(x) + \beta \sin(x) holds for any space unit quaternion \beta. Unlike in the complex case, we have now an entire 2-sphere which can be used as a choice for \beta. If u(0) and v(0) are real, then we stay in the plane spanned by 1 and \beta. If u(0) and v(0) are in different plane, then the wave will evolve inside a larger part of the quaternion algebra.
Also as before, the wave equation has not be put in artificially. It appears when letting the system move freely in its symmetry. In the limit of deformation we are given an anti-symmetric matrix B= \beta (b+b^*) and get a unitary evolution \exp(i B t). As we have used Pauli matrices to represent the quaternion algebra on C^2, a wave is now given as a pair (\psi(t),\phi(t)) of complex waves. Using pairs of complex vectors is nothing new in physics. It is the Jones calculus named after Robert Clark Jones (1916-2004) who developed this picture in 1941. Jones was a Harvard graduate who obtained his PhD in 1941 and after some postdoc time at Bell Labs, worked until 1982 at the Polaroid Corporation.
Why would a photography company emply a physisists dealing with quaternion valued waves? The Jones calculus deals with polarization of light. It applies if the electromagnetic waves F =(E,B) have a particular form where E,B are both in a plane and perpendicular to each other. Remember that light is described by a 2-form F=dA which has in 4 dimensions B(4,2)=6 components, three electric and three magnetic components. The Maxwell equations dF=0, d* F=0 are then in a Lorentz gauge d^*A=0 equivalent to a wave equation L A =0, where L is the Laplacian in the Lorentz space. Now, if light has a polarized form, one can describe it with a complex two vector \Psi=(u,v) rather than by giving the 6 components (E,B) of the electromagnetic field. How is this applied? Sun light arrives unpolarized but when scattering at a surface, it catches an amount of polarization. Polarized sunglasses filter out part of this light reducing the glare of reflected light. The effect is also used in LCD technology or for glasses worn in 3D movies. It can not only be used for light, but in radio wave technology, polarization can be used to “double book” frequency channels. And for radar waves, using polarized radar waves can help to avoid seeing rain drops. Even nature has made use of it. Octopi or cuttlefish are able to see polarization patterns. See the encylopedia entry for more. Mathematically the relation with quaternion is no suprise because the linear fibre of a 1-form A(x) at a point is 4-dimensional. Describing the motion of the electromagnetic field potential A (which satisfies the wave equation) is therefore equivalent to a quaternion valued field.
We have to stress however that the connection between a quaternion valued quantum mechanics and wave motion of the electromagnetic field is mostly a mathematical one. First of all, we work in a discrete setup over an arbitrary finite simplicial complex. We don’t even have to take the de Rham complex: any elliptic complex D=d+d* as discribed in a discrete Atiyah-Singer setup will do. The Maxwell equations even don’t need to be 1 forms. If E \oplus F=\oplus E_k + \oplus F_k is the arena of vector spaces on which D:E \to F, F \to E$ acts, then one can see for a given $j \in D_k$ the equations dF=0,d^*F=j as the Maxwell equation in that space. For F=dA and gauge d^*A=0, the Maxwell equations reduce to the Poisson equation D^2 A=j which in the case of an absense of “current” j gives the wave equation D^2 A=0 meaning that A is a harmonic k-form. Now, in a classical de Rham setup on a simplicial complex G, A is just an anti-symmetric function on k-dimensional simplices of the complex. Still, in this setup, when describing light on a space of k-forms, it is given by real valued functions. If we Lax deform the elliptic complex, then the exterior derivatives become complex but still, the harmonic forms do not change because the Laplacian does not change. Also note that we don’t incorporate time into the simplicial complex (yet). Time evolution is given by an external real quantity leading to a differential equation. The wave equation u_{tt}=Lu can be described as a Schrödinger equation u_t = i Du. We have seen that when placing three complex evolutions together that we can get a quaternion valued evolution. But the waves in that evolution have little to do with the just described Maxwell equations in vacuum, which just describes harmonic functions in the elliptic complex.
We will deal with the problematic of time elsewhere. Just to state now that describing a space time with a finite simplicial complex does not seem to work. It migth be beautiful and interesting to describe finite discrete space times but one can hardly solve the Kepler problem with it. Mathematically close to the Einstein equations is to describe simplicial complexes with a fixed number of simplices which have maximal or minimal Euler characteristic among all complexes. Anyway, describing physics with waves evolving on finite geometries is appealing because the mathematics of its quantum mechanics is identical to the mathematics of the quantum mechanics in the continuum, just that everything is finite dimensional. Yes there are certain parts of quantum mechanics which appear needing infinite dimensions but if one is interested in the PDE’s, the Schroedinger respectivly the wave equation on such a space there are many interesting problems already in finite dimensions. The question how fast waves travel is also iteresting in the nonlinear Lax set-up. See This HCRP project from 2016 of Annie Rak. In principle the mathematics of PDE’s on simplicial complexes (which are actually ordinary differential equations) has more resemblence with the real thing because if one numerically computes any PDE using a finite element method, one essentially does this.
Here is a photograph showing Robert Clark Jones:
Robert Jones Robert Jones
Source: Emilio Segrè Visual Archives.
There are other places in physics where complex vector-valued fields appear. In quantum mechanics it appears from SU(2) symmetries, two level systems, isospin or weak isospin. Essentially everywhere, where two quantities can be exchanged, the SU(2) symmetry appears. A quaternion valued field is also an example of a non-abelian gauge field. In that case, one is interested (without matter) in the Lagrangian |F|^2/2 with F=dA+A \wedge A, where A is the connection 1-form. Summing the Lagranging over space gives the functional. One is interested then in critical points. The satisfy d_A^* F=0, d_A F=0 meaning that they are “harmonic” similarly as in the abelian case, where harmonic functions are critical points of the quadratic Lagrangian. There are differences however. In the Yang-Mills case, one looks at SU(2) meaning that the fields are quaternions of length 1. When we look at the Lax (or asymptotically for large t, the Schrödinger evolution) of quaternion valued fields \psi(t), then for exach fixed simplex x, the field value \psi(t,x) is a quaternion, not necessarily a unit quaternion.
[Remark. A naive idea put forward in the “particle and primes allegory” is to see a particle realized if it has an integer value. The particles and prime allegory draws a striking similarity between structures in the standard model and combinatorics of primes in associative complete division algebras. The later is pure mathematics. As there are symmetry groups acting on the primes, it is natural to look at the equivalence classes. The symmetry groups in the division algebras are U(1) and SU(2) but there is also a natural SU(3) action due to the exchange of the space generators i,j,k in the quaternion algebra. This symmetry does not act linearly on the apace, but it produces an other (naturally called strong) equivalence relation. The weak (SU(2)) and strong equivalence relations combined lead to pictures of Mesons and Baryons among the Hadrons while the U(1) symmetry naturally leads to pictures of Electron-Positron pairs and Neutrini in the Lepton case. The nomenclature essentially pairs the particle structure seen in the standard model with the prime structure in the division algebras. As expected, the analogy does not go very far. The fundamental theorem of algebra for quaternions leads to some particle processes like pair creation and annihilation and recombination but not all. It does not explain for example a transition from a Hadron to a Lepton. The set-up also leads naturally to charges with values 1/3 or 2/3 but not all. Also, number theory has entered physics in many places, it is not clear why “integers” should appear at all in a quantum field theory. What was mentioned in the particles and primes allegory is the possibility to see particles only realized at a simplex x, if the field value is an integer there. As in a non-linear integrable Hamiltonian system like the Lax evolution soliton solutions are likely to appear and so, if the wave takes some integer value p at some time t and position x, it will at a later time have that value p at a different position. The particle has traveled. But as during the time it has jumped from one vertex to an other, it can have changed to a gauge equivalent particle. If the integer value is not prime, it decomposes as a product of primes. Taking a situation where space is a product of other spaces allows to model particle interactions. One can then ask why a particle like an electron modeled by some non-real prime is so stable and why if we model an electron-positron pair by a 4k+1 prime, the position of the electron and positron are different. A Fock space analogy is to view space as an element in the strong ring, where every part is a particle. Still the mathematics is the same, we have a geometric space G with a Dirac operator D. Time evolution is obtained by letting D go in its symmetry group.] |
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Additional resources for A collection of problems in atomic and nuclear physics
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C. magnetic field v o . 40 MHz. Deterr~llr:e the gyromagnetic ratio and nuclear magnetic moment. 29. The magnetic resonance method was used to study the magnetic properties of 7Li 19 F molecules whose el~ctron shells possess the zero angular momentum. c. magnetic field. The c~)lltro) expenmen~s showed that the peaks belong to lithium and fluo~me atom~ respectIvely. Find the magnetic moments of these nucleI. The spms of the nuclei are supposed to be known. 30. 'On the basis of that assumptiOn, evaluate the lughest kinetic energy of nucleons inside a nucleus.
10 22 cm-s. 25. 0 ~ 1. 7 /lm at very low temperatures. Calculate the temperature coefficient of resistance of this semiconductor at T = 300 K. 26. 2 ~ times, when the temperature is -2 \ raised from T 1 = 300 K to T 2 = \ = 400 K. 27. Figure 28 illustrates the ~ f-logarithmic electric conductance as -6 a function of reciprocal tempera[} 1 2 0 ture (T in Kelvins) for boron-doped Fig. 28 silicon (n-type semiconductor). Explain the shape of the graph. By means of the graph find the width of the forbidden band in silicon and activation energy of boron atoms.
25 Find the half-lives of both components and the ratio of radioactive nuclei of these components at the moment t = O. 13. A radionuclide A 1 with decay constant A1 transforms into a radionuclide A 2 with decay constant A2. Assuming that at the initial moment the preparation consisted of only N 10 nuclei of radionuclide A 1 , find: (a) the number of nuclei of radionuclide A 2 after a time interval t; (b) the time interval after which the number of nuclei of radionuclide A 2 reaches the maximum value; (c) under what condition the transitional equilibrium state can evolve, so that the ratio of the amounts of the radionuclides remains constant.
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97b0d231926114fc | Friday, July 4, 2014
A Derivation of Berry's Geometric Phase from the Geometric Potential
Excerpts from The Revolution of Matter
Following on from the previous derivation of the Higgs Mechanism from the Geometric Potential I'm going to show Berry's Geometric Phase is also derivable from the Geometric Potential.
Assume a global phase $\gamma$ for the Geometric Potential
$$\phi \to e^{i \gamma} \phi$$
Take the new Lagrangian for Euclidean space,
$$\mathcal{L}_O = \mathbb{T} (\phi) + \mathbb{U} (\phi) $$
this results in a new action $S_O$ for $\mathbb{R}^4$,
$$S_O = \int \mathcal{L}_O dt = \int \mathbb{T} (\phi) + \mathbb{U} (\phi) dt$$
for the ground state of $\mathbb{U} (\phi)$ the kinetic term $ \mathbb{T} (\phi) $ tends to zero, allowing the geometric phase to be determined in terms of $\mathbb{U} (\phi)$,
$$e^{i \gamma (t)} = e^{i \int \mathbb{U} (\phi)dt } $$
Taking time t as an independent variable, this phase successively becomes,
$$ \int \mathbb{U} dt = - \int - \mathbb{U} dt = - \int (- \frac{\partial \mathbb{U}}{\partial x}) dt dx$$
By Ehrenfest's Theorem and taking $\mathbb{P}$ as the geometric equivalent of momentum in the same way as $\mathbb{T}$ is the geometric equivalent of kinetic energy, (and this is the clever bit),
$$\frac{d \langle \mathbb{P}\rangle }{{dt}} = \langle - \frac{\partial \mathbb{U}}{\partial x} \rangle $$
$$ - \int (- \frac{\partial \mathbb{U}}{\partial x}) dt dx = - \int \frac{d \langle \mathbb{P}\rangle }{{dt}} = - \int \langle \mathbb{P}\rangle \cdot dx$$
Substitute the momentum operator for the n'th level of the infinite square potential in the vicinity of $\mathbb{U}_i$, even though $\hbar$ is dimensionless in $\mathbb{R}^4$ it is included for completeness,
$$ \int \mathbb{U} dt = - \int \langle \mathbb{P}\rangle \cdot dx = - \frac{\hbar }{i} \int \int \psi_n^* \frac{\partial }{\partial x} \psi_n dx \cdot dx$$
Simplify and use the Dirac notation,
$$ \int \mathbb{U} dt = - i \hbar \int \langle \psi_n \mid \nabla_x \mid \psi_n \rangle \cdot dx$$
to determine the phase $\gamma$ of the integral $\int \mathbb{U}$ dt divide by $\hbar$ and the $\hbar$ drops out, then integrate over all space,
$$ \gamma_n(t) = i \int \langle \psi_n \mid \nabla_R \mid \psi_n \rangle \cdot dR$$
$$e^{i \gamma_n(t)} = e^{ i i \int \langle \psi_n \mid \nabla_R \mid \psi_n \rangle \cdot dR}$$
so the phase is real,
$$i \gamma \in \mathcal{R}$$
giving the wavefunction in terms of a geometric phase,
$$\Psi = \int dx \Psi_0 e^{i \gamma}$$
the extra term applies globally to the potential $\mathbb{U}$ as $\Psi$ evolves, this is equivalent to a global geometric phase change $\gamma$,
Since this is a global phase change I expect it to apply in both $\mathbb{R}^4$ and $\mathbb{R}^{ (1,3) }$, returning to the new Lagrangian $\mathcal{L}_M$ to include the dynamic phase $e^{\int\mathcal{L}dt}$,
$$e^{\int\mathcal{L_M}dt} = e^{\int\mathbb{U} (\phi)dt} \quad e^{\int\mathcal{L}dt} $$
and finally the universal wave function can be written,
$$\Psi = \int \Psi_0 e^{[\gamma (t) - \theta (t)]}$$
It can be seen that integrating the new potential over time is identical to Berry's Geometric Phase factor from his work on the Adiabatic Theorem, where he showed from the geometrical properties of the parameter space the Hamiltonian of a cyclic quantal adiabatic process will acquire an additional phase $\gamma (C)$. This can be generalized by writing for a Hamiltonian $\hat{\mathcal{H}}$ (X(T)) on a parameter space R = (X,Y,Z...), where C is the circuit over R(T) = R(0), and quantal adiabatic limit T $\to$ $\infty$. Since the natural basis of discrete eigenstates under the Schrödinger equation with energies $E_n$(X) is,
$$\hat{\mathcal{H}} (R(t)) \mid n(R) \rangle = E_n(X) \mid n(R) \rangle $$
with dynamic phase,
$$\theta (T) = - \frac{i}{\hbar } \int _0^T dt E_n(R(t)) $$
and geometric phase over a closed cycle C,
$$ \gamma_n(C) = i \oint \langle \psi_n \mid \nabla_R \mid \psi_n \rangle \cdot dR$$
where Minkowski Spacetime is assumed to be a continuously transformable from Euclidean space and noting the geometric phase is a pure number, it is now possible without loss of generality to use the geometric phase as an additional factor of the wave function in $\mathbb{R}^{1,3}$ as it affects all points in $\mathbb{R}^{1,3}$ equally, allowing,
$$ \mid \psi (T) \rangle_{\mathbb{U}_i} = e^{i [\gamma (C) - \theta (t)] } \mid \psi (T(0)) \rangle$$
The idea that the dynamic phase disappears in the Euclidean domain is consistent with the idea the physical universe having a beginning in Time, where transforming from Euclidean space to Minkowski Spacetime under the Wick rotation at $\mathbb{U}_i$ is equivalent to the switching from a geometric system to a dynamic system, which is essentially the idea behind the Hartle-Hawking no boundary proposal, so remarkably the ideas of Hartle-Hawking and Berry can be combined into a single model.
Importantly this transformation is only possible for a cyclic space in its lowest energy level, and this will be of crucial importance in the construction of a Big Bang model to be addressed later in this paper, - very importantly this additional phase factor in $\mathbb{R}^{1,3}$ is homogeneous and isotropic and affects all particles equally and this crucial idea will be returned to in the section on Newton's First Law. |
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Degenerate energy levels
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In quantum mechanics, an energy level is degenerate if it corresponds to two or more different measurable states of a quantum system. Conversely, two or more different states of a quantum mechanical system are said to be degenerate if they give the same value of energy upon measurement. The number of different states corresponding to a particular energy level is known as the degree of degeneracy of the level. It is represented mathematically by the Hamiltonian for the system having more than one linearly independent eigenstate with the same energy eigenvalue.[1]:p. 48 In classical mechanics, this can be understood in terms of different possible trajectories corresponding to the same energy.
Degeneracy plays a fundamental role in quantum statistical mechanics. For an N-particle system in three dimensions, a single energy level may correspond to several different wave functions or energy states. These degenerate states at the same level are all equally probable of being filled. The number of such states gives the degeneracy of a particular energy level.
Degenerate states in a quantum system
The possible states of a quantum mechanical system may be treated mathematically as abstract vectors in a separable, complex Hilbert space, while the observables may be represented by linear Hermitian operators acting upon them. By selecting a suitable basis, the components of these vectors and the matrix elements of the operators in that basis may be determined. If A is a N × N matrix, X a non-zero vector, and λ is a scalar, such that , then the scalar λ is said to be an eigenvalue of A and the vector X is said to be the eigenvector corresponding to λ. Together with the zero vector, the set of all eigenvectors corresponding to a given eigenvalue λ form a subspace of Cn, which is called the eigenspace of λ. An eigenvalue λ which corresponds to two or more different linearly independent eigenvectors is said to be degenerate, i.e., and , where and are linearly independent eigenvectors.The dimensionality of the eigenspace corresponding to that eigenvalue is known as its degree of degeneracy, which can be finite or infinite. An eigenvalue is said to be non-degenerate if its eigenspace is one-dimensional. The eigenvalues of the matrices representing physical observables in quantum mechanics give the measurable values of these observables while the eigenstates corresponding to these eigenvalues give the possible states in which the system may be found, upon measurement. The measurable values of the energy of a quantum system are given by the eigenvalues of the Hamiltonian operator, while its eigenstates give the possible energy states of the system. A value of energy is said to be degenerate if there exist at least two linearly independent energy states associated with it. Moreover, any linear combination of two or more degenerate eigenstates is also an eigenstate of the Hamiltonian operator corresponding to the same energy eigenvalue.
Effect of degeneracy on the measurement of energyEdit
In the absence of degeneracy, if a measured value of energy of a quantum system is determined, the corresponding state of the system is assumed to be known, since only one eigenstate corresponds to each energy eigenvalue. However, if the Hamiltonian has a degenerate eigenvalue of degree gn, the eigenstates associated with it form a vector subspace of dimension gn. In such a case, several final states can be possibly associated with the same result , all of which are linear combinations of the gn orthonormal eigenvectors .
In this case, the probability that the energy value measured for a system in the state will yield the value is given by the sum of the probabilities of finding the system in each of the states in this basis, i.e.
Degeneracy in different dimensionsEdit
This section intends to illustrate the existence of degenerate energy levels in quantum systems studied in different dimensions. The study of one and two-dimensional systems aids the conceptual understanding of more complex systems.
Degeneracy in one dimensionEdit
In several cases, analytic results can be obtained more easily in the study of one-dimensional systems. For a quantum particle with a wave function moving in a one-dimensional potential , the time-independent Schrödinger equation can be written as
Since this is an ordinary differential equation, there are two independent eigenfunctions for a given energy at most, so that the degree of degeneracy never exceeds two. It can be proved that in one dimension, there are no degenerate bound states for normalizable wave functions. A sufficient condition on a piecewise continuous potential and the energy is the existence of two real numbers with such that we have .[3] In particular, is bounded below in this criterion.
Degeneracy in two-dimensional quantum systemsEdit
Two-dimensional quantum systems exist in all three states of matter and much of the variety seen in three dimensional matter can be created in two dimensions. Real two-dimensional materials are made of monatomic layers on the surface of solids. Some examples of two-dimensional electron systems achieved experimentally include MOSFET, two-dimensional superlattices of Helium, Neon, Argon, Xenon etc. and surface of liquid Helium. The presence of degenerate energy levels is studied in the cases of particle in a box and two-dimensional harmonic oscillator, which act as useful mathematical models for several real world systems.
Particle in a rectangular planeEdit
Consider a free particle in a plane of dimensions and in a plane of impenetrable walls. The time-independent Schrödinger equation for this system with wave function can be written as
The permitted energy values are
The normalized wave function is
So, quantum numbers and are required to describe the energy eigenvalues and the lowest energy of the system is given by
For some commensurate ratios of the two lengths and , certain pairs of states are degenerate. If , where p and q are integers, the states and have the same energy and so are degenerate to each other.
Particle in a square boxEdit
In this case, the dimensions of the box and the energy eigenvalues are given by
Since and can be interchanged without changing the energy, each energy level is at least twice as degenerate when and are different. Degenerate states are also obtained when the sum of squares of quantum numbers corresponding to different energy levels are the same. For example, the three states (nx = 7, ny = 1), (nx = 1, ny = 7) and (nx = ny = 5) all have and constitute a degenerate set.
1 1 2 1
2 2 8 1
3 3 18 1
Degrees of degeneracy of different energy levels for a particle in a square box
Finding a unique eigenbasis in case of degeneracyEdit
If two operators and commute, i.e. , then for every eigenvector of , is also an eigenvector of with the same eigenvalue. However, if this eigenvalue, say , is degenerate, it can be said that belongs to the eigenspace of , which is said to be globally invariant under the action of .
For two commuting observables A and B, one can construct an orthonormal basis of the state space with eigenvectors common to the two operators. However, is a degenerate eigenvalue of , then it is an eigensubspace of that is invariant under the action of , so the representation of in the eigenbasis of is not a diagonal but a block diagonal matrix, i.e. the degenerate eigenvectors of are not, in general, eigenvectors of . However, it is always possible to choose, in every degenerate eigensubspace of , a basis of eigenvectors common to and .
Choosing a complete set of commuting observablesEdit
If a given observable A is non-degenerate, there exists a unique basis formed by its eigenvectors. On the other hand, if one or several eigenvalues of are degenerate, specifying an eigenvalue is not sufficient to characterize a basis vector. If, by choosing an observable , which commutes with , it is possible to construct an orthonormal basis of eigenvectors common to and , which is unique, for each of the possible pairs of eigenvalues {a,b}, then and are said to form a complete set of commuting observables. However, if a unique set of eigenvectors can still not be specified, for at least one of the pairs of eigenvalues, a third observable , which commutes with both and can be found such that the three form a complete set of commuting observables.
It follows that the eigenfunctions of the Hamiltonian of a quantum system with a common energy value must be labelled by giving some additional information, which can be done by choosing an operator that commutes with the Hamiltonian. These additional labels required naming of a unique energy eigenfunction and are usually related to the constants of motion of the system.
Degenerate energy eigenstates and the parity operatorEdit
The parity operator is defined by its action in the representation of changing r to -r, i.e.
The eigenvalues of P can be shown to be limited to , which are both degenerate eigenvalues in an infinite-dimensional state space. An eigenvector of P with eigenvalue +1 is said to be even, while that with eigenvalue −1 is said to be odd.
Now, an even operator is one that satisfies,
while an odd operator is one that satisfies
Since the square of the momentum operator is even, if the potential V(r) is even, the Hamiltonian is said to be an even operator. In that case, if each of its eigenvalues are non-degenerate, each eigenvector is necessarily an eigenstate of P, and therefore it is possible to look for the eigenstates of among even and odd states. However, if one of the energy eigenstates has no definite parity, it can be asserted that the corresponding eigenvalue is degenerate, and is an eigenvector of with the same eigenvalue as .
Degeneracy and symmetryEdit
The physical origin of degeneracy in a quantum-mechanical system is often the presence of some symmetry in the system. Studying the symmetry of a quantum system can, in some cases, enable us to find the energy levels and degeneracies without solving the Schrödinger equation, hence reducing effort.
Mathematically, the relation of degeneracy with symmetry can be clarified as follows. Let us consider a symmetry operation associated with a unitary operator S. Under such an operation, the new Hamiltonian is related to the original Hamiltonian by a similarity transformation generated by the operator S, such that , since S is unitary. If the Hamiltonian remains unchanged under the transformation operation S, we have
Now, if is an energy eigenstate,
where E is the corresponding energy eigenvalue.
which means that is also an energy eigenstate with the same eigenvalue E. If the two states and are linearly independent (i.e. physically distinct), they are therefore degenerate.
In cases where S is characterized by a continuous parameter , all states of the form have the same energy eigenvalue.
Symmetry group of the HamiltonianEdit
The set of all operators which commute with the Hamiltonian of a quantum system are said to form the symmetry group of the Hamiltonian. The commutators of the generators of this group determine the algebra of the group. An n-dimensional representation of the Symmetry group preserves the multiplication table of the symmetry operators. The possible degeneracies of the Hamiltonian with a particular symmetry group are given by the dimensionalities of the irreducible representations of the group. The eigenfunctions corresponding to a n-fold degenerate eigenvalue form a basis for a n-dimensional irreducible representation of the Symmetry group of the Hamiltonian.
Types of degeneracyEdit
Degeneracies in a quantum system can be systematic or accidental in nature.
Systematic or essential degeneracyEdit
This is also called a geometrical or normal degeneracy and arises due to the presence of some kind of symmetry in the system under consideration, i.e. the invariance of the Hamiltonian under a certain operation, as described above. The representation obtained from a normal degeneracy is irreducible and the corresponding eigenfunctions form a basis for this representation.
Accidental degeneracyEdit
It is a type of degeneracy resulting from some special features of the system or the functional form of the potential under consideration, and is related possibly to a hidden dynamical symmetry in the system. It also results in conserved quantities, which are often not easy to identify. Accidental symmetries lead to these additional degeneracies in the discrete energy spectrum. An accidental degeneracy can be due to the fact that the group of the Hamiltonian is not complete. These degeneracies are connected to the existence of bound orbits in classical Physics.
Examples of systems with accidental degeneraciesEdit
The Coulomb and Harmonic Oscillator potentialsEdit
For a particle in a central 1/r potential, the Laplace–Runge–Lenz vector is a conserved quantity resulting from an accidental degeneracy, in addition to the conservation of angular momentum due to rotational invariance.
For a particle moving on a cone under the influence of 1/r and r2 potentials, centred at the tip of the cone, the conserved quantities corresponding to accidental symmetry will be two components of an equivalent of the Runge-Lenz vector, in addition to one component of the angular momentum vector. These quantities generate SU(2) symmetry for both potentials.
Particle in a constant magnetic fieldEdit
A particle moving under the influence of a constant magnetic field, undergoing cyclotron motion on a circular orbit is another important example of an accidental symmetry. The symmetry multiplets in this case are the Landau levels which are infinitely degenerate.
The hydrogen atomEdit
In atomic physics, the bound states of an electron in a hydrogen atom show us useful examples of degeneracy. In this case, the Hamiltonian commutes with the total orbital angular momentum , its component along the z-direction, , total spin angular momentum and its z-component . The quantum numbers corresponding to these operators are , , (always 1/2 for an electron) and respectively.
The energy levels in the hydrogen atom depend only on the principal quantum number n. For a given n, all the states corresponding to have the same energy and are degenerate. Similarly for given values of n and l, the , states with are degenerate. The degree of degeneracy of the energy level En is therefore : , which is doubled if the spin degeneracy is included.[1]:p. 267f
The degeneracy with respect to is an essential degeneracy which is present for any central potential, and arises from the absence of a preferred spatial direction. The degeneracy with respect to is often described as an accidental degeneracy, but it can be explained in terms of special symmetries of the Schrödinger equation which are only valid for the hydrogen atom in which the potential energy is given by Coulomb's law.[1]:p. 267f
Isotropic three-dimensional harmonic oscillatorEdit
It is a spinless particle of mass m moving in three-dimensional space, subject to a central force whose absolute value is proportional to the distance of the particle from the centre of force.
It is said to be isotropic since the potential acting on it is rotationally invariant, i.e. :
where is the angular frequency given by .
Since the state space of such a particle is the tensor product of the state spaces associated with the individual one-dimensional wave functions, the time-independent Schrödinger equation for such a system is given by-
So, the energy eigenvalues are
where n is a non-negative integer. So, the energy levels are degenerate and the degree of degeneracy is equal to the number of different sets satisfying
which is equal to
Only the ground state is non-degenerate.
Removing degeneracyEdit
The degeneracy in a quantum mechanical system may be removed if the underlying symmetry is broken by an external perturbation. This causes splitting in the degenerate energy levels. This is essentially a splitting of the original irreducible representations into lower-dimensional such representations of the perturbed system.
Mathematically, the splitting due to the application of a small perturbation potential can be calculated using time-independent degenerate perturbation theory. This is an approximation scheme that can be applied to find the solution to the eigenvalue equation for the Hamiltonian H of a quantum system with an applied perturbation, given the solution for the Hamiltonian H0 for the unperturbed system. It involves expanding the eigenvalues and eigenkets of the Hamiltonian H in a perturbation series. The degenerate eigenstates with a given energy eigenvalue form a vector subspace, but not every basis of eigenstates of this space is a good starting point for perturbation theory, because typically there would not be any eigenstates of the perturbed system near them. The correct basis to choose is one that diagonalizes the perturbation Hamiltonian within the degenerate subspace.
Physical examples of removal of degeneracy by a perturbationEdit
Some important examples of physical situations where degenerate energy levels of a quantum system are split by the application of an external perturbation are given below.
Symmetry breaking in two-level systemsEdit
A two-level system essentially refers to a physical system having two states whose energies are close together and very different from those of the other states of the system. All calculations for such a system are performed on a two-dimensional subspace of the state space.
If the ground state of a physical system is two-fold degenerate, any coupling between the two corresponding states lowers the energy of the ground state of the system, and makes it more stable.
If and are the energy levels of the system, such that , and the perturbation is represented in the two-dimensional subspace as the following 2×2 matrix
then the perturbed energies are
Examples of two-state systems in which the degeneracy in energy states is broken by the presence of off-diagonal terms in the Hamiltonian resulting from an internal interaction due to an inherent property of the system include:
• Benzene, with two possible dispositions of the three double bonds between neighbouring Carbon atoms.
• Ammonia molecule, where the Nitrogen atom can be either above or below the plane defined by the three Hydrogen atoms.
• H+
molecule, in which the electron may be localized around either of the two nuclei.
Fine-structure splittingEdit
The corrections to the Coulomb interaction between the electron and the proton in a Hydrogen atom due to relativistic motion and spin-orbit coupling result in breaking the degeneracy in energy levels for different values of l corresponding to a single principal quantum number n.
The perturbation Hamiltonian due to relativistic correction is given by
where is the momentum operator and is the mass of the electron. The first-order relativistic energy correction in the basis is given by
where is the fine structure constant.
The spin-orbit interaction refers to the interaction between the intrinsic magnetic moment of the electron with the magnetic field experienced by it due to the relative motion with the proton. The interaction Hamiltonian is
which may be written as
The first order energy correction in the basis where the perturbation Hamiltonian is diagonal, is given by
where is the Bohr radius. The total fine-structure energy shift is given by
Zeeman effectEdit
The splitting of the energy levels of an atom when placed in an external magnetic field because of the interaction of the magnetic moment of the atom with the applied field is known as the Zeeman effect.
Taking into consideration the orbital and spin angular momenta, and , respectively, of a single electron in the Hydrogen atom, the perturbation Hamiltonian is given by-
where and . Thus,
Now, in case of the weak-field Zeeman effect, when the applied field is weak compared to the internal field, the spin-orbit coupling dominates and and are not separately conserved. The good quantum numbers are n, l, j and mj, and in this basis, the first order energy correction can be shown to be given by
, where
is called the Bohr Magneton.Thus, depending on the value of , each degenerate energy level splits into several levels.
Lifting of degeneracy by an external magnetic field
In case of the strong-field Zeeman effect, when the applied field is strong enough, so that the orbital and spin angular momenta decouple, the good quantum numbers are now n,l,ml and ms. Here, Lz and Sz are conserved, so the perturbation Hamiltonian is given by-
assuming the magnetic field to be along the z-direction. So,
For each value of ml, there are two possible values of ms, .
Stark effectEdit
The splitting of the energy levels of an atom or molecule when subjected to an external electric field is known as the Stark effect.
For the hydrogen atom, the perturbation Hamiltonian is-
if the electric field is chosen along the z-direction.
The energy corrections due to the applied field are given by the expectation value of in the basis. It can be shown by the selection rules that when and .
The degeneracy is lifted only for certain states obeying the selection rules, in the first order. The first-order splitting in the energy levels for the degenerate states and , both corresponding to n=2, is given by .
See alsoEdit
1. ^ a b c Merzbacher, Eugen (1998). Quantum Mechanics (3rd ed.). New York: John Wiley. ISBN 0471887021.
2. ^ Levine, Ira N. (1991). Quantum Chemistry (4th ed.). Prentice Hall. p. 52. ISBN 0-205-12770-3.
3. ^ a b Messiah, Albert (1967). Quantum mechanics (3rd ed.). Amsterdam, NLD: North-Holland. pp. 98–106. ISBN 0471887021.
Further readingEdit |
1da8a99c93eb4637 |
Department of Physics
PHYS2511 Foundations of Physics 2 (2010/11)
Elements of Quantum Mechanics
Dr J. Jäckel
8 lectures + 3 examples classes in Michaelmas Term
Syllabus: Wavefunction, probability density and expectation value of position. Time-dependent and time-independent Schrödinger equations. Stationary states. Solution of the Schrödinger equation for model potentials. Operators and Hamiltonian. Eigenfunctions and eigenvalues. Orthogonality and completeness of eigenfunctions. Expansion of a wavefunction in terms of a complete set and significance of expansion coefficients. Measurement and the reduction of the wavefunction. Compatible and incompatible observables. Schrödinger equation in three dimensions using spherical polar co-ordinates.
Quantum Physics, R. Eisberg and R. Resnick (Wiley), R
Quantum Physics, S. Gasiorowicz (Wiley), R
Introduction to Quantum Mechanics, B.H. Bransden and C.J. Joachain (Longman), B
Physics of Atoms and Molecules, B.H. Bransden and C.J. Joachain (Longman), B
Prof D.P. Hampshire
19 lectures + 6 examples classes in Michaelmas and Epiphany Terms
Syllabus: Review: Gauss' Law, Faraday's Law of Induction, Lenz Law, Ampère's Circuital Law, Lorentz force, grad, div, curl, phasor representation. Formulations of Maxwell's equations. Application to propagation of waves in free space. Dielectric media. Polarisation. Surface and volume charge densities and polarisation current density. Magnetic media. Surface and volume magnetisation current densities. Modified form of Maxwell's equations for Linear, Isotropic and Homogeneous (LIH) media. Propagation of waves in LIH media. Non-conductors and conductors. Skin depth. Complex refractive indices. Energy flow and Poynting's vector. Reflection and refraction of e.m. waves at the interface between two media. Boundary condition. Fresnel's equations. Brewster angle. Normal incidence. Total reflection. Dielectric/metallic interface. Plasmas. Plasma frequency, group and phase velocities and refractive index. Radiation. Oscillating electric dipole. Directivity, beam width and radiation resistance. Maxwell's equations and relativity. Lorentz transformation.
Electromagnetism, I.S. Grant and W.R. Phillips (McGraw-Hill), E
The Cambridge Handbook of Physics Formulae, G. Woan (CUP), R
Lectures in Physics Vol.2, R.P. Feynman (Addison-Wesley), R
Electricity and Magnetism, W. J. Duffin (McGraw-Hill), B
Electromagnetics, J. Edminster (Schaum), B
Classical Electrodynamics, J.D. Jackson (Wiley, 3rd Edition), B
Atomic Physics
Prof J.M.Girkin
11 lectures + 3 examples classes in Epiphany Term
Syllabus: Atomic nature of matter, atomic mass, size and charge. Basic features of atomic structure. Separation of variables for a spherically symmetric potential. Energy eigenfunctions of hydrogen atom: spherical harmonics, quantum numbers ml and l, radial dependence, quantum number n. Allowed values of n, l, ml and their physical significance. Radial distribution function and angular dependence of probability density. Angular momentum: operators of z-component and square of, their eigenfunctions and eigenvalues. Vector picture. Magnetic moment of one electron atom, classical model, Bohr magneton. Effect of uniform and inhomogeneous fields, Stern-Gerlach experiment and spatial quantization. Electron spin and associated quantum numbers s and ms. Spin g-factor. Total angular momentum, spin-orbit interaction. Quantum numbers j and mj. Fine structure. Landé g‑factor. Wavefunction for two electrons. Exchange, wavefunction symmetry, and exclusion principle. Ground state of the helium atom, effect of electron-electron interaction. Lowest excited states of helium. Singlet and triplet states. Coulomb and exchange integrals and exchange splitting. Multi-electron atoms. Central field approximation. Electronic configurations, the periodic table and the chemical properties of the elements. Angular momentum: Russell-Saunders or LS coupling and associated quantum numbers, cases of full subshell and two p-electrons in the same or different subshells. Spectroscopic notation. Hund's rules. Atomic magnetic moments. Hyperfine splitting. Binding energies of inner and outer electrons. Optical properties of alkali atoms. X-ray line spectra. This section covers many of the core areas of Quantum Physics as defined by the Institute of Physics.
Textbooks: As for Elements of Quantum Mechanics (see above)
3 lectures in Easter Term, one by each lecturer
Teaching methods
Lectures: 2 one-hour lectures per week.
Examples classes: These provide an opportunity to work through and digest the course material by attempting exercises and assignments assisted by direct interaction with the lecturers and demonstrators. Students will be divided into four groups, each of which will attend one two-hour class every two weeks. Over the course of the year the classes will be split as follows: 5 hours Elements of Quantum Mechanics + 10 hours Electromagnetism + 5 hours Atomic Physics.
Problem exercises: See |
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Early Physics with the Large Hadron Collider Thomas J. LeCompte High Energy Physics Division Argonne National Laboratory JLAB Users’ Meeting: 16 June 2008.
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Presentation on theme: "Early Physics with the Large Hadron Collider Thomas J. LeCompte High Energy Physics Division Argonne National Laboratory JLAB Users’ Meeting: 16 June 2008."— Presentation transcript:
2 2 First Order of Business Thanks very much for the invitation. I’ve wanted to visit Jefferson Lab for a long time, both for the rich scientific program, and because Nate Isgur was very kind to me when I was an ignorant graduate student.
3 3 Second Order of Business The HEP community likes Mont. You will too.
4 4 Outline The Standard Model –QCD –Electroweak Theory The Large Hadron Collider and Why You Might Want One –The problem with Electroweak Theory Detectors: ATLAS and CMS The problem with QCD More on the EWK problem Summary
5 5 The Traditional Opening Pitch Practically every HEP talk starts with this slide. This isn’t the way I want to start this talk.
6 6 Comparing Two Figures Both plots focus on the constituents of a thing, rather than their interactions. While there is meaning in both plots, it can be hard to see. –A plot of a composition by A. Schoenberg would look different A histogram of the notes used in Beethoven’s 5 th Symphony, first movement. I’d like to come at this from a different direction.
7 7 The Twin Pillars of the Standard Model Quantum Chromodynamics –Quarks carry a charge called “color” carried by gluons which themselves also carry color charge. –A strong force (in fact, THE strong force) –Confines quarks into hadrons Electroweak Unification –The electric force, the magnetic force and the weak interaction that mediates -decay are all aspects of the same “electroweak” force. –Only three constants enter into it: e.g. , G F and sin 2 ( w ). –A chiral theory: it treats particles with left-handed spin differently than particles with right-handed spin. A beautiful theory. Unfortunately, it’s broken.
8 8 Why Study The Standard Model? Understanding it is a necessary precondition for discovering anything beyond the Standard Model –Whatever physics you intend to do in 2011, you’ll be studying SM physics in 2008 Rate is also an issue It’s interesting in and of itself –It’s predictive power remains extraordinary (e.g. g-2 for the electron) We know it’s incomplete –It’s a low energy effective theory: can we see what lies beyond it? We’ve lived with the SM for ~25 years –Long enough so that features we used to find endearing are starting to become annoying Think of the LHC as “marriage counseling” for the SM
9 9 Local Gauge Invariance – Part I In quantum mechanics, the probability density is the square of the wavefunction: P ( x ) = | | 2 –If I change to – , anything I can observe remains unchanged P ( x ) = | | 2 can be perhaps better written as P ( x ) = * –If I change to e i anything I can observe still remains unchanged. –The above example was a special case ( = ) If I can’t actually observe , how do I know that it’s the same everywhere? –I should allow to be a function, (x,t). –This looks harmless, but is actually an extremely powerful constraint on the kinds of theories one can write down.
10 10 Local Gauge Invariance – Part II The trouble comes about because the Schrödinger equation (and its descendents) involves derivatives, and a derivative of a product has extra terms. At the end of the day, I can’t have any leftover ’s – they all have to cancel. (They are, by construction, supposed to be unobservable) If I want to write down the Hamiltonian that describes two electrically charged particles, I need to add one new piece to get rid of the ’s: a massless photon.
11 11 Massless? A massive spin-1 particle has three spin states ( m = 1,0,-1) A massless spin-1 particle has only two. –Hand-wavy argument: Massless particles move at the speed of light; you can’t boost to a frame where the spin points in another direction. To cancel all the ’s, I need just the two m = ± 1 states (“degrees of freedom”) –Adding the third state overdoes it and messes up the cancellations –The photon that I add must be massless m = ±1 “transverse” m = 0 “longitudinal” Aside: this has to be just about the most confusing convention adopted since we decided that the current flows opposite to the direction of electron flow. We’re stuck with it now.
12 12 A Good Theory is Predictive…or at least Retrodictive This is a theoretical tour-de-force: starting with Coulomb’s Law, and making it relativistically and quantum mechanically sound, and out pops: –Magnetism –Classical electromagnetic waves –A quantum mechanical photon of zero mass Experimentally, the photon is massless (< 10 -22 m e ) –10 -22 = concentration of ten molecules of ethanol in a glass of water Roughly the composition of “Lite” Beer –10 -22 = ratio of the radius of my head to the radius of the galaxy –10 -22 = probability Britney Spears won’t do anything shameless and stupid in the next 12 months
13 13 Let’s Do It Again A Hamiltonian that describe electrically charged particles also gives you: –a massless photon A Hamiltonian that describes particles with color charge (quarks) also gives you: –a massless gluon (actually 8 massless gluons) A Hamiltonian that describes particles with weak charge also gives you: –massless W +, W - and Z 0 bosons –Experimentally, they are heavy: 80 and 91 GeV Why this doesn’t work out for the weak force – i.e. why the W’s and Z’s are massive – is what the LHC is trying to find out.
14 14 Nobody Wants A One Trick Pony One goal: understand what’s going on with “electroweak symmetry breaking” –e.g. why are the W and Z heavy when the photon is massless Another goal: probe the structure of matter at the smallest possible distance scale –Small (= h/p ) means high energy Third goal: search for new heavy particles –This also means large energy ( E=mc 2 ) Fourth goal: produce the largest number of previously discovered particles (top & bottom quarks, W’s, Z’s …) for precision studies “What is the LHC for?” is a little like “What is the Hubble Space Telescope for?” – the answer depends on who you ask. A multi-billion dollar instrument really needs to be able to do more than one thing. All of these require the highest energy we can achieve.
15 15 The Large Hadron Collider The Large Hadron Collider is a 26km long circular accelerator built at CERN, near Geneva Switzerland. The magnetic field is created by 1232 superconducting dipole magnets (plus hundreds of focusing and correction magnets) arranged in a ring in the tunnel. Design Collision Energy = 14 TeV
16 16 Thermal Expansion and the LHC means that the LHC should shrink ~50 feet in radius when cooled down. The tunnel is only about 10 feet wide.
17 17 ATLAS = A Toroidal LHC ApparatuS Length = 44m Diameter = 22m Mass = 7000 t
18 18 CMS = Compact Muon Solenoid
19 19 How They Work Particles curve in a central magnetic field –Measures their momentum Particles then stop in the calorimeters –Measures their energy Except muons, which penetrate and have their momenta measured a second time. Different particles propagate differently through different parts of the detector; this enables us to identify them.
20 20 ATLAS Revisited
21 21 What ATLAS Looks Like Today
22 22 The ATLAS Muon Spectrometer – One Practical Issue We would like to measure a 1 TeV muon momentum to about 10%. –Implies a sagitta resolution of about 100 m. Thermal expansion is enough to cause problems. Instead of keeping the detector in position, we let it flex: –It’s easier to continually measure where the pieces are than to keep it perfectly rigid. Pictures from Jim Shank, Boston University Beam’s eye view: d= 22m
23 23 CMS: The Other LHC “Large” Detector Similar in concept to ATLAS, but with a different execution. Different detector technologies –e.g. iron core muon spectrometer vs. air core –Crystal calorimeter vs. liquid argon Different design emphasis –e.g. their EM calorimeter is optimized more towards precise measurement of the signal; ATLAS is optimized more towards background rejection
24 24 The Problem with QCD Calculations can be extraordinarily difficult – many quantities we would like to calculate (e.g. the structure of the proton) need to be measured.
25 25 QCD vs. QED QEDQCD Symmetry GroupU(1)SU(3) ChargeElectric chargeThree kinds of color Force carrier1 Photon – neutral8 Gluons - colored Coupling strength1/137 (runs slowly)~1/6 (runs quickly) changes by about 7% from Q=0 to Q=100 GeV. This will change the results of a calculation, but not the character of a calculation.
26 26 The Running of s At high Q 2, s is small, and QCD is in the perturbative region. –Calculations are “easy” At low Q 2, s is large, and QCD is in the non-perturbative region. –Calculations are usually impossible Occasionally, some symmetry principle rescues you –Anything we want to know here must come from measurement From I. Hinchliffe – this contains data from several kinds of experiments: decays, DIS, and event topologies at different center of mass energies.
27 27 An Early Modern, Popular and Wrong View of the Proton The proton consists of two up (or u) quarks and one down (or d) quark. –A u-quark has charge +2/3 –A d-quark has charge –1/3 The neutron consists of just the opposite: two d’s and a u –Hence it has charge 0 The u and d quarks weigh the same, about 1/3 the proton mass –That explains the fact that m(n) = m(p) to about 0.1% Every hadron in the Particle Zoo has its own quark composition So what’s missing from this picture?
28 28 Energy is Stored in Fields We know energy is stored in electric & magnetic fields –Energy density ~ E 2 + B 2 –The picture to the left shows what happens when the energy stored in the earth’s electric field is released Energy is also stored in the gluon field in a proton –There is an analogous E 2 + B 2 that one can write down –There’s nothing unusual about the idea of energy stored there What’s unusual is the amount: Thunder is good, thunder is impressive; but it is lightning that does the work. (Mark Twain) Energy stored in the field Atom10 -8 Nucleus1% Proton99%
29 29 The Modern Proton 99% of the proton’s mass/energy is due to this self- generating gluon field The two u-quarks and single d-quark –1. Act as boundary conditions on the field (a more accurate view than generators of the field) –2. Determine the electromagnetic properties of the proton Gluons are electrically neutral, so they can’t affect electromagnetic properties The similarity of mass between the proton and neutron arises from the fact that the gluon dynamics are the same –Has nothing to do with the quarks Mostly a very dynamic self-interacting field of gluons, with three quarks embedded. Like plums in a pudding. The Proton
30 30 The “Rutherford Experiment” of Geiger and Marsden particle scatters from source, off the gold atom target, and is detected by a detector that can be swept over a range of angles (n.b.) particles were the most energetic probes available at the time The electric field the experiences gets weaker and weaker as the enters the Thomson atom, but gets stronger and stronger as it enters the Rutherford atom and nears the nucleus.
31 31 Results of the Experiment At angles as low as 3 o, the data show a million times as many scatters as predicted by the Thomson model –Textbooks often point out that the data disagreed with theory, but they seldom state how bad the disagreement was There is an excess of events with a large angle scatter –This is a universal signature for substructure –It means your probe has penetrated deep into the target and bounced off something hard and heavy An excess of large angle scatters is the same as an excess of large transverse momentum scatters
32 32 Proton Collisions: The Ideal World 1. Protons collide 2. Constituents scatter 3. As proton remnants separate
33 33 What Really Happens You don’t see the constituent scatter. You see a jet: a “blast” of particles, all going in roughly the same direction. Calorimeter View Same Events, Tracking View 2 jets 3 jets 5 jets 2 2 3 5
34 34 Jets The force between two colored objects (e.g. quarks) is ~independent of distance –Therefore the potential energy grows (~linearly) with distance –When it gets big enough, it pops a quark-antiquark pair out of the vacuum –These quarks and antiquarks ultimately end up as a collection of hadrons We can’t calculate how often a jet’s final state is, e.g. ten ’s, three K’s and a . Fortunately, it doesn’t matter. –We’re interested in the quark or gluon that produced the jet. –Summing over all the details of the jet’s composition and evolution is A Good Thing. Two jets of the same energy can look quite different; this lets us treat them the same Initial quark Jet What makes the measurement possible & useful is the conservation of energy & momentum.
35 35 Jets after “One Week” Jet Transverse Energy 5 pb -1 of (simulated) data: corresponds to 1 week running at 10 31 cm -2 /s (1% of design) ATLAS This is in units of transverse momentum. Remember, large angle = large p T
36 36 Jets after “One Week” Number of events we expect to see: ~12 If new physics: ~50 Number we have seen to date worldwide: 0 Jet Transverse Energy 5 pb -1 of (simulated) data: corresponds to 1 week running at 10 31 cm -2 /s (1% of design) ATLAS New physics (e.g. quark substructure) shows up here.
37 37 Outrunning the Bear Present limits on 4-fermion contact interactions from the Tevatron are 2-4-2.7 TeV This may hit 3 TeV by LHC turn-on –Depends on how many people work on this If we shoot for 6 TeV at the LHC and only reach 5 TeV, we’ve already made substantial progress Note that there are ~a dozen jets that are above the Tevatron’s kinematic limit: a day at the LHC will set a limit that the Tevatron can never reach.
38 38 The Big Asterisk The first run will be at 10 TeV, not 14 TeV –Magnet training took longer than anticipated –CERN wisely decided to give the experiments something this year rather than to wait. This increases the running time for a given sensitivity by a factor of 3-4 –A week’s worth of good data in a 2-3 month initial run is much more likely than a month’s worth
39 39 Compositeness & The Periodic Table(s) The 9 lightest spin-0 particles The 8 lightest spin-1/2 particles Arises because atoms have substructure: electrons Arises because hadrons have substructure: quarks
40 40 Variations on a Theme? A good question – and one that the LHC would address Sensitivity is comparable to where we found “the next layer down” in the past. –Atoms: nuclei (10 5 :1) –Nuclei: nucleons (few:1) –Quarks (>10 4 :1) will become (~10 5 :1) There are some subtleties: if this is substructure, its nature is different than past examples. Does this arise because quarks have substructure?
41 41 The Complication Light quarks are…well, light. –Masses of a few MeV Any subcomponents would be heavy –At least 1000 times heavier Otherwise, we would have already discovered them Therefore, they would have to be bound very, very deeply. (binding energy ~ their mass) A -function potential has only one bound state – so the “particle periodic table” can’t be due to them being simply different configurations of the same components. Something new and interesting has to happen. I’m an experimenter. This isn’t my problem.
42 42 The Structure of the Proton Even if there is no new physics, the same kinds of measurements can be used to probe the structure of the proton. Because the proton is traveling so close to the speed of light, it’s internal clocks are slowed down by a factor of 7500 (in the lab frame) – essentially freezing it. We look at what is essentially a 2-d snapshot of the proton.
43 43 The Collision What appears to be a highly inelastic process: two protons produce two jets of other particles… (plus two remnants that go down the beam pipe) … is actually the elastic scattering of two constituents of the protons.
44 44 Parton Densities What looks to be an inelastic collision of protons is actually an elastic collision of partons: quarks and gluons. In an elastic collision, measuring the momenta of the final state particles completely specifies the momenta of the initial state particles. Different final states probe different combinations of initial partons. –This allows us to separate out the contributions of gluons and quarks. –Different experiments also probe different combinations. It’s useful to notate this in terms of x : – x = p (parton)/ p (proton) –The fraction of the proton’s momentum that this parton carries This is actually the Fourier transform of the position distributions. –Calculationally, leaving it this way is best.
45 45 Parton Density Functions in Detail One fit from CTEQ and one from MRS is shown –These are global fits from all the data Despite differences in procedure, the conclusions are remarkably similar –Lends confidence to the process –The biggest uncertainty is in the gluon The gluon distribution is enormous: –The proton is mostly glue, not mostly quarks
46 46 Improving the Gluon: Direct Photons DIS and Drell-Yan are sensitive to the quark PDFs. Gluon sensitivity is indirect –The fraction of momentum not carried by the quarks must be carried by the gluon. –Antiquarks in the proton must be from gluons splitting It would be useful to have a direct measurement of the gluon PDFs –This process depends on the (known) quark distributions and the (unknown) gluon distribution q q g Direct photon “Compton” process.
47 47 Identifying Photons – Basics of Calorimeter Design A schematic of an electromagnetic shower A GEANT simulation of an electromagnetic shower Not too much or too little energy here. Not too wide here. Not too much energy here. You want exactly one photon – not 0 (a likely hadron) or 2 (likely 0 ) One photon and not two nearby ones (again, a likely 0 ) Indicative of a hadronic shower: probably a neutron or K L.
48 48 Direct Photons & Backgrounds There are two “knobs we can turn” –Shower shape – does this look like a photon (last slide) –Isolation – if it’s a fake, it’s likely to be from a jet, and there is likely to be some nearby energy Different experiments (and analyses in the same experiment) can rely more on one method than the other. CMS Before event selection After event selection
49 49 More Variations on A Theme One can scatter a gluon off of a heavy quark in the proton as well as a light quark –This quark can be identified as a bottom or charmed quark by “tagging” the jet –This measures how much b (or c) is in the proton Determines backgrounds to various searches, like Higgs Turns out to have a surprisingly large impact on the ability to measure the W mass (ask me about this at the end, if interested) Replace the with a Z, and measure the same thing with different kinematics Replace the Z with a W and instead of measuring how much charm is in the proton, you measure how much strangeness there is …and so on…
50 50 Double Parton Scattering Two independent partons in the proton scatter: Searches for complex signatures in the presence of QCD background often rely on the fact that decays of heavy particles are “spherical”, but QCD background is “correlated” –This breaks down in the case where part of the signature comes from a second scattering. –Probability is low, but needed background reduction can be high We’re thinking about bbjj as a good signature –Large rate/large kinematic range 10 5 more events than past experiments –Relatively unambiguous which jets go with which other jets. might be better characterized by
51 51 Three Subtleties These densities are not quite universal –They depend on the wavelength of your probe of the proton. A large fraction of the proton’s momentum is carried by gluons at low x –There is a halo around the proton of large wavelength gluons (and quark- antiquark pairs) This sounds a lot like a particle physicist’s description of a pion cloud Measurements of heavy flavor in the proton can be interpreted as a cloud of flavored mesons (up to B’s) –It’s a little paradoxical – one needs the highest energy (i.e. shortest wavelength) to probe this large wavelength halo Double parton scattering delineates the breakdown of this simple model.
52 52 The Problem with Electroweak Theory Here we have the opposite problem than QCD – here calculations are easier, but there is a fundamental flaw in the underlying theory.
53 53 The “No Lose Theorem” Imagine you could elastically scatter beams of W bosons: WW → WW We can calculate this, and at high enough energies the cross-section violates unitarity –The probability of a scatter exceeds 1 - nonsense –The troublesome piece is (once again) the longitudinal spin state “High enough” means about 1 TeV –A 14 TeV proton-proton accelerator is just energetic enough to give you enough 1 TeV parton-parton collisions to study this The Standard Model is a low-energy effective theory. The LHC gives us the opportunity to probe it where it breaks down. Something new must happen.
54 54 Spontaneous Symmetry Breaking What is the least amount of railroad track needed to connect these 4 cities?
55 55 One Option I can connect them this way at a cost of 4 units. (length of side = 1 unit)
56 56 Option Two I can connect them this way at a cost of only 3 units.
57 57 The Solution that Looks Optimal, But Really Isn’t This requires only
58 58 The Real Optimal Solution This requires Note that the symmetry of the solution is lower than the symmetry of the problem: this is the definition of Spontaneous Symmetry Breaking. + n.b. The sum of the solutions has the same symmetry as the problem.
59 59 A Pointless Aside One might have guessed at the answer by looking at soap bubbles, which try to minimize their surface area. But that’s not important right now… Another Example of Spontaneous Symmetry Breaking Ferromagnetism: the Hamiltonian is fully spatially symmetric, but the ground state has a non-zero magnetization pointing in some direction.
60 60 The Higgs Mechanism Write down a theory of massless weak bosons –The only thing wrong with this theory is that it doesn’t describe the world in which we live Add a new doublet of spin-0 particles: –This adds four new degrees of freedom (the doublet + their antiparticles) Write down the interactions between the new doublet and itself, and the new doublet and the weak bosons in just the right way to –Spontaneously break the symmetry: i.e. the Higgs field develops a non-zero vacuum expectation value Like the magnetization in a ferromagnet –Allow something really cute to happen
61 61 The Really Cute Thing The massless w + and + mix. –You get one particle with three spin states Massive particles have three spin states –The W has acquired a mass The same thing happens for the w - and - In the neutral case, the same thing happens for one neutral combination, and it becomes the massive Z 0. The other neutral combination doesn’t couple to the Higgs, and it gives the massless photon. That leaves one degree of freedom left, and because of the non zero v.e.v. of the Higgs field, produces a massive Higgs. m = ±1 “transverse” m = 0 “longitudinal”
62 62 How Cute Is It? There’s very little choice involved in how you write down this theory. –There’s one free parameter which determines the Higgs boson mass –There’s one sign which determines if the symmetry breaks or not. The theory leaves the Standard Model mostly untouched –It adds a new Higgs boson – which we can look for –It adds a new piece to the WW → WW cross-section This interferes destructively with the piece that was already there and restores unitarity In this model, the v.e.v. of the Higgs field is the Fermi constant
63 63 Searching for the Higgs Boson H → ATLAS Simulation 100 fb -1 ATLAS Simulation 10 fb -1 H → ZZ → llll Because the theory is so constrained, we have very solid predictions on where to look and what to look for.
64 64 Two Alternatives Multiple Higgses –I didn’t have to stop with one Higgs doublet – I could have added two –This provides four more degrees of freedom: Manifests as five massive Higgs bosons: h 0, H 0, A 0, H +,H - – Usually some are harder to see, and some are easier –You don’t have to stop there either… New Strong Dynamics –Maybe the WW → WW cross-section blowing up is telling us something: The p → p cross-section also blew up: it was because of a resonance: the . Maybe there are resonances among the W’s and Z’s which explicitly break the symmetry Many models: LHC data will help discriminate among them.
65 65 The Higgs Triangle Direct Observation Loop Effects on m(W) Effect on 4W vertex W+W+ W-W- W+W+ W-W- Two of the three necessary measurements are SM measurements.
66 66 What is the Standard Model? The (Electroweak) Standard Model is the theory that has interactions like: W+W+ W+W+ Z0Z0 Z0Z0 but not Z0Z0 Z0Z0 W+W+ W-W- Z0Z0 W-W- W+W+ & but not: Z0Z0 Z0Z0 Z0Z0 & Z0Z0 Z0Z0 Only three parameters - G F, and sin 2 ( w ) - determine all couplings.
67 67 Portrait of a Troublemaker This diagram is where the SM gets into trouble. It’s vital that we measure this coupling, whether or not we see a Higgs. From Azuelos et al. hep-ph/0003275 100 fb-1, all leptonic modes inside detector acceptance W+W+ W-W- W+W+ W-W- Yields are not all that great
68 68 A Complication If we want to understand the quartic coupling… …first we need to measure the trilinear couplings We need a TGC program that looks at all final states: WW, WZ, W (present in SM) + ZZ, Z (absent in SM)
69 69 Semiclassically, the interaction between the W and the electromagnetic field can be completely determined by three numbers: –The W’s electric charge Effect on the E-field goes like 1/r 2 –The W’s magnetic dipole moment Effect on the H-field goes like 1/r 3 –The W’s electric quadrupole moment Effect on the E-field goes like 1/r 4 Measuring the Triple Gauge Couplings is equivalent to measuring the 2 nd and 3 rd numbers –Because of the higher powers of 1/r, these effects are largest at small distances –Small distance = short wavelength = high energy The Semiclassical W
70 70 Triple Gauge Couplings There are 14 possible WW and WWZ couplings To simplify, one usually talks about 5 independent, CP conserving, EM gauge invariance preserving couplings: g 1 Z, , Z, , Z –In the SM, g 1 Z = = Z = 1 and = Z = 0 Often useful to talk about g, and instead. Convention on quoting sensitivity is to hold the other 4 couplings at their SM values. –Magnetic dipole moment of the W = e(1 + + )/2M W –Electric quadrupole moment = -e( - )/2M W 2 –Dimension 4 operators alter g 1 Z, and Z : grow as s ½ –Dimension 6 operators alter and Z and grow as s These can change either because of loop effects (think e or magnetic moment) or because the couplings themselves are non-SM
71 71 Why Center-Of-Mass Energy Is Good For You The open histogram is the expectation for = 0.01 –This is ½ a standard deviation away from today’s world average fit If one does just a counting experiment above the Tevatron kinematic limit (red line), one sees a significance of 5.5 –Of course, a full fit is more sensitive; it’s clear that the events above 1.5 TeV have the most distinguishing power From ATLAS Physics TDR: 30 fb -1 Approximate Run II Tevatron Reach Tevatron kinematic limit
72 72 Not An Isolated Incident Qualitatively, the same thing happens with other couplings and processes These are from WZ events with g 1 Z = 0.05 –While not excluded by data today, this is not nearly as conservative as the prior plot A disadvantage of having an old TDR Plot is from ATLAS Physics TDR: 30 fb -1 Insert is from CMS Physics TDR: 1 fb -1
73 73 Not All W’s Are Created Equal The reason the inclusive W and Z cross-sections are 10x higher at the LHC is that the corresponding partonic luminosities are 10x higher –No surprise there Where you want sensitivity to anomalous couplings, the partonic luminosities can be hundreds of times larger. The strength of the LHC is not just that it makes millions of W’s. It’s that it makes them in the right kinematic region to explore the boson sector couplings. From Claudio Campagnari/CMS
74 74 TGC’s – the bottom line Not surprisingly, the LHC does best with the Dimension-6 parameters Sensitivities are ranges of predictions given for either experiment CouplingPresent ValueLHC Sensitivity (95% CL, 30 fb-1 one experiment) g1Zg1Z 0.005-0.011 0.03-0.076 Z 0.06-0.12 0.0023-0.0035 Z 0.0055-0.0073
75 75 Early Running Reconstructing W’s and Z’s quickly will not be hard Reconstructing photons is harder –Convincing you and each other that we understand the efficiencies and jet fake rates is probably the toughest part of this We have a built in check in the events we are interested in –The Tevatron tells us what is happening over here. –We need to measure out here. At high E T, the problem of jets faking photons goes down. –Not because the fake rate is necessarily going down – because the number of jets is going down.
76 76 Precision EWK:The W Mass I am not going to try and sell you on the idea that the LHC will reach a precision of [fill in your favorite number here]. Instead, I want to outline some of the issues involved.
77 77 CDF Results: The State of the Art These systematics are statistically limited. These systematics are not.
78 78 One Way Of Thinking About It 5 MeV 15 MeV 25 MeV If we shoot for 5 MeV, how close might we come? What needs to happen to get down to 5 (or 15, or 25) MeV? (If you shoot for 5, you might hit 10. If you shoot for 10, you probably won’t hit 5) 8 MeV is 100 parts per million. See Besson et al. arXiv:0805.2093v1 [hep-ex] arXiv:0805.2093v1
79 79 Difficulty 1: The LHC Detectors are Thicker Detector material interferes with the measurement. –You want to know the kinematics of the W decay products at the decay point, not meters later –Material modeling is tested/tuned based on electron E/p Thicker detector = larger correction = better relative knowledge of correction needed CMS material budget ATLAS material budget X~16.5%X 0 (red line on lower plots)
80 80 Difficulty 2 – QCD corrections are more important No valence antiquarks at the LHC –Need sea antiquarks and/or higher order processes NLO contributions are larger at the LHC More energy is available for additional jet radiation At the Tevatron, QCD effects are already ¼ of the systematic uncertainty –Reminder: statistical and systematic uncertainties are comparable. To get to where the LHC wants to be on total m(W) uncertainty is going to require continuous effort on this front. q q W q g q W
81 81 Major Advantage – the W & Z Rates are Enormous The W/Z cross-sections at the LHC are an order of magnitude greater than the at the Tevatron The design luminosity of the LHC is ~an order of magnitude greater than at the Tevatron –I don’t want to quibble now about the exact numbers and turn-on profile for the machine, nor things like experimental up/live time Implications: –The W-to-final-plot rate at ATLAS and CMS will be ~½ Hz Millions of W’s will be available for study – statistical uncertainties will be negligible Allows for a new way of understanding systematics – dividing the W sample into N bins (see next slide) –The Z cross-section at the LHC is ~ the W cross-section at the Tevatron Allows one to test understanding of systematics by measuring m(Z) in the same manner as m(W) The Tevatron will be in the same situation with their femtobarn measurements: we can see if this can be made to work or not –One can consider “cherry picking” events – is there a subsample of W’s where the systematics are better?
82 82 Systematics – The Good, The Bad, and the Ugly Masses divided into several bins in some variable Masses are consistent within statistical uncertainties. Clearly there is a systematic dependence on this variable Provides a guide as to what needs to be checked. Point to point the results are inconsistent There is no evidence of a trend Something is wrong – but what? Good Bad Ugly
83 83 So, When Is This Going To Happen? The latest schedule shows the LHC ready for beam in about a month. Beam will be injected into sectors as soon as they are cold. The plan is to have collisions at 10 TeV for 2-3 months in 2008, train the magnets during the winter shutdown, and go to 14 TeV in 2009.
84 84 LHC Beam Stored Energy in Perspective Luminosity goes as the square of the stored energy. LHC stored energy at design ~700 MJ –Power if that energy is deposited in a single orbit: ~10 TW (world energy production is ~13 TW) –Battleship gun kinetic energy ~300 MJ It’s best to increase the luminosity with care USS New Jersey (BB-62) 16”/50 guns firing Luminosity Equation:
85 85 My Take on The Schedule If we only have the same old problems (i.e. no new ones) there will beam in fall. –Full energy will be in early 2009. We will turn on with very low luminosity and this will grow slowly as we learn to handle the stored energy –Luminosity grows as the square of stored energy After maybe a year, the luminosity will shoot up like a rocket –Luminosity grows as the square of stored energy
86 86 Apologies I didn’t cover even a tenth of the ATLAS physics program –Precision measurements –Top Quark Physics Orders of magnitude more events than at the Tevatron –Search for new particles Can we produce the particles that make up the dark matter in the universe? –Search for extra dimensions Why is gravity so much weaker than other forces? Are there mini-Black Holes? –B Physics and the matter-antimatter asymmetry Why is the universe made out of matter? –Heavy Ions What exactly has RHIC produced?
87 87 Summary Electroweak Symmetry Breaking is puzzling –Why is the W so heavy? Why is the weak force so weak? The Large Hadron Collider is in a very good position to shed light on this –The “no lose theorem” means something has to happen. Maybe it’s a Higgs, maybe it’s not. –Finding the Higgs is not enough. Precision electroweak measurements are needed to understand what’s going on. Any experiment that can do this can also answer a number of other questions –For example, addressing the structure of the proton –And the dozens I didn’t cover Thanks for inviting me!
88 The LHC: Ready or Not, Here It Comes
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f8831a65062617ca | The Time-Energy Uncertainty Relation
John Baez
April 10, 2010
In quantum mechanics we have an uncertainty relation between position and momentum:
(Δq) (Δp) ≥ /2
Now, as you probably know, time is to energy as position is to momentum, so it's natural to hope for a similar uncertainty relation between time and energy. Something like this:
(ΔT) (ΔE) ≥ /2
There's an energy operator in quantum mechanics, usually called the Hamiltonian and written H. But the problem is, there's no "time operator" in quantum mechanics! This makes people argue a lot about the time-energy uncertainty relation - whether it exists, what it would mean if it did exist, and so on.
A while back on sci.physics.research, Matthew Donald wrote something interesting about this subject. I'm editing it a little bit here:
Most treatments of the time-energy uncertainty principle point out that you do have to be careful to consider the meaning of t. t isn't an operator in quantum mechanics.
Uncertainty relations are mathematical theorems as well as physical statements so if we begin with a proof we should end up with an exact definition of what we are trying to understand.
There are probably several forms in which the time-energy uncertainty relation can be proved. Here's one (for the full details, see Messiah's Quantum Mechanics Section VIII.13).
Let H be the (time-independent) Hamiltonian of some non-relativistic system. Let ψ be a wavefunction and let A be some other observable. Write
<A> = <ψ, A ψ>
for the expectation value of A in the state ψ, write sqrt for square root, and define
ΔA = sqrt(<ψ, (A - <A>)2 ψ>).
ΔA is the standard deviation of the observable A in the state ψ.
Then, for all real numbers r, <ψ, (r (A - <A>) + i (H - <H>))(r (A - <A>) - i (H - <H>)) ψ> is non-negative.
So this quadratic (in r) cannot have two different real roots, and so, (cutting a long but standard story short)
2 (ΔA) (ΔH) ≥ |<[H,A]>| .
ΔH is the standard deviation of the energy E.
< [H,A] > = <ψ, [H,A] ψ>
is i times the time derivative at t = 0 of <ψ, A ψ>, as you can see if you note that the solution to the Schrödinger equation can be written in the form
U(t) ψ = exp(-itH/) ψ
<[H, A]> = i d<A>/dt
Putting everything together, we have the time-energy uncertainty relation in the form
(ΔA / (|d <A>/dt)|) (ΔH) ≥ /2.
Here the "uncertainty" in time is expressed as the average time taken, starting in state ψ, for the expectation of some arbitrary operator A to change by its standard deviation.
This is reasonable as a definition for time uncertainty, because it gives the shortest time scale on which we will be able to notice changes by using A in state ψ.
Hey, that's way cool! For some reason I'd never thought of it that way. But here's something related, which is well-known:
Suppose you could find an observable T which is canonically conjugate to the Hamiltonian H:
[H,T] = i
Then by one of the formulas you wrote, we'd have
d<T>/dt = 1
so the observable T would function as a "clock" - it would increase at the rate of one second per second. In other words, we could use it as a "time" observable... which is why I called it T.
From your uncertainty relation we then have
(ΔT) (ΔH) ≥ /2
the famous time-energy uncertainty relation that everyone keeps yearning for!
The problem is, for physically realistic Hamiltonians H one can prove there is no operator T with
[H,T] = i
In other words, there is no time observable!
The reason is this: by the Stone-von Neumann uniqueness theorem, any pair of operators satisfying the canonical commutation relations [H,T] = i can only be a slightly disguised version of the familiar operators p and q. These operators p and q are unbounded below - i.e., their spectra extend all the way down to negative infinity. But a physically realistic Hamiltonian must be bounded below!
(Here I am glossing over some mathematical nuances: if you read the precise statement of the Stone-von Neumann theorem, you'll see how to fill in these details.)
Crudely speaking, this theorem says that it's impossible to construct a clock that works perfectly no matter what its state is. That's not surprising - but it's sort of surprising that you can prove it, and it's sort of interesting to see what assumptions you need to prove it.
But what you're saying is: "So what? Let's use any operator A as a clock - we can't make d/dt = 1 in all states, but we can make it close to 1, or even equal to 1, in the state we're interested in! Then we can state the energy-time uncertainty relation even without having a time observable - we just say
Thanks - you taught me something cool about time, which is one of my favorite subjects, right up there with space.
Much later, Dmitry A. Arbatsky wrote:
You should mention the paper where the mathematically rigorous formulation of the time-energy uncertainty relation was first given. (It was given there even in nice finite form, not only infinitesimal. In 2005 it was generalized. It turned out that relations for energy and time in Mandelshtam-Tamm formulation, on the one hand, and for coordinate and momentum, on the other hand, are particular consequences of a more general approach.)
With best wishes,
Dmitry A. Arbatsky
Not till we are lost ... do we begin to find ourselves and realize where we are and the infinite extent of our relations. - Thoreau
© 2010 John Baez |
3ae35fe51f9acbe6 | Learning About Atoms
The Science PLC at our school is considering what students should know about atoms in 8th and 9th grade science classes (including Physics First). Just recently, Amber (Strunk) Henry posted on Twitter:
This is my attempt to arrange the ideas.
Map of the Territory of Things to Know
Next Generation Science Standards (NGSS)
Here are the progressions found in Appendix E of the standards. I do digress into talk of matter and substance when it supports later understanding of atoms. I’ve expanded these to list ideas explicitly and separately.
ESS1.A The universe and its stars
• (Grades 9-12) Light spectra from stars are used to determine their characteristics, processes, and lifecycles. Solar activity creates the elements through nuclear fusion. The development of technologies has provided the astronomical data that provide the empirical evidence for the Big Bang theory.
• Excited atoms/molecules emit light of particular frequencies and wavelengths (collectively called the emission spectrum of an atom/molecule).
• The frequencies and wavelengths of light emitted by atoms/molecules depend on the structure of the atom/molecules.
• Atoms/molecules absorb light at particular frequencies and wavelengths (collectively called the absorption spectrum of an atom/molecule).
PS1.A Structure of matter (includes PS1.C Nuclear processes)
• (Grades K-2) Matter exists as different substances that have observable different properties. Different properties are suited to different purposes. Objects can be built up from smaller parts.
• Matter can be made of different substances.
• Substances have many properties, each with their own uses.
• Objects are made of smaller parts.
• (Grades 3-5) Because matter exists as particles that are too small to see, matter is always conserved even if it seems to disappear. Measurements of a variety of observable properties can be used to identify particular materials.
• Indivisible particles of matter are too small to see.
• Measurements of properties characterize substances.
• (Grades 6-8) The fact that matter is composed of atoms and molecules can be used to explain the properties of substances, diversity of materials, states of matter, phase changes, and conservation of matter.
• Molecules are made of atoms.
• Matter is made of atoms and molecules.
• Different atoms and molecules explain different substances.
• Atoms and molecules behave differently in different states of matter.
• Atoms and molecules change their qualitative behavior at phase transitions.
• Matter is conserved because atoms are not destroyed in physical and chemical processes.
• (Grades 9-12) The sub-atomic structural model and interactions between electric charges at the atomic scale can be used to explain the structure and interactions of matter, including chemical reactions and nuclear processes. Repeating patterns of the periodic table reflect patterns of outer electrons. A stable molecule has less energy than the same set of atoms separated; one must provide at least this energy to take the molecule apart.
• An individual atom has structure explained by electromagnetic and nuclear interactions.
• The structure of the atom explains:
• arrangement of atoms into molecules
• chemical reactions
• nuclear processes
• trends in periodic table
• Energy is required to remove electrons from an atom.
• Energy is required to break molecular bonds.
PS1.B Chemical reactions
• (Grades K-2) Heating and cooling substances cause changes that are sometimes reversible and sometimes not.
• (Grades 3-5) Chemical reactions that occur when substances are mixed can be identified by the emergence of substances with different properties; the total mass remains the same.
• Mass is conserved in chemical reactions.
• Measurement of properties of substances identifies when chemical reactions have taken place.
• (Grades 6-8) Reacting substances rearrange to form different molecules, but the number of atoms is conserved. Some reactions release energy and others absorb energy.
• Chemical reactions result in different molecular arrangements of atoms.
• (Grades 9-12) Chemical processes are understood in terms of collisions of molecules, rearrangement of atoms, and changes in energy as determined by properties of elements involved.
• Chemical reactions occur when molecules collide and atoms rearrange.
• Changes in energy during a chemical reaction depend on properties of the atoms involved.
Let me know if you think I’ve forgotten anything here!
AAAS Science Assessment
The AAAS has a great website under the auspices of Project 2061 that lists ideas and misconceptions related to Atoms, Molecules, and States of Matter.
Arnold B. Arons. Teaching Introductory Physics
Arons identifies four lines of evidence necessary to build an early quantum model of the atom:
1. Bright line spectra of gases. This requires understanding of how accelerated charged particles can emit light, how charged particles can absorb light. It should include the Balmer-Rydberg formulae for hydrogen.
2. Radioactivity
3. Size of atoms (electron cloud and nuclear). Evidence from multiple sources.
4. Photoelectric effect and photon concept
How should this knowledge be arranged?
TODO: I’d like to work on a Learning Landscape, Knowledge Packet, or Learning Progression synthesizing these sources, but that will have to be added later.
Models of Atoms
1. BB Model of Atoms and Molecules (hard, indivisible balls)
• Needed to explain phases of matter.
2. Dalton Model of Atoms (hard, indivisible balls that can combine)
• Needed to explain chemical reactions in integer ratios.
3. Plum Pudding Model / Thompson Model (negatively charged electrons embedded in a positively charged medium)
• Needed to explain static electricity.
4. Planetary Model / Rutherford Model
• Needed to explain the Geiger-Marsden gold foil experiments.
5. Bohr Model / Rutherford-Bohr Model
• Needed to explain why electrons don’t fall into the nucleus after radiating EM waves.
• Needed to explain the Rydberg formula.
6. Bohr-Summerfeld Model
• Needed to allow elliptical orbits
7. Schrödinger Model / Electron Cloud Model
• Needed to explain more satisfactorily why electrons don’t fall into the nucleus after radiating EM waves
• Needed to explain atoms with more than one electron
• Needed to explain periodic table trends
• Needed to explain spectra of large Z atoms
• Needed to explain intensities of spectral lines
• Needed to explain Zeeman effect from magnetic fields
• Needed to explain spectral splittings (fine, although this could be done with Klein-Gordan equation and is really a hack onto the non-relativisticSchrödinger equation, and hyperfine) Note: I need to go back to my QM books on this one.
8. Swirles/Dirac Model
• Needed to explain spectra of large Z atoms better
• Needed to explain the color of gold and cesium
• Needed to explain chemical and physical property differences between the 5th and 6th periods
9. Quantum Field Theory Model
• Needed for ???
10. Nuclear Shell Model / Goeppert-Mayer et al. Model
• Needed to explain radioactivity
About the biggest controversy is disagreement over the need to teach pseudo-historically these models. This is leaving out all the really bad ones. However, the terrible picture that society has adopted as the meme for atom (see below) affects student perceptions of the atom.
Stylised Lithium Atom by Indolences, Rainer Klute on Wikimedia Commons. Note that this is only a model, based loosely on the Bohr model. Also, these 3 electrons couldn’t all occupy the same circular orbit.
It would be nicer if students came into classrooms with the following conception of an atom.
Helium Atom QM by Yzmo from Wikimedia Commons. This is a much better rendition of the electron cloud but might be as bad for the nucleus. However, it is nice that it shows scale.
Physics teachers tend to like the Bohr model in that it can quickly (although magically) explain the Rydberg formula. However, there are many reasons to dislike the Bohr model.
Classroom Experiments
TODO: What classroom experiments or simulations could help students to progress in their knowledge of atoms? |
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3. List of Calculation Modules
List of Calculation Modules
Molecular Mechanics Method
SCIGRESS Mechanics is a module for calculating the optimal structure and other properties of molecules by using molecular mechanics methods. Because molecular mechanics methods determine the pair potentials between the atomic nuclei, which are the particles in these methods, stable conformers of molecules can generally be found with quite good precisions by fast calculation in most cases.
The equations of molecular mechanics use the spring potential energy as represented by a pseudo-elastic force as the bonding force of the electrons. The force field energy of the molecule is represented by a theoretical energy called the steric energy that is found from the energy of the extension, compression, angular deviation, and rotation of bonds.
The potentials that are specific to molecular mechanics are determined by a force field. The bond length, bond angle, dihedral, strain, electrostatic force, van der Waals force, hydrogen bonding, and other properties form parameters of the force field. SCIGRESS uses extensions of the MM2 force field and MM3 force field of Professor Allinger.
The following extensions have been made in SCIGRESS.
1. Additional atomic types
• trigonal dipyramids
• square pgramid
• octahedron
• tetrahedron
2. Additional bonds
• weak bond
• ionic bond
• hydrogen bond
• coordination bond
3. The elements that can be used have been expanded to all of the elements on the periodic table by systematically applying empirical rules.
4. Whether or not the parameters that correspond to a π-electron system are applied is automatically determined by searching the structures.
Applications of molecular force field methods
• When optimizing the structure of a stable conformer of the ground state of a regular organic molecule
• When searching for a series of stable conformers
• When searching for a path from a given conformer to another conformer
• When estimating the steric interactions between molecules
• When investigating the structures of new substances and organometallic molecules
• When determining the initial structure as a precursor to quantum chemistry calculations
Dynamics (Molecular Dynamics Method)
SCIGRESS Dynamics is molecular dynamics module that is able to simulate the behavior of molecular models. SCIGRESS Dynamics calculates the potential energy using the same force field as SCIGRESS Mechanics. The kinetic energy is calculated from the atomic velocities in the molecular system that reflects the temperature being simulated.
Applications of SCIGRESS Dynamics
SCIGRESS Dynamics creates a trajectory according to the settings.
A trajectory is a collection of structures arranged sequentially in time. Each structure has a potential and a kinetic energy that are calculated according to the temperature.
The results of calculations can be displayed by linking the energy and structure in two windows shown side by side.
Furthermore, the following information can be obtained from this trajectory:
• The various conformations arising from the motion of the molecular model
• The relationship between the structure and energy of the molecular model
Extended Huckel (Molecular Orbital Method)
SCIGRESS Extended Huckel is an empirical molecular orbital calculation method that solves the Schrodinger equation. All of the elements in the periodic table are supported as calculation targets.
The following items can be calculated using the extended Huckel method.
• Bond order
• Partial charge
• Molecular orbitals
• Orbital energies
• Dipole moments
Extended Huckel
The extended Huckel parameters and the parameters collected by S. Alvarez are provided as the calculation parameters. The user is also able to create new parameter sets.
ZINDO (Molecular Orbital Method)
SCIGRESS ZINDO is a semiempirical molecular orbital method program. In order to solve the Schrodinger equation, CNDO/INDO can be used.
The following properties can be calculated using ZINDO.
• Partial charge
• Bond order
• Dipole moments
• Molecular orbital energies
• Ionization potentials
• Optimized structures
• Electron spectra (ultraviolet and visible absorption spectra)
• The molecular absorption spectra in the ultraviolet and visible regions can be calculated and visualized by performing C.I. calculations.
ZINDO incorporates a method for modeling polar solvent effects. This is called the SCRF (Self Consistent Reaction Field) method.
Limitations of ZINDO
• Because ZINDO only treats valence electrons, it cannot calculate properties that depend on changes in inner shell electrons.
• Note: In general, the molecules have large strains.
However, ZINDO treats small ring-shaped molecules as stable molecules.
• Number of atoms that can be calculated using ZINDO: 200 atoms Basis functions: 700
MO-G (Molecular Orbital Method)
MO-G is a semiempirical molecular orbital program. MINDO/3, MNDO, MNDO-d, AM1, PM3, and PM5 are included as the Hamiltonians for solving the Schrödinger equation.
The following properties can be calculated using MO-G.
• Partial charge
• Bond order
• Dipole moments
• Molecular orbital energies
• Ionization potentials
• Optimal structures
• Potential energy maps
• Structure of transition states
• Reaction coordinates
• Vibration spectra (IR)
MO-G main functions
• Linear Scaling SCF calculation using the MOZYME method
• Utility functions for inputting and outputting protein structures
• Structure optimization (EF, BFGS, NLLSQ, and SIGMA methods)
• Transition state calculation
• Energy decomposition
• Solvent effect calculation (COSMO method, TOMASI model)
• Internal reaction coordinate calculation (IRC)
• Dynamic reaction coordinate calculation (DRC)
• Analysis of intersystem crossing structure
• Hyperpolarizability calculation
• Automatic identification of symmetries (up to 8th-order point group representations)
• Infrared spectra calculation
• Ultraviolet/visible spectra calculation
• Normal vibration analysis
• Excited state calculation
• Open shell and radical calculation
• Calculation using periodic boundary conditions
• Parametric Molecular Electrostatic Potential
• Atomic charges using ESP calculation
Limitations on MO-G
• Because MO-G only treats valence electrons, it cannot calculate properties that depend on changes in inner shell electrons.
Note: MO-G was developed and productized by Fujitsu based on MOPAC2002
MO-S (Excited State Calculation)
MO-S is able to determine with high precision the ultraviolet and visible spectra of organic molecules. MO-S is also able to calculate the ultraviolet and visible absorption spectra of the ligand molecules in proteins by using QM/MM methods.
Example of calculating absorption spectra by using QM/MM methodsExample of calculating absorption spectra by using QM/MM methods
Elements that can be calculated by MO-S
• AM1, PM3, PM5: H, Li, Be, B, C, N, O, F, Na, Mg, Al, Si, P, S, Cl, K, Ca, Zn, Ga, Ge, As, Se, Br, Rb,Sr, Cd, In, Sn, Sb, Te, I, Cs, Ba, Hg, Tl, Pb, Bi
• INDO/S: H, Li, C, N, O, F, Mg, Si, P, S, Cl, Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn †
• CNDO/2: H, Li, Be, B, C, N, O, F, Na, Mg, Al, Si, P, S, Cl, Ge, As, Se, Br
• CNDO/S2: H, C, N, O, F, S, Cl, Br
• CNDO/S3: H, C, N, O, F, S, Cl, Br
CONFLEX3 (Search Stable Conformations Using Molecular Mechanics Methods)
SCIGRESS CONFLEX3 searches the conformation space of flexible molecules and is able to determine the optimal structure of the chemically important comformational isomers without missing any. In previous structural optimization programs (Mechanics, MO-G, etc.), only the locally optimal structure depending on the initial structure input by the user could be found, and it was difficult to identify the most stable structure. Because of this, only a limited amount of information could be obtained about the properties and behavior of flexible molecules. CONFLEX3 is a conformation search program that resolves these problems.
Features of CONFLEX3
1.Conformation search algorithm
The procedure for searching conformations is as follows:
(1) Select an initial structure from among the previously saved structures and create the corresponding structure
(2) Optimize the structure
(3) Compare to the previously obtained structures and save if it is a new conformation
This procedure is repeated until a termination condition is satisfied.
The biggest feature of CONFLEX3 is the creation of the structure in (1). Superior performance is obtained in particular by using Corner Flap and Edge Flip, which are applicable to ring-shaped sections.
2.Structure creation
Corner Flap
Creates a new starting structure by selecting a single constituent atom from a ring in the initial structure and moving it to the opposite side of the mean plane of the ring.
Corner Flap
Edge Flip
Performs a twisting operation by selecting two neighboring atoms from the constituent atoms of a ring in the initial structure and moving these to opposite sides of the mean plane of the ring.
In addition, an indenting operation is performed by moving the two atoms towards the inside of the ring.
Edge Flip
Stepwise Rotation
A new starting structure is created by performing a simple rotation operation on a side chain.
3.Reservoir filling algorithm
The above three operations are local searches based on the initial structure. However, CONFLEX3 also selects initial structures from among the found conformational isomers in order from the lowest energy structures. If a conformational isomer with a lower energy is found during this process, that conformation becomes the next initial structure. This makes the search region rapidly move towards lower energies. Next, once the most stable conformation has been found, the search region gradually expands to encompass higher energies. Because this resembles the way in which the surroundings become filled with water as water flows into a reservoir, it is called the “reservoir-filling algorithm”.
4.Structure optimization calculation
The structure optimization in the conformation search calculation is performed by using SCIGRESS Mechanics. This makes it possible to perform calculations that support all elements. Furthermore, structures that have already been found, enantiomers, geometric isomers, saddle-point structures, extremely unstable structures, etc. which frequently occur in the calculations are automatically deleted.
5.Simple input settings
Rings, asymmetric carbon atom configurations (R/S), and double-bond geometries (E/Z) within the target molecule are automatically identified. This allows the user to complete the preparations simply by automatically setting the “search labels” by creating molecules and using the “Generate Conformations” command.
Examples of CONFLEX3 calculations
Step 1: Draw using Workspace
The input files to CONFLEX3 are created using Workspace.
Molecule input screenFigure 1. Molecule input screen
Step 2: Set the search labels (optional)
Set the search labels if there are bonds that you want to rotate within the target molecule. When the “Geometry Label Wizard” command, which is one of the Workspace functions is selected, the dialog box shown in Figure 2 is displayed allowing you to easily set the labels. In Figure 2, the dihedral angles are set to -179, 61, and 120 degrees. In CONFLEX3 it is possible to search for rotamers that satisfy rotations in steps of 120 degrees. Furthermore, search labels can be displayed on the molecules after they are set as shown in Figure 3.
Generate Conformation dialog boxFigure 2. Generate Conformation dialog box
Displaying search labelsFigure 3. Displaying search labels
Step 3: Set CONFLEX3 parameters and execute
The Procedure Browser has an edit function that makes it easy to configure search conditions by accessing the following dialog box.
CONFLEX3 parameter settings dialog boxFigure 4. CONFLEX3 parameter settings dialog box
Step 4: Display the calculation results
The energy and structure of each conformer in the calculation results can be viewed simultaneously.
Calculation resultsFigure 5. Calculation results
CONFLEX3 is a program developed by Prof. Gotoh of the Toyohashi University of Technology.
Reference documents:
•J. Am. Chem. Soc., 1989, 111, 8950-8951;
•J. Chem. Soc., Perkin Trans. 2, 1993, 187-198;
(Note 1) CONFLEX is a registered trademark of the Conflex Corporation.
MD-ME (Molecular Dynamics Method)
This allows a variety of phenomena to be simulated from bulk effects to surfaces and boundaries by constructing crystal structures and aggregations of atoms and molecules using simple operations. Furthermore, graphs of various physical quantities and animations of atomic placement can be displayed. This is also able to handle large-scale high-speed calculations. This provides a wide variety of uses including research and development of various materials and education.
Polymer Modeling Function
This is a function for joining monomers to create polymers and dendrimers
Example of the polymers that can be createdExample of the polymers that can be created: a chain-shaped polymer (left) and a dendrimer (right)
MD Cell Modeling Function
This is a function for creating polymer aggregates (amorphous and infinite chain), liquid crystal structures, crystal structures (using templates), and randomly placed MD cells
Function for creating MD cells using crystal structure templatesFunction for creating MD cells using crystal structure templates
MD-ME features
Easy crystal model construction function
Equipped with a rich potential parameter library and functions. Suitable for a wide variety of materials research and simulations
MD-ME modeling functions
Crystal structure modeler
Function for constructing various crystal structures by expanding the space group
Cutting planes
Function for cutting planes by specifying the Miller indices
Dynamics models
Able to use potential models, constrained models, and rigid body models
Cutting plane function: Cut along the (111) plane of a sodium chloride crystalFigure 1:Cutting plane function: Cut along the (111) plane of a sodium chloride crystal
MD-ME interaction settings and potentials
MD-ME is equipped with the major potential functions used in molecular dynamics. In addition, it includes a library of 85 different potential parameters.
The dissimilar material interfaces (such as Si/SiO2) can be calculated by applying variable charge potentials where the charge varies depending on the environment. (Can only be used with Si)
[Potential functions]
Interatomic interactions (two-body: 16, three-body: 8, many-body: 14)
Intramolecular interactions (bonds: 4, angular: 7, dihedral: 4, out-of-plane: 5)
MD-ME calculation parameter settings and molecular dynamics calculations
Simulations can be performed by specifying experimental parameters such as temperature, pressure, heating or cooling, increasing or decreasing pressure, etc.
Simulations can also be executed by applying external fields such as electrostatic and electromagnetic fields.
A function that allows the temperature and pressure to be set in multiple stages, and a function that allows the cell lengths to vary during calculation (used for calculating elastic constants) are also provided.
A variety of systems can be freely simulated under the flexible calculation parameter settings like those above.
NEV, NTV, NPH, and NTP ensembles are available
[Stress calculation using variable length MD cells]
This function varies the cell side length during calculation
The stress can be obtained by applying a tensile or compressive strain to the cell, and the elastic constant of the material can then be obtained from the relationship between the stress and strain.
[External field application function]
This function applies a stress, electrostatic field, magnetostatic field, or gravitational field of the designated magnitude and direction
Particles are constrained within a sphere of the designated radius
[Control algorithms]
Temperature: Scaling method, Nose method
Pressure: Parrinello-Rahman method. Increased speed: Link-cell method, bookkeeping method, two-body force tables
[Time integration methods]
Gear method, Hernandez method
[Creating atoms and molecules]
This is a function for creating atoms and molecules while a molecular dynamics calculation is executing
This allows simulation of crystal growth and surface adsorption
(Patented Japanese patent number 3648033)
Calculation results of atomic and molecular creation calculationCalculation results of atomic and molecular creation calculation
MD-ME results display function
This function displays graphs of the time variations in physical quantities such as temperature, pressure, and internal energy, and displays animations of atomic arrangement.
MD-ME results display functionMD-ME results display function
[Calculation results]
Graph of internal energy, volume, pressure, and temperature (left of screen)
Variations in the various physical quantities obtained by the calculations can be displayed in a graph.
[Animation display (right of screen)]
Calculation results can be displayed as an animation.
[Secondary analysis functions]
These are functions for analyzing the calculation results of the molecular dynamics calculations
• Mean-square displacement
• Two-body correlation function
• Voronoi polyhedra
• Molecular internal coordinates
• Interference function
• Coefficient of viscosity
• Elastic constant
• Velocity auto-correlation function
• Rotation correlation function
Display of analysis results (two-body correlation function)Display of analysis results (two-body correlation function)
MD-ME high-speed parallel calculation
Molecular dynamics calculations are also able to support high-speed large-scale calculations by using the SCIGRESS MD (scale-up option) network-linked molecular dynamics calculation engine.
Creating the initial data, performing various analyses, and displaying the calculation and analysis results can be performed on the SCIGRESS for Materials (required) side and used without having to know about the remote environment. |
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108 Cards in this Set
• Front
• Back
Atoms and molecules are governed by same or different laws?
Atoms and molecules are not governed by the same physical laws as larger objects
Max Planck discovered?
Energy is continous, and Atoms and molecules emit energy only in certain discrete quantities or quanta.
What is a wave?
vibrating disturbance by which energy is transmitted
How do water waves travel?
Wave repeats itself at regular intervals.
Waves can be characterized by?
length, height, and number of waves per second
What is wavelength?
distance between identical points on successive waves
What is frequency?
number of waves that pass through a particular point in 1 second
What is amplitude?
What is speed?
Depends on the type of wave and nature of the medium. Speed is a product of wavelength and frequency
u = yV
What is wavelenght and frequency measured in?
m, cm, nm.
1 Hz = 1 cycle/second
Visbile light is made up of?
Electromagnetic waves
What are Electromagnetic waves?
Waves with an electrical field component and a magnetic field component
The two components of electromagnetic waves differ in?
Same speed, but travel in perpendicular planes
What is electromagnetic radiation?
Emission and transmission of energy in the form of electromagnetic waves
What speed do electromagnetic waves travel at?
3.00 * 10^8 m/s
Long waves
Are emitted from large antennas (radio, cell phones). Radio waves lowest frequency
Short waves
Have higher energy radiation. Gamma rays have the shortest wavelenght and highest frequency
What is quantum?
What is the wavelength (in m) of an electromagnetic wave whose frequency is 3.64 * 10^7 Hz?
What is the frequency (in Hz) of a wave whose speed is 713 m/s and wavelength is 1.14 m?
What is Planck's constant?
H= 6.63 * 10^-34
What is the photoelectric effect?
Electrons are ejected from the surface of certain metals exposed to light of at least a certain minimum frequency called the
Threshold frequency
Number of electrons depends on?
intensity of light but energy of electrons does not
Einstein theorized light is made up of?
A stream of particles called photons.
What do you need to break electrons free from a metal?
Requires light of sufficiently high frequency
What is KE and BE?
E = hn = KE + BE
KE = kinetic energy of electron
BE = binding energy of the electron in the metal
The higher the frequency the greater the KE
KE is dependent on?
Frequency of the light. Ejected
Light behaves as?
Both as a particle and wave depending on the property being measured. All matter actually exhibits this dual nature.
The energy of a photon is 5.87 * 10-20 J. What is the wavelength in nm? h = 6.63 * 10-34 J*s
A photon has a wavelength of 624 nm. Calculate the energy of the photon in J.
What is an emission spectra?
Either continous or line spectra of radiation emitted by substances
What is a line spectra?
The light emmission only at specific wavelenghts. Produced by atoms
Emission spectra of the sun or heated solids
Why is Bohr's model no accurate?
Because it does not explain the spectral lines
What is Rydberg's constant?
Rydberg constant (RH) = 2.18 * 10^-18 J
Energy of electron
Where are free electrons?
infinitely far from the nucleus
Why is there a negative sign in equation of Rydberg's constant?
Negative sign – is to assign a lower energy of electron in an atom than the energy of the free electron (arbitrarily assigned a value of zero)
What happens as the electron gets closer to the nucleus?
Becomes more stable and E becomes more negative, which corresponds to the most stable state.
What is the ground state?
The lowest energy state of a system or most stable
What is the excited state?
Higher energy than the ground state
When is radiant energy emitted?
When electrons drop from a higher energy orbital to a lower energy orbital
The quantity of energy produced is dependent only on what?
Initial and final states
If an electron starts at ni and drops to a lower energy state of nf the change in energy is given by Equation=
What happen in the states when energy is given off.
Ni>Nf change in energy is negative
Each line on the emission spectrum corresponds to?
To a transition in the H atom
When a large number of H atoms are examined all the lines of the spectrum are visible
Electrons bound to a nucleus behave like?
Electrons bound to a nucleus behave like a standing wave
Waves that can be generated by plucking a string
Some points on a standing wave are
Do not move at all
The amplitude at this point is zero
Nodes are located at the end of the string and maybe in the middle
De Broglie says that if an electron does not behave like a wave then?
De Broglie said that if an electron does behave like as wave then the wave must perfectly fit the circumference of the orbit
The circumference of the orbit is related to the wavelength by the equation
When can a particle be a wave and a wave be a particle?
A particle in motion can be treated as a wave
A wave can also exhibit properties of a particle
Protons can be accelerated to speeds near the speed of light in particle accelerators. Estimate the wavelength (in nm) of such a proton moving at 2.90 * 108 m/s. (mp = 1.673 * 10-27 kg)
A baseball has a mass of about 255 g. Calculate the wavelength of the baseball if it is thrown 100. mph.
What is Heisenberg uncertainity principle?
It is impossible to know simultaneously both the momentum p and the position of a particle
Applying the Heisenberg uncertainty principle to the H atom?
We see that the electron can not orbit the nucleus in a circular orbital.
If this were the case then we could know both the position and momentum of the electron at the same time.
Erwin schrodinger formulated what?
Formulated an equation to describe the behavior and energies of submicroscopic objects
This equation is very complicated and requires Calculus to solve
The equation incorporated?
The Schrödinger equation incorporates particle behavior of electrons in the form of mass and wave behavior in the form of wave function (y)
Why is the wave funciton significant?
Is that the square of the wave function (y2) is proportional to the probability of where the electron is located
Where is an electron most likely to be?
The most likely place an electron will be is where y2 is greatest
What tells Schrodinger equation tell us?
The Schrödinger equation gives possible energy states and identifies the wave function of the electrons.
These are characterized by quantum numbers. Quantum mechanics gives probability of an electron in a particular region of the atom (electron density)
In quantum mechanics the orbits are called?
Atomic orbitals
What is an atomic orbital?
The wave function of an electron in an atom.
This is to differentiate from the orbits in Bohr’s model.
Each atomic orbital has a characteristic energy and therefore a characteristic distribution of electron density.
An assumptions must be made
The difference between hydrogen and atoms with more than one electron is not that large.
What are quantum numbers?
Describe the distribution of electrons in hydrogen and other atoms.
What are three quantum numbers that describe the distributioni of electrons?
Principal quantum number
Angular momentum quantum number
Magnetic quantum number
What is the fourth quantum number that describes the behavior of a specific electron?
Spin quantum number
Principal quantum numbers
Represented by n
Can have values of integers
Relates the average distance from the electron to the nucleus in a particular orbital.
Larger the n what happens?
The greater distance of an electron in the orbital from the nucleus and therefor the larger the orbital.
Angular Momentum Quantum Number
Represent by l
Tells the shape of the orbital
Dependent on n
For any n, l = any integer from 0 to (n-1)
For n = 1, l = 0
For n = 3, l = 0, 1, or 2
l is normally designated by the letter that symbolizes the different atomic orbitals
Magnetic quantum number
Represented by ml
Depends on l
For any value l, there are (2l +1) values for ml
If l = o then ml = 0
If l = 1 then there are three possible values of ml, (-1, 0, 1)
The value of ml indicates the number of orbitals in the subshell with value l
Spin quantum numbers
Represented by ms
It was noticed that the application of a magnetic field could split the lines in an emission spectra
The only way this could be explain is if electron behave as tiny magnets
If the electrons are thought of as spinning on their own then the magnetic field can be explained
The spinning charge generates a magnetic field
Spin quantum numbers are always?
–½ or +½
Atomic orbital related to angular momentum
What is a shell?
A collection of orbitals with the same value of n
What are subshells?
Orbitals with the same n and l value are called subshells
For example n = 2
Two subshells: l = 0 and l = 1
Called the 2s and 2p subshells
What is the shape of the orbitals?
Shapes are not well defined
Wave function defining orbital extends from the nucleus to infinity
It is convenient to think of the orbitals as having a shape
It is especially helpful when talking about chemical bonds
What shape and size are s orbitals?
All s orbitals are spherical
The s orbitals do change in size however
The increase in size is the reason for the increase in the principal quantum number
Electron density
Falls off rapidly as the electron get farther from the nucleus.
p orbitals start with the principal quantum number? What happen when n=2, and l=1?
three possible orbital so there are 3 2p orbitals
What are the shape, and size of orbitals?
They are oriented along the axes of a 3-d plot
The three orbitals differ only in the orientation
They are identical in shape, size and energy
p orbital are thought of as two lobes on either side of the nucleus
d orbitals
n must equal at least 3 and l must equal 2 for the d orbitals to exist
How many d orbitals are there?
There are 5 d orbitals
The orbitals differ in orientation as well as one having a different shape.
All 3d orbitals have the same energy
d orbitals which have a larger n value have similarly shaped orbitals that are larger
Why are f orbitals important?
are important for accounting for the behavior element with atomic number > 57
In this class we are not concerned with orbitals having l > 3
Give the quantum numbers associated with the following orbitals:
Energy of orbitals increase as?
n increases
Does electron density change for 2s and 2p orbitals?
Although the electron density is different for 2s and 2p orbitals the energy remains the same
Which is orbital is the most stable?
The total energy depends on what?
The total energy depends on the sum of orbital energies as well as the repulsive forces
It turns ends up that the total energy is lower when the 4s orbital fills before the 3d
Order of atomic orbitals filling
2s 2p
3s 3p 3d
4s 4p 4d 4f
5s 5p 5d 5f
6s 6p 6d
7s 7p
2s quantum numbers
has no affect on energy, size, shape, or orientation of the orbital but determines how electrons are arranged in an orbital
What is electron configuration?
How the electrons of an atom are distributed among the orbitals
This is how the electrons are distributed in a ground state atom
The number of electrons in an atom is equal to?
Atomic number
Ground State H
Pauli exclusion principle?
If they have the same n, l, and ml, then they must have different ms.
These two electrons would be in the same orbital but have opposite spins.
Thus each orbital can contain only two electrons
Contain net unpaired spins and are attracted by a magnet
Do not contain unpaired spins and are slightly repelled by a magnet. A He atom with opposite spins in the orbital
What happens if the spins in an orbital do match up?
The magnetic fields reinforce each other. This would make an atom paramagnetic.
Odd and even numbered atoms have?
Odd numbered atoms always have one or more unpaired electrons
Even numbered atoms may or may not have unpaired electrons
Which orbital is filled first?
The 1s orbital is filled before electrons are start to fill the 2s or 2p orbital
2s and 2p orbitals
Both the 2s and 2p orbitals have electrons that spend more time away from the nucleus than electrons in the 1s orbital
The electrons of the 2s and 2p orbitals are shielded from the attractive forces of the nucleus by the 1s electrons
This reduces the electrostatic interactions between the nucleus and the 2s and 2p electrons
Experimentally the 2s orbital gives us a lower energy than the 2p
Although the 2s electron spend more time on average farther from the nucleus than a 2p electron; the denisty near the nucleus is greater for a 2s electron
So the 2s orbital is more penetrating
The 2s orbital is less shielded
For the same values of n, the penetrating power decreases as l increases
How is the stability of the electron is determined?
The stability of the electron is determined by the strength of attraction to the nucleus
Shielding effect
2s orbitals are less shielded than 2p orbitals so it follows that 2s orbitals have lower energy. Less energy is required to remove a 2p electron than a 2s electron
Hund's Rule
Rules for assigning electrons to orbitals
Each shell (or principal quantum number) n contains n subshells
Each subshell consists of quantum number l contain (2l + 1) orbitals
No more than 2 electrons can be placed in an orbital
The maximum number of electrons in principal level n is equal to 2n2
How many electrons can be present in the prinicpal level n=4?
What are the principal quantum numbers for the last electron in boron (B)?
Aufbau principle
As protons are added one by one to the nucleus to build up elements, electrons are similarly added to the atomic orbitals
Noble gas core
a method of showing electron configurations where the noble gas most nearly preceding the element being considered is used.The noble gas is followed by the electron configuration of the most highly filled subshells
Transition metals have either
Have either incompletely filled d orbitals or give rise to cations that have incompletely filled d subshells
Two irregularities in the fourth preiod
Chromium – [Ar]4s13d5
Copper – [Ar]4s13d10
The reason for this is that there is actually more stability in a half-filled or filled d orbital
Lanthanides and Actinide
Lanthanides – have incompletely filled 4f orbitals or readily give rise to cations with incompletely filled 4f subshells
Actinide series – last row of elements, most are not found in nature but have been synthesized
Write the ground-state electron configuration for Sr.
Write the ground-state electron configuration for Ga. |
14f4617ee78c2cfc | Hidden variable theory
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In physics, hidden variable theories are held by some physicists who argue that the state of a physical system, as formulated by quantum mechanics, does not give a complete description for the system; i.e., that quantum mechanics is ultimately incomplete, and that a complete theory would provide descriptive categories to account for all observable behavior and thus avoid any indeterminism. The existence of indeterminacy for some measurements is a characteristic of prevalent interpretations of quantum mechanics; moreover, bounds for indeterminacy can be expressed in a quantitative form by the Heisenberg uncertainty principle.
Albert Einstein, the most famous proponent of hidden variables, objected to the fundamentally probabilistic nature of quantum mechanics,[1] and famously declared "I am convinced God does not play dice".[2] Einstein, Podolsky, and Rosen argued that "elements of reality" (hidden variables) must be added to quantum mechanics to explain entanglement without action at a distance.[3][4] Later, Bell's theorem suggested that local hidden variables of certain types are impossible, or that they evolve non-locally. A famous non-local theory is De Broglie–Bohm theory.
Under the Copenhagen interpretation, quantum mechanics is non-deterministic, meaning that it generally does not predict the outcome of any measurement with certainty. Instead, it indicates what the probabilities of the outcomes are, with the indeterminism of observable quantities constrained by the uncertainty principle. The question arises whether there might be some deeper reality hidden beneath quantum mechanics, to be described by a more fundamental theory that can always predict the outcome of each measurement with certainty: if the exact properties of every subatomic particle were known the entire system could be modeled exactly using deterministic physics similar to classical physics.
In other words, it is conceivable that the standard interpretation of quantum mechanics is an incomplete description of nature. The designation of variables as underlying "hidden" variables depends on the level of physical description (so, for example, "if a gas is described in terms of temperature, pressure, and volume, then the velocities of the individual atoms in the gas would be hidden variables"[5]). Physicists supporting De Broglie–Bohm theory maintain that underlying the observed probabilistic nature of the universe is a deterministic objective foundation/property—the hidden variable. Others, however, believe that there is no deeper deterministic reality in quantum mechanics.[citation needed]
A lack of a kind of realism (understood here as asserting independent existence and evolution of physical quantities, such as position or momentum, without the process of measurement) is crucial in the Copenhagen interpretation. Realistic interpretations (which were already incorporated, to an extent, into the physics of Feynman[6]), on the other hand, assume that particles have certain trajectories. Under such view, these trajectories will almost always be continuous, which follows both from the finitude of the perceived speed of light ("leaps" should rather be precluded) and, more importantly, from the principle of least action, as deduced in quantum physics by Dirac. But continuous movement, in accordance with the mathematical definition, implies deterministic movement for a range of time arguments;[7] and thus realism is, under modern physics, one more reason for seeking (at least certain limited) determinism and thus a hidden variable theory (especially that such theory exists: see De Broglie–Bohm interpretation).
Although determinism was initially a major motivation for physicists looking for hidden variable theories, non-deterministic theories trying to explain what the supposed reality underlying the quantum mechanics formalism looks like are also considered hidden variable theories; for example Edward Nelson's stochastic mechanics.
"God does not play dice"[edit]
In June 1926, Max Born published a paper, "Zur Quantenmechanik der Stoßvorgänge" ("Quantum Mechanics of Collision Phenomena") in the scientific journal Zeitschrift für Physik, in which he was the first to clearly enunciate the probabilistic interpretation of the quantum wavefunction, which had been introduced by Erwin Schrödinger earlier in the year. Born concluded the paper as follows:
Here the whole problem of determinism comes up. From the standpoint of our quantum mechanics there is no quantity which in any individual case causally fixes the consequence of the collision; but also experimentally we have so far no reason to believe that there are some inner properties of the atom which conditions a definite outcome for the collision. Ought we to hope later to discover such properties ... and determine them in individual cases? Or ought we to believe that the agreement of theory and experiment—as to the impossibility of prescribing conditions for a causal evolution—is a pre-established harmony founded on the nonexistence of such conditions? I myself am inclined to give up determinism in the world of atoms. But that is a philosophical question for which physical arguments alone are not decisive.
Born's interpretation of the wavefunction was criticized by Schrödinger, who had previously attempted to interpret it in real physical terms, but Albert Einstein's response became one of the earliest and most famous assertions that quantum mechanics is incomplete:
Quantum mechanics is very worthy of regard. But an inner voice tells me that this is not yet the right track. The theory yields much, but it hardly brings us closer to the Old One's secrets. I, in any case, am convinced that He does not play dice.[8][9]
Niels Bohr reportedly replied to Einstein's later expression of this sentiment by advising him to "stop telling God what to do."[10]
Early attempts at hidden variable theories[edit]
Shortly after making his famous "God does not play dice" comment, Einstein attempted to formulate a deterministic counterproposal to quantum mechanics, presenting a paper at a meeting of the Academy of Sciences in Berlin, on 5 May 1927, titled "Bestimmt Schrödinger's Wellenmechanik die Bewegung eines Systems vollständig oder nur im Sinne der Statistik?" ("Does Schrödinger's wave mechanics determine the motion of a system completely or only in the statistical sense?").[11] However, as the paper was being prepared for publication in the academy's journal, Einstein decided to withdraw it, possibly because he discovered that, contrary to his intention, it implied non-separability of entangled systems, which he regarded as absurd.[12]
At the Fifth Solvay Congress, held in Belgium in October 1927 and attended by all the major theoretical physicists of the era, Louis de Broglie presented his own version of a deterministic hidden-variable theory, apparently unaware of Einstein's aborted attempt earlier in the year. In his theory, every particle had an associated, hidden "pilot wave" which served to guide its trajectory through space. The theory was subject to criticism at the Congress, particularly by Wolfgang Pauli, which de Broglie did not adequately answer. De Broglie abandoned the theory shortly thereafter.
Declaration of completeness of quantum mechanics, and the Bohr–Einstein debates[edit]
Also at the Fifth Solvay Congress, Max Born and Werner Heisenberg made a presentation summarizing the recent tremendous theoretical development of quantum mechanics. At the conclusion of the presentation, they declared:
[W]hile we consider ... a quantum mechanical treatment of the electromagnetic field ... as not yet finished, we consider quantum mechanics to be a closed theory, whose fundamental physical and mathematical assumptions are no longer susceptible of any modification.... On the question of the 'validity of the law of causality' we have this opinion: as long as one takes into account only experiments that lie in the domain of our currently acquired physical and quantum mechanical experience, the assumption of indeterminism in principle, here taken as fundamental, agrees with experience.[13]
Although there is no record of Einstein responding to Born and Heisenberg during the technical sessions of the Fifth Solvay Congress, he did challenge the completeness of quantum mechanics during informal discussions over meals, presenting a thought experiment intended to demonstrate that quantum mechanics could not be entirely correct. He did likewise during the Sixth Solvay Congress held in 1930. Both times, Niels Bohr is generally considered to have successfully defended quantum mechanics by discovering errors in Einstein's arguments.
EPR paradox[edit]
The debates between Bohr and Einstein essentially concluded in 1935, when Einstein finally expressed what is widely considered his best argument against the completeness of quantum mechanics. Einstein, Podolsky, and Rosen had proposed their definition of a "complete" description as one that uniquely determines the values of all its measurable properties. Einstein later summarized their argument as follows:
Consider a mechanical system consisting of two partial systems A and B which interact with each other only during a limited time. Let the ψ function [i.e., wavefunction ] before their interaction be given. Then the Schrödinger equation will furnish the ψ function after the interaction has taken place. Let us now determine the physical state of the partial system A as completely as possible by measurements. Then quantum mechanics allows us to determine the ψ function of the partial system B from the measurements made, and from the ψ function of the total system. This determination, however, gives a result which depends upon which of the physical quantities (observables) of A have been measured (for instance, coordinates or momenta). Since there can be only one physical state of B after the interaction which cannot reasonably be considered to depend on the particular measurement we perform on the system A separated from B it may be concluded that the ψ function is not unambiguously coordinated to the physical state. This coordination of several ψ functions to the same physical state of system B shows again that the ψ function cannot be interpreted as a (complete) description of a physical state of a single system.[14]
Bohr answered Einstein's challenge as follows:
[The argument of] Einstein, Podolsky and Rosen contains an ambiguity as regards the meaning of the expression "without in any way disturbing a system." ... [E]ven at this stage [i.e., the measurement of, for example, a particle that is part of an entangled pair], there is essentially the question of an influence on the very conditions which define the possible types of predictions regarding the future behavior of the system. Since these conditions constitute an inherent element of the description of any phenomenon to which the term "physical reality" can be properly attached, we see that the argumentation of the mentioned authors does not justify their conclusion that quantum-mechanical description is essentially incomplete."[15]
Bohr is here choosing to define a "physical reality" as limited to a phenomenon that is immediately observable by an arbitrarily chosen and explicitly specified technique, using his own special definition of the term 'phenomenon'. He wrote in 1948:
As a more appropriate way of expression, one may strongly advocate limitation of the use of the word phenomenon to refer exclusively to observations obtained under specified circumstances, including an account of the whole experiment."[16][17]
This was, of course, in conflict with the definition used by the EPR paper, as follows:
Bell's theorem[edit]
In 1964, John Bell showed through his famous theorem that if local hidden variables exist, certain experiments could be performed involving quantum entanglement where the result would satisfy a Bell inequality. If, on the other hand, statistical correlations resulting from quantum entanglement could not be explained by local hidden variables, the Bell inequality would be violated. Another no-go theorem concerning hidden variable theories is the Kochen–Specker theorem.
Physicists such as Alain Aspect and Paul Kwiat have performed experiments that have found violations of these inequalities up to 242 standard deviations[18] (excellent scientific certainty). This rules out local hidden variable theories, but does not rule out non-local ones. Theoretically, there could be experimental problems that affect the validity of the experimental findings.
Gerard 't Hooft has disputed the validity of Bell's theorem on the basis of the superdeterminism loophole and proposed some ideas to construct local deterministic models.[19]
Bohm's hidden variable theory[edit]
Assuming the validity of Bell's theorem, any deterministic hidden-variable theory that is consistent with quantum mechanics would have to be non-local, maintaining the existence of instantaneous or faster-than-light relations (correlations) between physically separated entities. The currently best-known hidden-variable theory, the "causal" interpretation of the physicist and philosopher David Bohm, originally published in 1952, is a non-local hidden variable theory. Bohm unknowingly rediscovered (and extended) the idea that Louis de Broglie had proposed in 1927 (and abandoned) – hence this theory is commonly called "de Broglie-Bohm theory". Bohm posited both the quantum particle, e.g. an electron, and a hidden 'guiding wave' that governs its motion. Thus, in this theory electrons are quite clearly particles—when a double-slit experiment is performed, its trajectory goes through one slit rather than the other. Also, the slit passed through is not random but is governed by the (hidden) guiding wave, resulting in the wave pattern that is observed.
Such a view does not contradict the idea of local events that is used in both classical atomism and relativity theory as Bohm's theory (and quantum mechanics) are still locally causal (that is, information travel is still restricted to the speed of light) but allow nonlocal correlations. It points to a view of a more holistic, mutually interpenetrating and interacting world. Indeed, Bohm himself stressed the holistic aspect of quantum theory in his later years, when he became interested in the ideas of Jiddu Krishnamurti.
In Bohm's interpretation, the (nonlocal) quantum potential constitutes an implicate (hidden) order which organizes a particle, and which may itself be the result of yet a further implicate order: a superimplicate order which organizes a field.[20] Nowadays Bohm's theory is considered to be one of many interpretations of quantum mechanics which give a realist interpretation, and not merely a positivistic one, to quantum-mechanical calculations. Some consider it the simplest theory to explain quantum phenomena.[21] Nevertheless, it is a hidden variable theory, and necessarily so.[22] The major reference for Bohm's theory today is his book with Basil Hiley, published posthumously.[23]
A possible weakness of Bohm's theory is that some (including Einstein, Pauli, and Heisenberg) feel that it looks contrived.[24] (Indeed, Bohm thought this of his original formulation of the theory.[25]) It was deliberately designed to give predictions that are in all details identical to conventional quantum mechanics.[25] Bohm's original aim was not to make a serious counterproposal but simply to demonstrate that hidden-variable theories are indeed possible.[25] (It thus provided a supposed counterexample to the famous proof by John von Neumann that was generally believed to demonstrate that no deterministic theory reproducing the statistical predictions of quantum mechanics is possible.) Bohm said he considered his theory to be unacceptable as a physical theory due to the guiding wave's existence in an abstract multi-dimensional configuration space, rather than three-dimensional space.[25] His hope was that the theory would lead to new insights and experiments that would lead ultimately to an acceptable one;[25] his aim was not to set out a deterministic, mechanical viewpoint, but rather to show that it was possible to attribute properties to an underlying reality, in contrast to the conventional approach to quantum mechanics.[26]
Recent developments[edit]
In August 2011, Roger Colbeck and Renato Renner published a proof that any extension of quantum mechanical theory, whether using hidden variables or otherwise, cannot provide a more accurate prediction of outcomes, assuming that observers can freely choose the measurement settings.[27] Colbeck and Renner write: "In the present work, we have ... excluded the possibility that any extension of quantum theory (not necessarily in the form of local hidden variables) can help predict the outcomes of any measurement on any quantum state. In this sense, we show the following: under the assumption that measurement settings can be chosen freely, quantum theory really is complete".
In January 2013, GianCarlo Ghirardi and Raffaele Romano described a model which, "under a different free choice assumption [...] violates [the statement by Colbeck and Renner] for almost all states of a bipartite two-level system, in a possibly experimentally testable way".[28]
See also[edit]
1. ^ The Born-Einstein letters: correspondence between Albert Einstein and Max and Hedwig Born from 1916–1955, with commentaries by Max Born. Macmillan. 1971. p. 158. , (Private letter from Einstein to Max Born, 3 March 1947: "I admit, of course, that there is a considerable amount of validity in the statistical approach which you were the first to recognize clearly as necessary given the framework of the existing formalism. I cannot seriously believe in it because the theory cannot be reconciled with the idea that physics should represent a reality in time and space, free from spooky actions at a distance.... I am quite convinced that someone will eventually come up with a theory whose objects, connected by laws, are not probabilities but considered facts, as used to be taken for granted until quite recently".)
2. ^ private letter to Max Born, 4 December 1926, Albert Einstein Archives reel 8, item 180
3. ^ a b Einstein, A.; Podolsky, B.; Rosen, N. (1935). "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?". Physical Review. 47 (10): 777–780. Bibcode:1935PhRv...47..777E. doi:10.1103/PhysRev.47.777.
4. ^ "The debate whether Quantum Mechanics is a complete theory and probabilities have a non-epistemic character (i.e. nature is intrinsically probabilistic) or whether it is a statistical approximation of a deterministic theory and probabilities are due to our ignorance of some parameters (i.e. they are epistemic) dates to the beginning of the theory itself". See: arXiv:quant-ph/0701071v1 12 Jan 2007
5. ^ Senechal M, Cronin J (2001). "Social influences on quantum mechanics?-I". The Mathematical Intelligencer. 23 (4): 15–17. doi:10.1007/BF03024596.
6. ^ Individual diagrams are often split into several parts, which may occur beyond observation; only the diagram as a whole describes an observed event.
7. ^ For every subset of points within a range, a value for every argument from the subset will be determined by the points in the neighbourhood. Thus, as a whole, the evolution in time can be described (for a specific time interval) as a function, e.g. a linear one or an arc. See Continuous function#Definition in terms of limits of functions
8. ^ The Born–Einstein letters: correspondence between Albert Einstein and Max and Hedwig Born from 1916–1955, with commentaries by Max Born. Macmillan. 1971. p. 91.
9. ^ Cache of the Einstein section of the American Museum of Natural History[permanent dead link]
11. ^ Albert Einstein Archives reel 2, item 100
12. ^ Baggott, Jim (2011). The Quantum Story: A History in 40 Moments. New York: Oxford University Press. pp. 116–117.
13. ^ Max Born and Werner Heisenberg, "Quantum mechanics", proceedings of the Fifth Solvay Congress.
14. ^ Einstein A (1936). "Physics and Reality". Journal of the Franklin Institute. 221.
15. ^ Bohr N (1935). "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?". Physical Review. 48 (8): 700. Bibcode:1935PhRv...48..696B. doi:10.1103/physrev.48.696.
16. ^ Bohr N. (1948). "On the notions of causality and complementarity". Dialectica. 2 (3–4): 312–319 [317]. doi:10.1111/j.1746-8361.1948.tb00703.x.
17. ^ Rosenfeld, L. (). 'Niels Bohr's contribution to epistemology', pp. 522–535 in Selected Papers of Léon Rosenfeld, Cohen, R.S., Stachel, J.J. (editors), D. Riedel, Dordrecht, ISBN 978-90-277-0652-2, p. 531: "Moreover, the complete definition of the phenomenon must essentially contain the indication of some permanent mark left upon a recording device which is part of the apparatus; only by thus envisaging the phenomenon as a closed event, terminated by a permanent record, can we do justice to the typical wholeness of the quantal processes."
18. ^ Kwiat P. G.; et al. (1999). "Ultrabright source of polarization-entangled photons". Physical Review A. 60 (2): R773–R776. arXiv:quant-ph/9810003Freely accessible. Bibcode:1999PhRvA..60..773K. doi:10.1103/physreva.60.r773.
19. ^ G 't Hooft, The Free-Will Postulate in Quantum Mechanics [1]; Entangled quantum states in a local deterministic theory [2]
20. ^ David Pratt: "David Bohm and the Implicate Order". Appeared in Sunrise magazine, February/March 1993, Theosophical University Press
21. ^ Michael K.-H. Kiessling: "Misleading Signposts Along the de Broglie–Bohm Road to Quantum Mechanics", Foundations of Physics, volume 40, number 4, 2010, pp. 418–429 (abstract)
22. ^ "While the testable predictions of Bohmian mechanics are isomorphic to standard Copenhagen quantum mechanics, its underlying hidden variables have to be, in principle, unobservable. If one could observe them, one would be able to take advantage of that and signal faster than light, which – according to the special theory of relativity – leads to physical temporal paradoxes." J. Kofler and A. Zeiliinger, "Quantum Information and Randomness", European Review (2010), Vol. 18, No. 4, 469–480.
23. ^ D. Bohm and B. J. Hiley, The Undivided Universe, Routledge, 1993, ISBN 0-415-06588-7.
24. ^ Wayne C. Myrvold (2003). "On some early objections to Bohm's theory" (PDF). International Studies in the Philosophy of Science. 17 (1): 8–24. doi:10.1080/02698590305233. Archived from the original on 2014-07-02.
25. ^ a b c d e David Bohm (1957). Causality and Chance in Modern Physics. Routledge & Kegan Paul and D. Van Nostrand. p. 110. ISBN 0-8122-1002-6.
27. ^ Roger Colbeck; Renato Renner (2011). "No extension of quantum theory can have improved predictive power". Nature Communications. 2 (8): 411. arXiv:1005.5173Freely accessible. Bibcode:2011NatCo...2E.411C. doi:10.1038/ncomms1416.
28. ^ Giancarlo Ghirardi; Raffaele Romano (2013). "Onthological models predictively inequivalent to quantum theory". Physical Review Letters. 110 (17): 170404. arXiv:1301.2695Freely accessible. Bibcode:2013PhRvL.110q0404G. doi:10.1103/PhysRevLett.110.170404. PMID 23679689.
• Albert Einstein, Boris Podolsky, and Nathan Rosen, "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" Physical Review 47, 777–780 (1935).
• John Stewart Bell, "On the Einstein–Podolsky–Rosen paradox", Physics 1, (1964) 195–200. Reprinted in Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, 2004.
• Wolfgang Pauli, letter to M. Fierz dated 10 August 1954, reprinted and translated in K. V. Laurikainen, Beyond the Atom: The Philosophical Thought of Wolfgang Pauli, Springer-Verlag, Berlin, 1988, p. 226.
• Werner Heisenberg, Physics and Beyond: Encounters and Conversations, translated by A. J. Pomerans, Harper & Row, New York, 1971, pp. 63–64.
• Claude Cohen-Tannoudji, Bernard Diu and Franck Laloë, Mecanique quantique (see also Quantum Mechanics translated from the French by Susan Hemley, Nicole Ostrowsky, and Dan Ostrowsky; John Wiley & Sons 1982) Hermann, Paris, France. 1977.
• P. S. Hanle, Indeterminacy before Heisenberg: The Case of Franz Exner and Erwin Schrödinger, Historical Studies in the Physical Sciences 10, 225 (1979).
• Asher Peres and Wojciech Zurek, "Is quantum theory universally valid?" American Journal of Physics 50, 807 (1982).
• Wojciech Zurek "Environment-induced superselection rules" Physical Review D 26 1862. 1982.
• Max Jammer, "The EPR Problem in Its Historical Development", in Symposium on the Foundations of Modern Physics: 50 years of the Einstein–Podolsky–Rosen Gedankenexperiment, edited by P. Lahti and P. Mittelstaedt (World Scientific, Singapore, 1985), pp. 129–149.
• Arthur Fine, The Shaky Game: Einstein Realism and the Quantum Theory, University of Chicago Press, Chicago, 1986.
• Thomas Kuhn. Black-Body Theory and the Quantum Discontinuity, 1894–1912 Chicago University Press. 1987.
• Asher Peres, Quantum Theory: Concepts and Methods, Kluwer, Dordrecht, 1993.
• Carlton M. Caves and Christopher A. Fuchs, "Quantum Information: How Much Information in a State Vector?", in The Dilemma of Einstein, Podolsky and Rosen – 60 Years Later, edited by A. Mann and M. Revzen, Ann. Israel Physical Society 12, 226–257 (1996).
• Carlo Rovelli. "Relational quantum mechanics" International Journal of Theoretical Physics 35 1637–1678. 1996.
• Roland Omnès, Understanding Quantum Mechanics, Princeton University Press, 1999.
• Roman Jackiw and Daniel Kleppner, "One Hundred Years of Quantum Physics", Science, Vol. 289 Issue 5481, p. 893, August 2000.
• Orly Alter and Yoshihisa Yamamoto (2001). Quantum Measurement of a Single System (PDF). Wiley-Interscience. 136 pp. doi:10.1002/9783527617128. ISBN 9780471283089. Slides. Archived from the original (PDF) on 2014-02-03.
• Erich Joos, et al., Decoherence and the Appearance of a Classical World in Quantum Theory, 2nd ed., Berlin, Springer, 2003.
• Wojciech Zurek (2003). "Decoherence and the transition from quantum to classical — Revisited", arXiv:quant-ph/0306072 (An updated version of Physics Today, 44:36–44 (1991) article)
• Wojciech Zurek, "Decoherence, einselection, and the quantum origins of the classical" in Reviews of Modern Physics, vol.75, (715).
• Asher Peres and Daniel Terno, "Quantum Information and Relativity Theory", Reviews of Modern Physics 76 (2004) 93.
• Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe, Alfred Knopf 2004.
• Maximilian Schlosshauer, "Decoherence, the Measurement Problem, and Interpretations of Quantum Mechanics", in Reviews of Modern Physics, vol.76, pages 1267–1305, 2005.
• Federico Laudisa and Carlo Rovelli. "Relational Quantum Mechanics" The Stanford Encyclopedia of Philosophy (Fall 2005 Edition).
• Marco Genovese, "Research on hidden variable theories: a review of recent progresses", in Physics Reports, vol.413, 2005.
External links[edit] |
0dc7510fd7ad4f0b | Pin Me
What is the Science Behind God's Fist - The Research on Rogue Waves
written by: Dr. Crystal Cooper • edited by: Ricky • updated: 6/29/2011
Scientists are not content with the mere observation of the mysterious phenomena that are rogue waves, but are carrying out intensive research. This article examines how optics, neural networks, and quantum physics are related to rogue wave research.
• slide 1 of 4
We started talking about the physics of rogue waves in our previous article and will continue the discussion here. The old, linear models of hydrodynamics neither account for nor predict the existence of monster waves.ig20 hurricane 04 02 The theory that they are just combinations or superpositions of small ones that form during storms does not explain their sudden appearance in calm waters, for example. Physicists, mathematicians, engineers, and oceanographers now have several different newer models to explain their existence.
NOAA once had a program where rogue waves were studied in great detail. Researchers there found that the lack of predictability happens due to the fact that most measurements are based on ocean wave models as stationary random Gaussian processes. A stationary process has a probability distribution which is the same for all times and all positions. A Gaussian process has a probability of occurrence that is based on the Gaussian or normal distribution, also known as a "bell curve". The NOAA researchers studied rogue wave data at various places around the world and proposed different non-linear models.
• slide 2 of 4
Current Research
What follows is an overview of current day research into the science of extreme waves.
• Nonlinear Models: Researchers at the University of Massachusetts believe that sea-floor topography, near surface currents, and the wind are a major factor in their genesis. They have constructed several nonlinear models which they use as the basis for numerical simulations.
• Neural Networks: At Texas A&M, researchers have found a mathematical model, based on neural networks and data from buoys, that is able to make predictions of wave heights off the coast of the United States for up to 24 hours. This model is used in coastal areas of Maine and Texas, and also in Alabama.
• Schrödinger's equation: This is used in quantum physics to explain the behavior of atoms and particles. Norway researchers have successfully used the nonlinear version to model the physical characteristics and sudden appearance of monster waves.
• Optics: Some researchers are studying optics to come up with viable models. At UCLA's Henry Samueli School of Engineering and Applied Science, they have developed experiments and mathematical models for optical rogue waves that they believe are applicable to hydrodynamics. Their work also uses the nonlinear Schrödinger equation as a basis. Optical rogue waves are easier to create experimentally and detect than their ocean counterparts.
In the next part of this series, we will examine monster waves and naval architecture.
• slide 3 of 4 |
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Eigenequation and eigenvalue
1. Jan 27, 2005 #1
what is an eigenequation? what is the purpose of the eigenvalue? how does this fit into the schrodinger equation (particle in a box problem) ?
2. jcsd
3. Jan 27, 2005 #2
An eigenequation is for example the following:
M x = b x
where M is a Matrix (for example a 3x3), x is a vector (3 components)
and b is a real number (could also be complex number).
You see that the Matrix doesn't change the direction of x, only it's length (right hand side of the equation).
x is called eigenvector and b eigenvalue of M.
Now in Quantum mechanics you have operators (instead of matrices)
and so called state vectors,
for example:
H |Psi> = E |Psi>
( M x = b x )
H is the Hamilton-Operator, |Psi> is your eigenvector and E the eigenvalue.
Whats the meaning of the equation above?
It just says that you got a system represented by the vector |Psi>
(for example electron in the Hydrogen atom).
And then you want to measure the energy. This is done by
'throwing' the operator H on your vector |Psi>. What comes out
is your eigenvalue E which is the energy.
Now whats the Schrödinger equation?
Suppose you want to examine the energy of the electron in the hydrogen atom. So you just apply H on |Psi> and get the energy E on the right hand side of the eigenequation.
The PROBLEM is, you dont know how your |Psi> looks like.
So here's where the SCHRÖDINGER equation comes into the play.
The Schrödinger equation is a differential equation,
which you have to solve in order to get your |Psi>. (solving the differential equation means you get a solution |Psi>)
You put your potential (square well potential for particle in a box, or Coloumb potential for hydrogen atom) into the Schrödinger equation and solve it. You get your |Psi> from it.
I hope I could help you.
4. Jan 28, 2005 #3
thanks alot! |
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