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6bfdcf65a0b67f2f | Electron magnetic dipole moment
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"Electron spin" redirects here. See also Electron spin resonance and Spin (physics).
In atomic physics, the electron magnetic dipole moment is the magnetic moment of an electron caused by its intrinsic property of spin.
Magnetic moment of an electron[edit]
The electron is a charged particle of charge (−1e), where e is the unit of elementary charge. Its angular momentum comes from two types of rotation: spin and orbital motion. From classical electrodynamics, a rotating electrically charged body creates a magnetic dipole with magnetic poles of equal magnitude but opposite polarity. This analogy holds as an electron indeed behaves like a tiny bar magnet. One consequence is that an external magnetic field exerts a torque on the electron magnetic moment depending on its orientation with respect to the field.
If the electron is visualized as a classical charged particle literally rotating about an axis with angular momentum L, its magnetic dipole moment μ is given by:
\boldsymbol{\mu} = \frac{-e}{2m_e}\, \mathbf{L}.
where me is the electron rest mass. Note that the angular momentum L in this equation may be the spin angular momentum, the orbital angular momentum, or the total angular momentum. It turns out the classical result is off by a proportional factor for the spin magnetic moment. As a result, the classical result is corrected by multiplying it with a dimensionless correction factor g is known as the g-factor;
\boldsymbol{\mu} = g \frac{-e}{2m_e} \mathbf{L}.
It is usual to express the magnetic moment in terms of the reduced Planck constant ħ and the Bohr magneton μB:
\boldsymbol{\mu} = -g \mu_B \frac{\mathbf{L}}{\hbar}.
Since the magnetic moment is quantized in units of μB, correspondingly the angular momentum is quantized in units of ħ.
Spin magnetic dipole moment[edit]
The spin magnetic moment is intrinsic for an electron.[1] It is:
\boldsymbol{\mu}_S=- g_S \mu_B \frac{\mathbf{S}}{\hbar}.
Here S is the electron spin angular momentum. The spin g-factor is approximately two: gs ≈ 2. The magnetic moment of an electron is approximately twice what it should be in classical mechanics. The factor of two implies that the electron appears to be twice as effective in producing a magnetic moment as the corresponding classical charged body.
The spin magnetic dipole moment is approximately one μB because g ≈ 2 and the electron is a spin one-half particle: S = ħ/2.
\mu_S\approx 2\frac{e}{2m_e}\frac{\frac{\hbar}{2}}{\hbar}=\mu_B.
The z component of the electron magnetic moment is:
(\boldsymbol{\mu}_S)_z=-g_S \mu_B m_S
where mS is the spin quantum number. Note that μ is a negative constant multiplied by the spin, so the magnetic moment is antiparallel to the spin angular momentum.
2.00231930419922 ± (1.5 × 10−12).[2]
-928.476377 × 10−26 ± 0.000023 × 10−26 J·T−1.[3]
The classical theory of g-factor[edit]
The Dirac theory is not necessary to explain the g-factor for the electron. The deviation of the electron g-factor from that of the rigid sphere can be readily explained assuming that the charge distribution inside electron is different from the mass distribution. The electron can still be assumed a rigid body. Assuming for example the simplest and the most physical spherical Gaussian distributions for the charge and the mass separately:
\rho_e(r)=e N_e e^{-r^2/r_e^2}
\rho_m(r)=m_e N_m e^{-r^2/r_m^2}
where r_m is the mass radius of the electron and r_e is the charge radius we can obtain the tunable g-factor as the ratio
g=\left ( \frac{r_e}{r_m} \right )^8.
For the electron g=2 they differ therefore very slightly, namely
\left ( \frac{r_e}{r_m} \right )\approx 1.09051.
Orbital magnetic dipole moment[edit]
The revolution of an electron around an axis through another object, such as the nucleus, gives rise to the orbital magnetic dipole moment. Suppose that the angular momentum for the orbital motion is L. Then the orbital magnetic dipole moment is:
\boldsymbol{\mu}_L = -g_L\mu_B \frac{\mathbf{L}}{\hbar}.
Here gL is the electron orbital g-factor and μB is the Bohr magneton. The value of gL is exactly equal to one, by a quantum-mechanical argument analogous to the derivation of the classical gyromagnetic ratio.
Total magnetic dipole moment[edit]
The total magnetic dipole moment resulting from both spin and orbital angular momenta of an electron is related to the total angular momentum J by a similar equation:
\boldsymbol{\mu}_J =g_J \mu_B \frac{\mathbf{J}}{\hbar}.
The g-factor gJ is known as the Landé g-factor, which can be related to gL and gS by quantum mechanics. See Landé g-factor for details.
Example: hydrogen atom[edit]
For a hydrogen atom, an electron occupying the atomic orbital Ψn, ℓ, m, the magnetic dipole moment is given by:
Here L is the orbital angular momentum, n, ℓ and m are the principal, azimuthal and magnetic quantum numbers respectively. The z-component of the orbital magnetic dipole moment for an electron with a magnetic quantum number m is given by:
(\mathbf{\mu_L})_z=-\mu_B m_\ell.\,
Electron spin in the Pauli and Dirac theories[edit]
Main articles: Pauli equation and Dirac 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 non-uniform 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 3 parts, corresponding to atoms with Lz = −1, 0, and +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 [ \boldsymbol{\sigma}\cdot \left ( \mathbf{p} - \frac{e}{c}\mathbf{A} \right ) \right ]^2 + e\phi.
Here A is the magnetic potential and ϕ the electric potential representing the electromagnetic field, and σ = (σx, σy, σz) 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 ( \mathbf{p} - \frac{e}{c}\mathbf{A} \right )^2 + e\phi - \frac{e\hbar}{2mc}\boldsymbol{\sigma}\cdot \mathbf{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 incorporating relativity into quantum mechanics. On introducing the external electromagnetic 4-potential into the Dirac equation in a similar way, known as minimal coupling, it takes the form (in natural units ħ = c = 1)
\left [ -i\gamma^\mu\left ( \partial_\mu + ieA_\mu \right ) + m \right ] \psi = 0\,
where \scriptstyle \gamma^\mu are the gamma matrices (aka Dirac matrices) and i is the imaginary unit. 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. 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:
(mc^2 - E + e \phi) & c\sigma\cdot \left (\mathbf{p} - \frac{e}{c}\mathbf{A} \right ) \\ -c\boldsymbol{\sigma}\cdot \left ( \mathbf{p} - \frac{e}{c}\mathbf{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\boldsymbol{\sigma}\cdot \left( \mathbf{p} - \frac{e}{c}\mathbf{A} \right) \psi_{-} = mc^2 \psi_+
-(E - e\phi) \psi_{-} + c\boldsymbol{\sigma}\cdot \left( \mathbf{p} - \frac{e}{c}\mathbf{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 reducing 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} \boldsymbol{\sigma}\cdot \left ( \mathbf{p} - \frac{e}{c}\mathbf{A} \right ) \psi_+
See also[edit]
1. ^ A. Mahajan and A. Rangwala. Electricity and Magnetism, p. 419 (1989). Via Google Books.
2. ^ http://physics.nist.gov/cgi-bin/cuu/Value?eqae%7Csearch_for=electron+magnetic+moment
3. ^ http://physics.nist.gov/cgi-bin/cuu/Value?muem%7Csearch_for=magnetic+moment+electron
4. ^ Source: Journal of Mathematical Physics, 52, 082303 (2011) (http://jmp.aip.org/resource/1/jmapaq/v52/i8/p082303_s1 or http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf ) |
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Open Access Nano Express
The effects of porosity on optical properties of semiconductor chalcogenide films obtained by the chemical bath deposition
Yuri V Vorobiev1*, Paul P Horley2, Jorge Hernández-Borja1, Hilda E Esparza-Ponce2, Rafael Ramírez-Bon1, Pavel Vorobiev1, Claudia Pérez1 and Jesús González-Hernández2
Author Affiliations
1 CINVESTAV-IPN Unidad Querétaro, Libramiento Norponiente 2000, Fracc. Real de Juriquilla, Querétaro, Qro, CP 76230, México
2 CIMAV Chihuahua/Monterrey, Avenida Miguel de Cervantes 120, Chihuahua, Chih, CP 31109, México
For all author emails, please log on.
Nanoscale Research Letters 2012, 7:483 doi:10.1186/1556-276X-7-483
Received:16 April 2012
Accepted:4 August 2012
Published:29 August 2012
© 2012 Vorobiev et al.; licensee Springer.
This paper is dedicated to study the thin polycrystalline films of semiconductor chalcogenide materials (CdS, CdSe, and PbS) obtained by ammonia-free chemical bath deposition. The obtained material is of polycrystalline nature with crystallite of a size that, from a general point of view, should not result in any noticeable quantum confinement. Nevertheless, we were able to observe blueshift of the fundamental absorption edge and reduced refractive index in comparison with the corresponding bulk materials. Both effects are attributed to the material porosity which is a typical feature of chemical bath deposition technique. The blueshift is caused by quantum confinement in pores, whereas the refractive index variation is the evident result of the density reduction. Quantum mechanical description of the nanopores in semiconductor is given based on the application of even mirror boundary conditions for the solution of the Schrödinger equation; the results of calculations give a reasonable explanation of the experimental data.
polycrystalline films; chalcogenide materials; nanopores; quantum confinement in pores
Chemical bath deposition (CBD) is a cheap and energy-efficient method commonly used for the preparation of semiconductor films for sensors, photodetectors, and solar cells. It was one of the traditional methods to obtain chalcogenide semiconductors including CdS and CdSe [1-6]. However, large-scale CBD deposition of CdS films raises considerable environmental concerns due to utilization of highly volatile and toxic ammonia. On the other hand, the volatility of ammonia modifies pH of the reacting solution during the deposition process, causing irreproducibility of thin film properties for the material obtained in different batches [1,3].
We manufacture CdS, CdSe, and PbS films using the CBD process to minimize the production cost and energy consumption. Ammonia-free CBD process was used to avoid negative environmental impact (see [7] reporting an example of CBD-made solar cell with structure glass/ITO/CdS/PbS/conductive graphite with quantum efficiency of 29% and energy efficiency of 1.6% ). All these materials have the melting temperatures above 1,000°C, remaining stable during the deposition process. It is also known that PbS is very promising for solar cell applications, confirmed by the recent discovery of multiple exciton generation in their nanocrystals [8].
Chemical bath-deposited films [9] have a particular structure. As a rule, at initial deposition stages, small (3 to 5 nm) nanocrystals are formed. They exhibit strong quantum confinement leading to large blueshift of the fundamental absorption edge. Historically, blueshift was first discovered namely in CBD-made CdSe films [9,10]. At later stages, the crystallite size becomes larger so that the corresponding blueshift decreases. Another feature characteristic to the process is a considerable porosity [3,9] inherent to the growth mechanism, which takes place ion by ion or cluster by cluster depending on the conditions or solution used (see also [11,12]). The degree of porosity decreases for larger deposition time because the film becomes denser. At the initial stage, the porosity can be up to 70% [9], and at final stages, it will be only about 5% to 10% .
In this paper, we present the experimental results for the investigation of porosity effects for relatively large deposition times upon the optical characteristics of CBD-made semiconductor materials such as CdS, CdSe, and PbS. We show that the nanoporosity can blueshift the absorption edge, leading to the variation observed for material with pronounced nanocrystallinity. For theoretical study of nanopores in a semiconductor, we use mirror boundary conditions to solve the Schrödinger equation, which were successfully applied to nanostructures of different geometries [13-15]. We show that the same treatment of pores allows to achieve a good correlation between theorical and experimental data.
The authors successfully developed ammonia-free CBD technology for polycrystalline CdS, CdSe, and PbS films, described in detail elsewhere [4-7,11,12]. We characterize the obtained structures by composition, microstructure (including average grain size), and morphology using X-ray diffraction, SEM, and EDS measurements. Optical properties were investigated with UV–vis and FTIR spectrometers. All experimental methods are described in the aforementioned references, together with the detailed results of this complex material study. Here, we would like to discuss optical phenomena characteristic to the entire group of semiconductor film studied, skipping the technological details that are given in [4-7,11,12].
Results and discussion
For CBD-made materials obtained after long deposition time (which resulted into dense films with crystallite size of about 20 nm), we observed a blueshift of the fundamental absorption edge relative to the bulk material data [16] in all cases with the following shift values: 0.06 eV for CdS [7], 0.15 eV for CdSe [6] (see also Figure 1), and 0.1 to 0.4 eV for different samples of PbS (Figure 2). This effect was accompanied by reduction of refractive index n (in comparison with bulk crystal data, see Figure 3 for CdSe and Figure 4 for PbS). This reduction is larger for samples obtained with small deposition times, but it is always present in the films discussed here. We connect both effects with pronounced porosity of the films obtained by CBD method. In particular, the blueshift in the dense CBD films is attributed to the quantum confinement in pores.
thumbnailFigure 1 . Transmission spectrum of 0.5-μm thick CdSe film.
thumbnailFigure 2 . Diagram used to determine bandgap of PbS CBD sample with growth time of 3 h. The value of D corresponds to optical density.
thumbnailFigure 3 . Refractive index of CdSe. Squares indicate the data for the bulk material adapted from [14], and circles correspond to CBD film.
thumbnailFigure 4 . Optical constantsn, kof PbS CBD films with different deposition times.
Figure 1 presents the transmission spectrum of 0.5-μm-thick CdSe films (deposition time of 4 h) displaying a clear interference pattern, characterized with transmission maxima at 2dn = and minima at 2dn = (N − 1/2) λ. Here, λ is the wavelength, d is the film thickness, and N is an integer defining the order of interference pattern. With these expressions, we calculated the spectrum of refractive index (Figure 3, circles). The squares in the same figure present the data for the bulk material [17] displaying a considerable drop of refractive index for the film in comparison with bulk material.
Figure 2 presents the diagram for PbS allowing to determinate the bandgap via direct interband transitions observed for all the materials studied by plotting the squared product of optical density and photon energy as a function of the latter. The similar diagrams for CdS and CdSe were given in [6,7].
The case of PbS requires more attention. Figure 5 presents the dependence of the crystallite size upon the deposition time. Figure 4 shows the spectra for optical constants (refractive index n and extinction coefficient k) measured for four PbS films deposited with growth time ranging from 1 to 4 h; in the latter case, the result was a 100-nm-thick film. It is clear that for larger deposition time, the film becomes denser so that refraction index and extinction coefficients increase. Their spectral behavior follows qualitatively the corresponding curves of the bulk material, but the values are essentially lower, even when deposited film has a considerable thickness. For example, the refractive index for film is 4 at most for the wavelength 450 nm, whereas for the bulk material, the corresponding value is 4.3. As for extinction coefficient k, the maximum of 2.75 is achieved at the wavelength of 350 nm, with the corresponding bulk value of 3.37.
thumbnailFigure 5 . Dependence of the grain size of PbS CBD samples on growth time. The line is given as eye guide only.
We assume that the pores in a dense CBD film correspond to the spaces between crystallites' boundaries. Therefore, in cubic crystals, the pores most probably will be of prismatic shape, defined by plane boundaries of the individual grains. These prismatic pores most probably will have a length (height) equal to the grain size, with quadratic or rectangular triangle cross-section. As pores and crystallites are considered to be of equal height, the question of a volume fraction of pores reduces to two dimensions by being equal to the ratio of pore cross-sectional area to the total cross-section of the film, assuming that in the average there will be one pore per one crystallite. The dimensions of the pore will define the blueshift observed, which can be seen from the following theoretical consideration.
Electron confined in pores: quantum mechanical approach
It was proposed (see [13-15]) to treat semiconductor quantum dots (QDs) as ‘mirror-wall boxes’ confining the particle, resulting in mirror boundary conditions for analytical solution of the Schrödinger equation in the framework of the effective mass approximation. The basic assumption is that a particle (an electron or a hole) is specularly reflected by a QD boundary, which sets the boundary conditions as equivalence of particle's Ψ-function in an arbitrary point r inside the semiconductor (Ψr) with wave function in the image point im (Ψim). It must be mentioned that Ψ-function in real and image points can be equated by its absolute values since the physical meaning is connected with |Ψ|2, so that mirror boundary conditions can have even and odd forms (Ψr = Ψim in the former case, and Ψr = −Ψim in the latter). The ‘odd’ case is equivalent to the impenetrable boundary conditions and strong confinement because Ψ-function vanishes at the boundary. The milder case of even mirror boundary conditions represents weak confinement and occurs when a particle is allowed to have tunnel probability inside the boundary.
It is evident that our basic assumption is favorable for effective mass approximation as it increases the length of effective path for a particle in a semiconductor material. Besides, in high symmetry case, the assumption of mirror boundary conditions forms a periodic structure filling the space. We have shown [15] that the use of even mirror boundary conditions gives the same solution as Born-von Karman boundary conditions applied to a periodic structure. The treatment performed in [13-15] of the QDs with different shapes (rectangular prism, sphere and square base pyramid) yielded the energy spectra that have a good agreement with the published experimental data achievable without any adjustable parameters.
Let us consider an inverted system: a pore formed by a void surrounded by a semiconductor material. The reflection accompanied with a partial tunneling into QD boundary (for the case of even mirror boundary conditions) can be described as equivalence of Ψ-function values in a real point in the vicinity of the boundary and a reflection point in a mirror boundary. Hence, the solution of the Schrödinger equation for a pore within semiconductor material will be the same as that for a QD of equal geometry with an equal expression for the particle's energy spectrum.
Table 1 summarizes the expressions for energy spectra obtained for QDs of several basic shapes with application of even mirror boundary conditions. All spectra have the same character, with a quadratic dependence on quantum numbers (all integers or odd numbers for a particular case of spherical QD [15]) and an inverse quadratic dependence on QD's dimensions. Besides, the position of energy levels has an inverse dependence on the effective mass [18,19].
Table 1. Energy spectra of different QDs
Comparison with the experiment
In the following discussion, we take into account that typical pores in CBD materials have a characteristic size a of several nanometers [3,9], being much smaller than the Bohr radius αB for an exciton, α/2 < < αB, which is especially important for the case of exciton formation under the action of a light beam incident on semiconductor. The energy difference defines the blueshift of absorption edge. In all semiconductors studied, the value of αB exceeds 15 nm according to the expression below:
<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/483/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/483/mathml/M5">View MathML</a>
Here, me,h is the electron/hole effective mass, ϵ is the dielectric constant of the material, and ϵ0 is a permittivity constant. Following the argumentation given in [18,19], we see that one can directly apply the expressions for energy spectra because the separation between the quantum levels proportional to ħ2/ma2 is large compared to the Coulomb interaction between the carriers which is proportional to e2/ϵϵ0α. Therefore, Coulomb interaction can be neglected, and the energy levels could be found from quantum confinement effect alone. Accordingly, we shall calculate the emission/absorption photon energy for transitions corresponding to the exciton ground state, which is given by n = 0 for spherical QD and n = 1 for other geometries. From Table 1, it follows that the lowest energy value can be obtained for a spherical QD, whereas for a prism with quadratic section, the energy value is twice larger. For all other geometries, the energy has the latter order of magnitude. For the estimation of porosity effects, we will use the expression for a prismatic QD with a square base, assuming that the fundamental absorption edge corresponds to generation of an exciton with ground state energy:
<a onClick="popup('http://www.nanoscalereslett.com/content/7/1/483/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.nanoscalereslett.com/content/7/1/483/mathml/M6">View MathML</a>
with the semiconductor bandgap Eg.
In the case of CdSe (exciton reduced mass of 0.1 m0) using the expression (2) and the band edge shift ħωminEg = 0.15 eV (1.88 − 1.73), we calculate the pore size of 7 nm. For the average crystallite dimension of 22 nm, the pore fraction, thus, would be (7/22)2 ≈ 10% , which is twice as big as the relative reduction of refractive index found (Figure 3).
To explain the edge shift observed in CdS (exciton reduced mass 0.134 m0[16]), one obtains the pore size of 8 nm. Here, the crystallite size is 20.1 nm, making the total pore fraction of approximately 12% . The observed reduction of refractive index changes from 2.5 for the bulk material [7,16] to 2.3 for 600-nm-thick film, yielding the pore fraction of 9% that is close to our predictions.
The reduced mass for PbS is 0.0425 m0[16], and the observed edge shift is 0.4 eV, yielding the average pore size of 6.5 nm. Having the crystallite size of 20 nm, it will give the pore fraction of 10% (observed reduction of refractive index in [7] was 8% , and from Figure 4 we obtain the value of 7.5% ). We see that in all cases, the volumetric percentage of pores calculated using the blueshift values renders the correct order that is verified from the refractive index reduction. However, the latter value is always smaller that may mean that the pores' height is about 30% to 40% less than that of the grains.
It should be noted that in cases of PbS, due to high value of dielectric constant (17) and small exciton reduced mass, the Bohr radius for an exciton (21 nm) appears to be the same order of magnitude as the grain size. It means that the quantum confinement effect can be observed even without taking into account the porosity of the material. This effect was studied experimentally in [20] for PbS spherical quantum dots. It was found that in PbS quantum dots with diameter of 3.5 nm, the blue band edge shift of 1.05 eV is observed. Taking into account that the blueshift due to quantum confinement is reversely proportional to the square of the dot's diameter, we find that the shift caused by the crystallite size of 20 nm will be equal to 0.03 eV, which is about 10 times smaller than the observed values. We also note that the smaller crystallite size observed in our experiments at early stages of CBD process (variation from 8 to 18 nm, see Figure 5) does not allow to explain the experimentally observed blueshift. Thus, we conclude the mandatory accounting of nanopores, which offers improved agreement between theoretical and experimental data.
We report on ammonia-free CBD method that provides cheap, efficient, and environmentally harmless production of CdS, CdSe, and PbS films. Material porosity inherent to CBD technique can be used to fine-tune the material bandgap towards the required values, paving promising ways for solar cell applications. The theoretical description of porosity based on the solution of the Schrödinger equation with even mirror boundary conditions provides a good correlation of theoretical and experimental data.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
YVV suggested the treatment of pores as inverted quantum dots. PPH realized the theoretical description and drafted the manuscript. JHB conducted the experiments on CdS and PbSe. HEEP made the experiments on CdSe. RRB adjusted the chemical part of CBD method and helped in drafting the manuscript. PV performed modeling of a porous semiconductor. CP realized the experiments with PbS. JGH supervised all the study. All authors read and approved the final manuscript.
The authors are grateful to Editor Prof. Andres Cantarero for the support and encouragement in the revision of the manuscript. PV and CP wish to thank CONACYT for their scholarships.
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77cc6948c1fc7446 | About this Journal Submit a Manuscript Table of Contents
Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 532610, 6 pages
Research Article
Stable One-Dimensional Periodic Wave in Kerr-Type and Quadratic Nonlinear Media
Department of Constructive and Technological Engineering—Lasers and Fibre Optic Communications, National Institute of R&D for Optoelectronics INOE 2000, 409 Atomistilor Street, P.O. Box MG-5, 077125 Magurele, Ilfov, Romania
Received 6 December 2011; Revised 9 February 2012; Accepted 13 February 2012
Academic Editor: Cristian Toma
We present the propagation of optical beams and the properties of one-dimensional (1D) spatial solitons (“bright” and “dark”) in saturated Kerr-type and quadratic nonlinear media. Special attention is paid to the recent advances of the theory of soliton stability. We show that the stabilization of bright periodic waves occurs above a certain threshold power level and the dark periodic waves can be destabilized by the saturation of the nonlinear response, while the dark quadratic waves turn out to be metastable in the broad range of material parameters. The propagation of (1+1) a dimension-optical field on saturated Kerr media using nonlinear Schrödinger equations is described. A model for the envelope one-dimensional evolution equation is built up using the Laplace transform.
1. Introduction
The discrete spatial optical solitons have been introduced and studied theoretically as spatially localized modes of periodic optical structures [1]. A standard theoretical approach in the study of the discrete spatial optical solitons is based on the derivation of an effective discrete nonlinear Schrödinger equation and the analysis of its stationary localized solitons-discrete localized modes [1, 2].
The spatial solitons may exist in a broad branch of nonlinear materials, such as cubic Kerr, saturable, thermal, reorientation, photorefractive, and quadratic media, and periodic systems. Furthermore, the existence of solitons varies in topologies and dimensions [3].
The theory of spatial optical solitons has been based on the nonlinear Schrödinger (NLS) equation with a cubic nonlinearity, which is exactly integrable by means of the inverse scattering (IST) technique. From the physical point of view, the integrable NLS equation describes the -dimensional beams in a Kerr (cubic) nonlinear medium in the framework of the so-called paraxial approximation [4].
Bright solitons are formed due to the diffraction or dispersion compensated by self-focusing nonlinearity and appear as an intensity hump in a zero background. Solitons, which appear as intensity dips with a CW background, are called dark soliton [3].
Kerr solitons rely primarily on a physical effect, which produces an intensity-dependent change in refractive index [3].
The periodic wave structures play an important role in the nonlinear wave domain so that they are core of instability modulation development and optics chaos on continuous nonlinear media, modes of quasidiscrete systems or discrete system on mechanic and electric domain. Thus, periodic wave structures are unstable in the propagation process. For example, photorefractive crystals accept relatively high nonlinearity of saturated character at an already known intensity for He-Ne laser in continuous regime.
2. Methodology
The propagation of the optical radiation in dimensions in saturable Kerr-type medium is described by the nonlinear Schrödinger equation for the varying field amplitude [5]:
The transverse and the longitudinal coordinates are scaled in terms of the characteristic pulse (beam) width and dispersion (diffraction) length, respectively; is the saturation parameter; stands for focusing (defocusing) media [5]
The simplest periodic stationary solutions of (2.1) have the following form: where is the propagation constant.
By replacing the field in such a form into (2.1), one gets
To perform the linear stability analysis of periodic waves in the saturable medium, we use the mathematical formalism initially developed for periodic waves in cubic nonlinear media [5].
We consider an analytic model, which used the Laplace transform of (2.4):
With the boundary conditions,
From (2.5) we get the Laplace transform of the field:(i)direct form: (ii)inverse transformation form: where is a finite number.
For the integration on real () and imaginary poles, we calculated the complex amplitude of nonlinear equation such as
For the harmonic case integration form of the complex amplitude is
By using the integration, we get or
The total phase of the optical field envelope is as follows:
We assume a frequency () as a speed variation of total phase such as
We have the complex amplitude of envelope field with the following form:
The hyperbolic secant plays this equation resulting in a conservative effect. The longitudinal component is
Some numerical simulations of the complex amplitude of the nonlinear equation and the total phase of the optical field depending on the propagation constant and an integer number are illustrated in Figure 1.
Figure 1: Numerical simulations of complex amplitude and phase.
Figure 1 represents the model amplitude and the phase functions of the complex total number, which explained the theoretical model presented. Thanks to the complex model, the initial solution includes the hyperbolic secant and the conjugate complex part
3. Conclusions
We have described the propagation in quadratic nonlinear media of the periodic waves in saturated Kerr type. The analytic solution for one-dimensional, bright and dark spatial solitons was found. To describe the spatial optical solitons in saturated Kerr type and the quadratic nonlinear media, we propose an analytical model based on Laplace transform. The theoretical model consists in solving analytically the Schrödinger equation with photonic network using Laplace transform. The propagation properties were found by using different forms of saturable nonlinearity. However, an exact analytic solution of the propagation problem presented herein creates possibilities for further theoretical investigation. As a result, it is a useful structure, which obtains one-dimensional “bright” and “dark” solitons with transversal structure and transversal one-dimensional periodic waves.
1. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” Journal of the Optical Society of America B, vol. 14, no. 11, pp. 2980–2993, 1997. View at Scopus
2. F. Lederer, S. Darmanyan, and A. Kobyakov, Spatial Solitons, springer, Berlin, Germany, 2001.
3. X. u. Zhiyong, All-optical Soliton Control in Photonic Lattices, Master work, Universitat Politècnica de Catalunya, Barcelona, Spain, 2007.
4. Y. S. Kivshar, “Bright and dark spatial solitons in non-Kerr media,” Optical and Quantum Electronics, vol. 30, no. 7–10, pp. 571–614, 1998. View at Scopus
5. Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, “Stable one-dimensional periodic waves in Kerr-type saturable and quadratic nonlinear media,” Journal of Optics B, vol. 6, no. 5, pp. S279–S287, 2004. View at Publisher · View at Google Scholar · View at Scopus |
db51bde206b040a3 | Tuesday, May 26, 2009
The Schrödinger Equation
Update: A corrected and improved version of this post is now up: http://behindtheguesses.blogspot.com/2009/06/schrodinger-equation-corrections.html
notElon asked me to discuss, and to try and derive the Schrödinger equation, so I'll give it a shot. This derivation is partially based on Sakurai,[1] with some differences.
A brief walk through classical mechanics
Say we have a function of and we want to translate it in space to a point . To do this, we'll find a ``space translation'' operator which, when applied to , gives . That is,
We'll expand in a Taylor series:
which can be simplified using the series expansion of the exponential1 to
from which we can conclude that
where is the -component of the angular momentum.
where is the generator of this particular transformation.2 See [2] for an example with Lorentz transformations.
From classical to quantum
In classical dynamics, the time derivative of a quantity is given by the Poisson bracket:
where is the classical Hamiltonian of the system and is shorthand for a messy equation.[3] In quantum mechanics this equation is replaced with
where the square brackets signify a commutation relation and is the quantum mechanical Hamiltonian.[4] This holds true for any quantity , and is a number which commutes with everything, so we can argue that the quantum mechanical Hamiltonian operator is related to the classical Hamiltonian by
So, using (4) the quantum mechanical space translation operator is given by
and, using (5), the rotation operator by
or, from (6) any arbitrary (unitary) transformation, , can be written as
where is (an Hermitian operator and is) the classical generator of the transformation.
Time translation of a quantum state
From our previous discussion we know that if we know the classical generator of time translation we can write using (13). Well, classically, the generator of time translations is the Hamiltonian![5] So we can write
and (14) becomes
This holds true for any time translation, so we'll consider a small time translation and expand (16) using a Taylor expansion3 dropping all quadratic and higher terms:
Moving things around gives
In the limit the righthand side becomes a partial derivative giving the Schrödinger equation
[1] J.J. Sakurai. Modern Quantum Mechanics. Addison-Wesley, San Francisco, CA, revised edition, 1993.
[3] L.D. Landau and E.M. Lifshitz. Mechanics. Pergamon Press, Oxford, UK.
[4] L.D. Landau and E.M. Lifshitz. Quantum Mechanics. Butterworth-Heinemann, Oxford, UK.
1. That's an interesting approach, and very clearly written and understandable.
It seems to me however somewhat against the grain of "behind the guesses". In particular, equation 8, motivating the -i/h difference between the quantum and classical Hamiltonians, appears to be to have been pulled out of a magic hat. Although it obviously works to get the desired equation, a natural motivation for this factor or the quantum version of the poisson bracket would be desirable.
Also, it isn't clear to me that the generalization in (10) to "any" quantum operator from it's classical counterpart is justifyable or if it is the details of what that mean exactly are not clear (consider the one dimensional position operator with no i/hbar factors until you go to the momentum space representation).
2. Peeter,
Thanks for your comment. I agree that Eq. (8) is a bit "out of the blue," but going through that derivation is really a bit more off topic. Maybe I'll cover it in another blog post :-) But see the Landau and Lifshitz book (ref [4]), section 9, pp. 27-27 and footnote for an excellent discussion (they differ by a factor of i).
Regarding your latter point, I think you are right -- I made a mistake (I'll correct it in a new post soon). Briefly, though, as I noted, any classical Poisson bracket can be transferred over to the QM commutator with the iℏ. This means that there's a difference of iℏ for only one of the operators. For the Hamiltonian equation (8) f can be anything, so my argument holds. But for the x and p operators there's a vagueness. So in one representation the p's get those factors, in another the x's do.
3. Perhaps offtopic, but what would you recommend as the best QM book(s), or online references, for self study. I've currently got Bohm's Quantum theory, French's Introductory QM, Pauli's Wave Mechanics, and Feynman's volume 3, but am lacking any text with a modern treatment.
I've been working my way through these kind of lock step. Bohm's text is very well laid out and the problems are helpful. Feynman's doesn't have problems which makes it fairly hard to actively use (will probably be better once I know the subject). Pauli's is fairly dense and takes a lot of puzzling out, but has helpful bits, and French's I plan to revisit in more detail later (having covered some of it eons ago when I did my engineering undergrad). I wasn't impressed with Liboff's book which I borrowed from the public library (too many magic hats used there).
4. Hi,
You cited a couple books, but do you happen to have the pages/chapters in which the specific examples appear?
5. Lucas,
In Sakurai: pp. 68-72
Jackson: pp. 543-548
Landau Lifshitz CM: pp. 135-138
Landau Lifshitz QM: pp. 27-27
Goldstein: 407-408
Contact me via email: elansey@gmail.com
One of these days I'll make a recommended book section on the sidebar. |
b88a5353dda71ecf | Density functional theory
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Density functional theory (DFT) is a computational quantum mechanical modelling method used in physics, chemistry and materials science to investigate the electronic structure (principally the ground state) of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by using functionals, i.e. functions of another function, which in this case is the spatially dependent electron density. Hence the name density functional theory comes from the use of functionals of the electron density. DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry.
Despite recent improvements, there are still difficulties in using density functional theory to properly describe intermolecular interactions, especially van der Waals forces (dispersion); charge transfer excitations; transition states, global potential energy surfaces, dopant interactions and some other strongly correlated systems; and in calculations of the band gap and ferromagnetism in semiconductors.[1] Its incomplete treatment of dispersion can adversely affect the accuracy of DFT (at least when used alone and uncorrected) in the treatment of systems which are dominated by dispersion (e.g. interacting noble gas atoms)[2] or where dispersion competes significantly with other effects (e.g. in biomolecules).[3] The development of new DFT methods designed to overcome this problem, by alterations to the functional and inclusion of additional terms to account for both core and valence electrons [4] or by the inclusion of additive terms,[5][6][7][8] is a current research topic.
Overview of method[edit]
Although density functional theory has its conceptual roots in the Thomas–Fermi model, DFT was put on a firm theoretical footing by the two Hohenberg–Kohn theorems (H–K).[9] The original H–K theorems held only for non-degenerate ground states in the absence of a magnetic field, although they have since been generalized to encompass these.[10][11]
The first H–K theorem demonstrates that the ground state properties of a many-electron system are uniquely determined by an electron density that depends on only 3 spatial coordinates. It lays the groundwork for reducing the many-body problem of N electrons with 3N spatial coordinates to 3 spatial coordinates, through the use of functionals of the electron density. This theorem can be extended to the time-dependent domain to develop time-dependent density functional theory (TDDFT), which can be used to describe excited states.
The second H–K theorem defines an energy functional for the system and proves that the correct ground state electron density minimizes this energy functional.
Within the framework of Kohn–Sham DFT (KS DFT), the intractable many-body problem of interacting electrons in a static external potential is reduced to a tractable problem of non-interacting electrons moving in an effective potential. The effective potential includes the external potential and the effects of the Coulomb interactions between the electrons, e.g., the exchange and correlation interactions. Modeling the latter two interactions becomes the difficulty within KS DFT. The simplest approximation is the local-density approximation (LDA), which is based upon exact exchange energy for a uniform electron gas, which can be obtained from the Thomas–Fermi model, and from fits to the correlation energy for a uniform electron gas. Non-interacting systems are relatively easy to solve as the wavefunction can be represented as a Slater determinant of orbitals. Further, the kinetic energy functional of such a system is known exactly. The exchange-correlation part of the total-energy functional remains unknown and must be approximated.
Another approach, less popular than KS DFT but arguably more closely related to the spirit of the original H-K theorems, is orbital-free density functional theory (OFDFT), in which approximate functionals are also used for the kinetic energy of the non-interacting system.
Derivation and formalism[edit]
As usual in many-body electronic structure calculations, the nuclei of the treated molecules or clusters are seen as fixed (the Born–Oppenheimer approximation), generating a static external potential V in which the electrons are moving. A stationary electronic state is then described by a wavefunction \Psi(\vec r_1,\dots,\vec r_N) satisfying the many-electron time-independent Schrödinger equation
\hat H \Psi = \left[{\hat T}+{\hat V}+{\hat U}\right]\Psi = \left[\sum_i^N \left(-\frac{\hbar^2}{2m_i}\nabla_i^2\right) + \sum_i^N V(\vec r_i) + \sum_{i<j}^N U(\vec r_i, \vec r_j)\right] \Psi = E \Psi
where, for the \ N -electron system, \hat H is the Hamiltonian, \ E is the total energy, \hat T is the kinetic energy, \hat V is the potential energy from the external field due to positively charged nuclei, and \hat U is the electron-electron interaction energy. The operators \hat T and \hat U are called universal operators as they are the same for any \ N -electron system, while \hat V is system dependent. This complicated many-particle equation is not separable into simpler single-particle equations because of the interaction term \hat U .
There are many sophisticated methods for solving the many-body Schrödinger equation based on the expansion of the wavefunction in Slater determinants. While the simplest one is the Hartree–Fock method, more sophisticated approaches are usually categorized as post-Hartree–Fock methods. However, the problem with these methods is the huge computational effort, which makes it virtually impossible to apply them efficiently to larger, more complex systems.
Here DFT provides an appealing alternative, being much more versatile as it provides a way to systematically map the many-body problem, with \hat U , onto a single-body problem without \hat U . In DFT the key variable is the particle density n(\vec r), which for a normalized \,\!\Psi is given by
n(\vec r) = N \int{\rm d}^3r_2 \cdots \int{\rm d}^3r_N \Psi^*(\vec r,\vec r_2,\dots,\vec r_N) \Psi(\vec r,\vec r_2,\dots,\vec r_N).
This relation can be reversed, i.e., for a given ground-state density n_0(\vec r) it is possible, in principle, to calculate the corresponding ground-state wavefunction \Psi_0(\vec r_1,\dots,\vec r_N). In other words, \,\!\Psi is a unique functional of \,\!n_0,[9]
\,\!\Psi_0 = \Psi[n_0]
and consequently the ground-state expectation value of an observable \,\hat O is also a functional of \,\!n_0
O[n_0] = \left\langle \Psi[n_0] \left| \hat O \right| \Psi[n_0] \right\rangle.
In particular, the ground-state energy is a functional of \,\!n_0
E_0 = E[n_0] = \left\langle \Psi[n_0] \left| \hat T + \hat V + \hat U \right| \Psi[n_0] \right\rangle
where the contribution of the external potential \left\langle \Psi[n_0] \left|\hat V \right| \Psi[n_0] \right\rangle can be written explicitly in terms of the ground-state density \,\!n_0
V[n_0] = \int V(\vec r) n_0(\vec r){\rm d}^3r.
More generally, the contribution of the external potential \left\langle \Psi \left|\hat V \right| \Psi \right\rangle can be written explicitly in terms of the density \,\!n,
V[n] = \int V(\vec r) n(\vec r){\rm d}^3r.
The functionals \,\!T[n] and \,\!U[n] are called universal functionals, while \,\!V[n] is called a non-universal functional, as it depends on the system under study. Having specified a system, i.e., having specified \hat V, one then has to minimize the functional
E[n] = T[n]+ U[n] + \int V(\vec r) n(\vec r){\rm d}^3r
with respect to n(\vec r), assuming one has got reliable expressions for \,\!T[n] and \,\!U[n]. A successful minimization of the energy functional will yield the ground-state density \,\!n_0 and thus all other ground-state observables.
The variational problems of minimizing the energy functional \,\!E[n] can be solved by applying the Lagrangian method of undetermined multipliers.[12] First, one considers an energy functional that doesn't explicitly have an electron-electron interaction energy term,
E_s[n] = \left\langle \Psi_s[n] \left| \hat T + \hat V_s \right| \Psi_s[n] \right\rangle
where \hat T denotes the kinetic energy operator and \hat V_s is an external effective potential in which the particles are moving, so that n_s(\vec r)\ \stackrel{\mathrm{def}}{=}\ n(\vec r).
Thus, one can solve the so-called Kohn–Sham equations of this auxiliary non-interacting system,
\left[-\frac{\hbar^2}{2m}\nabla^2+V_s(\vec r)\right] \phi_i(\vec r) = \epsilon_i \phi_i(\vec r)
which yields the orbitals \,\!\phi_i that reproduce the density n(\vec r) of the original many-body system
n(\vec r )\ \stackrel{\mathrm{def}}{=}\ n_s(\vec r)= \sum_i^N \left|\phi_i(\vec r)\right|^2.
The effective single-particle potential can be written in more detail as
V_s(\vec r) = V(\vec r) + \int \frac{e^2n_s(\vec r\,')}{|\vec r-\vec r\,'|} {\rm d}^3r' + V_{\rm XC}[n_s(\vec r)]
where the second term denotes the so-called Hartree term describing the electron-electron Coulomb repulsion, while the last term \,\!V_{\rm XC} is called the exchange-correlation potential. Here, \,\!V_{\rm XC} includes all the many-particle interactions. Since the Hartree term and \,\!V_{\rm XC} depend on n(\vec r ), which depends on the \,\!\phi_i, which in turn depend on \,\!V_s, the problem of solving the Kohn–Sham equation has to be done in a self-consistent (i.e., iterative) way. Usually one starts with an initial guess for n(\vec r), then calculates the corresponding \,\!V_s and solves the Kohn–Sham equations for the \,\!\phi_i. From these one calculates a new density and starts again. This procedure is then repeated until convergence is reached. A non-iterative approximate formulation called Harris functional DFT is an alternative approach to this.
NOTE1: The one-to-one correspondence between electron density and single-particle potential is not so smooth. It contains kinds of non-analytic structure. E_s[n] contains kinds of singularities, cuts and branches. This may indicate a limitation of our hope for representing exchange-correlation functional in a simple analytic form.
NOTE2: It is possible to extend the DFT idea to the case of Green function G instead of the density n. It is called as Luttinger-Ward functional (or kinds of similar functionals), written as E[G]. However,G is determined not as its minimum, but as its extremum. Thus we may have some theoretical and practical difficulties.
NOTE3: There is no one-to-one correspondence between one-body density matrix n({\vec r},{\vec r}') and the one-body potential V({\vec r},{\vec r}'). (Remember that all the eigenvalues of n({\vec r},{\vec r}') is unity). In other words, it ends up with a theory similar as the Hartree-Fock (or hybrid) theory.
Approximations (exchange-correlation functionals)[edit]
The major problem with DFT is that the exact functionals for exchange and correlation are not known except for the free electron gas. However, approximations exist which permit the calculation of certain physical quantities quite accurately. In physics the most widely used approximation is the local-density approximation (LDA), where the functional depends only on the density at the coordinate where the functional is evaluated:
E_{\rm XC}^{\rm LDA}[n]=\int\epsilon_{\rm XC}(n)n (\vec{r}) {\rm d}^3r.
The local spin-density approximation (LSDA) is a straightforward generalization of the LDA to include electron spin:
E_{\rm XC}^{\rm LSDA}[n_\uparrow,n_\downarrow]=\int\epsilon_{\rm XC}(n_\uparrow,n_\downarrow)n (\vec{r}){\rm d}^3r.
Highly accurate formulae for the exchange-correlation energy density \epsilon_{\rm XC}(n_\uparrow,n_\downarrow) have been constructed from quantum Monte Carlo simulations of jellium.[13]
Generalized gradient approximations[14][15][16] (GGA) are still local but also take into account the gradient of the density at the same coordinate:
E_{XC}^{\rm GGA}[n_\uparrow,n_\downarrow]=\int\epsilon_{XC}(n_\uparrow,n_\downarrow,\vec{\nabla}n_\uparrow,\vec{\nabla}n_\downarrow)
Using the latter (GGA) very good results for molecular geometries and ground-state energies have been achieved.
Potentially more accurate than the GGA functionals are the meta-GGA functionals, a natural development after the GGA (generalized gradient approximation). Meta-GGA DFT functional in its original form includes the second derivative of the electron density (the Laplacian) whereas GGA includes only the density and its first derivative in the exchange-correlation potential.
Functionals of this type are, for example, TPSS and the Minnesota Functionals. These functionals include a further term in the expansion, depending on the density, the gradient of the density and the Laplacian (second derivative) of the density.
Difficulties in expressing the exchange part of the energy can be relieved by including a component of the exact exchange energy calculated from Hartree–Fock theory. Functionals of this type are known as hybrid functionals.
Generalizations to include magnetic fields[edit]
The DFT formalism described above breaks down, to various degrees, in the presence of a vector potential, i.e. a magnetic field. In such a situation, the one-to-one mapping between the ground-state electron density and wavefunction is lost. Generalizations to include the effects of magnetic fields have led to two different theories: current density functional theory (CDFT) and magnetic field density functional theory (BDFT). In both these theories, the functional used for the exchange and correlation must be generalized to include more than just the electron density. In current density functional theory, developed by Vignale and Rasolt,[11] the functionals become dependent on both the electron density and the paramagnetic current density. In magnetic field density functional theory, developed by Salsbury, Grayce and Harris,[17] the functionals depend on the electron density and the magnetic field, and the functional form can depend on the form of the magnetic field. In both of these theories it has been difficult to develop functionals beyond their equivalent to LDA, which are also readily implementable computationally. Recently an extension by Pan and Sahni [18] extended the Hohenberg-Kohn theorem for non constant magnetic fields using the density and the current density as fundamental variables.
C60 with isosurface of ground-state electron density as calculated with DFT.
In general, density functional theory finds increasingly broad application in the chemical and material sciences for the interpretation and prediction of complex system behavior at an atomic scale. Specifically, DFT computational methods are applied for the study of systems exhibiting high sensitivity to synthesis and processing parameters. In such systems, experimental studies are often encumbered by inconsistent results and non-equilibrium conditions. Examples of contemporary DFT applications include studying the effects of dopants on phase transformation behavior in oxides, magnetic behaviour in dilute magnetic semiconductor materials and the study of magnetic and electronic behavior in ferroelectrics and dilute magnetic semiconductors.[19][20]
In practice, Kohn–Sham theory can be applied in several distinct ways depending on what is being investigated. In solid state calculations, the local density approximations are still commonly used along with plane wave basis sets, as an electron gas approach is more appropriate for electrons delocalised through an infinite solid. In molecular calculations, however, more sophisticated functionals are needed, and a huge variety of exchange-correlation functionals have been developed for chemical applications. Some of these are inconsistent with the uniform electron gas approximation, however, they must reduce to LDA in the electron gas limit. Among physicists, probably the most widely used functional is the revised Perdew–Burke–Ernzerhof exchange model (a direct generalized-gradient parametrization of the free electron gas with no free parameters); however, this is not sufficiently calorimetrically accurate for gas-phase molecular calculations. In the chemistry community, one popular functional is known as BLYP (from the name Becke for the exchange part and Lee, Yang and Parr for the correlation part). Even more widely used is B3LYP which is a hybrid functional in which the exchange energy, in this case from Becke's exchange functional, is combined with the exact energy from Hartree–Fock theory. Along with the component exchange and correlation funсtionals, three parameters define the hybrid functional, specifying how much of the exact exchange is mixed in. The adjustable parameters in hybrid functionals are generally fitted to a 'training set' of molecules. Unfortunately, although the results obtained with these functionals are usually sufficiently accurate for most applications, there is no systematic way of improving them (in contrast to some of the traditional wavefunction-based methods like configuration interaction or coupled cluster theory). Hence in the current DFT approach it is not possible to estimate the error of the calculations without comparing them to other methods or experiments.
Thomas–Fermi model[edit]
The predecessor to density functional theory was the Thomas–Fermi model, developed independently by both Thomas and Fermi in 1927. They used a statistical model to approximate the distribution of electrons in an atom. The mathematical basis postulated that electrons are distributed uniformly in phase space with two electrons in every h^{3} of volume.[21] For each element of coordinate space volume d^{3}r we can fill out a sphere of momentum space up to the Fermi momentum p_f [22]
\frac43\pi p_f^3(\vec{r}).
Equating the number of electrons in coordinate space to that in phase space gives:
Solving for p_{f} and substituting into the classical kinetic energy formula then leads directly to a kinetic energy represented as a functional of the electron density:
t_{TF}[n] = \frac{p^2}{2m_e} \propto \frac{(n^\frac13)^2}{2m_e} \propto n^\frac23(\vec{r})
T_{TF}[n]= C_F \int n(\vec{r}) n^\frac23(\vec{r}) d^3r =C_F\int n^\frac53(\vec{r}) d^3r
where C_F=\frac{3h^2}{10m_e}\left(\frac{3}{8\pi}\right)^\frac23.
As such, they were able to calculate the energy of an atom using this kinetic energy functional combined with the classical expressions for the nuclear-electron and electron-electron interactions (which can both also be represented in terms of the electron density).
Although this was an important first step, the Thomas–Fermi equation's accuracy is limited because the resulting kinetic energy functional is only approximate, and because the method does not attempt to represent the exchange energy of an atom as a conclusion of the Pauli principle. An exchange energy functional was added by Dirac in 1928.
However, the Thomas–Fermi–Dirac theory remained rather inaccurate for most applications. The largest source of error was in the representation of the kinetic energy, followed by the errors in the exchange energy, and due to the complete neglect of electron correlation.
Teller (1962) showed that Thomas–Fermi theory cannot describe molecular bonding. This can be overcome by improving the kinetic energy functional.
The kinetic energy functional can be improved by adding the Weizsäcker (1935) correction:[23][24]
T_W[n]=\frac{\hbar^2}{8m}\int\frac{|\nabla n(\vec{r})|^2}{n(\vec{r})}d^3r.
Hohenberg–Kohn theorems[edit]
1.If two systems of electrons, one trapped in a potential v_1(\vec r) and the other in v_2(\vec r), have the same ground-state density n(\vec r) then necessarily v_1(\vec r)-v_2(\vec r) = const.
Corollary: the ground state density uniquely determines the potential and thus all properties of the system, including the many-body wave function. In particular, the "HK" functional, defined as F[n]=T[n]+U[n] is a universal functional of the density (not depending explicitly on the external potential).
2. For any positive integer N and potential v(\vec r), a density functional F[n] exists such that E_{(v,N)}[n] = F[n]+\int{v(\vec r)n(\vec r)d^3r} obtains its minimal value at the ground-state density of N electrons in the potential v(\vec r). The minimal value of E_{(v,N)}[n] is then the ground state energy of this system.
The many electron Schrödinger equation can be very much simplified if electrons are divided in two groups: valence electrons and inner core electrons. The electrons in the inner shells are strongly bound and do not play a significant role in the chemical binding of atoms; they also partially screen the nucleus, thus forming with the nucleus an almost inert core. Binding properties are almost completely due to the valence electrons, especially in metals and semiconductors. This separation suggests that inner electrons can be ignored in a large number of cases, thereby reducing the atom to an ionic core that interacts with the valence electrons. The use of an effective interaction, a pseudopotential, that approximates the potential felt by the valence electrons, was first proposed by Fermi in 1934 and Hellmann in 1935. In spite of the simplification pseudo-potentials introduce in calculations, they remained forgotten until the late 50's.
Ab initio Pseudo-potentials
A crucial step toward more realistic pseudo-potentials was given by Topp and Hopfield and more recently Cronin, who suggested that the pseudo-potential should be adjusted such that they describe the valence charge density accurately. Based on that idea, modern pseudo-potentials are obtained inverting the free atom Schrödinger equation for a given reference electronic configuration and forcing the pseudo wave-functions to coincide with the true valence wave functions beyond a certain distance rl_.. The pseudo wave-functions are also forced to have the same norm as the true valence wave-functions and can be written as
R_{\rm l}^{\rm pp}(r)=R_{\rm nl}^{\rm AE}(r).
\int_{0}^{rl}dr|R_{\rm l}^{\rm PP}(r)|^2r^2=\int_{0}^{rl}dr|R_{\rm nl}^{\rm AE}(r)|^2r^2.
where R_{\rm l}(r). is the radial part of the wavefunction with angular momentum l_., and pp_. and AE_. denote, respectively, the pseudo wave-function and the true (all-electron) wave-function. The index n in the true wave-functions denotes the valence level. The distance beyond which the true and the pseudo wave-functions are equal, rl_., is also l_.-dependent.
Software supporting DFT[edit]
DFT is supported by many Quantum chemistry and solid state physics software packages, often along with other methods.
See also[edit]
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3. ^ Vondrášek, Jiří; Bendová, Lada; Klusák, Vojtěch; Hobza, Pavel (2005). "Unexpectedly strong energy stabilization inside the hydrophobic core of small protein rubredoxin mediated by aromatic residues: correlated ab initio quantum chemical calculations". Journal of the American Chemical Society 127 (8): 2615–2619. doi:10.1021/ja044607h. PMID 15725017.
4. ^ Grimme, Stefan (2006). "Semiempirical hybrid density functional with perturbative second-order correlation". Journal of Chemical Physics 124 (3): 034108. Bibcode:2006JChPh.124c4108G. doi:10.1063/1.2148954. PMID 16438568.
5. ^ Zimmerli, Urs; Parrinello, Michele; Koumoutsakos, Petros (2004). "Dispersion corrections to density functionals for water aromatic interactions". Journal of Chemical Physics 120 (6): 2693–2699. Bibcode:2004JChPh.120.2693Z. doi:10.1063/1.1637034. PMID 15268413.
6. ^ Grimme, Stefan (2004). "Accurate description of van der Waals complexes by density functional theory including empirical corrections". Journal of Computational Chemistry 25 (12): 1463–1473. doi:10.1002/jcc.20078. PMID 15224390.
7. ^ Von Lilienfeld, O. Anatole; Tavernelli, Ivano; Rothlisberger, Ursula; Sebastiani, Daniel (2004). "Optimization of effective atom centered potentials for London dispersion forces in density functional theory". Physical Review Letters 93 (15): 153004. Bibcode:2004PhRvL..93o3004V. doi:10.1103/PhysRevLett.93.153004. PMID 15524874.
8. ^ Tkatchenko, Alexandre; Scheffler, Matthias (2009). "Accurate Molecular Van Der Waals Interactions from Ground-State Electron Density and Free-Atom Reference Data". Physical Review Letters 102 (7): 073005. Bibcode:2009PhRvL.102g3005T. doi:10.1103/PhysRevLett.102.073005. PMID 19257665.
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10. ^ Levy, Mel (1979). "Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem". Proceedings of the National Academy of Sciences (United States National Academy of Sciences) 76 (12): 6062–6065. Bibcode:1979PNAS...76.6062L. doi:10.1073/pnas.76.12.6062.
11. ^ a b Vignale, G.; Mark Rasolt (1987). "Density-functional theory in strong magnetic fields". Physical Review Letters (American Physical Society) 59 (20): 2360–2363. Bibcode:1987PhRvL..59.2360V. doi:10.1103/PhysRevLett.59.2360. PMID 10035523.
12. ^ Kohn, W.; Sham, L. J. (1965). "Self-consistent equations including exchange and correlation effects". Physical Review 140 (4A): A1133–A1138. Bibcode:1965PhRv..140.1133K. doi:10.1103/PhysRev.140.A1133.
13. ^ John P. Perdew, Adrienn Ruzsinszky, Jianmin Tao, Viktor N. Staroverov, Gustavo Scuseria and Gábor I. Csonka (2005). "Prescriptions for the design and selection of density functional approximations: More constraint satisfaction with fewer fits". Journal of Chemical Physics 123 (6): 062201. Bibcode:2005JChPh.123f2201P. doi:10.1063/1.1904565. PMID 16122287.
14. ^ Perdew, John P; Chevary, J A; Vosko, S H; Jackson, Koblar, A; Pederson, Mark R; Singh, D J; Fiolhais, Carlos (1992). "Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation". Physical Review B 46 (11): 6671. doi:10.1103/physrevb.46.6671.
15. ^ Becke, Axel D (1988). "Density-functional exchange-energy approximation with correct asymptotic behavior". Physical Review A 38 (6): 3098. doi:10.1103/physreva.38.3098.
16. ^ Langreth, David C; Mehl, M J (1983). "Beyond the local-density approximation in calculations of ground-state electronic properties". Physical Review B 28 (4): 1809. doi:10.1103/physrevb.28.1809.
17. ^ Grayce, Christopher; Robert Harris (1994). "Magnetic-field density-functional theory". Physical Review A 50 (4): 3089–3095. Bibcode:1994PhRvA..50.3089G. doi:10.1103/PhysRevA.50.3089. PMID 9911249.
18. ^ Viraht, Xiao-Yin (2012). "Hohenberg-Kohn theorem including electron spin". Physical Review A 86. Bibcode:1994PhRvA.86.042502. doi:10.1103/physreva.86.042502.
19. ^ Segall, M.D.; Lindan, P.J (2002). "First-principles simulation: ideas, illustrations and the CASTEP code". Journal of Physics: Condensed Matter 14 (11): 2717. Bibcode:2002JPCM...14.2717S. doi:10.1088/0953-8984/14/11/301.
20. ^ "Ab initio study of phase stability in doped TiO2". Computational Mechanics 50 (2): 185–194. 2012. doi:10.1007/s00466-012-0728-4.
21. ^ (Parr & Yang 1989, p. 47)
22. ^ March, N. H. (1992). Electron Density Theory of Atoms and Molecules. Academic Press. p. 24. ISBN 0-12-470525-1.
23. ^ Weizsäcker, C. F. v. (1935). "Zur Theorie der Kernmassen". Zeitschrift für Physik 96 (7–8): 431–58. Bibcode:1935ZPhy...96..431W. doi:10.1007/BF01337700.
24. ^ (Parr & Yang 1989, p. 127)
Key papers[edit]
External links[edit] |
3ccefea68ce1ac4c | Why Probability in Quantum Mechanics is Given by the Wave Function Squared
One of the most profound and mysterious principles in all of physics is the Born Rule, named after Max Born. In quantum mechanics, particles don’t have classical properties like “position” or “momentum”; rather, there is a wave function that assigns a (complex) number, called the “amplitude,” to each possible measurement outcome. The Born Rule is then very simple: it says that the probability of obtaining any possible measurement outcome is equal to the square of the corresponding amplitude. (The wave function is just the set of all the amplitudes.)
Born Rule: \mathrm{Probability}(x) = |\mathrm{amplitude}(x)|^2.
The Born Rule is certainly correct, as far as all of our experimental efforts have been able to discern. But why? Born himself kind of stumbled onto his Rule. Here is an excerpt from his 1926 paper:
Born Rule
That’s right. Born’s paper was rejected at first, and when it was later accepted by another journal, he didn’t even get the Born Rule right. At first he said the probability was equal to the amplitude, and only in an added footnote did he correct it to being the amplitude squared. And a good thing, too, since amplitudes can be negative or even imaginary!
The status of the Born Rule depends greatly on one’s preferred formulation of quantum mechanics. When we teach quantum mechanics to undergraduate physics majors, we generally give them a list of postulates that goes something like this:
1. Quantum states are represented by wave functions, which are vectors in a mathematical space called Hilbert space.
2. Wave functions evolve in time according to the Schrödinger equation.
3. The act of measuring a quantum system returns a number, known as the eigenvalue of the quantity being measured.
4. The probability of getting any particular eigenvalue is equal to the square of the amplitude for that eigenvalue.
5. After the measurement is performed, the wave function “collapses” to a new state in which the wave function is localized precisely on the observed eigenvalue (as opposed to being in a superposition of many different possibilities).
It’s an ungainly mess, we all agree. You see that the Born Rule is simply postulated right there, as #4. Perhaps we can do better.
Of course we can do better, since “textbook quantum mechanics” is an embarrassment. There are other formulations, and you know that my own favorite is Everettian (“Many-Worlds”) quantum mechanics. (I’m sorry I was too busy to contribute to the active comment thread on that post. On the other hand, a vanishingly small percentage of the 200+ comments actually addressed the point of the article, which was that the potential for many worlds is automatically there in the wave function no matter what formulation you favor. Everett simply takes them seriously, while alternatives need to go to extra efforts to erase them. As Ted Bunn argues, Everett is just “quantum mechanics,” while collapse formulations should be called “disappearing-worlds interpretations.”)
Like the textbook formulation, Everettian quantum mechanics also comes with a list of postulates. Here it is:
That’s it! Quite a bit simpler — and the two postulates are exactly the same as the first two of the textbook approach. Everett, in other words, is claiming that all the weird stuff about “measurement” and “wave function collapse” in the conventional way of thinking about quantum mechanics isn’t something we need to add on; it comes out automatically from the formalism.
The trickiest thing to extract from the formalism is the Born Rule. That’s what Charles (“Chip”) Sebens and I tackled in our recent paper:
Self-Locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics
Charles T. Sebens, Sean M. Carroll
A longstanding issue in attempts to understand the Everett (Many-Worlds) approach to quantum mechanics is the origin of the Born rule: why is the probability given by the square of the amplitude? Following Vaidman, we note that observers are in a position of self-locating uncertainty during the period between the branches of the wave function splitting via decoherence and the observer registering the outcome of the measurement. In this period it is tempting to regard each branch as equiprobable, but we give new reasons why that would be inadvisable. Applying lessons from this analysis, we demonstrate (using arguments similar to those in Zurek’s envariance-based derivation) that the Born rule is the uniquely rational way of apportioning credence in Everettian quantum mechanics. In particular, we rely on a single key principle: changes purely to the environment do not affect the probabilities one ought to assign to measurement outcomes in a local subsystem. We arrive at a method for assigning probabilities in cases that involve both classical and quantum self-locating uncertainty. This method provides unique answers to quantum Sleeping Beauty problems, as well as a well-defined procedure for calculating probabilities in quantum cosmological multiverses with multiple similar observers.
Chip is a graduate student in the philosophy department at Michigan, which is great because this work lies squarely at the boundary of physics and philosophy. (I guess it is possible.) The paper itself leans more toward the philosophical side of things; if you are a physicist who just wants the equations, we have a shorter conference proceeding.
Before explaining what we did, let me first say a bit about why there’s a puzzle at all. Let’s think about the wave function for a spin, a spin-measuring apparatus, and an environment (the rest of the world). It might initially take the form
(α[up] + β[down] ; apparatus says “ready” ; environment0). (1)
This might look a little cryptic if you’re not used to it, but it’s not too hard to grasp the gist. The first slot refers to the spin. It is in a superposition of “up” and “down.” The Greek letters α and β are the amplitudes that specify the wave function for those two possibilities. The second slot refers to the apparatus just sitting there in its ready state, and the third slot likewise refers to the environment. By the Born Rule, when we make a measurement the probability of seeing spin-up is |α|2, while the probability for seeing spin-down is |β|2.
In Everettian quantum mechanics (EQM), wave functions never collapse. The one we’ve written will smoothly evolve into something that looks like this:
α([up] ; apparatus says “up” ; environment1)
+ β([down] ; apparatus says “down” ; environment2). (2)
This is an extremely simplified situation, of course, but it is meant to convey the basic appearance of two separate “worlds.” The wave function has split into branches that don’t ever talk to each other, because the two environment states are different and will stay that way. A state like this simply arises from normal Schrödinger evolution from the state we started with.
So here is the problem. After the splitting from (1) to (2), the wave function coefficients α and β just kind of go along for the ride. If you find yourself in the branch where the spin is up, your coefficient is α, but so what? How do you know what kind of coefficient is sitting outside the branch you are living on? All you know is that there was one branch and now there are two. If anything, shouldn’t we declare them to be equally likely (so-called “branch-counting”)? For that matter, in what sense are there probabilities at all? There was nothing stochastic or random about any of this process, the entire evolution was perfectly deterministic. It’s not right to say “Before the measurement, I didn’t know which branch I was going to end up on.” You know precisely that one copy of your future self will appear on each branch. Why in the world should we be talking about probabilities?
Note that the pressing question is not so much “Why is the probability given by the wave function squared, rather than the absolute value of the wave function, or the wave function to the fourth, or whatever?” as it is “Why is there a particular probability rule at all, since the theory is deterministic?” Indeed, once you accept that there should be some specific probability rule, it’s practically guaranteed to be the Born Rule. There is a result called Gleason’s Theorem, which says roughly that the Born Rule is the only consistent probability rule you can conceivably have that depends on the wave function alone. So the real question is not “Why squared?”, it’s “Whence probability?”
Of course, there are promising answers. Perhaps the most well-known is the approach developed by Deutsch and Wallace based on decision theory. There, the approach to probability is essentially operational: given the setup of Everettian quantum mechanics, how should a rational person behave, in terms of making bets and predicting experimental outcomes, etc.? They show that there is one unique answer, which is given by the Born Rule. In other words, the question “Whence probability?” is sidestepped by arguing that reasonable people in an Everettian universe will act as if there are probabilities that obey the Born Rule. Which may be good enough.
But it might not convince everyone, so there are alternatives. One of my favorites is Wojciech Zurek’s approach based on “envariance.” Rather than using words like “decision theory” and “rationality” that make physicists nervous, Zurek claims that the underlying symmetries of quantum mechanics pick out the Born Rule uniquely. It’s very pretty, and I encourage anyone who knows a little QM to have a look at Zurek’s paper. But it is subject to the criticism that it doesn’t really teach us anything that we didn’t already know from Gleason’s theorem. That is, Zurek gives us more reason to think that the Born Rule is uniquely preferred by quantum mechanics, but it doesn’t really help with the deeper question of why we should think of EQM as a theory of probabilities at all.
Here is where Chip and I try to contribute something. We use the idea of “self-locating uncertainty,” which has been much discussed in the philosophical literature, and has been applied to quantum mechanics by Lev Vaidman. Self-locating uncertainty occurs when you know that there multiple observers in the universe who find themselves in exactly the same conditions that you are in right now — but you don’t know which one of these observers you are. That can happen in “big universe” cosmology, where it leads to the measure problem. But it automatically happens in EQM, whether you like it or not.
Think of observing the spin of a particle, as in our example above. The steps are:
1. Everything is in its starting state, before the measurement.
2. The apparatus interacts with the system to be observed and becomes entangled. (“Pre-measurement.”)
3. The apparatus becomes entangled with the environment, branching the wave function. (“Decoherence.”)
4. The observer reads off the result of the measurement from the apparatus.
The point is that in between steps 3. and 4., the wave function of the universe has branched into two, but the observer doesn’t yet know which branch they are on. There are two copies of the observer that are in identical states, even though they’re part of different “worlds.” That’s the moment of self-locating uncertainty. Here it is in equations, although I don’t think it’s much help.
You might say “What if I am the apparatus myself?” That is, what if I observe the outcome directly, without any intermediating macroscopic equipment? Nice try, but no dice. That’s because decoherence happens incredibly quickly. Even if you take the extreme case where you look at the spin directly with your eyeball, the time it takes the state of your eye to decohere is about 10-21 seconds, whereas the timescales associated with the signal reaching your brain are measured in tens of milliseconds. Self-locating uncertainty is inevitable in Everettian quantum mechanics. In that sense, probability is inevitable, even though the theory is deterministic — in the phase of uncertainty, we need to assign probabilities to finding ourselves on different branches.
So what do we do about it? As I mentioned, there’s been a lot of work on how to deal with self-locating uncertainty, i.e. how to apportion credences (degrees of belief) to different possible locations for yourself in a big universe. One influential paper is by Adam Elga, and comes with the charming title of “Defeating Dr. Evil With Self-Locating Belief.” (Philosophers have more fun with their titles than physicists do.) Elga argues for a principle of Indifference: if there are truly multiple copies of you in the world, you should assume equal likelihood for being any one of them. Crucially, Elga doesn’t simply assert Indifference; he actually derives it, under a simple set of assumptions that would seem to be the kind of minimal principles of reasoning any rational person should be ready to use.
But there is a problem! Naïvely, applying Indifference to quantum mechanics just leads to branch-counting — if you assign equal probability to every possible appearance of equivalent observers, and there are two branches, each branch should get equal probability. But that’s a disaster; it says we should simply ignore the amplitudes entirely, rather than using the Born Rule. This bit of tension has led to some worry among philosophers who worry about such things.
Resolving this tension is perhaps the most useful thing Chip and I do in our paper. Rather than naïvely applying Indifference to quantum mechanics, we go back to the “simple assumptions” and try to derive it from scratch. We were able to pinpoint one hidden assumption that seems quite innocent, but actually does all the heavy lifting when it comes to quantum mechanics. We call it the “Epistemic Separability Principle,” or ESP for short. Here is the informal version (see paper for pedantic careful formulations):
ESP: The credence one should assign to being any one of several observers having identical experiences is independent of features of the environment that aren’t affecting the observers.
That is, the probabilities you assign to things happening in your lab, whatever they may be, should be exactly the same if we tweak the universe just a bit by moving around some rocks on a planet orbiting a star in the Andromeda galaxy. ESP simply asserts that our knowledge is separable: how we talk about what happens here is independent of what is happening far away. (Our system here can still be entangled with some system far away; under unitary evolution, changing that far-away system doesn’t change the entanglement.)
The ESP is quite a mild assumption, and to me it seems like a necessary part of being able to think of the universe as consisting of separate pieces. If you can’t assign credences locally without knowing about the state of the whole universe, there’s no real sense in which the rest of the world is really separate from you. It is certainly implicitly used by Elga (he assumes that credences are unchanged by some hidden person tossing a coin).
With this assumption in hand, we are able to demonstrate that Indifference does not apply to branching quantum worlds in a straightforward way. Indeed, we show that you should assign equal credences to two different branches if and only if the amplitudes for each branch are precisely equal! That’s because the proof of Indifference relies on shifting around different parts of the state of the universe and demanding that the answers to local questions not be altered; it turns out that this only works in quantum mechanics if the amplitudes are equal, which is certainly consistent with the Born Rule.
See the papers for the actual argument — it’s straightforward but a little tedious. The basic idea is that you set up a situation in which more than one quantum object is measured at the same time, and you ask what happens when you consider different objects to be “the system you will look at” versus “part of the environment.” If you want there to be a consistent way of assigning credences in all cases, you are led inevitably to equal probabilities when (and only when) the amplitudes are equal.
What if the amplitudes for the two branches are not equal? Here we can borrow some math from Zurek. (Indeed, our argument can be thought of as a love child of Vaidman and Zurek, with Elga as midwife.) In his envariance paper, Zurek shows how to start with a case of unequal amplitudes and reduce it to the case of many more branches with equal amplitudes. The number of these pseudo-branches you need is proportional to — wait for it — the square of the amplitude. Thus, you get out the full Born Rule, simply by demanding that we assign credences in situations of self-locating uncertainty in a way that is consistent with ESP.
We like this derivation in part because it treats probabilities as epistemic (statements about our knowledge of the world), not merely operational. Quantum probabilities are really credences — statements about the best degree of belief we can assign in conditions of uncertainty — rather than statements about truly stochastic dynamics or frequencies in the limit of an infinite number of outcomes. But these degrees of belief aren’t completely subjective in the conventional sense, either; there is a uniquely rational choice for how to assign them.
Working on this project has increased my own personal credence in the correctness of the Everett approach to quantum mechanics from “pretty high” to “extremely high indeed.” There are still puzzles to be worked out, no doubt, especially around the issues of exactly how and when branching happens, and how branching structures are best defined. (I’m off to a workshop next month to think about precisely these questions.) But these seem like relatively tractable technical challenges to me, rather than looming deal-breakers. EQM is an incredibly simple theory that (I can now argue in good faith) makes sense and fits the data. Now it’s just a matter of convincing the rest of the world!
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96 Responses to Why Probability in Quantum Mechanics is Given by the Wave Function Squared
1. Eric Winsberg says:
One problem I have with all of these attempts to get the born rule (the problem applies equally to your approach and to the Deutsch-Wallace approach) is that they all go like this.
1. Assume decoherence gets you branches in some preferred basis.
2. Give an argument that the Born rule applied to the amplitudes of these branches yields something worthy of the name ‘probability.’
The problem is that these steps happen in the reverse order that one would like them to happen. Look at step one. Decoherence arguments involve steps
1.a) showing that as the system+detector gets entangled with the environment, the reduced density matrix of this entangled pair evolves such that all the off-diagonal elements get very close to zero,
1.b)reasoning that therefore, each diagonal element corresponds to an emergent causally inert “branch.”
But step 1.b is fishy insofar as it happens before step 2. Who cares if the little numbers on the off-diagonals are very close to zero, until I know what their physical interpretation is? Not all very small numbers in physics can be interpreted as standing in front of unimportant things. Now, if we could accomplish step 2, then we could discard the off-diagonal elements, because we know that very small _probabilities_ are unimportant. But the cart has been put in front of the horse. We can’t conclude that the “Branches” are real and causally inert and have independent “obsevers” in them _until_ we have a physical interpretation of the off-diagonal elements being small. But all of these Everettian moves do 1.b first, and only afterwards do 2.
2. Rationalist says:
– This is where I lose the thread of the argument. This is the key problem with MWI and I would appreciate an intuitive explanation. It’s like you’ve just proved that 2+2=5 but the key step is “straightforward but technical”
3. Sean Carroll says:
Eric, I’m not sure I follow the worry. The fact that the off-diagonal elements are small tells us that the different branches don’t interfere with each other in terms of their future evolution. I.e., I could evolve a branch forward in time, and the result is completely independent of the existence of the other branches. That doesn’t seem to rely directly on any probability interpretation, but maybe I’m missing something.
4. Sean Carroll says:
Rationalist– Have a look at the paper. Sometimes arguments just have to be technical. 🙂
5. Stewart says:
Even if the off-diagonal elements are small, they are nonzero, so technically the branches still interfere, correct? How does this happen, and how can this be measured?
6. Peter Donis says:
The argument you give here shows how, given a particular wave function, the Born Rule gives the right credences for observing various outcomes. But how do you know the wave function in the first place?
In the real world, we know wave functions by observing relative frequencies of outcomes. For example, if you tell me that the device in your lab produces electrons with the (spin) wave function 1/sqrt(2) |up> + 1/sqrt(2) |down>, and I ask you how you know that, you’re not going to show me a mathematical derivation of what credence you should assign to up vs. down; you’re going to show me data from the test runs you made of the device, that recorded equal numbers of up electrons and down electrons.
But it seems to me that, if the MWI is true, we can’t draw that conclusion from the test data, because if the MWI is true, *any* wave function with nonzero amplitude for both |up> and |down> will produce a “world” in which equal numbers of up and down electrons are observed. So I don’t see how your argument justifies assigning equal amplitudes to |up> and |down> based on such test data.
7. Sean Carroll says:
Stewart– In principle, yes. But the numbers are incredibly super-tiny — you’d be better off looking at a glass of cool water and waiting for it to spontaneously evolve into an ice cube in a glass of warm water.
Peter– That’s something else we discuss in the paper. We show that the ordinary rules for Bayesian inference and hypothesis-testing are perfectly well respected by EQM. Of course unlikely things will happen, but that’s not what one should expect. It’s a big multiverse, so someone is going to be unlucky and experience very low-probability series of events (just as they would in a big classical universe).
8. Stewart says:
In principles, yes, so isn’t there anything sort of wrong about that? Anything that can happen, in quantum mechanics, will happen. So since the off-diagonals are nonzero, how will this interference take place when it happens? Can this be measured?
9. trivialknot says:
I admit I looked at the paper just so I could see your solution to the quantum sleeping beauty problem. Looks great! Makes me feel like some philosophical dilemmas really do have answers.
10. Eric Winsberg says:
Hi Sean,
OK, maybe that helps. But let me be clear about what you are saying. Suppose for simplicity that my system plus detector evolves into only two “branches”. You say “I could evolve a branch forward in time, and the result is completely independent of the existence of the other branches” I take it you really mean, as you say in response to Steward, that the degree to which they are not completely independent is represented by numbers that are incredibly super tiny. But I still have no physical interpretation of those tiny effects. you say “you’d be better off looking at a glass of cool water and waiting for it to spontaneously evolve into an ice cube in a glass of warm water” but I don’t know how you can say what impact those small numbers have on what I am likely to see until you have interpreted them as relating to probabilities.
11. Anon says:
The problem with the Everettian interpretation is that it assumes that QM is fundamentally correct. I think this is a fairly unsafe assumption and we have lots of (indirect) evidence to tell us that QM is incomplete.
Sure, if QM is complete as we know it, MWI is the simplest explanation. But it seems much more likely that QM is in fact not complete, and therefore any conclusions derived by assuming it is complete are meaningless.
12. kashyap vasavada says:
Whether to do the QM experiment at this moment is entirely arbitrary human decision. Is the branching taking place in observer’s mind or is it real? In either case this sounds like too much metaphysical. If the branching was decided before and the observer is just picking up a branch at random, that is even worse. In fact I am astonished that you can get away with such arguments when you openly attack religious and metaphysical arguments! Well, life is unfair!! Isn’t it?
13. D says:
Sean, you say that “Quantum probabilities are really credences … rather than statements about truly stochastic dynamics or frequencies in the limit of an infinite number of outcomes.” I don’t totally understand what you’re saying here — in the MWI, aren’t the probabilities both credences and frequencies? If I do a long sequence of approximately identical experiments, the quantum probabilities tell me the frequencies with which the different outcomes will be present in my branch of the wavefunction.
14. Keith Allpress says:
What about quantum recombination? Surely any physicality of all those copies is impossible if you intend to reconstitute a superposition in the same apparatus? Now you have a super super position of being in many worlds and not being in many worlds at the same time.
15. Reader297 says:
Several remarks.
The first is that classical physics does indeed allow us to describe multiple worlds provided that we interpret classical probabilities according to something like David Lewis’s modal realism. When studying the evolution of classical probability distributions, all the states “are just there” in the formalism, so why not simply accept that they exist in reality, as one does in EQM?
My second remark is about axioms. All logical claims consist of premises (axioms), arguments that follow from those premises, and conclusions, and EQM is no different. Proponents often suggest that EQM doesn’t need as many axioms as the traditional interpretations. But the trouble with EQM is that although it seems at first like you don’t need very many axioms, the truth is that you do. Simply insisting that we don’t mess with the superposition rule isn’t enough. Quantum-information instrumentalism (say, QBism) doesn’t mess with superpositions either, and allows arbitrarily large systems to surperpose. Declaring that we must interpret the elements of a superpositions as physical parallel universes is therefore an affirmative, nontrivial axiom about the ontology of the theory, even if some people might regard it as an “obvious” axiom.
The pointer-variable argument also implicitly assumes axioms as well. We have to declare that something singles out a preferred basis (for the cat, this means that we need to single out the alive vs. dead basis, rather than, say, the alive+dead vs. alive-dead basis). You can keep adding additional systems and environments, but at some point you have to declare that once you’ve added enough, you can shout “stop!” and pick your preferred basis. And what is our criterion for picking that basis? That’s going to be another axiom! And if you pick locality or something like that for specifying your preferred-basis-selection postulate, you have to contend with the fact that locality may not be a fundamental feature of reality once we figure out quantum gravity, so if we do add locality as part of our axiom for picking the preferred basis, the EQM interpretation is now sensitive to features of quantum gravity that we don’t know yet.
Finally, are you assuming that there’s some big universal wave function that evolves unitarily? Given all we know about eternal inflation, is this a reliable assumption anymore? Even if you’re willing to accept it, it represents another axiom to add on.
The problem with EQM is that this process of adding axioms keeps going on (your “epistemic separability principle,” for example, is another axiom, and far from an obvious one!), and even then we still have to contend with the serious trouble of trying to make sense of the concept of probability starting from deterministic assumptions, a serious philosophical problem on par with the is-ought problem of Hume.
So, to summarize, you can’t justifiably start by saying “Hey, I only need two axioms!” and then inserting additional axioms (some implicitly) as you proceed. At the end of the day, you’ll have as many axioms as (or more than), say, instrumentalism, but then you still have the weirdness of deriving probability from non-probability.
16. Avattoir says:
kashyap vasavada: Shorter – “Squirrel!”
17. Stuart says:
True, Quantum theory is incomplete.The complete form is called Quantum Gravity. In the complete theory a quantum of energy is associated with a wave packet of spacetime called a graviton.The MWI can be considered as the spliting of the spacetime wave packet.There 10^60 possible states for the spacetime wave packet each separated by an energy gap E=hH from which the cosmological constant =3(E/hc)^2. Here H is the Hubble constant,h Planck’s const. and c the speed of light.That is Quantum gravity should be able to inteprate quantum theory and General Relativity in one single framework.
18. Charlie says:
OK, biologist here so take it easy on me.
I’m still having trouble with the cartoon representation of MWI where you have a film splitting into two films. Is this supposed to apply only in (simple) cases of binary events? I understand the value of focusing on simple examples (spin-up/down), but what is the cartoon representation for a continuous range of possibilities (electron position)? Does the film split into an infinity of films? (A film shmear?)
[Asked in previous post but too late for answer.] Am I allowed to think of MWI as many superpositions rather than many universes? When the cartoon-filmstrip splits, I imagine all mass/energy doubling. However, when I think of Schrödinger’s Cat, it never occurred to me that you had 10 lbs of cat (before box closed), then somehow 20 lbs of cat (during superposition), then 10 lbs again when I observe it. It’s always been called Schrödinger’s Cat (singular) rather than Schrödinger’s Cats (plural) even before collapse. So why now must we have many worlds rather than one world in superposition?
And I have to ask (even though the answer seems obvious): Are there more worlds today than there were yesterday?
“There are still puzzles to be worked out, no doubt, especially around the issues of exactly how and when branching happens, …” I thought that a major appeal of this approach is that nothing “happens”. We have continuous evolution rather than “collapses” or any other magic moments.
19. M.Black says:
Thank you. I am a complete and abject layperson whose skill is reading English, not grasping the mathematics of quantum mechanics. But grasping English alone can you get you a little ways with a message as clear and consistent and easily stated as the Everettian premise: The sophisticated mathematical construction called the wave function, which to date matches quantum observations perfectly, describes the physical superposition of macroworlds. Then I read something like this article — a series of ideas, formulations, qualifications, theories and axioms dedicated to untying knots that, golly, just weren’t there in the beginning when I was promised the breath of simplicity itself — and nothing is quite so plain as the fact that there is nothing at all obvious about the “Many Worlds” interpretation, and that, for all the “evidence” at hand, the physical reality, if any, represented by the wave function is as far from being glimpsed as it has ever been.
20. Sean Carroll says:
About to run away, so some selective responses–
Eric– I think there is a fair point here, and I’m not sure I’ve thought it through completely. My feeling would be that it’s correct to say (1) off-diagonal terms are small, so branches evolve almost-independently, therefore (2) we can assign probabilities to branches, and once we do that we can (3) ask about the probability of the off-diagonal terms growing large and witnessing interference between branches. At the very least it seems like a self-consistent story.
D– The probabilities are credences at each individual branching. Of course they can lead to frequencies if you do many individual trials of some kind of experiment.
Charlie– The detailed process of branching is a technical problem worthy of more study, no doubt. As you say, there aren’t really any problems with energy conservation, once you understand how it works in regular quantum mechanics. (If you like, the thing that is conserved is the energy times the amplitude squared.)
21. Stewart says:
Sean, so you said:
So, “almost-independently” isn’t the same as “actually independently”, but let’s leave that aside for now. If the off-diagonal terms do grow large, then that invalidates the “off-diagonal terms are small” assumption, therefore the branches are definitely not independent, therefore you cannot assign probabilities. There’s nothing circular/inconsistent about this?
OK…so I probably don’t understand this too well. Heck, I never even read your paper.
22. Tom Andersen says:
I agree completely that the MWI is the simplest form of QM, and like the “disappearing-worlds interpretations.”
The real question is not whether MWI is a better way of looking at QM. The real question is whether QM is correct. Every physical law found to date has either proved itself an approximation, or is waiting for its day. QM is exceedingly likely just another law waiting for its day to end.
So if you assume that QM is in someway wrong, will infinite dimensional Hilbert spaces and perfect linearity remain? Because without those things MWI is a non starter.
MWI is built upon the one part of QM that is weakest – the collapse.
Most of the alternative ‘explanations’ of QM have an obvious place where collapse occurs due to limited bandwidth (any non linearity). It will have to be experiment that proves QM wrong, as it is pretty firmly entrenched in the Physics Community.
23. FWIW, here are my comments, believe-it-or-not derived independently just last evening, though I acknowledge having been recently strongly influenced by your take on the Many Worlds Interpretation of quantum mechanics:
Coherence consists of all possible relations
emanating from every point in continually evolving spacetime.
Present moment decoherence
descries an organic universe
exploding from a localizing identity
ceaselessly redefining present experience.
Boundless such states are invariably emergent
within endlessly evolving relations.
Each and every relation within the organic multiverse
resonates with all others;
its influence exponentially attenuating with spacetime remoteness.
It’s useful to recognize the synchronicity of coherence
on scales ranging from quantum to cosmic.
All portrayals of experience
are exquisitely sensitive to
localizing identity.
The experience of the organism
is largely determined by
the point of view, or perspective,
of its identity.
Whatever may be perceived is rooted in organic remembrance,
reflecting naught but current decoherence—
entanglement evolving as natively cognizing environment
energizes present experience
as an ever more discrete subset of boundless probabilities.
I’d love to see a formulaic reduction of these ideas.
24. vmarko says:
“There are still puzzles to be worked out, no doubt, especially around the issues of exactly how and when branching happens, and how branching structures are best defined. […] But these seem like relatively tractable technical challenges to me, rather than looming deal-breakers.”
I would really like to see how the pointer basis problem can be considered a technical challenge, let alone a tractable one. At best, you’ll need an additional set of axioms in the theory, which should fix the choice of the basis. But the looming feeling is that the task of actually formulating these axioms is equivalent to resolving the measurement problem and the Schrodinger’s cat paradox. And that may prove to be much more difficult than a mere technical challenge — just remember that people like von Neumann tried, failed and gave up on that challenge — so it’s certainly not going to be easy.
Best, 🙂
25. Milkshake says:
Sean, you have a weird accent where are you from? |
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1. Part 2 Growth of Thin Films and Low-Dimensional Structures
2. 12 Controlled Growth of C-Oriented AlN Thin Films: Experimental Deposition and Characterization Manuel García-Méndez Centro de Investigación en Ciencias Físico-Matemáticas, FCFM de la UANL Manuel L. Barragán S/N, Cd. Universitaria, México 1. Introduction Nowadays, the science of thin films has experienced an important development and specialization. Basic research in this field involves a controlled film deposition followed by characterization at atomic level. Experimental and theoretical understanding of thin film processes have contributed to the development of relevant technological fields such as microelectronics, catalysis and corrosion. The combination of materials properties has made it possible to process thin films for a variety of applications in the field of semiconductors. Inside that field, the nitrides III-IV semiconductor family has gained a great deal of interest because of their promising applications in several technology-related issues such as photonics, wear-resistant coatings, thin-film resistors and other functional applications (Moreira et al., 2011; Morkoç, 2008). Aluminium nitride (AlN) is an III-V compound. Its more stable crystalline structure is the hexagonal würzite lattice (see figure 1). Hexagonal AlN has a high thermal conductivity (260 Wm-1K-1), a direct band gap (Eg=5.9-6.2 eV), high hardness (2 x 103 kgf mm-2), high fusion temperature (2400C) and a high acoustic velocity. AlN thin films can be used as gate dielectric for ultra large integrated devices (ULSI), or in GHz-band surface acoustic wave devices due to its strong piezoelectricity (Chaudhuri et al., 2007; Chiu et al., 2007; Jang et al., 2006; Kar et al., 2006; Olivares et al., 2007; Prinz et al., 2006). The performance of the AlN films as dielectric or acoustical/electronic material directly depends on their properties at microstructure (grain size, interface) and surface morphology (roughness). Thin films of AlN grown at a c-axis orientation (preferential growth perpendicular to the substrate) are the most interesting ones for applications, since they exhibit properties similar to monocrystalline AlN. A high degree of c-axis orientation together with surface smoothness are essential requierements for AlN films to be used for applications in surface acoustic wave devices (Jose et al., 2010; Moreira et al., 2011). On the other hand, the oxynitrides MeNxOy (Me=metal) have become very important materials for several technological applications. Among them, aluminium oxynitrides may have promissing applications in diferent technological fields. The addition of oxygen into a growing AlN thin film induces the production of ionic metal-oxygen bonds inside a matrix
3. 288 Modern Aspects of Bulk Crystal and Thin Film Preparation of covalent metal-nitrogen bond. Placing oxygen atoms inside the würzite structure of AlN can produce important modifications in their electrical and optical properties of the films, and thereby changes in their thermal conductivity and piezoelectricity features are produced too (Brien & Pigeat, 2008; Jang et al., 2008). Thus, the addition of oxygen would allow to tailor the properties of the AlNxOy films between those of pure aluminium oxide (Al2O3) and nitride (AlN), where the concentration of Al, N and O can be varied depending on the specific application being pursued (Borges et al., 2010; Brien & Pigeat, 2008; Ianno et al., 2002; Jang et al., 2008). Combining some of their advantages by varying the concentration of Al, N and O, aluminium oxynitride films (AlNO) can produce applications in corrosion protective coatings, optical coatings, microelectronics and other technological fields (Borges et al., 2010; Erlat et al., 2001; Xiao & Jiang, 2004). Thus, the study of deposition and growth of AlN films with the addition of oxygen is a relevant subject of scientific and technological current interest. Thin films of AlN (pure and oxidized) can be prepared by several techniques: chemical vapor deposition (CVD) (Uchida et al., 2006; Sato el at., 2007; Takahashi et al., 2006), molecular beam epitaxy (MBE) (Brown et al., 2002; Iwata et al., 2007), ion beam assisted deposition (Lal et al., 2003; Matsumoto & Kiuchi, 2006) or direct current (DC) reactive magnetron sputtering. Among them, reactive magnetron sputtering is a technique that enables the growth of c-axis AlN films on large area substrates at a low temperature (as low as 200C or even at room temperature). Deposition of AlN films at low temperature is a “must”, since a high-substrate temperature during film growth is not compatible with the processing steps of device fabrication. Thus, reactive sputtering is an inexpensive technique with simple instrumentation that requires low processing temperature and allows fine tuning on film properties (Moreira et al., 2011). In a reactive DC magnetron process, molecules of a reactive gas combine with the sputtered atoms from a metal target to form a compound thin film on a substrate. Reactive magnetron sputtering is an important method used to prepare ceramic semiconducting thin films. The final properties of the films depend on the deposition conditions (experimental parameters) such as substrate temperature, working pressure, flow rate of each reactive gas (Ar, O2, N2), power source delivery (voltage input), substrate-target distance and incidence angle of sputtered particles (Ohring, 2002). Reactive sputtering can successfully be employed to produce AlN thin films of good quality, but to achieve this goal requires controlling the experimental parameters while the deposition process takes place. In this chapter, we present the procedure employed to grow AlN and AlNO thin-films by DC reactive magnetron sputtering. Experimental conditions were controlled to get the growth of c-axis oriented films. The growth and characterization of the films was mainly explored by way of a series of examples collected from the author´s laboratory, together with a general reviewing of what already has been done. For a more detailed treatment of several aspects, references to highly-respected textbooks and subject-specific articles are included. One of the most important properties of any given thin film system relies on its crystalline structure. The structural features of a film are used to explain the overall film properties, which ultimately leads to the development of a specific coating system with a set of required properties. Therefore, analysis of films will be concerned mainly with structural characterization.
4. 289 Controlled Growth of C-Oriented AlN Thin Films: Experimental Deposition and Characterization Crystallographic orientation, lattice parameters, thickness and film quality were characterized through X-ray Diffraction (XRD) and UV-Visible spectroscopy (UV-Vis). Chemical indentification of phases and elemental concentration were characterized through X-ray photoelectron spectroscopy (XPS). From these results, an analysis of the interaction of oxygen into the AlN film is described. For a better understanding of this process, theoretical calculations of Density of States (DOS) are included too. The aim of this chapter is to provide from our experience a step wise scientific/technical guide to the reader interested in delving into the fascinating subject of thin film processing. Fig. 1. Würzite structure of AlN. Hexagonal AlN belongs to the space group 6mm with lattice parameters c=4.97 Å and a=3.11 Å. 2. Deposition and growth of AlN films The sputtering process consists in the production of ions within generated plasma, on which the ions are accelerated and directed to a target. Then, ions strike the target and material is ejected or sputtered to be deposited in the vicinity of a substrate. The plasma generation and sputtering process must be performed in a closed chamber environment, which must be maintained in vacuum. To generate the plasma gas particles (usually argon) are fed into the chamber. In DC sputtering, a negative potential U is applied to the target (cathode). At critical applied voltage, the initially insulating gas turns to electrical conducting medium. Then, the positively charged Ar+ ions are accelerated toward the cathode. During ionization, the cascade reaction goes as follows:
5. 290 Modern Aspects of Bulk Crystal and Thin Film Preparation e- + Ar 2e- + Ar+ where the two additional (secondary) electrons strike two more neutral ions that cause the further gas ionization. The gas pressure “P” and the electrode distance “d” determine the breakdown voltage “VB” to set the cascade reaction, which is expressed in terms of a product of pressure and inter electrode spacing: APd VB (1) ln Pd B where A and B are constants. This result is known as Paschen´s Law (Ohring, 2002). In order to increase the ionization rate by emitted secondary electrons, a ring magnet (magnetron) below the target can be used. Hence, the electrons are trapped and circulate over the surface target, depicting a cycloid. Thus, the higher sputter yield takes place on the target area below this region. An erosion zone (trace) is “carved” on the target surface with the shape of the magnetic field. Equipment description: Films under investigation were obtained by DC reactive magnetron sputtering in a laboratory deposition system. The high vacuum system is composed of a pirex chamber connected to a mechanic and turbomolecular pump. Inside the chamber the magnetron is placed and connected to a DC external power supply. In front of the magnetron stands the substrate holder with a heater and thermocouple integrated. The distance target-substrate is about 5 cm and target diameter 1”. The power supply allows to control the voltage input (Volts) and an external panel display readings of current (Amperes) and sputtering power (Watts) (see Figure 2). Fig. 2. Schematic diagram of the equipment utilized for film fabrication.
6. 291 Controlled Growth of C-Oriented AlN Thin Films: Experimental Deposition and Characterization Deposition procedure: A disc of Al (2.54 cm diameter, 0.317 cm thick, 99.99% purity) was used as a target. Films were deposited on silica and glass substrates that were ultrasonically cleaned in an acetone bath. For deposition, the sputtering chamber was pumped down to a base pressure below 1x10-5 Torr. When the chamber reached the operative base pressure, the Al target was cleaned in situ with Ar+ ion bombardment for 20 minutes at a working pressure of 10 mTorr (20 sccm gas flow). A shutter is placed between the target and the substrate throughout the cleaning process. The Target was systematically cleaned to remove any contamination before each deposition. Sputtering discharge gases of Ar, N2 and O2 (99.99 % purity) were admitted separately and regulated by individual mass flow controllers. A constant gas mixture of Ar and N2 was used in the sputtering discharge to grow AlN films; a gas mixture of Ar, N2 and O2 was used to grow AlNO films. A set of eight films were prepared: four samples on glass substrates (set 1) and four samples on silica substrates (set 2). From set 1, two samples correspond to AlN (15 min of deposition time, labeled S1 and S2) and two to AlNO (10 min of deposition time, labeled S3 and S4). From set 2, three samples correspond to AlN (10 min of deposition time, labeled S5, S6 and S7) and one to AlNO (10 min of deposition time, labeled S8). All samples were deposited using an Ar flow of 20 sccm, an N2 flow of 1 sccm and an O2 flow of 1 sccm. In all samples (excluding the ones grown at room temperature.), the temperature was supplied during film deposition. Tables 1 (a) (set 1) and 1 (b) (set 2) summarize the experimental conditions of deposition. Calculated optical thickness by formula 4 is included in the far right column. Table 1a. Deposition parameters for DC sputtered films grown on glass substrates (set 1) Table 1b. Deposition parameters for DC sputtered films grown on silica substrates (set 2).
7. 292 Modern Aspects of Bulk Crystal and Thin Film Preparation 3. Structural characterization XRD measurements were obtained using a Philips X'Pert diffractometter equipped with a copper anode K radiation, =1.54 Å. High resolution theta/2Theta scans (Bragg-Brentano geometry) were taken at a step size of 0.005. Transmission spectra were obtained with a UV- Visible double beam Perkin Elmer 350 spectrophotometer. Figure 3 (a) and (b) display the XRD patterns of the films deposited on glass (set 1) and silica (set 2) substrates, respectively. The diffraction pattern of films displayed in figure 3 match with the standard AlN würzite spectrum (JCPDS card 00-025-1133, a=3.11 Å, c=4.97 Å) (Powder Diffraction file, 1998). The highest intensity of the (002) reflection at 2θ35.90 indicates an oriented growth along the c- axis perpendicular to substrate. From set 1, it can be observed that the intensity of (002) diffraction peak is the highest in S2. In this case, the temperature of 1000C increased the crystalline ordering of film. In S3 and S4 the intensity of (002) diffraction and grain size are very similar for both samples, which shows that applied temperature on S4 had not effect in improving its crystal ordering. From set 2, it can be observed that the intensity of (002) diffraction peak is the highest in S5. Generally, temperature gives atoms an extra mobility, allowing them to reach the lowest thermodynamically favored lattice positions hence, the crystal size becomes larger and the crystallinity of the film improves. However, the temperature applied to S6 and Ss makes no effect to improve their crystallinity. In this case, a substrate temperature higher than 100C can trigger a re-sputtering of the atoms that arrive at the substrate´s surface level and crystallinity of films experiences a downturn. From set 1 and set 2, S2 and S5, respectively, were the ones that presented the best crystalline properties. A temperature ranging from RT to 100C turned out to be the critical experimental factor to get a highly oriented crystalline growth. Fig. 3. XRD patterns of films deposited on (a) glass and (b) silica substrates. In terms of the role of oxygen, for S3, S4 and S8, the presence of alumina (-Al2O3: JCPDS file 29-63) or spinel (-AlON: JCPDS files 10-425 and 18-52) compounds in the diffraction patterns
8. 293 Controlled Growth of C-Oriented AlN Thin Films: Experimental Deposition and Characterization was not detected. However, it is known from thermodynamic that elemental aluminium reacts more favorably with oxygen than nitrogen: it is more possible to form Al2O3 by gaseous phase reaction of Al+(3/2)O2 than AlN of Al+(1/2)N since G(Al2O3)=-1480 KJ/mol and G(AlN)=-253 KJ/mol (Borges et al., 2010; Brien & Pigeat, 2007). Therefore, the existence of Al2O3 or even spinel AlNO phases in samples cannot be discarded, but maybe in such a small proportions as to be detected by XRD. S1, S2 and S5 show a higher crystalline quality than S3, S4 and S8. For these last samples, the extra O2 introduced to the chamber promotes the oxidation of the target-surface (target poisoning). In extreme cases when the target is heavily poisoned, oxidation can cause an arcing of the magnetron system. Formation of aluminium oxide on the target can act as an electrostatic shell, which in turn can affect the sputtering yield and the kinetic energy of species which impinge on substrate with a reduction of the sputtering rate: The lesser energy of species reacting on substrate, the lesser crystallinity of films. Also, the oxygen can enter in to the AlN lattice through a mechanism involving a vacancy creation process by substituting a nitrogen atom in the weakest Al-N bond aligned parallel to 0001 direction. During the process, the mechanism of ingress of oxygen into the lattice is by diffussion (Brien & Pigeat, 2007; Brien & Pigeat, 2008; Jose et al., 2010). On the other hand, the ionic radius of oxygen (rO=0.140 nm) is almost ten times higher than that of nitrogen (rN=0.01-0.02 nm) (Callister, 2006). Thus, the oxygen causes an expansion of the crystal lattice through point defects. As the oxygen content increases, the density of point defects increases and the stacking of hexagonal AlN arrangement is disturbed . It has been reported that the Al and O atoms form octahedral atomic configurations that eventually become planar defects. These defects usually lie in the basal 001 planes (Brien & Pigeat, 2008; Jose et al., 2010). As was mentioned, during the deposition of thin films, the oxygen competes with the nitrogen to form an oxidized Al-compound. The resulting films are then composed of separated phases of AlN and AlxOy domains. The presence of AlxOy domains provokes a disruption in the preferential growth of the film. For example, in S4, the applied temperature of 1200C can promote an even more efficient diffusive ingress of oxygen into the AlN lattice and such temperature was not a factor contributing to improve crystallinity. In S3 and S8, oxygen by itself was the factor that provoked a film´s low crystalline growth. By using the Bragg angle (b) as variable that satisfies the Bragg equation: 2dhklSenb=n (2) and the formula applied for hexagonal systems: 4 h 2 hk k 2 l 2 1 2 (3) 3 c 2 a2 dhkl the length of the lattice parameters “a” and “c” can then be obtained from the experimental data. As films crystallized in a hexagonal würzite structure, XRD patterns were processed with a software program in order to obtain the lattice parameters “a” and “c”. The AlN würzite structure from the JCPDS database (PDF file 00-025-1133, c= 4.97 Å, a=3.11 Å) was taken as a
9. 294 Modern Aspects of Bulk Crystal and Thin Film Preparation reference (Powder Diffraction File, 1998). For the fitting, input parameters of (h k l) planes with their corresponding theta-angle are given. By using the Bragg formula and the equation of distance between planes (for a hexagonal lattice), the lattice parameters are then calculated by using a multiple correlation analysis with a least squares minimization. The 2 angles were set fixed while lattice parameters were allowed to fit. Calculated lattice parameters “a” and “c” and grain size “L” by formula (4) are included in Table 2. Table 2. Lattice parameters “a” (nm) and “c” (nm) obtained from XRD measurements. The average grain size “L” is obtained through the Debye-Scherrer formula (Patterson, 1939): K L (4) B cos b where K is a dimensionless constant that may range from 0.89 to 1.30 depending on the specific geometry of the scattering object. For a perfect two dimenssional lattice, when every point on the lattice produces a spherical wave, the numerical calculations give a value of K=0.89. A cubic three dimensional crystal is best described by K=0.94 (Patterson, 1939). The measure of the peak width, the full width at half maximum (FWHM) for a given b is denoted by B (for a gaussian type curve). From table 2, it can be observed that the calculated lattice parameters differ slightly from the ones reported from the JCPDS database, mainly the “c” value, particularly for S3, S4 and S8. Introduction of oxygen into the AlN matrix along the {001} planes also modifies the lattice parameters. As expected, the “c” value is the most affected. The quality of samples can also be evaluated from UV-Visible spectroscopy (Guo et al., 2006). By analysing the measured T vs spectra at normal incidence, the absorption coefficient () and the film thickness can be obtained. If the thickness of the film is uniform, interference effects between substrate and film (because of multiple reflexions from the substrate/film interface) give rise to oscillations. The number of oscillations is related to the film thickness. The appearence of these oscillations on analized films indicates uniform thickness. If the thickness “t“ were not uniform or slightly tappered, all interference effects would be destroyed and the T vs spectrum would look like a smooth curve (Swanepoel, 1983). Oscillations are useful to calculate the thickness of films using the formula (Swanepoel, 1983; Zong et al., 2006):
10. 295 Controlled Growth of C-Oriented AlN Thin Films: Experimental Deposition and Characterization 1 t (5) 1 1 2n 2 1 Where t is the thickness of film, n the refractive index, 1 and 2 are the wavelength of two adjacent peaks. Calculated optical thickness of samples using the above mentioned formula, are included in Tables 1(a) and (b). Regarding the absorbance (), a T vs curve can be divided (grossly) into four regions. In the transparent region =0 and the transmitance is a function of n and t through multiple reflexions. In the region of weak absorption is small and the transmission starts to reduce. In the region of medium absorption the transmission experiences the effect of absoption even more. In the region of strong absorption the transmission decreases abruptly. This last region is also named the absorption edge. Near the absorption edge, the absorption coefficient can expressed as: h=( h-Eg) (6) where h is the photon energy, Eg the optical band gap and is the parameter measuring the type of band gap (direct or indirect) (Guerra et al., 2011; Zong et al., 2006). Thus, the optical band gap is determined by applying the Tauc model and the Davis and Mott model in the high absorbance region. For AlN films, the transmittance data provide the best linear curve in the band edge region, taking n=1/2, implying that the transition is direct in nature (for indirect transition n=2). Band gap is obtained by plotting (h)2 vs h by extrapolating the linear part of the absorption edge to find the intercept with the energy axis. By using UV-Vis measurements for AlNO films on glass sustrates, authors of ref. (Jang et al., 2008) found band gap values between 6.63 to 6.95 eV, depending the Ar:O ratio. From our measurements, figure 4 displays the optical spectra (T vs curve) graphs. The oscillations detected in the curves attest the high quality in homogeneity of deposited films. All the samples have oscillation regardless their degree of crystallinity. An important feature to note is that curves present differences in the “sharpness“, at the onset of the strong absorption zone. These differences are attributed to deposition conditions, where final density of films, presence of deffects and thickness, modify the shape of the curve at the band edge. A FESEM micrograph cross-section of S2 is displayed on figure 5. From figure, it is possible to identify a well defined substrate/film interface and a section of film with homogeneous thickness. Together with micrographs, in-situ EDAX analyses were conducted in two specific regions of the film. An elemental analysis by EDAX allows to distinguish the differences in elemental concentration depending on the analized zone. In the film zone , an elemental concentration of Al (54.7 %) and N (45.2 %) was detected, as expected for AlN film. Conversely, in the substrate zone, elemental concentration of Si and O with traces of Ca, Na, Mg was detected, as expected for glass. At this stage, we can establish that during the sputtering process, the oxygen diffuses in to the growing AlN films. Then, the oxygen attaches to available Al, forming AlxOy phases. Dominions of these phases, contained in the whole film, can induce defects. These defects are piled up along the c-axis. From X-ray diffractograms, a low and narrow intensity at the (0002) reflection indicates low crystallographic ordering. By calculating lattice parameters
11. 296 Modern Aspects of Bulk Crystal and Thin Film Preparation “a” and “c” and evaluating how far their obtained values deviate from the JCPDF standard (mainly the “c” distance), also provides evidence about the degree of crystalline disorder. In films, a low crystallographic ordering does not imply a disruption in the homogeneity, as was already detected by UV-Visible measurements. A more detailed analysis concerning the identification and nature of the phases contained in films were performed with a spectroscopic technique. 100 Absorption () Medium Strong Weak Transparent 80 Transmitance (%) 60 S1 S2 40 S3 S4 20 0 300 400 500 600 700 800 900 (nm) 100 80 Transmitance (%) 60 S5 S6 40 S7 S8 20 0 300 400 500 600 700 800 900 (nm) Fig. 4. Optical transmission spectra of deposited films.
12. 297 Controlled Growth of C-Oriented AlN Thin Films: Experimental Deposition and Characterization Fig. 5. Cross section FESEM micrograph of AlN film (S2). An homogeneous film deposition can be observed. In the right column an EDAX analysis of (a) film zone and (b) substrate zone is included.
13. 298 Modern Aspects of Bulk Crystal and Thin Film Preparation 4. Chemical characterization The process of oxidation is a micro chemical event that was not completely detected by XRD. Because of that, XPS analyses were performed in order to detect and identify oxidized phases. XPS measurements were obtained with a Perkin-Elmer PHI 560/ESCA-SAM system, equipped with a double-pass cylindrical mirror analyzer, and a base pressure of 110-9 Torr. To clean the surface, Ar+ sputtering was performed with 4 keV energy ions and 0.36 A/cm2 current beam, yielding to about 3 nm/min sputtering rate. All XPS spectra were obtained after Ar+ sputtering for 15 min. The use of relatively low current density in the ion beam and low sputtering rate reduces modifications in the stoichiometry of the AlN surface. For the XPS analyses, samples were excited with 1486.6 eV energy AlK X-rays. XPS spectra were obtained under two different conditions: (i) a survey spectrum mode of 0-600 eV, and (ii) a multiplex repetitive scan mode. No signal smoothing was attempted and a scanning step of 1 eV/step and 0.2 eV/step with an interval of 50 ms was utilized for survey and multiplex modes, respectively. The spectrometer was calibrated using the Cu 2p3/2 (932.4 eV) and Cu 3p3/2 (74.9 eV) lines. Al films deposited on the glass and silica substrates were used as additional references for Binding energy. In both kind of films, the BE of metallic (Al0) Al2p- transition gave a value of 72.4 eV respectively. On these films, the C1s-transition gave values of 285.6 eV and 285.8 eV for glass and silica substrates, respectively. These values were set for BE of C1s. The relative atomic concentration of samples was calculated from the peak area of each element (Al2p, O1s, N1s) and their corresponding relative sensitivity factor values (RSF). These RSF were obtained from software system analysis (Moulder, 1992). Gaussian curve types were used for data fitting. Figure 6 displays the XPS spectra of films. The elemental attomic concentration (atomic percent) calculated from the O1s, N1s and Al2p transitions is also included in the figure. Figure 6a shows the Al2p high-resolution photoelectron spectrum of S1. The binding energies (BE) from the acquired Al2p photoelectron transition are presented in table 3. The survey spectra show the presence of oxygen in all films, regardless of the fact that some samples were grown without oxygen during deposition. From the XPS analysis, S2 and S5, our films with the best crystalline properties, a concentration of oxygen of 26.3% and 21.6% atomic percent respectively, was measured. The highest measured concentration of oxygen was of about 36.6%, corresponding to S8. This occurrence of oxidation was not directly detected by the XRD analysis, since these oxidized phases can be spread in a low amount throughout the film. The nature of these phases can be inferred from the deconvoluted components of the Al2p transition. In Figure 6a, the Al2p core level spectrum is presented. This spectrum is composed of contributions of metallic Al (BE=72.4 eV), nitridic Al in AlN (BE=74.7 eV) and oxidic Al in Al2O3 (BE=75.6 eV). Despite the differences in experimental conditions, aluminium reacted with the nitrogen and the oxygen in different proportions. Even in S2, the thin film with the best crystalline properties, a proportion of about 30.6 % of aluminum reacted with oxygen to form an aluminium oxide compound. In S7, the relative contribution of Al in nitridic and oxidic state is almost similar, of 42.2% and 49.5%, respectively. A tendency, not absolute but in general, indicates that the higher the proportion of Al in oxidic state, the more amorphous the film.
14. 299 Controlled Growth of C-Oriented AlN Thin Films: Experimental Deposition and Characterization Fig. 6. XPS survey spectra of dc sputtered films. In this figure, the O1s, N1s and Al2p core- level principal peaks can be observed. Fig. 6a. Al2p XPS spectrum of S1. The Al2p peak is composed of contributions of metallic aluminium (AlO), aluminium in nitride (Al-N) and oxidic (Al-O) state.
15. 300 Modern Aspects of Bulk Crystal and Thin Film Preparation Table 3. Binding energy (eV) of metallic aluminium (AlO), aluminium in nitridic (Al-N) and oxidic (Al-O) state obtained from deconvoluted components of Al2p transition. Percentage (relative %) of Al bond to N and O is also displayed. For comparison purposes, some relevant literature concerning the binding energies of metallic-Al, AlN and Al2O3 has been reviewed and included in table 4. Aluminium in metallic state lies in the range of 72.5-72.8 eV. Aluminium in nitridic state lies in the range of 73.1- 74.6 eV, while aluminium in oxidic state lies in the range of 74.0-75.5 eV. Also, there is an Al- N-O spinel-like bonding, very similar in nature to oxidic aluminium with a BE value of 75.4 eV. Another criteria used by various authors for phase identification, is to take the difference (E) in BE of the Al2p transition corresponding to Al-N and Al-O bonds. This difference can take values of about 0.6 eV up to 1.1 eV (see Table 4). Table 4. Binding energy of (eV) of metallic aluminium (AlO), aluminium in nitridic (Al-N) and oxidic (Al-O) state obtained from literature
16. 301 Controlled Growth of C-Oriented AlN Thin Films: Experimental Deposition and Characterization In films, only small traces of metallic aluminium were detected in S1 at 72.4 eV. For S4 and S8, BE of Al in nitride gave a value of 74.4 eV, just below the BE of 74.7 eV, detected for the rest of the samples. This value of 74.4 eV can be attributed to a substoichiometric AlNx phase (Robinson et al., 1984; Stanca, 2004). On the other hand, the BE for aluminium in oxydic state varies from 75.1 eV to 75.7 eV. The lowest values of BE of about 75.1 eV and 75.2 eV, corresponding to S3 and S4, respectively, could be attributed to a substoichiometric AlxOy phase, although in our own experience, the reaction of aluminium with oxygen tends to form the stable -Al203 phase, which possesses somewhat higher value in BE. These finding agree with those reported in other works, where low oxidation states such as Al+1, Al+2 can be found at a BE lower than the one of Al+3 (Huttel et al., 1993; Stanca, 2004). Oxidation states lower than +3 confer an amorphous character to the aluminium oxide (Gutierrez et al., 1997). 5. Theoretical calculations Experimental results provided evidence that oxygen can induce important modifications in the structural properties of sputtered-deposited AlN films. In this way, theoretical calculations were performed to get a better understanding of how the position of the oxygen into the AlN matrix can modify the electronic properties of the film system. The bulk structure of hexagonal AlN was illustrated in Figure 1. Additionally, hexagonal AlN can be visualized as a matrix of distorted tetrahedrons. In a tetrahedron, each Al atom is surrounded by four N atoms. The four bonds can be categorized into two types. The first type is formed by three equivalent Al-Nx, (x=1,2,3) bonds, on which the N atoms are located in the same plane normal to the 0001 direction. The second type is the Al-N0 bond, on which the Al and N atoms are aligned parallel to the 0001 direction (see figure 7). This last bond is the most ionic and has a lower binding energy than the other three (Chaudhuri et al., 2007; Chiu et al., 2007; Zhang et al, 2005). When an AlN film is oxidized, the oxygen atom can substitute the nitrogen atom in the weakest Al-N0 bond while the displaced nitrogen atom can occupy an interstitial site in the lattice (Chaudhuri et al., 2007). For würzite AlN, there are four atoms per hexagonal unit cell where the positions of the atoms for Al and N are: Al(0,0,0), (2/3,1/3,1/2); N(0,0,u), (2/3,1/3, u+1/2), where “u” is a dimensionless internal parameter that represents the distance between the Al-plane and its nearest neighbor N-plane, in the unit of “c”, according to the JCPDS database (Powder diffraction file, 1998). The calculations were perfomed using the tight-binding method (Whangbo & Hoffmann, 1978) within the extended Hückel framework (Hoffmann, 1963) using the computer package YAeHMOP (Landrum, 1900). The extended Hückel method is a semiempirical approach that solves the Schrödinger equation for a system of electrons based on the variational theorem (Galván, 1998). In this approach, explicit correlation is not considered except for the intrinsic contributions included in the parameter set. For a best match with the available experimental information, experimental lattice parameters were used instead of optimized values. Calculations considered a total of 16 valence electrons corresponding to 4 atoms within the unit cell for AlN. Band structures were calculated using 51 k-points sampling the first Brillouin zone (FBZ). Reciprocal space integration was performed by k-point sampling (see figure 8). From band structure, the electronic band gap (Eg) was obtained.
17. 302 Modern Aspects of Bulk Crystal and Thin Film Preparation Fig. 7. Individual tetrahedral arrangement of hexagonal AlN. Fig. 8. Hexagonal lattice in k-space.
18. 303 Controlled Growth of C-Oriented AlN Thin Films: Experimental Deposition and Characterization Calculations were performed considering four scenarios: 1. A wurzite-like AlN structure with no oxygen in the lattice 2. An oxygen atom inside the interstitial site of the tetrahedral arrangement (interstitial) 3. An oxygen atom in place of the N atom in the weakest Al-N0 bond (substitution) 4. An oxygen atom on top of the AlN surface (at the surface). Theoretical band-gap calculations are summarized in Table 5. Values are given in electron volts (eV). Table 5. Calculated energy gaps for pure AlN (würzite) and with oxygen in different atomic site positions. For AlN hexagonal, a direct band gap of 7.2 eV at M was calculated (see Figure 9). When oxygen was taken into account in the calculations, the band gap value undergoes a remarkable change: 1.3 eV for AlN with intercalated oxygen (2) and 0.8 eV for AlN with oxygen substitution (3). In terms of electronic behavior, the system transformed from insulating (7.2 eV) to semiconductor (1.3 eV), and then from semiconductor (1.3 eV) to semimetal (0.82 eV). This change in the electronic properties is explained by the differences between the ionic radius of Nitrogen (rN) and Oxygen (rO). The ionic radius of the materials involved was: rN=0.01-0.02 nm, rO=0.140 nm (Callister, 2006). Comparing these values, it can be noted that rO is almost ten times higher than rN. This fact would imply that when the oxygen atom takes the place of the nitrogen atom (by substitution o intercalation of O), the crystalline lattice expands because of the larger size of oxygen. Any change in the distance among atoms and the extra valence electron of the oxygen will alter the electronic interaction and in consequence, the band gap value In calculation (4), the atoms of Al and N are kept in their würzite atomic positions while the oxygen atom is placed on top of the AlN lattice. In this case, the calculated band gap (6.31 eV) is closer in value to pure AlN (7.2 eV) than the calculated ones for interstitial (1.3 eV) and substitution (0.82 eV). In this case, theoretical results predicts that when the oxygen is not inside the Bravais lattice, the band gap will be close in value to the one of hexagonal AlN; conversely, the more the oxygen interacts with the AlN lattice, the more changes in electronical properties are expected; However, in energetic terms, competition between N and O atoms to get attached to the Al to form separated phases of AlN and AlxOy is the most probable configuration, as far as experimental results suggests. Theoretical calculations of band structure for würzite AlN have been performed using several approaches; For comparison purposes, some of them are briefly described in Table 6.
19. 304 Modern Aspects of Bulk Crystal and Thin Film Preparation Fig. 9. Band structure for 2H-AlN hexagonal, sampling the first Brillouin zone (FBZ). Energy Method/Procedure Reference band gap (eV) 6.05 Local density approximation (LDA) within the (Ferreira et al., density functional theory (DFT) with a correction 2005) g, using a quasi-particle method: LDA+g 6.2 Empirical pseudopotential method (EPM). An (Rezaei et al., analytical function using a fitting procedure for both 2006) symmetric and antisymmetric parts, and a potential is constructed 4.24 Full potential linear muffin-tin orbital (FPLMTO) (Persson et al., 2005) FPLMTO with a corrected band gap g 6.15 (Persson et al., 2005) Table 6. AlN energy band gap values obtained from theoretical calculations.
Theo dõi chúng tôi
Đồng bộ tài khoản |
c5becb3fde24b6b0 | Tao, TerrenceTerence ( born July 17, 1975 , Adelaide, Austl.Australian AustraliaAustralian mathematician awarded a Fields Medal in 2006 “for his contributions to partial differential equations, combinatorics, harmonic analysis and additive number theory.”
Tao received a bachelor’s and a master’s degree from Flinders University of South Australia and a doctorate from Princeton University (1996), after which he joined the faculty at the University of California, Los Angeles.
Tao’s work is characterized by a high degree of originality and a diversity that crosses research boundaries, together with an ability to work in collaboration with other specialists. His main field is the theory of partial differential equations. These Those are the principal equations used in mathematical physics. For example, the nonlinear Schrödinger equation models light transmission in fibre optics. Despite the ubiquity of partial differential equations in physics, it is usually difficult to obtain or rigorously prove that such equations have solutions or that the solutions have the required properties. Along with that of several collaborators, Tao’s work on the nonlinear Schrödinger equation established crucial existence theorems. He has also done important work on the gravitational waves predicted by Albert Einstein’s theory of general relativity. This work is ongoing, but it has already reawakened interest in a subject that was long thought too difficult for further progress.
In work with the British mathematician Ben Green, Tao showed that the set of prime numbers contains arithmetic progressions of any length. For example, 109, 219, 329, 439, 549 is an arithmetic progression of five prime numbers, where successive numbers differ by 110. Standard arguments had indicated that arithmetic progressions in the set of primes might not be very long, so the discovery that they can be arbitrarily long was a profound discovery about the building blocks of arithmetic. Tao’s other awards include a Salem Prize (2000) and an American Mathematical Society Bocher Memorial Prize (2002). |
d0b40df2a6e3c7fd | Graphene is a single layer of carbon atoms organized in a honeycomb lattice. Scientists now know that particles, like electrons, moving through such a structure behave as though they have no mass and travel through the material at near light speeds. These particles are called massless Dirac fermions and their behaviour could be exploited in a host of applications, including transistors that are faster than any that exist today.
The new "molecular" graphene, as it is has been dubbed, is similar to natural graphene except that its fundamental electronic properties can be tuned much more easily. It was made using a low-temperature scanning tunnelling microscope whose tip – made of iridium atoms – can be used to individually position carbon monoxide molecules on a perfectly smooth, conducting copper substrate. The carbon monoxide repels the freely moving electrons on the copper surface and "forces" them into a honeycomb pattern, where they then behave like massless graphene electrons, explains team leader Hari Manoharan.
"We confirmed that the graphene electrons are massless Dirac fermions by measuring the conductance spectrum of the electrons travelling in our material," he told "We showed that the results match the two-dimensional Dirac equation for massless particles moving at the speed of light rather than the conventional Schrödinger equation for massive electrons."
The researchers then succeeded in tuning the properties of the electrons in the molecular graphene by moving the positions of the carbon monoxide molecules on the copper surface. This has the effect of distorting the lattice structure so that it looks as though it has been squeezed along several axes – something that makes the electrons behave as though they have been exposed to a strong magnetic or electric field, although no actual such field has been applied. The team was also able to tune the density of the electrons on the copper surface by introducing defects or impurities into the system.
More control over Dirac fermions
"Studying such artificial lattices in this way may certainly lead to technological applications but they also provide a new level of control over Dirac fermions and allow us to experimentally access a set of phenomena that could only be investigated using theoretical calculations until now," says Manoharan. "Introducing tunable interactions between the electrons could allow us to make spin liquids in graphene, for instance, and observe the spin quantum Hall effect if we can succeed in introducing spin-orbit interactions between the electrons."
He adds that molecular graphene is just the first of this type of “designer” quantum structure and hopes to make other nanoscale materials with such exotic topological properties using similar bottom-up techniques.
The work was detailed in Nature. |
6d98fa4814227db7 | My long, complexity-theoretic journey
So, what was I doing these past few weeks that could possibly take precedence over writing ill-considered blog entries that I’d probably regret for the rest of my life?
1. On the gracious invitation of Renato Renner, I visited one of Al Einstein’s old stomping-grounds: ETH Zürich. There I gave a physics colloquium called How Much Information Is In A Quantum State?, as well as a talk on my paper Quantum Copy-Protection and Quantum Money, which has been more than three years in the procrastinating. Though I was only in Switzerland for three days, I found enough time to go hiking in the Swiss Alps, if by “Swiss Alps” you mean a 200-foot hill outside the theoretical physics building. I’m quite proud of having made it through this entire trip—my first to Switzerland—without once yodeling or erupting into cries of “Riiiiiiicola!” On the other hand, what with the beautiful architecture, excellent public transportation, and wonderful hosts, it was a struggle to maintain my neutrality.
2. On the plane to and from Switzerland, I had the pleasure of perusing Computational Complexity: A Modern Approach, by Sanjeev Arora and Boaz Barak, which has just been published after floating around the interweb for many years. If you’re a hardcore complexity lover, I can recommend buying a copy in the strongest terms. The book lives up to its subtitle, concentrating almost entirely on developments within the last twenty years. Classical complexity theorists should pay particular attention to the excellent quantum computing chapter, neither of whose authors has the slightest background in the subject. You see, people, getting quantum right isn’t that hard, is it? The book’s only flaw, an abundance of typos, is one that can and should be easily fixed in the next edition.
3. I then visited the National Institute of Standards and Technology—proud keepers of the meter and the kilogram—at their headquarters in Gaithersburg, MD. There I gave my talk on Quantum Complexity and Fundamental Physics, a version of the shtick I did at the QIS workshop in Virginia. Afterwards, I got to tour some of the most badass experimental facilities I’ve seen in a while. (Setting standards and making precision measurements: is there anything else that sounds so boring but turns out to so not be?) A highlight was the Center for Neutron Research, which houses what’s apparently the largest research reactor still operating in the US. This thing has been operating since 1967, and it shoots large numbers of slow-moving neutrons in all directions so that archaeologists, chemists, physicists, etc. can feed off the trough and do their experiments. The basic physics that’s been done there recently has included setting bounds on possible nonlinearities in the Schrödinger equation (even though any nonlinearity, no matter how small, could be used to send superluminal signals and solve NP-complete problems in polynomial time), as well as observing the photons that the Standard Model apparently predicts are emitted 2% of the time when a neutron decays. I also got to see one of the world’s least jittery floors: using dynamical feedback, they apparently managed to make this floor ~107 times less jittery than a normal floor, good enough that they can run a double-slit experiment with slow neutrons on top of it and see the interference pattern. (Before you ask: yes, I wanted to jump on the floor, but I didn’t. Apparently I would’ve messed it up for a day.)
I have to add: the few times I’ve toured a nuclear facility, I felt profoundly depressed by the “retro” feel of everything around me: analog dials, safety signs from the 60s… Why are no new reactors being built in the US, even while their value as stabilization wedges becomes increasingly hard to ignore? Why are we unwilling to reprocess spent fuel rods like France does? Why do people pin their hopes on the remote prospect of controlled fusion, ignoring the controlled fission we’ve had for half a century? Why, like some horror-movie character unwilling to confront an evil from the past, have we decided that a major technology possibly crucial to the planet’s survival must remain a museum piece, part of civilization’s past and not its future? Of course, these are rhetorical questions. While you can be exposed to more radiation flying cross-country than working at a nuclear reactor for months, while preventing a Chernobyl is as easy as using shielding and leaving on the emergency cooling system, human nature is often a more powerful force than physics.
4. Next I went to STOC’2009 in Bethesda, MD. Let me say something about a few talks that are impossible not to say something about. First, in what might or might not turn out to be the biggest cryptographic breakthrough in decades, Craig Gentry has proposed a fully homomorphic encryption scheme based on ideal lattices: that is, a scheme that lets you perform arbitrary computations on encrypted data without decrypting it. Currently, Gentry’s scheme is not known to be breakable even by quantum computers—despite a 2002 result of van Dam, Hallgren, and Ip, which said that if a fully homomorphic encryption scheme existed, then it could be broken by a quantum computer. (The catch? Van Dam et al.’s result applied to deterministic encryption schemes; Gentry’s is probabilistic.)
Second, Chris Peikert (co-winner of the Best Paper Award) announced a public-key cryptosystem based on the classical worst-case hardness of the Shortest Vector Problem. Previously, Regev had given such a cryptosystem based on the assumption that there’s no efficient quantum algorithm for SVP (see also here for a survey). The latter was a striking result: even though Regev’s cryptosystem is purely classical, his reduction from SVP to breaking the cryptosystem was a quantum reduction. What Peikert has now done is to “dequantize” Regev’s security argument by thinking very hard about it. Of course, one interpretation of Peikert’s result is that classical crypto people no longer have to learn quantum mechanics—but a better interpretation is that they do have to learn QM, if only to get rid of it! I eagerly await Oded Goldreich‘s first paper on quantum computing (using it purely as an intellectual tool, of course).
Third, Robin Moser (co-winner of the Best Paper Award and winner of the Best Student Paper Award) gave a mindblowing algorithmic version of the Lovász Local Lemma. Or to put it differently, Moser gave a polynomial-time algorithm that finds a satisfying assignment for a k-SAT formula, assuming that each clause intersects at most 2k-2 other clauses. (It follows from the Local Lemma that such an assignment exists.) Moser’s algorithm is absurdly simple: basically, you repeatedly pick an unsatisfied clause, and randomly set its variables so that it’s satisfied. Then, if doing that has made any of the neighboring clauses unsatisfied, you randomly set their variables so that they’re satisfied, and so on, recursing until all the damage you’ve caused has also been fixed. The proof that this algorithm actually halts in polynomial time uses a communication argument that, while simple, seemed so completely out of left field that when it was finished, the audience of theorists sort of let out a collective gasp, as if a giant black “QED” box were hovering in the air.
Fourth, Babai, Beals, and Seress showed that if G is a matrix group over a finite field of odd order, then the membership problem for G can be solved in polynomial time, assuming an oracle for the discrete logarithm problem. This represents the culmination of about 25 years of work in computational group theory. I was all pumped to announce an important consequence of this result not noted in the abstract—that the problem is therefore solvable in quantum polynomial time, because of Shor’s discrete log algorithm—but Laci, alas, scooped me on this highly nontrivial corollary in his talk.
5. Finally, I took the train up to Princeton, for a workshop on “Cryptography and Complexity: Status of Impagliazzo’s Worlds”. (For the insufficiently nerdy: the worlds are Algorithmica, where P=NP; Heuristica, where P≠NP but the hard instances of NP-complete problems are hard to find; Pessiland, where the hard instances are easy to find but none of them can be used for cryptographic one-way functions; Minicrypt, where one-way functions do exist, enabling private-key cryptography, but not the trapdoor one-way functions needed for public-key cryptography; and Cryptomania, where trapdoor one-way functions exist, and cryptography can do pretty anything you could ask.) I gave a talk on Impagliazzo’s worlds in arithmetic complexity, based on ongoing join work with Andy Drucker (where “ongoing” means we’re pretty sure more of our results are correct than would be expected by random guessing).
Tell you what: since it’s been a long time, feel free to ask whatever you feel like in the comments section, whether related to my journeys or not. I’ll try to answer at least a constant fraction of questions.
56 Responses to “My long, complexity-theoretic journey”
1. Sean Carroll Says:
And yet, Scott has been awesome enough to find time to read a book draft for me and offer excellent comments. Which is to say, very awesome.
2. Brian Says:
Good to see you’re back, Scott. It’s always fun to see what you have to say – and occasionally I understand some of it!
3. asterix Says:
Why was Gentry’s work less significant than Peikert’s work? I don’t mean this in a competitive way. I am not a crypto expert, but to me they both sounds like astounding results, and I’m wondering why one is considered more of a breakthrough than the other. I assume Peikert’s may have some easy to explain idea behind it (like Moser’s) whereas Gentry’s is more technical? Is homomorphic encryption less important than using worst-case SVP problems to get crypto systems? Thanks.
4. John Sidles Says:
Scott asserts: Any nonlinearity [in the Schrödinger equation], no matter how small, could be used to send superluminal signals and solve NP-complete problems in polynomial time.
Gee whiz … Scott …. for this to be true … don’t you have add a pretty lengthy conditional clause: “in the Schrödinger equation on vector state-spaces having exponentially large dimension.”
One reason for focusing on this stipulation is that in the world of practical calculations (meaning, PSPACE and PTIME resources) and also in the world of practical experiments (meaning, finite-temperature and/or noisy laboratories), it is commonplace to compute on tensor network state-spaces … which definitely are not vector spaces, but rather are Kähler manifolds.
In working through this manifold-oriented framework for QIS/QIT, our QSE Group translated chapters 2 and 8 of Nielsen and Chuang’s textbook into the language of Kähler manifolds as contrasted with Hilbert space. This was a fun exercise (in which Ashtekar and Schilling preceded us by a decade, I will mention).
The resulting QIS/QIT mathematical framework proved sufficiently compact that we could summarize it on one page …
… and this compactness proves to be very convenient for organizing large-scale QIS/QIT calculations.
Now, it is true that when QIS/QIT is formulated as a manifold theory, the mathematical focus naturally shifts from the (linear) Hamiltonian dynamics of superposition to the (nonlinear) concentration dynamics that is generic to Lindbladians. But isn’t this a good thing? … when it helps us to efficiently simulate large-scale quantum systems?
As far as our QSE Group knows, there is nothing in orthodox Lindbladian quantum dynamics that permits “sending superluminal signals and solving NP-complete problems.” Because isn’t Lindbladian dynamics so constructed as to rule-out these possibilities? Even when the Lindbladian dynamics concentrates quantum trajectories onto a non-vector manifold?
The point being, noise and nonlinearity are valuable QIS/QIT resources—to be treasured, not scorned! :)
5. Domenic Denicola Says:
Wow, thanks for enlightening us bystanders; those results from point #4 are really cool!
6. Arne Peeters Says:
Unrelated (but you said it’s ok): It’s now 4 years since, so what’s your updated opinion on “waste papers”? Would you still write that post (given a sufficient reason to procrastinate ;-) ) and if not, what would you write instead?
7. Carl Says:
If I die… Tell my wife… “Hello…”.
8. Scott Says:
Why was Gentry’s work less significant than Peikert’s work?
asterix: Please be assured, you’re not the only one to have asked that question! Since I wasn’t privy to the decision, and have no strong personal feelings about it, I suppose it’s OK for me to say what I know (and if it isn’t, and I end up using this blog to blab about something I shouldn’t one more time … well, who’s counting? :-) ). My understanding is that the PC had some concerns about the correctness of at least part of Gentry’s paper (and maybe about the underlying assumptions—though I’m just guessing there), and didn’t want to risk looking foolish by (e.g.) giving a Best Paper Award for a cryptosystem that might be broken half a year later. The trouble, of course, is that in a situation like this one, the PC runs that risk no matter what it does! :-)
What I can say with confidence is the following:
(1) Both Peikert’s and Moser’s papers would be clear contenders for the Best Paper Award in an ordinary year.
(2) Gentry’s might someday be seen as the best paper ever to have been passed over for the Best Paper Award.
9. Scott Says:
John: You’re right, of course; I was talking about adding a nonlinear term to the Schrödinger equation while keeping the “rest” of QM (the state space, the measurement rule, etc.) unchanged. In reality, though, my view is that any nonlinear term (no matter how small) would amount to a complete collapse of QM—so that conditioned on finding such a term, the state space and everything else would seem like fair game as well.
10. anon Says:
Yeah, what’s the deal? Could some crypto experts give their opinion on the extent to which the Gentry paper is correct? If it’s 100% correct then it’s completely amazing, right?
11. harrison Says:
Scott, I haven’t read either of the papers, so forgive me if I’m missing something trivial, but it seems like if you used a sufficiently strong PRNG with Gentry’s cryptosystem, wouldn’t it then be (quantum) breakable by van Dam et al.? And therefore, either no PRNG exists which can fool a quantum computer, or Gentry’s argument is flawed? What am I missing here?
12. Aspiring Blogger Says:
Scott, I love your blog and have been following it for a long time. Sometimes I entertain the thought of creating a forum as entertaining and intellectually rich as this one, but I ask myself — how much effort would it take?
Since you opened the floor for questions, can you comment on this? How much time do you spend (or how much sleep do you lose!) on your blog? Any tips or admonitions for aspiring bloggers?
13. Scott Says:
Harrison, as I understand it, the main issue is whether you can efficiently recognize an encryption of the all-0 string. Van Dam et al. assume you can (as would be the case, in particular, for any deterministic encryption scheme), while in Gentry’s scheme you presumably can’t. In which case, if you replaced the randomness in Gentry’s scheme by the output of a cryptographic PRG (with random seed), it would still be hard to recognize an encryption of the all-0 string (since otherwise, you’d get a polynomial-time algorithm for distinguishing the PRG output from true randomness, contrary to assumption). I trust others will correct me if I’m wrong.
14. Scott Says:
Aspiring Blogger: Like many questions I’ve gotten over the years and never answered, yours really deserves a post of its own! Briefly: right now the blog doesn’t take much of my time, since I hardly ever update it. Back when I updated it every other day, it took maybe half my time. But that’s a statement more about me than about blogging: I understand many other bloggers are able to dash off a decent entry in 20 minutes; I’m more than an order of magnitude less efficient.
Now to watch Colbert…
15. Bram Cohen Says:
That paper should be called ‘Walksat finds the Lovasz Local Minima in polynomial time’. That result has of course been known for the special case of 2-clauses for a long time.
There’s another possible world – in which P=NP in circuit complexity but there’s no finite TM solves arbitrarily large NP complete problems in polynomial time.
16. Martin Schwarz Says:
Hi Scott,
I’m glad you’re back blogging! By the way, why didn’t you make it to your Vienna, Austria, talk this week? I would have enjoyed watching one of your terrific talks live and meeting you in person.
best regards,
17. Scott Says:
Bram: There’s actually a huge number of possible worlds not covered by Russell’s classification (the one where P=NP but the algorithm takes n10000 time; the one where there’s a uniform algorithm that solves SAT in polynomial time on particular input lengths only; the one where P≠NP but NP⊆BQP; the one where public-key crypto is possible using “lossy” trapdoor one-way functions, but ordinary TDOWFs don’t exist…). Indeed, Russell pointed out at the workshop that for the foreseeable future, the worlds are in far more danger of proliferating than they are of collapsing!
Incidentally, I can easily imagine an alternative history of theoretical computer science, where instead of using complexity classes as our basic concept, we directly used the Impagliazzo-worlds (which are basically possibilities for collapses of complexity classes). Of course it might get cumbersome, as in principle there could be exponentially more of the latter than of the former. So maybe people who complain about the size of the current Complexity Zoo should count their blessings! On the other hand, I conjecture that Cryptomania, Pessiland, etc. would be a much easier public-relations sell than BQP and PSPACE.
18. Nagesh Says:
Hi Scott,
Nice to hear a summary of your recent travels. I myself wanted to summarize my own travels I have been doing lately even though most of it for family reasons :)
So as for the questions can I ask what did you propose to do in your NSF career proposal? Can you share the title (and proposal) if it’s ok?
19. Scott Says:
Nagesh, the title of my CAREER proposal was “Limits on Efficient Computation in the Physical World” (same as my thesis title). The main things I proposed to work on, besides education, outreach, and diversifying, were (1) BQP vs. the polynomial hierarchy, (2) the need for structure in quantum speedups (e.g., quantum query complexity lower bounds for almost-total Boolean functions), and (3) non-relativizing techniques in quantum complexity theory. I understand these things are generally not made public, but email me if you want a copy.
20. Aspiring Complexity Researcher Says:
Dear Scott,
I am an advanced doctoral student in a systems area at small school but have fallen in love with topics in complexity, tcs, discrete math and such. (And I am not a natural genius, but I am creative and enthusiastic about tcs!)
What is your best suggestion for me as to ways and means by which I could make valuable contributions to any of the areas I mentioned above? So far, I have made tiny contributions and have been busy trying to squeeze time to read more about these things. But, for example, isn’t having a theorist to interact with / mentor essential? After graduation, what could I do to achieve my goal best? How could I, say, become a postdoc with a theory group without having done a lot of theory work?
Right now, with the economy and what not, the future seems bleak for my love interest.
Any thoughtful (as usual) guidance would be welcome.
Of course, I enjoy your blog and all the remarkable work you are doing for tcs. I have also heard you talk at Harvard once. You inspire many like me. Please know that I am always wishing the very best for you.
Thank you!
21. Scott Says:
Martin: My apologies! I’d never been to Vienna, and had really wanted to go. Alas, I ended up having back-to-back travel in the two weeks prior, and urgently needed to get back to MIT as I had five summer students to meet and get started on their projects. I hope the workshop went well!
Warning to All Workshop Organizers: For whatever reason, I have an extraordinarily hard time saying ‘no’ to anyone, until I’m forced to by circumstances.
22. Scott Says:
Aspiring Complexity Researcher: Given how far you’ve already gone in your studies, it sounds to me like your best bet is to pursue whatever career you were going to pursue in systems (whether that’s in industry or academia), and then look for connections with theory and for theorists to talk with. Quite a few scientists gradually change areas over time, so that what they eventually end up doing might be completely unrelated to what they got their PhD in. But they usually start by doing what they got their PhD in. :-) And the paths between different parts of CS happen to be particularly well-trodden ones: we really do talk to each other!
23. Nagesh Says:
Thanks a lot for responding! I will email you now :)
24. Bobby Says:
Regarding result #4, how is that different than having the input data encrypted with a public key, and the “computation” being the process of appending a message that encapsulates the computation to the end of the input data?
The decryption process in this case would include both decrypting the input data and applying the computation message.
I can see that it’s possible that this asymmetric encryption + messages process would be susceptible to “easy” quantum cracking, or that it may be provable that the method given in paper #4 is quicker to decrypt. Possibly that’s all that’s new that the paper is illuminating.
However, given that the computation can easily add information to the encrypted message, it must be the case that the output data after the computation can be larger than the input data, so the process of computation must be able to lengthen the time it takes to decrypt the data. Even worse, if you can’t derive information about the encrypted data by the computation process, if the computation conditionally would add information to the encrypted data depending on the encrypted values, it must always add information to the encrypted data *even if the computation has no actual effect on the original data*.
I.e. if the computation is “f(y) is y if the y
25. Bobby Says:
Hrm, last post got truncated. Continuation:
I.e. if the computation is “f(y) is y if y < 40, otherwise f(y) is the result of a lookup in some arbitrary table using y as the index”, then the process of applying the computation to the encrypted data must encode all the information in the table into the encrypted result, even if the encrypted y is 39.
Did I misunderstand what the paper is demonstrating?
(BTW, I think the truncation was because I included a < that I didn’t encode as <)
26. komponisto Says:
Would you consider doing a Bloggingheads diavlog with Eliezer Yudkowsky?
27. Scott Says:
Komponisto: LOL! By exploiting my backwards-in-time causal powers, I just did a diavlog with Eliezer this afternoon. It will be up shortly.
28. Responder Says:
Hi Bobby,
The point is that you don’t trust the person doing the computation, so you don’t want him to be able to decrypt the input, compute, then append the answer, because then he’d see the input. Imagine cloud computing. I have confidential data that I’d like to process, but I have very little computing power. I could pay amazon.com, send them the data encrypted, and have them process it on their million computers and send me the results, while being assured that amazon hasn’t learned anything about my confidential data.
29. d Says:
Ha ha funny:
30. Anon Says:
How many of the “ten most annoying questions in quantum computing” (http://scottaaronson.com/blog/?p=112) are still unsolved?
31. Scott Says:
Anon, here’s the status of questions 1-9 (10 not being a real question):
Solved by Montenaro and Shepherd.
2. Can we get any upper bound on QMIP (quantum multi-prover interactive proofs with unlimited prior entanglement)?
Under the conjecture that the provers only need to share a finite-dimensional Hilbert space, Doherty et al. prove a recursive upper bound, which is already highly nontrivial. Without that conjecture, no upper bound is known.
In my paper with Beigi et al., we solved this problem assuming a weak version of the Additivity Conjecture from quantum information theory (the general Additivity Conjecture is now known to be false, but our version still seems plausible). No unconditional result is known.
Solved by Sheridan, Maslov, and Mosca (contrary to my conjecture, the answer is yes)
I’m not sure whether this particular question is still open or not—does anyone else? What I know is that Anup Rao solved a closely related problem, by proving a concentration bound for parallel repetition of the CHSH game.
6. Forget about an oracle relative to which BQP is not in PH. Forget about an oracle relative to which BQP is not in AM. Is there an oracle relative to which BQP is not in SZK?
I realized shortly after posing this problem that an affirmative answer follows from, e.g., the Childs et al. conjoined tree problem.
Still open
8. How many mutually unbiased bases are there in non-prime-power dimensions?
Still open, as far as I know
Still open, though I have a plausible strategy for closing it that I haven’t pursued—thanks for reminding me! :-)
32. milkshake Says:
Nuclear reactors: an awesome one-stop-source for all questions nuclear is Garwin’s Archive. He explains why re-processing currently does not make any economic sense (it saves less than 20% of uranium at a ridiculous cost, and it actually increases the volume of radioactive waste without much reducing its radioactivity, the possibility of separated reactor plutonium theft is a serious proliferation risk etc). France and Japan got into the re-processing business in anticipation of breeder reactors which never really took off. Even if you have pure, already separated weapons-grade plutonium surplus that you want to dispose off, it is actually cheaper and less problematic to mix it with some highly radioactive waste and burry it in a mine depository rather than blend it into a reactor fuel to save uranium.
Retro-looking reactors: Freeman Dyson has a wonderful reminiscence in Disturbing the Universe about his time in 50s with Teller and Freddy deHoffman at General Atomics, designing the reactors. I think the NIST facility uses a different (heavy-water) design than the TRIGA and the graphite-moderated reactors that Dyson worked on but I think his fond memories of the little red-brick schoolhouse where they run reactor calculations over the summer captures well the initial momentum, the enthusiasm which gradually evaporated as the accountants, MBAs and government regulators took over the industry.
Nuclear accidents: safety is expensive and private companies operating nuclear reactors in US and Japan were cutting corners and skimping on upgrades in the past. Plus there is a normal human stupidity and complacency. There were few horribly close calls, not just the Three Mile Island accident.
33. Anon Says:
Thanks for the update!
What does Question 7 mean? What sort of oracle are we looking for?
34. computational simplicity Says:
Hi, a general question:
Can you give us a list of 10 weblogs that you truly enjoy and read regularly ? They don’t have to be CS-oriented, just what you enjoy the most. Also, they don’t have to be *blogs* (a news site, webcomic, or anything like that is okay). You can give more that 10 if you feel like it.
35. Scott Says:
Anon, we’re looking for a classical oracle: that is, one that maps each classical basis state |x⟩ to (-1)f(x)|x⟩, for some Boolean function f. There’s certainly a quantum oracle, namely U itself!
36. Scott Says:
Computational Simplicity: Look at the blogroll to the right!
A few others that I occassionally read: Andrew Sullivan, FiveThirtyEight, Lubos Motl, Bitch PhD, I Blame the Patriarchy.
37. John Sidles Says:
Scott, your above-linked Zurich talk How Much Information Is In A Quantum State? is really excellent, and the numerical experiments you are doing with Eyal Dechter provide a good striking example of Terry Tao’s maxim that “progress generally starts in the concrete and then flows to the abstract.”
This inspires us to flex the narrative to be even more concrete … without (AFAICT) changing any of the fundamental mathematics. We can accomplish this by (1) altering Alice’s motive from conveying information to concealing her activities (which is always more fun!) and (2) altering the quantum informatic framework from informatic/algebraic to stochastic/geometric (which provides a broader perspective on how these ideas work).
We suppose that Bob has in his laboratory an ion-trap containing (say) 100 trapped ions. Every morning, Bob performs just one tomographic measurement on these ions (it’s a long-running experiment). And every afternoon, Bob prepares (the same) quantum state for the following day’s tomographic measurements. Thus Bob’s life is pretty boring — every afternoon the (identical) state preparation, every morning a tomographic measurement of the previous day’s state.
Alice is spying on Bob. Every night, Alice sneaks into Bob’s lab and measures his carefully-prepared state (thereby destroying it). To conceal her activities, Alice performs covert measurement-and-control operations on Bob’s ions, leaving behind a state ρ that Bob will measure the following morning. Alice’s goal is that Bob’s daytime tomographic measurements reliably yield (as your lecture puts it) “Tr(Eρ) for most measurements E drawn from some probability measure D.”
So Alice is secretly “dry-labbing” Bob’s experiment … leaving behind states that are informatically indistinguishable (by Bob) from undisturbed states.
Of course, Alice has finite resources in information and time … she has to be done preparing the ions before Bob arrives the following morning. Obviously it’s a challenging task — she has 100 ions to entangle! To make Alice’s life harder, we assume that she does not know Bob’s experimental protocol (otherwise she could just duplicate it), but instead only knows the specified outcome distribution D.
How can Alice achieve her quantum deception goal? Or is it impossible?
Equally interesting—and not addressed in the lecture—does Alice really have to physically restore Bob’s quantum state? Could Alice instead install a (classical) “root kit” on Bob’s tomographic measurement software; a root kit that could reliably simulate every morning (with classical resources) any tomographic measurement that Bob might specify that morning?
—- End of Part 1 —-
38. Bobby Says:
Responder: I apologize, I meant to mention in my post that it seemed theoretically possible that the method given in the paper would allow the decryption process to perform better, in that the computation work would have been done by the other computers.
However, without something explicit in the paper which asserts the complexity of the decryption is bounded, and what’s more that the increase in size of the encrypted package by the computation process is bounded, it seems to me that there are no guarantees.
Also, honestly, my initial reaction to seeing the paper’s description is that it would have some profound implications, beyond letting you play performance games. After I recognized that you could do the same thing modulo performance by conventional methods, I thought I would see if someone saw a flaw in my reasoning.
39. Bram Cohen Says:
If you really want your head to explode, go read up on liquid flouride thorium reactors. There are vastly safer and cheaper ways of getting nuclear energy than we have now, which have the unfortunate downsides of being radically different than what’s used currently, so noone’s an expert in them, and far too inherently safe to have any weapons use at all, so the military won’t fund them, and politics has basically killed them for the last fifty years.
40. Jonathan Vos Post Says:
“If you really want your head to explode” — and who wouldn’t want that?
41. Jr Says:
What do you think is the most important open mathematical problem outside of TCS?
In science, outside of math and computer science?
42. Jr Says:
Also, why do you think religion exists?
43. Scott Says:
I was going to say the Riemann hypothesis, but we all know that’s just a derandomization problem. :-) So maybe the Langlands conjectures? But those, too, could conceivably turn out to be relevant to circuit lower bounds via Mulmuley’s program…
Whatever the answer is, “important” presumably means that a significant fraction of mathematicians would need to agree. So old standbys like 3x+1, twin primes, and the transcendence of π+e are presumably out… :-)
In science, outside of math and computer science?
In fundamental science, here are the first four things that popped into my head:
1. Why sex, sleep, and homosexuality exist
2. Extraterrestrial life (or even “earth-like” extrasolar planets, or non-DNA/RNA-based life on earth)
3. Physics beyond the Standard Model (wherever progress turns out to happen—electroweak symmetry breaking, Λ, ultra-high-energy cosmic rays, the Pioneer anomaly?)
4. Not clear whether there’s anything new and compelling with actual technical content to say about consciousness, free will, the anthropic principle, or the quantum measurement problem, but if there were, that would certainly count
In applied science (similarly, first four things that popped into my head):
1. Cheaper, more efficient solar cells (likewise, cheaper, safer nuclear reactors)
2. Mass manufacturing of wacky materials like carbon nanotubes, so we can haz SPACE ELEVATORS!
3. The ability to google and edit your own genome
4. Batteries that last
44. Scott Says:
Also, why do you think religion exists?
Once you accept that for almost all of history, and in most of the world today, the “purpose of life” has been to maintain cohesive tribes in which the men valiantly fight the rival tribes, the women stay faithful and raise children, etc.—and that uncovering the true nature of the physical universe only ever enters the picture insofar as it directly advances those goals—the question becomes, why shouldn’t religion exist?
45. John Sidles Says:
A Google search for “space elevator elastic energy” will find a literature replete with sobering engineering realities …
… that’s why I work in quantum spin microscopy instead … where the realities are sobering, but not as sobering.
As Pope put it: “Shallow draughts intoxicate the brain, but drinking largely sobers us again.”
Here “largely” has the seventeenth century meaning given in Samuel Johnson’s dictionary: “amply, widely, copiously”
46. Jonathan Vos Post Says:
I like Scott’s Comment #43 (half of a twin prime pair).
Massively compressing my impressions:
1. Why sex, sleep, and homosexuality exist — as problems in reconstructing path-dependent models embedded in Evolution by Natural Selection. Otherwise we can take the myth that humans were once 4-armed, 4-legged, unisexual, and were bifurctaed, always seeking our other halves.
2. Extraterrestrial life (or even “earth-like” extrasolar planets, or non-DNA/RNA-based life on earth) — interesting recent publications extrapolating back to BEFORE the RNA World. And a Strong Gaia hypothesis suggests doubling the Drake Equation approximation.
3. Physics beyond the Standard Model — and the Cosmology that dervives from that, via Quantum Cosmology arguments.
4. consciousness — I’ve been speaking with Cristoph Koch, whose 20 years of work with Francis Crick has yielded some amazing experimental results, in multiple processes competing in the human brain’s visual/semantic subsystems, with the interference nicely measurable, but below conscious awareness. Again, old Bayesian rationalists deny the demonstrable circuitry of the human brain.
Crick & Koch asked what are the neural Correlates of Consciousness, including: is there a minimum complexity of a system for it to be able to be a substrate for consciousness? And why, if our immune system, or entereic nervous system, or genome exceeds that threshhold, is the immune, gut, or genetic network NOT conscious?
47. Jonathan Vos Post Says:
Likewise, educated first impressions:
1. Cheaper, more efficient solar cells [I keep in touch with Dr. Geoffrey Landis, a real expert; and with the IdeaLab solar companies] (likewise, cheaper, safer nuclear reactors [also: smaller, down to scale of individual business or home])
2. Mass manufacturing of wacky materials like carbon nanotubes, so we can haz SPACE ELEVATORS! [some stranger molecules being investigated; meanwhile Space Elevators already feasible for the Moon]
3. The ability to google and edit your own genome [The humanist ethic begins with the belief that humans
are an essential part of nature. Through human minds
the biosphere has acquired the capacity to steer its
own evolution, and now we are in charge. Humans have
the right and the duty to reconstruct nature so that
humans and biosphere can both survive and prosper. For
humanists, the highest value is harmonious coexistence
between humans and nature. The greatest evils are
poverty, underdevelopment, unemployment, disease and
hunger, all the conditions that deprive people of
opportunities and limit their freedoms. -- HERETICAL THOUGHTS ABOUT SCIENCE AND SOCIETY
By Freeman Dyson]
4. Batteries that last [Cowan's Heinlein Concordance: Shipstone
1. Common power source. It involved intensive solar collection and energy storage but was not otherwise described. It apparently replaced almost all other sources of energy. The name also applied to the conglomerate that apparently owned most of the corporations on and off Earth.... In effect, Shipstone controlled the entire economy. A feud among different factions resulted in the overthrow and disruption of many Earth governments, particularly in North America. (Friday)
2. [mentioned in passing] Power source used for automobiles (and probably other devices).
(To Sail Beyond the Sunset)
[Compare D. D. Harriman's extensive holdings and economic influence in earlier stories, and the more benevolent depiction of an unlimited power source in "Let There Be Light".]
48. Bram Cohen Says:
Scott, does survey propogation constitute a full-blown exhaustive search algorithm, as opposed to just a stochastic search problem (I think the answer is ‘yes’, but just checking) and if so, would it apply naturally to algorithm X type problems, and if it does do you think it would on some instances be faster than dancing links in practice?
49. coder Says:
the homomorphic cryptosystem reminds me of McEliece and other coding cryptosystems. these too resist quantum attacks but require copious entropy.
poor entropy sources are the bane of a cryptographer’s existence; the concerns in #11 are relevant but well understood. many computers have hardware entropy sources these days…
50. Zack Says:
Is there a way to construct quantum money that allows one to make change? That is, I have a quantum banknote worth $A, I want to be able to convert it into two banknotes worth $X and $Y, where X + Y = A is enforced, without communicating with the issuing authority. Conversely, I would also like to be able to merge banknotes worth $X and $Y into a single banknote worth $(X+Y), again without communication.
51. Raoul Ohio Says:
While of course your religion is the one true religion, it is interesting to speculate about what’s the deal with all the wrong ones.
One of the weisenheimer columnists in Scientific American has an interesting model: The standard human mental kit fails to include a working BS detector. If true, this explains lots of other curious things. He fleshes the model into a mini theory by speculating how evolution produced this state of affairs. His guess is that higher brain functions are largely pattern recognition, and overreacting to any plausible threat pays off when there are lions about. This is an advantage, natural selection wise, leading to most people not thinking critically about whatever.
The virus theory, which I first read about in “Godel, Escher, and Bach”, is also good for a few laughs. An entertaining variation has religion as kind of a Ponzi scheme, the various priesthoods may or may not be in on the joke. Putting on your optimization hat, can anyone think of a better scam than trading infinite bliss in the next life for money and obedience right now?
52. Scott Says:
Zack: That’s an extremely interesting question!
“Merging” two banknotes can in some sense be trivially done, by just putting the banknotes side-by-side. It’s “splitting” a banknote that’s the problem—at least, assuming you don’t want the number of qubits in a banknote to grow linearly with its value (in which case we could just let an $n banknote consist of n $1 notes).
Another simple observation is that ordinary cash does not provide the functionality you ask for, and yet we seem to make do anyway, mostly by choosing denominations ($100, $20, $10, $5, $1…) in such a way that if there are enough people at the restaurant table, then w.h.p. it’s possible to make change.
Probably the first step should be to find one quantum money scheme with reasonable evidence for its security, that at least provides the same functionalities as ordinary cash! Then we can worry about additional functionalities like the one you ask for.
[Note: Using the ideas from my CCC paper, it shouldn't be hard to construct a quantum oracle relative to which a quantum money scheme with the "splitting" and "merging" functionalities exists. The hard part, as usual, is to find an explicit scheme, one that works even with no oracle.]
53. Scott Says:
Bram: Survey propagation is basically a local search algorithm; it doesn’t find refutations for unsatisfiable instances. I’m sorry I don’t know the answers to your other questions.
54. Bram Cohen Says:
Hrmph, my reading of survey propogation was way off. Is there an intro to it for non-mathematicians? I can read reference code, but not speak math.
55. Zack Says:
It’s true that regular old cash is not splittable, but I observe that cash is falling into disuse (replaced by debit and credit cards) and speculate that not having to deal with change is a major reason for this. Debit and credit cards, of course, do require communication with the bank.
If quantum banknotes could be split and merged, then they would solve a practical problem with cash (unsplittability) as well as one that’s not really a problem in practice (unforgeability)…
56. Bobby Says:
Can someone answer my question from comment #24?
In short:
I understand that the paper does demonstrate a system with at least one property that the classical system I give above doesn’t have. Excerpted from comment 24:
My question is, does the method discussed in the paper provide anything else that a classical system of concatenating messages listing the operations to be performed wouldn’t provide? |
f523c8d34b06e431 | zbMATH — the first resource for mathematics
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
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any anywhere an internal document identifier
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Global existence of small solutions to a relativistic nonlinear Schrödinger equation. (English) Zbl 0948.81025
Summary: We study the Cauchy problem associated to a nonlinear Schrödinger equation modelling the self-channeling of a high power, ultra-short laser pulse in matter. The new nonlinear terms arise from relativistic effects and from the ponderomotive force. We prove global existence and uniqueness of small solutions in transverse space dimensions 2 and 3, and local existence without any smallness condition in transverse space dimension 1.
81V80Applications of quantum theory to quantum optics
35Q55NLS-like (nonlinear Schrödinger) equations
35B60Continuation of solutions of PDE |
234c35a543346e98 | HOME » PROGRAMS/ACTIVITIES » Annual Thematic Program
IMA Annual Program Year Tutorial
Mathematical and Computational Approaches to Quantum Chemistry
September 26-27, 2008
Eric Cances CERMICS, Ecole Nationale des Ponts et Chaussées
Juan C. Meza Lawrence Berkeley National Laboratory
Electronic structure calculations have become an indispensable tool in chemistry, molecular biology, materials science, and nanotechnology. The density functional theory (DFT) of Hohenberg, Kohn and Sham is an approach for computing the ground-state density and energy of a many-electron system by solving a constrained minimization problem whose first order optimality conditions, the Kohn-Sham equations, can be written as a nonlinear eigenvalue problem. Used almost exclusively in condensed matter physics since the 1970's, DFT became popular in quantum chemistry in the 1990's due to the development of more accurate approximations. Today, DFT is the most widely used ab initio method in material simulations. DFT can be used to calculate the electronic structure, the charge density, the total energy, and the atomic forces of a material system; and with the advance of new algorithms and supercomputers, DFT can now be used to study thousand-atom systems. There are many challenges remaining though, especially for large systems (more than 100,000 atoms), problems requiring many total energy calculation steps (molecular dynamics or atomic relaxations), or systems with open-shell character. More accurate and better-justified approximations to the density functional for the exchange-correlation energy are also continually being developed, requiring new exact constraints and presenting new computational challenges.
Wave function methods have also known spectacular development in recent years. These methods allow, in principle, the construction of increasingly refined approximations to the many-electron Schrödinger equation. They outperform conventional DFT with respect to accuracy, but at the price of a dramatic increase in computational cost. Reducing the computational cost of wave function methods, while preserving its accuracy is one of the major challenges in quantum chemistry. Important steps in this direction have been taken with the introduction of linear scaling algorithms. Other important challenges include systems with electronic degeneracies and calculations of a wider range of properties and experimental observables.
This tutorial will focus on presenting some of the fundamental concepts and techniques currently used in electronic structure calculations. The first day will introduce some of the key ideas of quantum mechanics and wave function methods, including coupled cluster methods and DFT. This will be followed on the second day by an introduction to some of the major mathematical techniques used in the formulation and solution of electronic structure problems. We will also discuss some commonly used computational methods for solving these problems. Throughout, we will present some of the mathematical and computational challenges in developing accurate, efficient, and robust algorithms for electronic structure calculations of large systems. |
7311f645c7527112 |
Argumentation about de Broglie-Bohm pilot wave theory
The most important thing: Measurement theory
Your equations about \(X\) are completely irrelevant for the measurement of the spin. The problem is not when one wants to measure \(X\). Indeed, the measurement of \(X\) might occur analogously to its measurement in the spinless case. The problem occurs when one actually wants to measure the spin itself.
The projection of the spin \(j_z\) is an observable that can have two values, in the spin \(1/2\) case, either \(+1/2\) or \(-1/2\). It is a basic and completely well-established feature of QM that one of these values must be measured if we measure it.
How is your 17th century deterministic theory supposed to predict this discrete value? Like with \(X\), it must already have a classical value for this quantity. Except that in this case, it has to be discrete, so it can't be described by any continuous equation. ...
Preemptively: you might also argue that any actual measurement of the spin reduces to a measurement of \(X\). But it's not true. I can design gadgets that either absorb or not absorb the electron depending on its \(j_z\). So they measure \(j_z\) directly. deBB theories of all kinds will inevitably fail, not being able to predict that with some probability, the electron is absorbed, and with others, they're not. This has nothing to do with \(X\) or some driving ways. It is about the probability of having the spin itself.
First, there is some interaction of the wave function of the electron with the wave function of the measurement device. (There is of course also an equation for the position of the electron \(q_{el}\) – the \(X\) in lumo's text – but it is completely irrelevant, not only at this stage, but in the whole process.) The result of the measurement is, as usual, a wave function of type\[
|\psi\rangle = \alpha_1|{\rm up}\rangle|q_1\rangle + \alpha_2|{\rm down}\rangle|q_2\rangle
\] This exploitation of standard QT is not enough – now decoherence will be exploited in an equally shameless way. We leave it to decoherence considerations to decide which observables of the measurement device become amplified or macroscopic. Assume the quantum states \(|q_1\rangle, |q_2\rangle\) are decoherence-preferred. In this case, decoherence amplifies the microscopic measurement results \(|q_1\rangle, |q_2\rangle\) into classical, macroscopically different states \(|c_1\rangle, |c_2\rangle\). After finishing this hard job, it presents the following state:\[
|\psi\rangle = \alpha_1|{\rm up}\rangle|c_1\rangle + \alpha_2|{\rm down}\rangle|c_2\rangle
\] Now, everything is prepared, it remains to make the really important decision which of the wave packets is the best one ;-). At this moment a hidden variable enters the scene. But, surprise, it is not the hidden variable of the electron \(q_{el}\) (lumo's X), but that of the classical measurement device \(q_c\).
The job of \(q_c\) is not a really hard one. After driving around (no, being driven around by quantum guides) in an almost unpredictable way, it simply takes the wave packet prepared for him by the quantum operators at the point of arrival ;-). In other words, we simply have to put the actual value of \(q_c(t)\) into the full wave function \(|\Psi\rangle\) to obtain the (unnormalized) effective wave function:\[
\psi(q_e) = \Psi(q_e, q_c(t))
\]What we need for this scheme to work as an ideal quantum measurement is not much. We need that the two states of the macroscopic device \(|c_1\rangle, |c_2\rangle\) do not (significantly) overlap as functions of the hidden variable \(q_c\). In this case, whatever the value of \(q_c\), the result \(\psi(q_e)\) will be a unique choice between two effective wave functions, namely between \(|{\rm up}\rangle\) if \(q_c\) is in the support of \(|c_1\rangle\), and \(|{\rm down}\rangle\) otherwise. And we need the quantum equilibrium assumption for \(q_c\) to obtain the probabilities for these two choices as \(|\alpha_1|^2\) resp. \(|\alpha_2|^2\).
About the zeros of the wave function
How do we know that \(m=l_z/\hbar\) must be an integer? Well, it is because the wave function \(\psi(x,y,z)\) of the m-eigenstates depends on \(\phi\), the longitude (one of the spherical or axial coordinates), via the factor \(\exp(i\cdot m\cdot\phi)\) which must be single-valued. Only in terms of the whole \(\psi\), we have an argument.
However, when you rewrite the complex function \(\psi(r,\theta,\phi)\) in the polar form, as \(R\exp(iS)\), the condition for the single-valuedness of \(\psi\) becomes another condition for the single-valuedness of S up to integer multiples of \(2\pi\). If you write the exponential as \(\exp(iS/\hbar)\), the "action" called S here must be well-defined everywhere up to jumps that are multiples of \(h = 2\pi\hbar\).
More generally, something very singular seems to be happening near the \(R=0\) strings in the Bohmian model of space.
About relativistic symmetry and the preferred frame
I'm happy to answer this question: The preferred coordinates are harmonic. Given, additionally, the global CMBR frame, with time after big bang as the time coordinate, this prescription is already unique. For a corresponding theory of gravity, mathematically almost exactly GR on flat background in harmonic gauge, physically with preferred frame and ether interpretation, see my generalization of the Lorentz ether to gravity.
Then, to postulate a fundamental Poincare symmetry is, of course, a technically easy way if one wants to obtain a theory with Poincare symmetry. But what is the purpose of a postulated global Poincare symmetry in a situation where the observable symmetry is different, depends on the physics, as in general relativity? Whatever the representation of the \(g_{\mu\nu}(x)\) on the Minkowski background – it will (except for simple conformally trivial cases) have a different light cone almost everywhere. If the Minkowski background lightcone is the smaller one, one has somewhere to violate the background Poincare symmetry. It may be always the other way. But in this case, the axioms of the theory give only restrictions for the background Minkowski light cone, not for the physical light cone. Thus, tensions with the physical Lorentz invariance may arise in the same way, because the theory only looks like one which, in the particular point \(x\), has the Lorentz invariance for the metric \(g_{\mu\nu}(x)\). But really it is a theory with Lorentz invariance for a different metric \(\eta_{\mu\nu}\), with a larger light cone, thus, allows for superluminal information transfer relative to \(g_{\mu\nu}(x)\).
... and the ether ...
The similarity with the luminiferous aether seems manifest. ...
About signs of the heavens
It is not surprising in any way that the new, Bohmian equation for \(X(t)\) can be written down: it is clearly always possible to rewrite the Schrödinger equation as one real equation for the squared absolute value (probability density) and one for the phase (resembling the classical Hamilton-Jacobi equation). And it is always possible to interpret the first equation as a Liouville equation and derive the equation for \(X(t)\) that it would follow from. There's no "sign of the heavens" here.
How to distinguish useful improvements from unnecessary superconstructions
[Einstein] called the picture an unnecessary superconstruction.
What are the fundamental beables?
In the simplest case of a scalar field, the natural candidate for the "primitive property" or the "beable" is simply the field \(\phi(x)\). This is a very old idea, proposed already by Bohm. But the effective fields of the standard model are also bad candidates for really fundamental beables. They are, last but not least, only effective fields, not fundamental fields. In my opinion, one needs a more fundamental theory to find the true beables.
My proposal for such more fundamental beables can be found in my paper about the cell lattice model arXiv:0908.0591. Even if pilot wave theory is not mentioned at all in this paper, it is quite obvious that the canonical quantization proposal for fermion fields I have made there allows to apply the standard formalism of pilot wave theory to obtain a pilot wave version of this theory.
Problems with spin and with particle ontology in quantum field theories
About the "segregation" among observables
And this construction is actually very unnatural because it picks \(X\) as a preferred observable in whose basis the wave vector should be (artificially) separated into the probability densities and phases
Clearly, some quantities in the real world look more classical than others. But what are the rules of the game that separates them? The Bohmists assume that everything that "smells" like \(X\) or \(P\) is classical while other things are not. ...
About history
About decoherence and the classical limit
Essentially, you can measure every operator, together with every other, if the accuracy of the common measurement is below the boundaries of the uncertainty relations. And in the classical \(\hbar \to 0\) limit they all like to behave classically.
Last but not least, some funny but unimportant polemics
And, indeed, the "experimental evidence" presented by lumo was (in his polarizer argument, and similar ones about spins) based on the common error not to take into account the measurement device, or (in his quantization argument) not applicable to de Broglie's version of pilot wave theory. About the theoretical evidence judge yourself.
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reader Luboš Motl said...
Dear Ilja, thanks for this almost professionally constructed reply - with a nice formatting, formulae etc.
Unfortunately, almost no part of the content of this blog entry is correct. ;-)
Perhaps, a valid point is that the "pilot wave theory" is more accurate than "Bohmian mechanics". However, when you said that the original de Broglie theory is preferred to solve the non-single-valuedness problem of mine, I had to laugh out loud because a few paragraphs earlier, you wrote that this theory was abandoned by de Broglie because of another argument of mine, more or less.
Concerning some other points, it's amusing that you say that "decoherence solves everything" because decoherence only works in proper quantum mechanics. The pilot wave theory isn't quantum mechanics and indeed, the very main point of this theory is that it replaces the genuine dynamical quantum mechanism selecting the "preferred observable and bases" - decoherence - by something totally different, namely predetermined observables that also have classical values aside from the pilot wave that guides them.
So if you need decoherence in the pilot wave theory, it won't work and it will become yet another crushing argument against the pilot wave theory because decoherence is incompatible with the actual pilot-wave-based mechanisms that select what will be observed. Do you agree with that?
When you say it's just a "rotor", you don't actually show that the theory gives the right prediction - S is single-valued up to additive integer multiples of 2.pi. You don't show that because you can't - this correct constraint doesn't really follow from the pilot wave theory. Incidentally, the divergent velocity isn't harmless, either. It's experimentally more or less demonstrable that there's nothing special happening near the places where psi=0. In particular, the relativistic corrections don't get any stronger because of these points. In the pilot wave theory, as you admit, the "Bohmian trajectory's" velocity goes to infinity which does indicate that relativity should play an increased role there. But it doesn't.
reader Luboš Motl said...
Second part. I used the term "dramatically nonlocal" because the ability to influence remote regions belongs to the very basic built-in properties of the objects in the pilot wave theory. I mean that there doesn't exist any glimpse of an argument that these effects should be small - so they won't be small unless one tries to fine-tune everything. The pilot wave theory contains classical waves that are functions of several position vectors and the evolution equation directly guides the positions of particles depending on the immediate values of these multilocal objects anywhere in the configuration space. Those guiding waves are affected by other particles, e.g. those freshly created ones if you assume that the theory *is* able to produce new particles, which it's not, so there is a heavily, dramatically, lethally nonlocal action in both directions. The result must look like a completely generic nonlocal evolution, in contrast with all observations of the 20th century physics.
reader and said...
with the risk of sounding like LM 's echo: there is *nothing* valid about "boemian" pilot wave theory...
reader John H Duffield said...
Lubos, well done for offering this guest blog. I'm broadly in agreement with Ilja, and I think you should more closely into this subject and try to set aside your hostility. Einstein reintroduced an aether for GR , the optical Fourier transform is an analogy for wavefunction-wavefunction interaction, see work by Aephraim Steinberg et al and Jeff Lundeen et al re "wavefunction is real", check out Percy Hammond re electromagnetic geometry, look at The Other Meaning of Special Relativity by Robert Close, see , and , to think of the electron as a Dirac's belt standing-wave photon-field structure. Etc etc. There's elements of TQFT and even an underlying "stringiness" to this. Don't dismiss it all because somebody can't get the maths right.
reader Mephisto said...
I personally like Bohm. He was a nice person with great strength of character, a political victim of the McCarthy era. But from what I know about the Bohmian mechanics, I do not believe it to be true
Arguments against it
1) Bohm theorists believe that the quantum wave is real. It is easy in 1 particle case. But if you have N particles, you need a wave function in 3N+1 dimensions. Are these 3N+1 dimensional wave functions also real?
2) Spin. Lumo made the point: "If de Broglie and Bohm claim that a particle should also have a well-defined position and velocity, it should naturally have a well-defined z-projection of spin, too. But once you adopt such an assumption, you clearly break the rotational symmetry. Particles would only have classical projections of spin with respect to the z axis so the z axis is preferred and you can measure its direction, at least in principle, uncovering anisotropy of space. The rotational symmetry of a theory including spinors heavily depends on the probabilistic nature of quantum mechanics. If you give up the equal treatment of position and spin and decide to treat spin differently and give an electron well-defined binary-valued projections of spin with respect to all axes, you will also encounter problems. Bell's inequality will show you very sharply that the required dynamics is completely non-local but you will also have problems with the Lorentz invariance and the precise rules for the evolution of the discrete function of the direction. The probabilistic meaning of the spinorial wave functions is completely essential for us to be able to translate a physical arrangement to any convention, including an arbitrary choice of the z-axis."
Spin needs to be understood within the framework of relativistic quantum field theory. In QFT, every particle species is associated with a quantum field and the quantum field Lorentz transforms in a partiular way - we have spin 0,1,2 and spin 1/2,3/2 etc fields. It turns out that all these fields are related to representations of the Poincare group (Wigner classification). There is a deep connection between the relativistic symmetry of spacetime (Poincare group) and the spin of quantum fields (representations of the group). This connection is imho very elegant and powerful and the Bohmian mechanics is ugly mess in comparison.
3) My personal issue with QM. I agree that wave function is not real. Collapse of the wave function is just a change in our knowledge. Most misunderstandings of quantum theory come from incorrect use of language and the use of vaguely defined concepts like "local reality". Lumo says that in an entangled pair, the particles do not communicate in any way. I agree. It is the only meaningful way how to avoid terrible paradoxes with space-like sepated entangled particles. But I have issues with the following claim "The moon is not there if nobody is looking". Where and how does nature store information about the correlation of the particles (how does nature remember the correlation), if the particles DO NOT EXIST prior to measurement. Only the quantum fields of bubbling probabilities exist before the measurement. If the particle and its spin is created (comes into existence) by the act of measurement at detector A, how does nature know that other particle at detector B should be created in such a way that it is correlated. This in my oppinion seems to invalidate the claim that nothing exists prior to measurement. (position of the Copenhagen school)
reader Mephisto said...
And when discussing someone's theory, it is always best to go the to source
David Bohm - The de Broglie Pilot Wave Theory
reader Luboš Motl said...
Well, if and when the original theory doesn't work, it doesn't help one much to go to the original source.
reader Ilja said...
1.) In dBB, yes. I don't like this too, and I think it is possible to get rid of this, using dBB theory as a starting point. See arXiv:1103.3506
2.) As explained, I also prefer field-theoretic variants.
3.) To reject realism is of course consistent, as consistent as "God moves in mysterious ways". If you accept realism, you have to accept its nonlocal variant, given the violation of Bell's inequality. So if you want causality without causal loops, you need a hidden preferred frame.
reader lucretius said...
Bohm can be called a "victim of the McCarthy era" but he can hardly be called "an innocent victim". I no more sympathetic to communist victims of McCarthy during the Stalin period than I am to the 740 members of the British Union of Fascist who were interned in Britain from 1940 till the end of the war.
I would like to add that I find this discussion fascinating (thank you Lubos) although I don't want at this point in time to take clear sides. I agree, however, that there most people have psychological difficulties with the Copenhagen interpretation and that this is only natural. Unfortunately the Bohmian approach, ,does not seem to me to be significantly better in this respect (although I need to think more about it, when I find the time). Personally I still prefer to think of QM as a computing tool. In this sense the key issue would seem to me to be: does the Bohmian mechanics really enhance computation? It seems unlikely.
As for Mephisto's question about "where the information is stored" - clearly the information about the correlation needs to be "remembered" by Nature. It seems indeed strange that the information about correlation could be "remembered" if the correlated particles do not exist but one could also ask: where are the "laws of nature" themselves stored? I seems to me we can't expect intuitive ideas acquired from our daily experience to apply to these sort of matters.
reader Rehbock said...
Interesting piece. But the premise:
"... I think there should be really good evidence to justify the rejection of such simple, general, fundamental and beautiful principles like realism. "
seems flawed.
Are not experimental outcomes confirming always QM during last 100 years "good evidence", if needed, that nature is not required to share our primitive view of reality. Also, our personal subjective construction(s) of reality does not established realism as any of those glowing adjectives.Why should realism be so fundamental?
reader Ilja said...
Experimental outcomes of QM are in agreement with dBB theory, which is realistic, so are not a problem for realism. And one should, of course, distinguish our particular primitive realistic models from realism, that means, the general hypothesis that such a model (however complicate) exists in principle.
See my home page for some arguments in favour of realism.
One can, in principle, use rigorous positivism, we observe correlations, and have formulas to compute them, that's all, no idea why the formulas work. I don't think it is a good idea.
reader Luboš Motl said...
Dear Ilja, the probabilistic distribution of X for non-relativistic QM models for one or several spinless particles may be "emulated" in this "realistic" dBB pictures but that's far from enough to do physics today and all opinions that the theory agrees with more than that are flawed ideas based on wishful thinking, neverending promises, and lies.
The pilot wave theory can never deal with quantum field theory or any other relativistic theory. It's not just the absence of the Lorentz symmetry. It's also the existence of observables with discrete spectra that appear everywhere and that can't be given dBB "actual value supplements".
Moreover, dBB is inevitably incompatible with the particle production - creation and annihilation of pairs in QFT. This is also easy to see.
dBB also fails to account for the actual macroscopic quantum behavior of large systems, contradicts decoherence, and I am not even discussing the aesthetic flaws that show, to a person with a good physics intuition, that it is just a completely fabricated attempt to deny the important insights that the quantum revolution has made.
reader Ilja said...
About the Wallstrom objection
It does not indicate. The Bohmian trajectory is unobservable, but relativistic symmetry is about observables only. (For Bohmian field theories, which is what is preferable in the relativistic case, it is irrelevant anyway, because it is \(\dot{\phi}\) and not a velocity in space which becomes infinite.)
Again, if one starts with the wave function as being fundamental - as modern dBB or "pilot wave" theory does - this is as unproblematic as in quantum theory.
It becomes problematic only if one goes beyond standard dBB theory and prefers, instead, to consider R and S, or \(R^2\) and v, as fundamental. This is what I prefer. But I also have a way how to solve this problem, see arXiv:1101.5774. This approach also regularizes the infinity of the velocity.
It's ugly? Ok, I think it is a good idea to look for more beautiful interpretations. It seems, the difference is about the criteria for comparison. I think giving up realism is stupid, equivalent to "Nature moves in mysterious ways". With realism and loop-free causality we need a preferred frame. That's my starting point. The next inacceptable thing are infinities. I don't have anything against hidden variables. Ok, its not nice that they are hidden, so let's try to find them, for example by looking where they become very large or infinite - this may be the place where the theory is wrong and they become visible. Symmetry is something very important, but not as important as realism and finiteness. As simple and as symmetric as possible.
reader Ilja said...
The dBB scheme works for arbitrary configuration spaces Q, no need to restrict it to particles. The first example of a relativistic quantum field theory (EM) is already part of Bohm's paper. All what you need for observables with discrete spectra see Bohms original paper or the text here in the blog. The classical limit in dBB theory is much easier.
reader Justin Glick said...
What about having a non-local interaction without a preferred frame? e.g. preserving Einsteinian relativity.
reader Ilja said...
A realistic Einstein-causal theory cannot give the violations of Bell's inequality predicted by quantum theory. So this is rather hopeless.
(I also don't like that "local" is used instead of "Einstein-causal", but this is how "local" is used today.)
reader Justin Glick said...
And if you read what I wrote, I did not say the theory should be local, or "Einstein-causal" as you like to write. I wrote that it should 1. be non-local
2. preserve Einstein causality
Also, by Einstein causality I mean no preferred frame, but also Lorentz invariant.
Before you say this is impossible, let me remind you that before 1905 everybody in the world thought that
1. Inertial frames are equivalent
2. speed of light is frame independent
were incompatible.
reader Ilja said...
Sounds like I misunderstood you, but, whatever, I see no reasonable chance to make realism compatible with preservation of Einstein causality.
reader Ilja said...
What's wrong about a hidden preferred frame?
The most horrible point: It's been really known not to exist since 1887 when its inevitable prediction of the aether wind was falsified by Morley and Michelson. That's the end of the story. A theory without it had to be designed. Einstein shown that the Lorentz invariance was needed for every theory that avoids the pathological because falsified prediction of the aether wind. Sounds like the type of argument from "relativists" who have not even heard about the Lorentz interpretation, which has a hidden preferred frame. So, relativity 101, there have been two interpretations of the Lorentz-Einstein theory, the Minkowski interpretation, without ether but a spacetime, and the Lorentz interpretation, with absolute time, and an ether which distorts rulers and clocks, in such a way that one cannot measure absolute time, and so the preferred frame remains hidden. Above variants predict Lorentz symmetry for all observables, and the same result for Michelson Morley. So, the MMX does not falsify the Lorentz interpretation.
Ok, maybe lumo used a polemical way to point out that the Lorentz interpretation has a problem to explain why the preferred frame is hidden? That would be fine, because this is really an interesting problem. How to solve it? The next failure in lumos answer: Without an infinite amount of fine-tuning, you just can't get it. Really no other way? Just an idea: It is quite typical that the symmetry groups of a fundamental theory and its approximation are different. Fine, lumo even has some nice theory about this: Quite generally, the recipe for "partially valid" symmetries in particle physics goes in the opposite direction. They're preserved at short distances, in the fundamental equations, and broken at long distances where symmetry-breaking mechanisms become important.
Oh, really only in this direction? Ever heard about a lattice theory? The fundamental theory has a discrete symmetry, its large distance approximation, instead, a continuous symmetry group. A nice example is the silicon lattice. It, of course, has some preferred planes. But if one considers its mechanical properties in the large distance and the lowest order, these preferred planes become unobservable and we obtain rotational symmetry. So, the other direction exists too. Approximation means loss of information, and the result of a loss of distinguishing information may be an increase in symmetry.
Let's clarify: These are only simple common sense arguments, appropriate for a blog, to show where lumo's arguments fail. The problem remains: To explain why we have Lorentz symmetry for the observable effects. Fortunately, it has been solved in arXiv:gr-qc/0205035. In this paper, I have derived the Lagrangian of my theory from some simple first principles, and this gives, as a side effect, the Einstein equivalence principle, thus, local Lorentz symmetry. So, yes, there is a problem, but it is not unsolvable, as lumo claims with obviously weak arguments, but already solved.
Another rather trivial example of a higher symmetry obtained by approximation is equilibrium. In the simplest example of global thermodynamic equilibrium we obtain, instead of a lot of inhomogeneous non-equilibrium solutions, only homogeneous equilibrium solutions, thus, we obtain translational symmetry not present in non-equilibrium theory. Something similar happens in dBB theory. We start from a nonlocal theory, and consider quantum equilibrium. And the theory reduces effectively to quantum theory, with quite different symmetries - those of the Hamiltonian. In particular, if the Hamiltonian has the appropriate relativistic symmetry, the predictions about observables will show relativistic symmetry, and it becomes impossible to use the nonlocal fundamental features for information transfer.
reader and said...
Sorry, but No! I tried again but I still think LM is right... You just cannot change the place of the "hidden variable" and think you cured everything. You remain with the same problem explained for the spin 1/2 electron...
reader Ilja said...
Sorry, please explain, I don't understand your point.
Do you mean the choice of the beable, particles vs. fields? This changes a lot because you don't have to handle particle creation.
Do you think about how to handle Dirac particles in dBB field theory? A completely different and nontrivial question, there are various ideas about this. My own approach gives only pairs of Dirac fermions together with a massive scalar field, to be interpreted as electroweak doublet together with some dark matter. See arXiv:0908.0591 for how to reduce this to a simple scalar field with strange potential. How to handle a scalar field in dBB is well-known and simple.
reader Justin Glick said...
OK, let me clarify. I agree with what you just wrote. You can't have traditional notions of causality. But, relativity doesn't say that we can't modify notions of causality. It only says C is constant in all frames, and inertial frames are equivalent. Now, if we modify our traditional ideas about causality, then we can save Einsteinian relativity and also preserve realism. No preferred frame is necessary.
reader anna v said...
As an experimentalist , in particle physics, I hope I am a student of reality. Quantum mechanics is a beautiful self contained mathematical framework that works for all the known data and at the same time an intuition can be developed about how nature behaves in the microcosm, which helps in looking for new unexpected effects.
For an experimentalist, a new microcosm mathematical framework which gives the exact same measurable predictions is not interesting or relevant to reality, it is a mathematical game. Are there any predictions of this new mathematical framework, supposing that all of Lumo's objections are met, which diverge from the predictions of the standard QM mathematical framework? Is there an experiment that can show it up?
If not the adjective "real" cannot really be applied to mathematics, except if one is talking about the form of written formulae . In my books, "real" in physics means "measurable".
reader Mephisto said...
It sounds kind of boring to be an experimentalist - no matter how it works, why it works, what it means, I am happy if I can feed it with numbers and get predictions for my experiments. Fortunately not all experimentalists have this attitute. I read a book from Zeilinger (Einsteins Schleier). He is an experimentalist and he is interested in what it means. The quest for meaning of quantum mechanics was probably his driving motive for his career choice and his work.
There are various formulations and various interpretations of QM and every formulations gives you an unique perspective. Through every formulation you understand the underlying theory better. I remember Feynman talking about the same thing in one of his lectures (The Character of Physical Law)
Quantum mechanics is very interesting for philosophers. The questions of reality, ontology, knowledge etc. were always the traditional domain of philosophy and QM can tell us much about these things. Unfortunately, not many philosophers understand QM, since to understand it, you have spend years studying physics.
reader anna v said...
The various formulations of QM are all within the same framework/postulates. This proposal adds another level/ a meta level of complexity without giving any physics results different than the simpler in complexity levels, except philosophical preferences. I am interested in the physics not the philosophy.
reader Ilja said...
Fine. But this is simply a direction of research which I would not follow. I think there are a lot of other people following these directions, while I'm almost alone in the other direction,
Of course lumo is right if he argues that giving up some symmetry is not nice. I argue only that giving up realism or causality is even worse. But the point is not even what is worse, because it is clearly reasonable to look in different directions. I have found a quite nice one, with no competition, because to research in this direction is anathema. Quite comfortable, if one does not need a job. Interesting problems with reasonably simple solutions abound, because nobody looks for them.
reader lucretius said...
“Every formulation offers a unique perspective” sounds like a truism. Of course for philosophers such truisms can be interesting, especially if you agree with Wittgenstein that “philosophy leaves everything as it is”. But physics does not leave everything as it is: the point of physics is not to describe the same thing again in a new way but to discover new phenomena, explain things that have not been previously explained, suggest new experiments etc.
If you have two formulations of a theory that are formally equivalent (in the sense that they can be used to derive the same mathematical formulas in all areas of applicability of the theory) they may still differ in their convenience and effectiveness. Things that take simple form in one formulation may become complex and convoluted in the other. I think this is much more important to a physicist than the purely psychological comfort of being able to retain “realism”.
Philosophers, who generally don’t compute things or apply mathematics to resolve confusing physical puzzles (like the recent discussion of black-hole “firewall”) have different priorities but for most physicists the key issue should be: how good is bohmian mechanics compared with standard Copenhagen approach as a computational tool? Even if Lubos’s objections can be overcome, the record suggest that very few new phenomena have been discovered by means of bohemian mechanics and most of the work done within this formalism is “parasitic” on the standard QM formalism. The only area in which this is not true is, I think, quantum chemistry. It would be interesting to hear some suggest some explanation of this fact.
reader Luboš Motl said...
Dear Anna, I would personally not endorse the algorithm of theory selection that you propose - it's Occam's razor ad absurdum.
Of course that in the development of science, there are often moments in which the newer theory *is* or at least *looks* more complicated than the older one, but it must still be accepted and this necessity becomes more manifest later when further unification or addition of new sectors or applications arrives.
The only legitimate way to rule out a theory in science is falsification - a proof of incompatibility of theory's predictions either with themselves or with the empirical data. The pilot wave theory may be falsified in this way but if it couldn't, your vague philosophical observations of complexity wouldn't be a solid enough proof to abandon the framework.
reader Luboš Motl said...
Dear Ilja, this favorite verb of yours, "giving up", just doesn't belong to science. Your usage of it proves that you are not thinking about these things rationally, scientifically, impartially.
Science is not about "preserving" or "giving up" something. These labels mean nothing else than some bias, an emotional attachment to some belief. Science is about finding the truth about Nature.
Darwinism has to "give up" God, at least some previously believed essential parts of this construct. Heliocentrism has to give up the "natural" (blah blah, propaganda) assumption that the body we inhabit is the center of the Universe. A kinetic theory of heat "gives up" the idea (of phlogiston) that everything we can feel by our skin is a material with a particular atomic composition. And so on, and so on.
But it's right to "give up" these assumptions because they are simply wrong. The case of realism behind foundations of classical/quantum mechanics is *totally* analogous. One must "give up" - without any crying - the assumptions behind classical physics (and the pilot wave theory) because science has demonstrated them to be wrong. If you cry or whine, you're just not an honest scientist.
The real problem (one of very numerous problems) of the de Broglie theory isn't that it "gives up" a symmetry. It's that the theory gives wrong predictions for experiments that show that the symmetry is actually there - in some cases, an absolute contradiction that can't be fixed by any improvement; in other cases, a soft contradiction which means that the pilot wave theory has to be unacceptably fine-tuned or fudged to account for the observations. The first situation is a straight and immediate falsification of the theory; the latter is a gradual disfavoring of the theory that may become arbitrarily strong and urgent.
reader Ilja said...
For me, "possibly measurable in 500 years" means also "real".
The problem of infinite velocities in dBB theory near the zeros of the wave function suggests (if one assumes that there are no infinities in Nature) that there has to be some regularization. As a consequence, in the regularized subquantum theory there would be no point with exactly zero probability. See arXiv:1101.5774.
I think this is a general scheme, and a reason to consider different interpretations. Different interpretations may have different weak points, which suggest modifications (regularizations) of these interpretations, which are, then, already different theories making different predictions. Atomic theory has been, at the start, only an interpretation. Predictions came later.
From this point of view interpretations which propose hidden variables seem especially good ideas, because "hidden" in a normal situation does not mean "without problems". A preferred frame, even if hidden in the Solar system, becomes problematic if one considers solutions with causal loops, but possibly already in more harmless situations. Which? These may be the places where new physics appear.
So my theory of gravity arXiv:gr-qc/0205035 identifies such places as the big bang (replaced by a very rigid inflation with a big bounce, and an additional dark energy term which would shift the expansion toward a'=0) and black holes near the horizon, a place where according to GR nothing strange happens. Now there are discussions about firewalls at the same place.
By the way, I would not name QM (at least in the Copenhagen interpretation) self-contained.
reader Ilja said...
You have forgotten Bell's inequality. Bell was at that time almost the only proponent of dBB, so this suggests good output per man-year.
Technically, the classical limit is much easier in dBB - you don't have to consider wave packets, instead, already in a wide packet (rho close to const) the Bohmian trajectories are almost classical. This may explain why it may be useful in chemistry.
reader Mephisto said...
Looking at things from different perspectives can give you better understanding of the problem, can help you train your intuition and this can help you later to look for future research directions and advance physics.
Why study the Hamilton-Jacobi theory of classical mechanics? It gives you nothing new except of better understanding of classical mechanics. Later it unexpectredly helped Schrödinger to invent his equation
And the same applies to string theory. First various formulations were discovered. Later partly unified into M-theory. By studying the various perspectives (versions) of string theory, you gain a better understanding of the whole, of the structure underlying all of the versions. So even in physics, it always a good idea to study a problem from all available perspectives, because it helps you to understand a problem better and if you understand a problem better, you have a better chance of coming up with new solutions.
reader Ilja said...
Scientists are human beings with human errors and emotions, me too, not a problem as long as other scientists follow other emotions and make different errors. Giving up or preserving some principles are strategies for the search of new theories, and I insist that it is useful if different scientists follow different strategies. Most of them will fail, that's the risk.
Wrong predictions for experiments are not (yet) a problem of an interpretation with an equivalence theorem with QM.
Not having an explanation for an observable symmetry is, without doubt, a serious problem. See my other reply for how I propose to solve it.
How many man-years have been spend for versions of string theory unable to handle fermions? I doubt you think this was wrong. dBB theory is in a better situation now if we count open problems.
reader lucretius said...
I completely agree that it is worth to study a problem or a phenomenon from all available perspectives - I don't think many people would disagree with such a general statement. If (and that is a big if) bohmian mechanics is really capable of offering different (and correct) insights into quantum mechanics than by all means people should study it ( I don't think even Lubos would disagree with this conditional statement). However, I don't think that "preserving reality" alone provides sufficient justification - and that seemed to be a key element of Ilja's original argument.
reader Luboš Motl said...
Dear Ilja, nope, your opinion that a researcher's bias is "compensated" by other emotions of someone else is completely and fundamentally wrong. There is absolutely no reason why the "average emotions" of all the researchers should be close to the truth, why the errors caused by the emotions should "cancel".
The opinion that they cancel is precisely the idiotic meme that e.g Feynman beautifully attacked in his Judging Books By Their Covers:
Search for Emperor of China's nose.
A string theory without fermions was never argued to be a right description of phenomena that obviously do contain fermions - it would indeed be as preposterous as what you're doing.
reader Mephisto said...
I haven't studied the Bohmian mechanics enough to be able to make strict judgements.
From what I gathered, in the interpretation of quantum mechanics, we either need to give up locality or reality. Some interpretations give up reality (copenhagen), some locality (Bohm), some both. I personaly believe, that it is probably necessary to modify the concept of reality. The Bohmian mechanics is very non-local (the quantum potential spreads instantly ftl). But these FTL influences lead to time paradoxes. The preference for various interpretations is a problem of psychology - what you find more tolerable to give up.
reader Ilja said...
It is clearly nonsense to compose various contributions. Of course, not. Research directions which fail contribute nothing to the final results. But we don't know in advance which research directions will be successfull (with you as an exception for string theory, of course) and which will fail. If all scientists would follow the same strategy, there would be a much larger possibility that all would fail. If different scientists follow different strategies, most of them will fail, but there will be, with higher probability, some of them who make the correct choice.
The advantage of science is that it has a method to evaluate the final results of the work of different people in very different directions, following different strategies.
Not by nonsensical averaging, counting papers and taxpayer's money spend for them (here string theory wins), but by identifying the single one which was not a complete failure.
reader Ilja said...
No, dBB does not lead to time paradoxes, because it assumes a preferred frame. A hidden one, so no problem with relativistic predictions for observables.
Time paradoxes are a problem of GR, not of quantum theory or dBB.
reader Luboš Motl said...
Dear Ilja, the only problem is that some theories have already failed - theories containing any Lorentz-violating aether failed in 1887, for example.
reader Ilja said...
Correct. So what? That's what I have said - most theories have failed and will fail in future too. Nobody proposes to go back to pre-relativistic ether falsified 1887. What I propose in the direction of ether theory is a generalization of the Lorentz ether to gravity arXiv:gr-qc/0205035 which gives a metric theory of gravity with GR equations in a limit, and an ether model which gives fermions and gauge fields of the SM arXiv:0908.0591. It's something very different from old ether theory, which tried to explain only the EM field. What it shares with the old ether is the preferred frame of Lorentz and the attempt to use condensed matter models to explain the observable fields. I don't see a reason to reject these ideas in general forever only because the old ether has failed to explain the EM field.
reader Rehbock said...
In the cited paper you say "Giving up realism means giving up the search for realistic explanations of observable phenomena."
One can instead accept experimental evidence - 'observable phenomena' beautifully described by QM (leaving aside the aether) as better revealing true reality.
reader Ilja said...
The observable phenomena reveal nothing. Or at least not much. A correlates with B. This explains and reveals nothing. Is A cause of B, or B cause of A, or is there another cause C which causes A and B? This is what is interesting, and this is what is not revealed simply by observation.
reader Justin Glick said...
What if in entanglement, A and B both exert a mutual influence on each other? Then, there would be no causal paradoxes, and a complete symmetry of description which would save relativity.
reader Ilja said...
Feel free to try this way. I would not follow you, I think the preservation of classical causality is the better way. |
5636197cf2827648 | Theoretical chemistry
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Theoretical chemistry seeks to provide explanations to chemical and physical observations. Should the properties derived from the quantum theory give a good account of the above mentioned phenomena, we derive consequences using the same theory. Should the derived consequences fall too far from the experimental evidence, we go to a different theory. G. Lewis proposed that chemical properties originated from the electrons of the atom's valence shell, ever since the theoretical chemistry has dealt with modelling of the outer electrons of interacting atoms or molecules in a reaction. Theoretical chemistry includes the fundamental laws of physics Coulomb's law, Kinetic energy, Potential energy, the Virial Theorem, Planck's Law, Pauli exclusion principle and many others to explain but also predict chemical observed phenomena. The term quantum chemistry which comes from Bohr's quantized model of electron in the atom, applies to both the time independent Schrödinger and the time dependent Dirac formulations.
In general one has to distinguish, theoretical approach (theory level such as Hartree–Fock (HF), Coupled cluster, Relativistic, etc.) from mathematical formalism, plane wave, spherical harmonics, Bloch wave periodic potential. Methods that solve iteratively the energies (Eigenvalues) of stationary state waves in a potential include Restricted Hartree–Fock (RHF), Multi-configurational self-consistent field (CASSCF or MCSCF) but the theory pertains to Schrödinger. Related areas in theoretical chemistry include the mathematical characterization of bulk materials (e.g. the study of electronic band structure in solid state physics) using the theory of electronic band structure in a periodic crystal lattice. Different theoretical approaches are molecular mechanics and topology. The study of the applicability of well established mathematical theories to chemistry is crucial to metals (i.e. topology in the study of small bodies explains the elaborate electronic structures of clusters). This later area of theoretical chemistry originates from the so-called mathematical chemistry. Time-dependent quantum molecular dynamics,[1] is a modern approach to the interaction of light with molecules that vibrate and drive reactions in a desired direction.
Time independent or non-relativistic quantum chemistry is the most widely used formalism of quantum mechanics to solve electronic problems in chemistry. This part of theoretical chemistry may be broadly divided into electronic structure, dynamics, and statistical mechanics. The relativistic quantum chemistry Dirac equation on the other hand explains electron phenomena in heavy atoms with complex electronic interactions, i.e. spin-orbit coupling and relativistic corrections observed for heavy elements such as Re, Os, Ir, Pt, Au, Hg and Pb. Both relativistic quantum chemistry and non-relativistic quantum chemistry are used to solve the problem of predicting chemical reactivity which depends on electrons.
Some chemical theoreticians Car-Parrinello apply molecular dynamics to provide a thorough bridge between the electronic phenomena and the displacement phenomena, this includes properties within organized systems. Currently, many experimental chemists are using Hybrid Gradient Corrected Density Functionals (e.g. B3LYP) to explain the magnetic properties of metals with unpaired electrons; however, a rigorous theoretical examination of this, shows a misuse of the DFT approach, as the electronic spin appears only in Dirac time dependent equations.[citation needed] One way to avoid a full 4e Dirac calculation, is to use TD-DFT method which includes several electronic states for the "same ground geometry". This approach leads to overemphasize on the orbital part wave function to deduce the electronic spin properties, without considering the spin equations, or that the geometry of excited and ground state differ.
Theoretical attempts on chemical problems go back to before 1926, but until the formulation of the Schrödinger equation by the Austrian physicist Erwin Schrödinger in that year, the techniques available were rather crude and approximated. Currently, much more sophisticated theoretical approaches, based on quantum field theory and Non-equilibrium Green's Function Theory are very popular. Green's function theory provides a much closer explanation of electronic transitions than the Hartree–Fock formalism.
In order to explain an observable one has to choose the "appropriate level of theory". For example, some theoretical methods (DFT) may not be appropriate to solve magnetic coupling or electron transitions properties. Instead, there are serious reports like Multireference configuration interaction (MRCI), which accurately and thoroughly explain the observed phenomena by means of the fundamental interactions. Major components include quantum chemistry, the application of quantum mechanics to the understanding of valence, molecular dynamics, statistical thermodynamics and theories of electrolyte solutions, reaction networks, polymerization, catalysis, molecular magnetism and spectroscopy.
Branches of theoretical chemistry[edit]
Quantum chemistry
The application of quantum mechanics or fundamental interactions to chemical and physico-chemical problems. Spectroscopic and magnetic properties are between the most frequently modelled.
Computational chemistry
The application of computer codes to chemistry, involving approximation schemes such as Hartree–Fock, post-Hartree–Fock, density functional theory, semiempirical methods (such as PM3) or force field methods. Molecular shape is the most frequently predicted property. Computers can also predict vibrational spectra and vibronic coupling, but also acquire and Fourier transform Infra-red Data into frequency information. The comparison with predicted vibrations supports the predicted shape.
Molecular modelling
Methods for modelling molecular structures without necessarily referring to quantum mechanics. Examples are molecular docking, protein-protein docking, drug design, combinatorial chemistry. The fitting of shape and electric potential are the driving factor in this graphical approach.
Molecular dynamics
Application of classical mechanics for simulating the movement of the nuclei of an assembly of atoms and molecules. The rearrangement of molecules within an ensemble is controlled by Van der Waals forces and promoted by temperature.
Molecular mechanics
Modelling of the intra- and inter-molecular interaction potential energy surfaces via empirical potentials. The latter are usually parameterized from ab initio calculations.
Mathematical chemistry
Discussion and prediction of the molecular structure using mathematical methods without necessarily referring to quantum mechanics. Topology is a branch of mathematics that allows to predict properties of flexible finite size bodies like clusters.
Theoretical chemical kinetics
Theoretical study of the dynamical systems associated to reactive chemicals, the activated complex and their corresponding differential equations.
Cheminformatics (also known as chemoinformatics)
The use of computer and informational techniques, applied to crop information to solve problems in the field of chemistry.
Closely related disciplines[edit]
Historically, the major field of application of theoretical chemistry has been in the following fields of research:
• Atomic physics: The discipline dealing with electrons and atomic nuclei.
• Molecular physics: The discipline of the electrons surrounding the molecular nuclei and of movement of the nuclei. This term usually refers to the study of molecules made of a few atoms in the gas phase. But some consider that molecular physics is also the study of bulk properties of chemicals in terms of molecules.
• Physical chemistry and chemical physics: Chemistry investigated via physical methods like laser techniques, scanning tunneling microscope, etc. The formal distinction between both fields is that physical chemistry is a branch of chemistry while chemical physics is a branch of physics. In practice this distinction is quite vague.
Hence, the theoretical chemistry discipline is sometimes seen[by whom?] as a branch of those fields of research. Nevertheless, more recently, with the rise of the density functional theory and other methods like molecular mechanics, the range of application has been extended to chemical systems which are relevant to other fields of chemistry and physics like biochemistry, condensed matter physics, nanotechnology or molecular biology.
See also[edit]
• Attila Szabo and Neil S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover Publications; New Ed edition (1996) ISBN 0-486-69186-1, ISBN 978-0-486-69186-2
• Robert G. Parr and Weitao Yang, Density-Functional Theory of Atoms and Molecules, Oxford Science Publications; first published in 1989; ISBN 0-19-504279-4, ISBN 0-19-509276-7
• D. J. Tannor, V. Kazakov and V. Orlov, Control of Photochemical Branching: Novel Procedures for Finding Optimal Pulses and Global Upper Bounds, in Time Dependent Quantum Molecular Dynamics, J. Broeckhove and L. Lathouwers, eds., 347-360 (Plenum, 1992)
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Question: How does quantum mechanics contradict common sense?
David Albert: Here's the deal: quantum mechanics allows physical systems -- and the easiest systems in which to observe phenomena like this are very tiny systems like subatomic particles, electrons or neutrons or protons -- quantum mechanics apparently allows for the existence of physical conditions of material objects like electrons in which questions about where the electron is located in space seem to fail to make sense. Let me back up a little bit and explain this a little more slowly. There are experiments we can do where an electron passes through a certain apparatus, is fed into one end of an apparatus, comes out the other side of the apparatus. And the apparatus has several routes inside of it which the electron could potentially have taken from the input to the output. And there are experiments we can do with pieces of apparatus like this which, taken together, make a compelling case that although the electron went from here to there, it didn't go by route A, it also didn't go by route B, and it didn't go in any intelligible sense by both routes; that is, it didn't split in half, with one half taking one route and one half taking the other route; and it also didn't take neither route, okay?
And what's puzzling about that is that that would seem to be all the logical possibilities that there are. These experiments, you know, are now very routine experiments to do in physics laboratories. We've been good at doing experiments like this for something on the order of 70 years now. We're very good at doing them now. The results are very, very compelling. And after enormous soul-searching and puzzlement and confusion and so on and so forth, the sort of standard consensus understanding that evolved in physics of situations like this is that electrons could apparently be in situations where asking a question of the form "which route did the electron take?" was something like asking a question of the form "what is the marital status of the number five?" Or "what are the political affiliations of this tuna sandwich?" or something like that. These are questions that philosophers often refer to as category mistakes, okay? The very raising of a question about the political affiliations of a tuna sandwich or the marital status of the number five indicates that there's something basic that you're misunderstanding of what it is that you're asking a question about.
And the strikingly strange thing about quantum mechanics -- and indeed it seems to me a case could be made that this is the strangest and most unsettling result to come out of the natural sciences since the scientific revolution of the Renaissance -- is that even things like particles can be in conditions where it simply radically fails to make sense even to ask where the thing is located in space. What's particularly strange about this is that of course there are other circumstances where it does make sense to ask those questions about where it's located in space. The electron determinately goes into the box over here and comes out over there, okay? But we can give good arguments from these experiments that while it's inside, it's not merely that we don't know where it is, it's something much more radical and much more unsettling than that: that the very act of raising a question about where it is represents some kind of misunderstanding of what mode of being that electron is participating in while it's going through this device.
David Albert: Anyway, here's a further fact about electrons: if we go -- if we do one of these experiments that I just described and stop it in the middle, rip the box open, okay, and go look for the electron, as a matter of fact we always find it in some determinate position, okay? We have equations, we have basic laws of motion: the Schrödinger equation in the case of nonrelativistic quantum mechanics; in the case of relativistic quantum mechanics the fundamental equations are the Dirac equation or the Klein-Gordon equation. Anyway, we have these fundamental laws of motion for things like electrons; indeed, for all material things. And these laws are very successful at predicting when these strange -- let me back up and say these conditions in which it fails to make sense to ask whether the electron is here and here -- in a long, distinguished tradition of facing a mystery that one doesn't understand by at least making up a name for it, a name has been made up for this condition. People speak of electrons in such circumstances as being in a superposition of going along route A and going along route B. And although it's very difficult for us to get our heads around what this word means, we are very adept at treating these situations mathematically. We have very reliable equations that tell us when and under what circumstances these superpositions are going to arise and when they're going to go away, and blah blah blah blah blah. Good.
Further empirical fact: when we rip open these boxes and look for these electrons, we always find them in one position or another, okay? So that somehow the act of looking at them makes these superpositions go away, okay? Good. On the other hand, we could perform the following exercise: take these fundamental equations that we have discovered and which we have very good reason to believe are reliable at predicting when superpositions are going to arise and when they're going to go away and so on and so forth; use those equations to predict what ought to occur when we rip the box open and look inside. That may sound like a very difficult calculation to do. It involves this macroscopic human being and his brain and so on and so forth. Actually, there's a mathematical trick for getting this calculation done, as miraculous as that sounds.
And it's very easy to show that what these equations predict ought to occur when we rip this box open is that we ourselves go into a superposition of seeing the electron on route A and seeing the electron on route B, okay? That is, that we ourselves go into some condition in which not only does it fail to make sense to ask where the electron is; it fails to make sense even to ask about our beliefs about where the electron is, okay? Or it fails to make sense to ask whether we're in the brain state corresponding to believing that the electron is on route A or in the brain state corresponding to believing that the electron is on route B. Good. God knows what the hell that would feel like, okay? But the usual way of setting up this measurement problem is merely to observe that whatever it is that would feel like, that's not what happens to us when we rip open these boxes. When we rip open these boxes, there is always a perfectly determinate matter of fact about where we take the electron to be. Sometimes we see it on route A; sometimes we see it on route B; it's never the case that anything else is going on. It's never the case that it looks fuzzy, or we get nauseous, or we become disoriented, or in any sense that one can put one's finger on there fails to be a fact about where we see the electron. Good.
So we have a flat-out contradiction. And this is the more explicit version of the way this story about the glass breaks down. We have a flat-out contradiction between, on the one hand, the predictions of the fundamental quantum mechanical equations of motion about what ought to happen when we rip open these boxes, and our everyday introspective experience of what's going on when we rip open these boxes, which is that in each of those occasions we either see an electron there, or we see an electron there. These two claims flatly contradict one another. Of course, you know, the empirical claim is the one that's true; that's the one that we see from our observations. There's something wrong with these equations.
On the other hand, we also know that there's an enormous amount that's right about these equations. These equations are where, you know, indescribably vast swathes of 20th century science and technology come from. These equations get an enormous amount right. On the other hand, it couldn't be more obvious from our everyday experience of the world and of ourselves that something is wrong with them. There's a problem about how to put these two facts together. There's a problem more specifically about how to modify the theory in such a way that this contradiction goes away without ruining the rest of the good predictions of the theory. This problem, once again, is called the measurement problem.
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How the Green's functions and the Quantum Mechanics are related? Do they can be used to solve the Schrödinger equation of an particle subjet to some potential that is not a Dirac's delta? And the proprieties of some Green's functions that are symmetrical, i.e. $ G(x|\xi) = G(\xi|x)^{\ast} $, has some relation with the propriety of the inner product $ \langle \alpha \vert \beta \rangle = \langle \beta \vert \alpha \rangle^{\ast} $?
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Schrödinger equation is a linear partial differential equation, so sure, you can use the usual formalism of Green's functions to solve it.
First let's recall how the stuff works. Suppose $L$ is the linear operator and $D$ are the boundary conditions and we want to solve equations $Lu = f$ and $Du = 0$ for $u$. Using the identity property of the convolution $g*\delta = g$ one is motivated to solve the simpler equation $LG = \delta$ and then one finds $u = G*f$ because $$L(G*f) = (LG)*f = \delta*f = f$$.
Now, for the time-independent Schrödinger equation the following should be useful. If the operator (understood also with the given boundary conditions) also has a complete basis of eigenvectors $\left\{\left|\phi_n\right>\right\}$ corresponding to eigenvalues $\left\{\lambda_n\right\}$ then the Green's function can easily be seen to be $$G(x, x') = \sum_n {\phi_n(x)^* \phi_n(x') \over \lambda_n}$$ (just apply the operator $L$ to it and use that $L \left|\phi_n\right> = \lambda_n \left|\phi_n\right>$. So again we can see that $G$ is in a sense an inverse of $L$ (and indeed it is often written simply as $L^{-1}$).
Now, it turns out there is a deeper connection between Green's functions and quantum mechanics via Feynman's path integral if we pass to the time dependent Schrödinger equation. I am not going to derive all the stuff here but suffice it to say that Green's function takes on the meaning of a propagator of the particle. Namely, the probability amplitude that the particle gets from the event (t, x) to the event (t', x') is a Green's function of the time-dependent Schrödinger equation $G(x,t;x',t') = \left<x\right| U(t,t') \left|x'\right>$. So yes, the fact that the Green's function is symmetric is precisely because it can be interpreted as an inner product.
This stuff generalizes further to quantum field theory and Green's functions are among the basic objects of study there.
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Nice job buddy +1 – user346 Feb 5 '11 at 11:10
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In more 'down-to-earth' QM, you use Green's functions to find the density of states. I'm deprived of my books so at a loss for giving a good reference, but the idea is to calculate $$G(x,x';E) = \langle x, (E - H)^{-1} x' \rangle,$$ where $H$ is the system's Hamiltonian. You can then define a spectral function $F(x,x',E) = -\frac{1}{\pi} lim_{\epsilon \rightarrow 0} \text{Im } G(x,x',E+i\epsilon),$ whose trace is the density of states: $$\mathcal{N}(E) = \int F(x,x,E) dx.$$ Finally, you can also use this formalism to calculate other expectation values, with formulas like (modulo an incorrect prefactor) $\langle A \rangle = -\frac{1}{\pi} \text{Im Tr}(AG).$ So yes, they are symmetrical, but they can not really be used to 'solve' a Schrödinger's equation, only on a formal level. That's why they're useful though: they're used all the time in many-body QM/solid state physics, where you'll never 'solve' the problem but can learn lots of interesting stuff by indirect approaches, as the one used above.
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+1 all correct except for your statement that they can not really be used to 'solve' a Schrödinger's equation, only on a formal level. For scattering from a potential the Green's function is exactly calculable for many important cases and encodes the full physical content of the solutions. – user346 Feb 5 '11 at 9:36
@space_cadet: forgot about that entirely, good job mentioning it. – Gerben Feb 5 '11 at 17:25
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An interesting system to study to understand this is the Poincare half disk or half plane. ${\cal H}^2$. It also illustrates the role of Laplacian operators, Green’s functions and resolvents. The Laplace-Beltrami operator on a Riemannian manifold with a Gaussian metric is $$ \Delta~=~\sum_{ij}{1\over\sqrt{g}}{\partial\over{\partial x^i}}\Big(\sqrt{g}g^{ij}{\partial\over{\partial x^j}}\Big), $$ for $g~=~|det(g_{ij}|$. The Laplacians for the Poincare half plane and disk are $$ \Delta_{{\cal H}^2}~=~y^2\Big({{\partial^2}\over{\partial x^2}}~+~{{\partial^2}\over{\partial y^2}}\Big),~\Delta_{{\cal D}}~=~(\alpha^2~-~x^2~-~y^2)^2\Big({{\partial^2}\over{\partial x^2}}~+~{{\partial^2}\over{\partial y^2}}\Big). $$ The Laplacian commutes with all group elements $g~\in~Iso({\cal H}^2)$ $\Delta T_g~=~T_g\Delta$, so the metric is invariant under these isometries. The Laplacian satisfies the the differential equation $(\Delta~+~\lambda)f(z)~=~0$ defined as the kernel of the resolvent $(\Delta~+~\lambda)^{-1}$ by the equation $$ (\Delta~+~\lambda)^{-1}f(z)~=~\int (z,~z^\prime,~\lambda)f(z^\prime)d\mu(z^\prime), $$ with the harmonic condition $(\Delta~+~\lambda)G(z,~z^\prime,~\lambda)~=~\delta(z,~z^\prime)$.
This space is such that the Laplaican $\Delta$ has eigenvalue $-2$, or a negative Gaussian curvature. This space is a model for the $AdS_2$ spacetime. I hope that with this example you can see answers to you questions, such as the symmetry under interchange.
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Perhaps, in the interests of clarity, you could make explicit the definition of the Green's function. I can see why someone might find this answer unhelpful. @Rodrigo asks about Green's functions in QM and the Schrodinger equation and you start off with the Poincare half-plane. Technically, of course, your answer is helpful for someone who actually bothers to read it. +1 – user346 Feb 5 '11 at 8:10
Hm, not that this isn't interesting stuff but I don't see any connection with the question whatsoever. -1 – Marek Feb 5 '11 at 8:55
@Rodrigo asks How the Green's functions and the Quantum Mechanics are related?. @Lawrence provides a concrete example of how to calculate a Green's function, albeit for a free field. The Laplace-Beltrami operator is nothing more than the kinetic term of the Schrodinger equation. How is that not related to the question? The question has other parts, but every answer does not have to answer every single subpart of a question. @Lawrence's answers tend to have more analytical content and are superior to many "hand-wavy" answers. – user346 Feb 5 '11 at 9:30
@space_cadet: be so kind and point to me the place where there is any quantum mechanics in this example. Also show me the place where there is an explicit connection between QM and GF. – Marek Feb 5 '11 at 9:34
@Marek did you not read my previous comment? – user346 Feb 5 '11 at 9:37
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Spontaneous emission is the process in which a quantum mechanical system (such as an atom, molecule or subatomic particle) transitions from an excited energy state to a lower energy state (e.g., its ground state) and emits a quantum in the form of a photon. Spontaneous emission is ultimately responsible for most of the light we see all around us; it is so ubiquitous that there are many names given to what is essentially the same process. If atoms (or molecules) are excited by some means other than heating, the spontaneous emission is called luminescence. For example, fireflies are luminescent. And there are different forms of luminescence depending on how excited atoms are produced (electroluminescence, chemiluminescence etc.). If the excitation is affected by the absorption of radiation the spontaneous emission is called fluorescence. Sometimes molecules have a metastable level and continue to fluoresce long after the exciting radiation is turned off; this is called phosphorescence. Figurines that glow in the dark are phosphorescent. Lasers start via spontaneous emission, then during continuous operation work by stimulated emission.
Spontaneous emission cannot be explained by classical electromagnetic theory and is fundamentally a quantum process. The first person to derive the rate of spontaneous emission accurately from first principles was Dirac in his quantum theory of radiation,[1] the precursor to the theory which he later coined quantum electrodynamics.[2] Contemporary physicists, when asked to give a physical explanation for spontaneous emission, generally invoke the zero-point energy of the electromagnetic field.[3][4] In 1963 the Jaynes-Cummings model[5] was developed describing the system of a two-level atom interacting with a quantized field mode (i.e. the vacuum) within an optical cavity. It gave the nonintuitive prediction that the rate of spontaneous emission could be controlled depending on the boundary conditions of the surrounding vacuum field. These experiments gave rise to cavity quantum electrodynamics (CQED), the study of effects of mirrors and cavities on radiative corrections.
If a light source ('the atom') is in an excited state with energy , it may spontaneously decay to a lower lying level (e.g., the ground state) with energy , releasing the difference in energy between the two states as a photon. The photon will have angular frequency and an energy :
where is the reduced Planck constant. Note: , where is the Planck constant and is the linear frequency. The phase of the photon in spontaneous emission is random as is the direction in which the photon propagates. This is not true for stimulated emission. An energy level diagram illustrating the process of spontaneous emission is shown below:
If the number of light sources in the excited state at time is given by , the rate at which decays is:
where is the rate of spontaneous emission. In the rate-equation is a proportionality constant for this particular transition in this particular light source. The constant is referred to as the Einstein A coefficient, and has units .[6] The above equation can be solved to give:
where is the initial number of light sources in the excited state, is the time and is the radiative decay rate of the transition. The number of excited states thus decays exponentially with time, similar to radioactive decay. After one lifetime, the number of excited states decays to 36.8% of its original value ( -time). The radiative decay rate is inversely proportional to the lifetime :
Spontaneous transitions were not explainable within the framework of the Schrödinger equation, in which the electronic energy levels were quantized, but the electromagnetic field was not. Given that the eigenstates of an atom are properly diagonalized, the overlap of the wavefunctions between the excited state and the ground state of the atom is zero. Thus, in the absence of a quantized electromagnetic field, the excited state atom cannot decay to the ground state. In order to explain spontaneous transitions, quantum mechanics must be extended to a quantum field theory, wherein the electromagnetic field is quantized at every point in space. The quantum field theory of electrons and electromagnetic fields is known as quantum electrodynamics.
In quantum electrodynamics (or QED), the electromagnetic field has a ground state, the QED vacuum, which can mix with the excited stationary states of the atom.[2] As a result of this interaction, the "stationary state" of the atom is no longer a true eigenstate of the combined system of the atom plus electromagnetic field. In particular, the electron transition from the excited state to the electronic ground state mixes with the transition of the electromagnetic field from the ground state to an excited state, a field state with one photon in it. Spontaneous emission in free space depends upon vacuum fluctuations to get started.[7][8]
Although there is only one electronic transition from the excited state to ground state, there are many ways in which the electromagnetic field may go from the ground state to a one-photon state. That is, the electromagnetic field has infinitely more degrees of freedom, corresponding to the different directions in which the photon can be emitted. Equivalently, one might say that the phase space offered by the electromagnetic field is infinitely larger than that offered by the atom. This infinite degree of freedom for the emission of the photon results in the apparent irreversible decay, i.e., spontaneous emission.
In the presence of electromagnetic vacuum modes, the combined atom-vacuum system is explained by the superposition of the wavefunctions of the excited state atom with no photon and the ground state atom with a single emitted photon:
where and are the atomic excited state-electromagnetic vacuum wavefunction and its probability amplitude, and are the ground state atom with a single photon (of mode ) wavefunction and its probability amplitude, is the atomic transition frequency, and is the frequency of the photon. The sum is over and , which are the wavenumber and polarization of the emitted photon, respectively. As mentioned above, the emitted photon has a chance to be emitted with different wavenumbers and polarizations, and the resulting wavefunction is a superposition of these possibilities. To calculate the probability of the atom at the ground state ( ), one needs to solve the time evolution of the wavefunction with an appropriate Hamiltonian.[1] To solve for the transition amplitude, one needs to average over (integrate over) all the vacuum modes, since one must consider the probabilities that the emitted photon occupies various parts of phase space equally. The "spontaneously" emitted photon has infinite different modes to propagate into, thus the probability of the atom re-absorbing the photon and returning to the original state is negligible, making the atomic decay practically irreversible. Such irreversible time evolution of the atom-vacuum system is responsible for the apparent spontaneous decay of an excited atom. If one were to keep track of all the vacuum modes, the combined atom-vacuum system would undergo unitary time evolution, making the decay process reversible. Cavity quantum electrodynamics is one such system where the vacuum modes are modified resulting in the reversible decay process, see also Quantum revival. The theory of the spontaneous emission under the QED framework was first calculated by Weisskopf and Wigner.
In spectroscopy one can frequently find that atoms or molecules in the excited states dissipate their energy in the absence of any external source of photons. This is not spontaneous emission, but is actually nonradiative relaxation of the atoms or molecules caused by the fluctuation of the surrounding molecules present inside the bulk.[clarification needed]
Rate of spontaneous emissionEdit
The rate of spontaneous emission (i.e., the radiative rate) can be described by Fermi's golden rule.[9] The rate of emission depends on two factors: an 'atomic part', which describes the internal structure of the light source and a 'field part', which describes the density of electromagnetic modes of the environment. The atomic part describes the strength of a transition between two states in terms of transition moments. In a homogeneous medium, such as free space, the rate of spontaneous emission in the dipole approximation is given by:
where is the emission frequency, is the index of refraction, is the transition dipole moment, is the vacuum permittivity, is the reduced Planck constant, is the vacuum speed of light, and is the fine structure constant.[clarification needed] (This approximation breaks down in the case of inner shell electrons in high-Z atoms.) The above equation clearly shows that the rate of spontaneous emission in free space increases proportionally to .
In contrast with atoms, which have a discrete emission spectrum, quantum dots can be tuned continuously by changing their size. This property has been used to check the -frequency dependence of the spontaneous emission rate as described by Fermi's golden rule.[10]
Radiative and nonradiative decay: the quantum efficiencyEdit
In the rate-equation above, it is assumed that decay of the number of excited states only occurs under emission of light. In this case one speaks of full radiative decay and this means that the quantum efficiency is 100%. Besides radiative decay, which occurs under the emission of light, there is a second decay mechanism; nonradiative decay. To determine the total decay rate , radiative and nonradiative rates should be summed:
where is the total decay rate, is the radiative decay rate and the nonradiative decay rate. The quantum efficiency (QE) is defined as the fraction of emission processes in which emission of light is involved:
In nonradiative relaxation, the energy is released as phonons, more commonly known as heat. Nonradiative relaxation occurs when the energy difference between the levels is very small, and these typically occur on a much faster time scale than radiative transitions. For many materials (for instance, semiconductors), electrons move quickly from a high energy level to a meta-stable level via small nonradiative transitions and then make the final move down to the bottom level via an optical or radiative transition. This final transition is the transition over the bandgap in semiconductors. Large nonradiative transitions do not occur frequently because the crystal structure generally cannot support large vibrations without destroying bonds (which generally doesn't happen for relaxation). Meta-stable states form a very important feature that is exploited in the construction of lasers. Specifically, since electrons decay slowly from them, they can be deliberately piled up in this state without too much loss and then stimulated emission can be used to boost an optical signal.
See alsoEdit
1. ^ a b Dirac, Paul Adrien Maurice (1927). "The Quantum Theory of the Emission and Absorption of Radiation". Proc. Roy. Soc. A114 (767): 243–265. Bibcode:1927RSPSA.114..243D. doi:10.1098/rspa.1927.0039.
2. ^ a b Milonni, Peter W. (1984). "Why spontaneous emission?" (PDF). Am. J. Phys. 52 (4): 340. Bibcode:1984AmJPh..52..340M. doi:10.1119/1.13886.
3. ^ Weisskopf, Viktor (1935). "Probleme der neueren Quantentheorie des Elektrons". Naturwissenschaften. 23: 631–637. Bibcode:1935NW.....23..631W. doi:10.1007/BF01492012.
4. ^ Welton, Theodore Allen (1948). "Some observable effects of the quantum-mechanical fluctuations of the electromagnetic field". Phys. Rev. 74 (9): 1157. Bibcode:1948PhRv...74.1157W. doi:10.1103/PhysRev.74.1157.
5. ^ Jaynes, E. T.; Cummings, F. W. (1963). "Comparison of quantum and semiclassical radiation theories with application to the beam maser". Proceedings of the IEEE. 51 (1). doi:10.1109/PROC.1963.1664.
6. ^ R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford University Press Inc., New York, 2001).
9. ^ B. Henderson and G. Imbusch, Optical Spectroscopy of Inorganic Solids (Clarendon Press, Oxford, UK, 1989).
10. ^ A. F. van Driel, G. Allan, C. Delerue, P. Lodahl,W. L. Vos and D. Vanmaekelbergh, Frequency-dependent spontaneous emission rate from CdSe and CdTe nanocrystals: Influence of dark states, Physical Review Letters, 95, 236804 (2005).
External linksEdit |
c9a42d2ac7bde9ba | Condensed Matter
Negative Mass Created
Authors: George Rajna
Washington State University physicists have created a fluid with negative mass, which is exactly what it sounds like. Push it, and unlike every physical object in the world we know, it doesn't accelerate in the direction it was pushed. It accelerates backwards. [16] When matter is cooled to near absolute zero, intriguing phenomena emerge. These include supersolidity, where crystalline structure and frictionless flow occur together. ETH researchers have succeeded in realising this strange state experimentally for the first time. [15] Helium atoms are loners. Only if they are cooled down to an extremely low temperature do they form a very weakly bound molecule. In so doing, they can keep a tremendous distance from each other thanks to the quantum-mechanical tunnel effect. [14] Inside a new exotic crystal, physicist Martin Mourigal has observed strong indications of "spooky" action, and lots of it. The results of his experiments, if corroborated over time, would mean that the type of crystal is a rare new material that can house a quantum spin liquid. [13] An international team of researchers have found evidence of a mysterious new state of matter, first predicted 40 years ago, in a real material. This state, known as a quantum spin liquid, causes electrons-thought to be indivisible building blocks of nature-to break into pieces. [12] In a single particle system, the behavior of the particle is well understood by solving the Schrödinger equation. Here the particle possesses wave nature characterized by the de Broglie wave length. In a many particle system, on the other hand, the particles interact each other in a quantum mechanical way and behave as if they are "liquid". This is called quantum liquid whose properties are very different from that of the single particle case. [11] Quantum coherence and quantum entanglement are two landmark features of quantum physics, and now physicists have demonstrated that the two phenomena are "operationally equivalent"—that is, equivalent for all practical purposes, though still conceptually distinct. This finding allows physicists to apply decades of research on entanglement to the more fundamental but less-well-researched concept of coherence, offering the possibility of advancing a wide range of quantum technologies. [10] The accelerating electrons explain not only the Maxwell Equations and the Special Relativity, but the Heisenberg Uncertainty Relation, the Wave-Particle Duality and the electron's spin also, building the Bridge between the Classical and Quantum Theories. The Planck Distribution Law of the electromagnetic oscillators explains the electron/proton mass rate and the Weak and Strong Interactions by the diffraction patterns. The Weak Interaction changes the diffraction patterns by moving the electric charge from one side to the other side of the diffraction pattern, which violates the CP and Time reversal symmetry. The diffraction patterns and the locality of the self-maintaining electromagnetic potential explains also the Quantum Entanglement, giving it as a natural part of the relativistic quantum theory. The asymmetric sides are creating different frequencies of electromagnetic radiations being in the same intensity level and compensating each other. One of these compensating ratios is the electron – proton mass ratio. The lower energy side has no compensating intensity level, it is the dark energy and the corresponding matter is the dark matter.
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Atomic Shell Theory: Bohr-Sommerfeld model
1. Aug 8, 2011 #1
This is a very naive question. But I think, it's an important point that has been unattempted in textbooks. The question is:
How far should one trust the Bohr-Sommerfeld model or the atomic shell theory for all elements in the periodic table?
This question generally comes in mind, since we know that the Bohr's model was a kind of hypothesis or ansatz to explain the hydrogen atom spectra. And Sommerfeld modified the quantization condition in a spirit of generalization, using the analogy to the planetary motion under the central force due the sun. These all can be put together as the initial development of quantum mechanics and often regarded as theories of Old Quantum Mechanics.
Now we know that the H-atom can be exactly solved from the Schroedinger's equation. But what about atoms with higher atomic numbers (may be we can only expect hydrogen-like wavefunctions for alkali atoms)? Should Bohr-Sommerfeld model (apart from the relativistic correction) still be working ?
Then why do people say that in cuprates copper has d9 electronic state? Doesn't that sound imprecise?
2. jcsd
3. Aug 8, 2011 #2
It depends on the precision you want.
First of all, you must know that Schroedinger's equation cannot be solved exactly for atoms different from hydrogen atom.
Actually, one of the best representations of the atoms of higher atomic number is given by the Hartree theory, which tries to approximate Schroedinger's equation by making hypothesis about the potential seen by the electrons.
Next, what do you mean when you say "How far should one trust the Bohr-Sommerfeld model or the atomic shell theory for all elements in the periodic table?" ?
You know, the Bohr model is an oversimplified model of the atom, and in the new quantum mechanic it is replaced by the model deduced by Schroedinger's equation, in which the electrons are replaced by waveforms.
Moreover, Bohr's model cannot account for effects related to the filling of the subshells, that is important to build the periodic table of elements or for a finer structure of the spectrum of atoms than the one studied by Bohr, and to the spin-orbit interaction: an atom immersed in a magnetic field has different levels of energy than a free atom because of the spin-orbit interaction.
Bohr's model works for rough calculations but, as I said, for finer calculations and for to have a (better) explanation of a large variety of phenomena, not explained by Bohr's model because, as you said, it is just an hypothesis (that works) built upon experimental data.
Hope it answers :D
4. Aug 8, 2011 #3
I think, if you can answer my last question, it'll be clear.
How can one say Cu has d9 electronic state in cuprates? Or does it need to have d-state at all?
Let me put a few more questions?
1) Are there experiments that can see real electronic orbitals of H-atom? Note that the orbital photographs seen in most textbooks were engineeringly plotted by H. E. White (no real hydrogen atom experiments done).
Ref. http://prola.aps.org/abstract/PR/v37/i11/p1416_1
2) How can we define s, p, d orbitals for non-hydrogen atoms? We may have s',p', d' orbitals that may hold totally different kind of geometries.
5. Aug 8, 2011 #4
Actually it's the first time I hear about cuprate.
However, the fact is that the atomic configuration of an atom is not fixed, but it can change if a new configuration with lower energy is available.
I don't know (if you have some links please post me; I will search for myself) actually the structure of cuprate but, since the energies of the outer subshells narrows as you go away from hydrogen atom, it is possible that copper configuration in cuprate is more stable and less energetic if it is in 3d9 state.
As regards the photos or images of electron orbitals, last month a group of researchers at Politecnico di Milano (Italy) managed to "take a photo" of the orbitals in a molecule. I can't find the article in English, but http://www.galileonet.it/articles/4e2fdf4d72b7ab4b16000090" it is in Italian (maybe you can get it translated by google), with some photos.
I must remind you that after Schroedinger, it's almost wrong to think about orbitals as "paths": it is more correct if you think about where it is more probable to find an electron.
We can define the subshells from their energy. Dirac derived a formula that corrects Bohr's taking in account spin orbital interaction and other interactions in the second order. In this formula are present the quantum numbers n,l,j (where j is the total angular moment)
Actually, I've just read on Wikipedia that in cuprate copper is Cu++ and O--, so Cu++ is [Ar]3d9 because 2e- have been removed by the more electronegative oxygen.
Last edited by a moderator: Apr 26, 2017
6. Aug 8, 2011 #5
Actually it's the first time I hear about cuprate.
It's not only about cuprates.We always find the oxidation state and then decide the electronic configuration, purely from the chemistry point of view. My question is:
1) Is there any physics explanation behind the atomic configuration of non-hydrogen atoms?
As regards the photos or images of electron orbitals, last month a group of researchers at Politecnico di Milano (Italy) managed to "take a photo" of the orbitals in a molecule. I can't find the article in English, but here it is in Italian (maybe you can get it translated by google), with some photos.
This nature paper has the experimental detail. By orbitals I meant the probability density distribution, not paths (see the H. E. White's paper). And as I said, I wish to see the proof of electron's probability density distribution for the H-atom (apart from the indirect way from the spectroscopy results) that we solve through Schroedinger's equation in the textbooks.
Can you see the gap between chemistry and physics? I can believe atomic number, but how can I believe the atomic configuration and hence the s,p,d,f nomenclature?
Did you mean subshells for the Bohr-Sommerfeld theory, i.e. the old quantum mechanics?
I know Dirac's relativistic correction and that is for the H-atom.
Again, this is ridiculous. Electrons are not removed. We can say, there's no significant probability density. But in what region? I can think of chemical bonding as overlap or distribution of electronic wavefunction.
Now the questions:
1) How can I calculate the atomic wavefunction or the energy (i. e. eigenvalue) of a non-hydrogen atom?
2) Even if I succeed to do for atoms, how should I calculate the redistribution of wavefunctions when they form a molecule?
7. Aug 8, 2011 #6
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Both of those are done using techniques of Quantum Chemistry. This is basically hardcore numerical quantum many body physics, and there are (large and very complex) software packages which can do such calculations (e.g., Molpro, CFOUR and MRCC). Using these techniques, relative energies and properties can be calculated to around 0.1 kJ/mol ... 4 kJ/mol (depending on your patience. 1 kJ/mol is about 0.010 eV) for sufficiently friendly atoms and small molecules (e.g., energy differences between states, between products, transition states, and educts of a chemical reaction etc.). Unfortunatelly, understanding how these many-body method work require an intimate knowledge of quantum mechanics, many-body theory, and numerics, and this is not easily explained. Some terms to get you started are "Coupled cluster theory", "Multirefernce configuration interaction", "correlation consistent basis set" and "basis set extrapolation".
8. Aug 9, 2011 #7
As far as I know, the quantum chemistry methods use Hartree-Fock or post-Hartree-Fock methods that use linear combinations of atomic orbitals and again those orbitals are treated like s, p, d kind of orbitals, which are hard to accept.
Hope you got my point. Again my question is very simple:
How can we believe that s, p, d, ... orbitals exist beyond hydrogen atom? Any experimental or theoretical evidence?
9. Aug 9, 2011 #8
Well, one can also ask: "How do we know that these orbitals exist for H atom?". The only answer which comes to mind is that they are predicted using theory and that no experimental evidence is available against it. The theory explains well the experimental facts. This is true not only in the present case but for any theory in general.
For atoms with Z > 1, as mentioned previously, HF theory is used (other approaches are also available) which assumes existence of such orbitals. The results of these calculations predict some results. If those results are proved experimentally, then the theory stands.
In short, if the experimental results can be explained using the assumption of existence of such orbitals, then those experimental results can be treated as the proof for the existence of these orbitals.
10. Aug 9, 2011 #9
If subshells exists for multielectron atoms, it will result in atomic spectra: there are transition rules that allow only certain decay. You can look very carefully at the spectrum and calculate the energy of the gap.
I'm quite sure it happens, and they already found evidence, but I cannot be sure 100% having not my book under hand :D
Moreover, probably this is a stronger evidence: they found two types of helium, ortohelium and parahelium, with different physical properties. This is due to the fact that parahelium has both the electron in state 1s2 (a state of singulet), whilst ortohelium is in the state 1s1 2s2 (a state of triplet). The transition rules shows that is impossible for optical transition to transform a singulet state to a triplet state and vice versa.
I think this is a proof of the existence of subshells, or not?
Another proof could that the energy levels within each subshell are very narrow, while between shells are broader, and you can see this when you ionize the atom.
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/helium.html" [Broken]
http://ibchem.com/IB/ibnotes/full/ato_htm/12.2.htm" [Broken]
I'm sorry for my poor speech: "because 2e- have been removed by the more electronegative oxygen." Obviously electrons aren't removed in the meaning we give to this word :D
You are right, sorry :D
Actually I can't, probably because I don't like chemistry too much :D
When I treated Hartree-Fock method, we studied that it was based on Schroedinger's equation. That is, Hartree hypothesized the effective potential Z(r) and put it into Schroedinger's equation. Through iterations, he arrived to a waveform that was in accord with experimental evidence. In this waveform, we have subshells.
Last edited by a moderator: May 5, 2017
11. Aug 9, 2011 #10
Sorry. A severe objection. We can think of subshells. But they don't need to have the same geometry of the hydrogen atom s,p,d orbitals.
Moreover, transition doesn't show the nature of the orbitals. It just reflects the difference between two energy levels.
So it seems that we are still using the Bohr's hypothesis (a bit modification due to Sommerfeld) to decide the electronic configuration of elements. So it's an empirical fitting of the spectroscopy results to the Bohr-Sommerfeld energy spectra formula, which is true only for the hydrogen atom.
It's strange to see that shell theory (old quantum mechanics) still works for many elements. And there exists no true quantum theory to support or deny the theory.
So I can say, we still trust the old quantum theory (since in modern quantum mechanics, non-hydrogen atom problem is non-trivial). In that sense it's as good/bad as we have the Boyle's law to describe gases.
Hope you'll agree with my point.
12. Aug 9, 2011 #11
See, the Bohr's model as an ansatz gives correct values for the hydrogen atom spectra (apart from the fine-structure correction). And solving Schroedinger equation for the H-atom reproduces Bohr's formula. In that sense, we have a microscopic theory that supports Bohr's formula for the H-atom. But there's no such quantum theory that support Bohr-Sommerfeld formula for other atoms. However, Bohr-Sommerfeld formula seems to work well (I hope so though I never have tested). It's similar to the shell theory that works in nuclear physics.
So my point is that we still use an ansatz (which is not true quantum mechanics) to infer
the atomic orbitals and hence the elctronic configurations.
13. Aug 9, 2011 #12
No, it doesn't work well. There are many things which are not explained: http://en.wikipedia.org/wiki/Bohr_model#Shortcomings"
This is also not correct. We are not using the atomic orbitals because of the BS model. These orbitals follow from quantum mechanics. Please read what @DiracRules have written in this regard.
Transition do show the nature of the orbitals. They tell you about the symmetry of the orbitals by utilizing the selection rules, which in most cases is good enough to infer the initial and final state orbitals.
No ways. Spectroscopy cannot be described using the BS model and no one actually use it in the field of spectroscopy, apart from obtaining some initial guesses. This is specially true for elements with Z > 1.
Last edited by a moderator: Apr 26, 2017
14. Aug 9, 2011 #13
I think, s,p,d orbitals were defined only for the hydrogen or hydrogen-like (alkali) atoms.
How do you determine probability densities for other atoms?
I guess, the wiki link tells about shortcoming of the Bohr's model, not about the BS model.
How can you be sure there are no other orbitals 'except' s,p,d,f,g, etc? Note that all these orbital geometries are defined from hydrogen atom wavefunctions. Also remember that in the hydrogen atom problem in quantum mechanics, we can actually separate the differential equation into radial and spherical harmonics part. It may not be possible to do the same for other atoms.
In atomic spectroscopy, I guess, we look at emission lines. Or may be absorption in some cases, I'm not sure. Now probably I can calculate the wavelengths of those lines and hence differences between energy levels. Now could you tell me, what formula I'm going to use next and how I connect to symmetry of orbitals from these?
Last edited by a moderator: Apr 26, 2017
15. Aug 9, 2011 #14
I have to correct what I said previously and say that you are right saying that we cannot do the same for multielectron atoms if we want to solve it exactly. The problem is the presence of the potential interaction between electrons.
However, this problem is, in a certain way, overcome by making hypothesis about the potential that affects the electrons ( Z(r) ). By making this hypothesis, we can split the equation in an angular and radial part.
I realize this cannot satisfy you, but it's the way things are going since 1930! If it was wrong, it would have been corrected :D
If you do not think that approximation is a sort of trickery that hides true reality, think of this: physicists approximated the potential interaction between electrons in the way I told you also to treat identical particles, and to take into account the spin. And it works very well too :D
But I think that the one of the best way to "believe" into the existence of subshells comes from spectroscopy.
From spectroscopy, you can see that each level is split into other sublevel: for example, transition rules says that you can't have a transition 3s->2s but 3p->2s or 3d->2p.
Now, since [itex]\Delta E_{3p\rightarrow 2s}[/itex] is different from [itex]\Delta E_{3d\rightarrow 2p}[/itex], you can't explain this if you don't consider subshells into your model.
Now you can say, why do we not consider subshells as proper shells?
The mathematical answer can be "because they share the same principal quantum number n".
The experimental answer can be "because going up through the periodic table, we see that removing certain electrons requires more energy than the trend. We say thus that we passed into another shell".
16. Aug 9, 2011 #15
Now the discussion has become pretty interesting.
If an approximation works, then there should be arguments for it. If we lack reasoning, then it's just a fitting formula.
I find no reasoning to extend the same hydrogen atomic orbitals (electronic probability densities) and corresponding energy eigen values for other atoms.
It may be satisfactory for chemists, but not for physicists.
Question is : While calculating the energy of the levels (s, p, d, whatever), are we using the hydrogen-atom energy eigenvalue formula? Don't we require to modify that since many electrons are talking to each other now?
Again filling the shells by putting electrons one by one doesn't make any physics sense.
Electrons are always interacting and they are in a cloud, not in isolated islands.
I believe, this is a serious issue which has been neglected since 1930. We should try to solve two, three, four atoms problems numerically and figure out true atomic orbitals for them.
And even if a large community accepts a theory, without a justification, the theory cannot be regarded as the correct or true theory.
17. Aug 9, 2011 #16
There is not much difference between the two.
It has been working so far. Do not freak out if someone gives you this answer. This is true with every theory/model.
This has already been answered by @DiracRules
Most of the theories (if not all) are "fitting formula". If the fitting works for all/most cases, then it is a good theory/model.
This is done in all the multi-fermion cases. If this does not make sense to you in atomic theory, then this should not make sense to you in all branches of Physics involving fermions.
People have tried, they do not know enough mathematics.
Again, answer to all your questions seems to be that "it works".
18. Aug 9, 2011 #17
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you are confusing lots of things. (I admit it's simple with this topic, as there is lots of information around which is misleading or plainly wrong, even in textbooks). Let's clarify some things:
1) The spherical harmonics (s,p,d,f,...) have /nothing/ to do with the hydrogen atom. These are the solutions of the homogenous angular Laplace equation, and they form (exactly) the angular part of any Laplace problem with a spherical potential. If a potential is slightly non-spherical (think of a atom in a molecule), it is often a good idea to expand the solutions into spherical harmonics anyway, because these form a convenient complete orthonormal set of functions on the sphere (i.e., for angular functions). Note that this is /NOT/ an approximation as long as the expansion is not truncated (complete orthonormal set!). In practice that means for example that the orbitals of the O atom in H2O have not only s and p character, but also a bit of d, of f, of g etc character, with increasingly negligible weight. This is realized in practical calculations by using systematic series of basis sets for expaning the one-particle wave functions (for example, the mentioned correlation consistent basis sets). By doing that you can approach the infinite basis limit to any degree you like.
2) There are no "true orbitals" for systems with more than one electron. Orbitals are, by definition, ONE PARTICLE wave functions. They are DEFINED in terms of some kind of Hartree-Fock or Kohn-Sham mean field picture (there are also "natural orbitals", which are something different, but let's ignore them for now). Orbitals are not true wave functions either[1]. Rather, they are used to BUILD wave functions by plugging them into Slater determinants or configuration state functions.
3) Of course 3--6 electron systems can be calculated effectively exactly, and this has been done long time ago. Some people are still doing it now. The most accurate methods for that are typically of the variational monte carlo or iterative complent class. For systems with more than 6 electrons the mentioned quantum chemistry methods come into play. Note that also these methods are accurate to more than 0.01% in total energies, often much more (1 kJ/mol is a VERY VERY SMALL energy compared to the total energies of the systems calculated! Even Hartree-Fock typically gets total energies right to around 1%).
4) The aufbau principle and shell filling you mention is based on the Hund rules, which were originally empirically derived. Nevertheless, the theoretical reasoning behind them is sound (go look up Hund rules in Wikipedia), and almost always works. If you don't like the reasoning, there is no reason to believe it: You can /calculate/ the energies of the different states (using quantum chemistry) and see that it works. You don't have to believe it, you can test it.
5) Ab initio quantum chemistry is not an empirical science. It calculates properties of molecules using /nothing/ as input execept for the Schroedinger equation and fundamental constants (hbar, electron mass, electron charge etc.). It can calculate true wave functions to any desired degree of accuracy, the limits are only given by the computational power which you are willing to put into it.
[1] (and never probability densities. There abs square gives a density, but not the orbitals themselves),
19. Aug 10, 2011 #18
I think, hydrogen atom and one electron atom are equivalent. Now once you put one extra electron, trouble enters. Now the Schroedinger equation has two coordinates: [itex]r_1[/itex] and [itex]r_2[/itex], and due to Coulomb repulsion they are coupled, i.e. we have a term like [tex]e^2/|r_{1}-r_{2}|[/tex]. So unless we get rid of electronic repulsion, we cannot do separation of variables and hence cannot have the spherical harmonics for the angular part.
Before we go to H2O, I am curious about knowing orbitals of O (oxygen).
I agree. But again the single-body wavefunction in the Hartree-Fock or the Slater determinants are ambiguous (even the shape).
I must underline the word effectively. I'm just curious to know, did the results show (effective) orbitals? How do they look like? Could you provide some references where I can see that?
I agree that it was originally empirically derived. But probably Schroedinger or Hartree-Fock equation cannot take care of that.
How does the ab initio method distinguish a hydrogen and a oxygen atom? Note that I'm only interested about orbitals in an atom, not in a molecule.
I didn't get what you wanted to mean here.
20. Aug 10, 2011 #19
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before we go on with this discussion, you should look up the following points:
1) What is a Slater determinant?
2) What is the difference between a N-electron wave function (e.g., a Slater determinant) and a one-electron wave function (e.g., a Hartree-Fock orbital)
3) What is Hartree-Fock and how does it allow you to get an approximate N-electron wave function (a Slater determinant) using one-particle wave functions (occupied HF orbitals)?
I'm not saying this to tease you. Understanding these points is absolutely essential to understand the answers to the questions you are asking, or to understand why other questions you are asking don't make sense.
I just told you that the spherical harmonics have nothing to do with hydrogen or the number of electrons. They are related to the spherical symmetry of the problem, nothing more. In particular, they form the angular part of solutions of any one-particle Laplace equation (e.g., a one-particle Schrödinger equation) with a spherical potential. The one-particle Schrödinger equations in this case are the Hartree-Fock equations for the orbitals, which for spherical systems (e.g. atoms like N) feature an spherically symmetric potential: The sum of the nuclear attraction potential (a spherically symmetric one-particle potential) and the mean field repulsion of the electrons (also a spherically symmetric one-particle potential)[1]. Both these potentials and the orbitals themselves are determined self-consistently in the Hartree-Fock procedure. Please, forget everything about hydrogen or "hydrogen-like atoms". These solutions do not help in any way in understanding the many-electron problem.
[1] In reality it's a bit more complicated due to the presence of the exchange potential, but this is not essential for understanding this point.
No, they are not. They are determined uniquely (up to unitary transformations amongst themselves) by the Hartree-Fock solution. In particular, for any ground state spherical atom, there is only one set of canonical Hartree-Fock orbitals. The orbitals are determined by Hartree-Fock, they are the OUTPUT of the Hartree-Fock calculation.
Being able to calculate something to any desired accuracy is equivalent to being able to calculate it exactly. And no, these most accurate methods for 2--6 electron systems do not use orbital expansions but rather direct real space wave function ansaetze. Asking about orbitals for such wave functions is not the right question.
Of course they can. I just told you that. If you don't believe the empirical occupation rules, you can calculate the state energies for the other, non-Hund occupations, and then see whether or not they are lower in energy than the Hund-occupation solutions. In almost all cases they are not.
H and O have different nuclear potentials (one 1/r, the other 8/r) and a different number of electrons. That are (formally) the only two inputs to HF.
21. Aug 11, 2011 #20
1) I know the Slater determinant. It is a many-body wave function constructed upon single-particle wave function so that the Pauli exclusion principle (or the exchange interaction) is taken care (using the property of exchanging row/column of a determinant).
2) and 3) I think, we get Hartree-Fock (HF) equation by minimizing the Slater determinant and HF wave functions satisfy the HF equations.
Tell me anything further I need to know.
Did you mean self-consistent Hartree-Fock? And do you start with trial single-body wave functions, e,g, Gaussians?
Sorry, how do you check the accuracy? I thought, once you know the wave function, you can construct the orbitals (sorry again H-atom analogy).
Once we have Hartree-Fock wavefunctions, are not their energy-levels defined? But Hunds rule is not included inside the Hartree-Fock!
22. Aug 16, 2011 #21
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There are many different ways of arriving at a set of ground state Hartree-Fock orbitals, but there is only one canonical set of such orbitals for a given system (modulo degeneracy etc). The SCF method (the one with repeated diagonalization of a Fock operator) is the most common way of getting these orbitals.
The Gaussians you mention are not trial functions, but basis functions. In most molecular HF programs the orbitals are determined as a linear combination of Gaussian-time-solid-harmonic basis functions, using optimized pre-determined standard basis sets. However, note the following:
i) The solid harmonics form a complete basis set for the angular variables around a nucleus, and the Gaussians form a complete basis for the radial functions around a nucleus. That means that ANY possible orbital shape, no matter how it looks, could be represented by a suitably large Gaussian basis set.
ii) There is no need to use Gaussian basis sets. This is only done because it is by far the most efficient way of handling molecules. But there are other programs using numerical grids, finite elements, or even plane waves to represent the orbitals. In the complete basis limit, all of those approaches produce identical orbitals and total energies.
Quantum chemistry knows systematic ways of making the calculations more accurate (and more expensive at the same time). You typically estimate the accuracy by the changes between successive approximation levels becoming smaller and smaller.
Orbitals are objects associated with one-particle theories, like Hartree-Fock or Kohn-Sham. They are used in most many-body methods, but only as initial approximation and as guide for constructing the real many-body wave function by operators acting on the determinant (or other reference function). I.e., in practice, the orbitals are the input for many-body wave function calculations, not its output.
Also, if you directly parameterize a real-space many-body wave function, like it is done in the monte carlo and iterative complement methods which give the most accurate numbers for up to six electron systems, there is no need for orbitals at all.
You can still construct a set of so called "natural orbitals" from a correlated wave function (by diagonalizing its first order density matrix), but these orbitals do not have a meaning in the sense of representing an effective Slater determinant wave function.
The orbitals have energy eigenvalues, but these eigenvalues do not have that much meaning (the only thing you get from them are approximate ionization potentials via Koopman's theorem). Also, each orbital and each eigenvalue depends on all the other orbitals and eigenvalues and on the electronic state in question (i.e., if you optimize different states (e.g., different orbital occupations), you will get different orbital energies).
About Hund rules: The Hund rules (some of them, anyway) are sometimes used in HF programs to determine the initial occupations for the initial guess orbitals. In later iterations, usually only the aufbau principle or overlap consistency criteria are used to determine which orbitals are occupied and which are not (diagonalizing a Fock operator only gives you a set of orbitals, it does not tell you which of them are supposed to be occupied before self-consistence is reached) |
8cfdf1e64a589180 | Quasicrystals: anticipating the unexpected
The following guest entry is contributed by Peter Kramer
Dan Shechtman received the Nobel prize in Chemistry 2011 for the experimental discovery of quasicrystals. Congratulations! The press release stresses the unexpected nature of the discovery and the struggles of Dan Shechtman to convince the fellow experimentalists. To this end I want to contribute a personal perspective:
From the viewpoint of theoretical physics the existence of icosahedral quasicrystals as later discovered by Shechtman was not quite so unexpected. Beginning in 1981 with Acta Cryst A 38 (1982), pp. 257-264 and continued with Roberto Neri in Acta Cryst A 40 (1984), pp. 580-587 we worked out and published the building plan for icosahedral quasicrystals. Looking back, it is a strange and lucky coincidence that unknown to me during the same time Dan Shechtman and coworkers discovered icosahedral quasicrystals in their seminal experiments and brought the theoretical concept of three-dimensional non-periodic space-fillings to live.
More about the fascinating history of quasicrystals can be found in a short review: gateways towards quasicrystals and on my homepage.
Time to find eigenvalues without diagonalization
Solving the stationary Schrödinger (H-E)Ψ=0 equation can in principle be reduced to solving a matrix equation. This eigenvalue problem requires to calculate matrix elements of the Hamiltonian with respect to a set of basis functions and to diagonalize the resulting matrix. In practice this time consuming diagonalization step is replaced by a recursive method, which yields the eigenfunctions for a specific eigenvalue.
A very different approach is followed by wavepacket methods. It is possible to propagate a wavepacket without determining the eigenfunctions beforehand. For a given Hamiltonian, we solve the time-dependent Schrödinger equation (i ∂t-H) Ψ=0 for an almost arbitrary initial state Ψ(t=0) (initial value problem).
The reformulation of the determination of eigenstates as an initial value problem has a couple of computational advantages:
• results can be obtained for the whole range of energies represented by the wavepacket, whereas a recursive scheme yields only one eigenenergy
• the wavepacket motion yields direct insight into the pathways and allows us to develop an intuitive understanding of the transport choreography of a quantum system
• solving the time-dependent Schrödinger equation can be efficiently implemented using Graphics Processing Units (GPU), resulting in a large (> 20 fold) speedup compared to CPU code
Aharnov-Bohm Ring conductance oscillations
The Zebra stripe pattern along the horizontal axis shows Aharonov-Bohm oscillations in the conductance of a half-circular nanodevice due to the changing magnetic flux. The vertical axis denotes the Fermi energy, which can be tuned experimentally. For details see our paper in Physical Review B.
The determination of transmissions requires now to calculate the Fourier transform of correlation functions <Ψ(t=0)|Ψ(t)>. This method has been pioneered by Prof. Eric J. Heller, Harvard University, and I have written an introductory article for the Latin American School of Physics 2010 (arxiv version).
Recently, Christoph Kreisbeck has done a detailed calculations on the gate-voltage dependency of the conductance in Aharonov-Bohm nanodevices, taking full adventage of the simultaneous probing of a range of Fermi energies with one single wavepacket. A very clean experimental realization of the device was achieved by Sven Buchholz, Prof. Saskia Fischer, and Prof. Ulrich Kunze (RU Bochum), based on a semiconductor material grown by Dr. Dirk Reuter and Prof. Anreas Wieck (RU Bochum). The details, including a comparison of experimental and theoretical results shown in the left figure, are published in Physical Review B (arxiv version).
Cosmic topology from the Möbius strip
Fig 1. The Möbius twist.
The following article is contributed by Peter Kramer.
Einstein’s fundamental theory of Newton’s gravitation relates the interaction of masses to the curvature of space. Modern cosmology from the big bang to black holes results from Einstein’s field equations for this relation. These differential equations by themselves do not yet settle the large-scale structure and connection of the cosmos. Theoretical physicists in recent years tried to infer information on the large-scale cosmology from Cosmic microwave background radiation (CMBR), observed by satellite observations. In the frame of large-scale cosmology, the usual objects of astronomy from solar systems to galaxy clusters are smoothed out, and conditions imprinted in the early stage of the universe dominate.
Fig 2: The planar Möbius crystal cm
In mathematical language one speaks of cosmic topology. Topology is often considered to be esoteric. Here we present topology from the familiar experience with the twisted Möbius strip. This strip on one hand can be seen as a rectangular crystallographic lattice cell whose copies tile the plane, see Fig. 2. The Möbius strip is represented as a rectangular cell, located between the two vertical arrows, of a planar crystal. A horizontal dashed line through the center indicates a glide-reflection line. A glide reflection is a translation along the dashed line by the horizontal length of the cell, followed by a reflection in this line. The crystallographic symbol for this planar crystal is cm. In three-dimensional space the planar Möbius crystal (top panel of Fig. 1) is twisted (middle panel of Fig. 1). The twist is a translation along the dashed line, combined with a rotation by 180 degrees around that line. A final bending (bottom panel of Fig. 1) of the dashed line and a smooth gluing of the arrowed edges yields the familiar Möbius strip.
Fig 3: Cubic twist N3.
Given this Möbius template in two dimension, we pass to manifolds of dimension three. We present in Fig. 3 a new cubic manifold named N3. Three cubes are twisted from an initial one. A twist here is a translation along one of the three perpendicular directions, combined with a right-hand rotation by 90 degrees around this direction. To follow the rotations, note the color on the faces. The three neighbor cubes can be glued to the initial one. If the cubes are replaced by their spherical counterparts on the three-sphere, the three new cubes can pairwise be glued with one another, with face gluings indicated by heavy lines. The complete tiling of the three-sphere comprises 8 cubes and is called the 8-cell. The gluings shown here generate the so-called fundamental group of a single spherical cube on the three-sphere with symbol N3. This spherical cube is a candidate for the cosmic topology inferred from the cosmic microwave background radiation. A second cubic twist with a different gluing and fundamental group is shown in Fig. 4. Here, the three twists combine translations along the three directions with different rotations.
The key idea in cosmic topology is to pass from a topological manifold to its eigen- or normal modes. For the Möbius strip, these eigenmodes are seen best in the planar crystal representation of Fig. 2. The eigenmodes can be taken as sine or cosine waves of wave length \lambda which repeat their values from edge to edge of the cell. It is clear that the horizontal wavelength \lambda of these modes has as upper bound the length L of the rectangle. The full Euclidean plane allows for infinite wavelength, and so the eigenmodes of the Möbius strip obey a selection rule that characterizes the topology. Moreover the eigenmodes of the Möbius strip must respect its twisted connection.
Fig 4: Cubic twist N2.
Similarly, the eigenmodes of the spherical cubes in Fig. 3 must repeat themselves when going from cube to neighbor cube. It is intuitively clear that the cubic eigenmodes must have a wavelength smaller than the edge length of the cubes. The wave length of the eigenmodes of the full three-sphere are bounded by the equator length of the three-sphere. Seen on a single cube, the different twists and gluings of the manifolds N2 and N3 shown in Figs. 3 and 4 form different boundary value problems for the cubic eigenmodes.
Besides of these spherical cubic manifolds, there are several other competing polyhedral topologies with multiple connection or homotopy. Among them are the famous Platonic polyhedra. Each of them gives rise to a Platonic tesselation of the three-sphere. Everitt has analyzed all their possible gluings in his article Three-manifolds from platonic solids in Topology and its applications, vol 138 (2004), pp. 253-263. In my contribution Platonic topology and CMB fluctuations: Homotopy, anisotropy, and multipole selection rules, Class. Quant. Grav., vol. 27 (2010), 095013 (freely available on the arxiv) I display them and present a full analysis of their corresponding eigenmodes and selection rules.
Since terrestrial observations measure the incoming radiation in terms of its spherical multipoles as functions of their incident direction, the eigenmodes must be transformed to a multipole expansion as done in my work. New and finer data on the CMB radiation are expected from the Planck spacecraft launched in 2009. These data, in conjunction with the theoretical models, will promote our understanding of cosmic space and possible twists in its topology.
Hot spot: the quantum Hall effect in graphene
Hall potential in a graphene device due to interactions and equipotential boundary conditions at the contacts.
An interesting and unfinished chapter of condensed mater theory concerns the quantum Hall effect. Especially the integer quantum Hall effect (IQHE) is actually not very well understood. The fancy cousin of the IQHE is the fractional quantum Hall effect (FQHE). The FQHE is easier to handle since there is agreement about the Hamiltonian which is to be solved (although the solutions are difficult to obtain): the quantum version of the very Hamiltonian used for the classical Hall effect, namely the one for interacting electrons in a magnetic field. The Hamiltonian is still lacking the specification of the boundary conditions, which can completely alter the results for open and current carrying systems (as in the classical Hall effect) compared to interacting electrons in a box.
Surprisingly no agreement about the Hamiltonian underlying the IQHE exists. It was once hoped that it is possible to completely neglect interactions and still to obtain a theoretical model describing the experiments. But if we throw out the interactions, we throw out the Hall effect itself. Thus we have to come up with the correct self-consistent solution of a mean field potential which incorporates the interactions and the Hall effect.
Is it possible to understand the integer quantum Hall effect without including interactions – and if yes, how does the effectively non-interacting Hamiltonian look like?
Starting from a microscopic theory we have constructed the self-consistent solution of the Hall potential in our previous post for the classical Hall effect. Two indispensable factors caused the emergence of the Hall potential:
1. repulsive electronic interactions and
2. equipotential boundary conditions at the contacts.
The Hall potential which emerges from our simulations has been directly imaged in GaAs Hall-devices under conditions of a quantized conductance by electro-optical methods and by scanning probe microscopy using a single electron transistor. Imaging requires relatively high currents in order to resolve the Hall potential clearly.
In graphene the dielectric constant is 12 times smaller than in GaAs and thus the Coulomb repulsion between electrons are stronger (which should help to generate the Hall potential). The observation of the FQHE in two-terminal devices has led the authors of the FQHE measurments to conjecture that hot-spots are also present in graphene devices [Du, Skachko, Duerr, Luican Andrei Nature 462, 192-195 (2009)].
These observations are extremely important, since the widely used theoretical model of edge-state transport of effectively non-interacting electrons is not readily compatible with these findings. In the edge-state model conductance quantization relies on the counter-propagation of two currents along the device borders, whereas the shown potential supports only a unidirectional current from source to drain diagonally across the device.
Moreover the construction of asymptotic scattering states is not possible, since no transverse lead-eigenbasis exists at the contacts. Electrons moving strictly along one side of the device from one contact to the other one would locally increase the electron density within the contact and violate the metallic boundary condition (see our recent paper on the Self-consistent calculation of electric potentials in Hall devices [Phys. Rev. B, 81, 205306 (2010)]).
Are there models which support a unidirectional current and at the same time support a quantized conductance in units of the conductivity quantum?
We put forward the injection model of the quantum Hall effect, where we take the Hall potential as being the self-consistent mean-field solution of the interacting and current carrying device. On this potential we construct the local density of states (LDOS) next to the injection hot spot and calculate the resulting current flow. In our model, the conductivity of the sample is completely determined by the injection processes at the source contact where the high electric field of the hot spots leads to a fast transport of electrons into the device. The LDOS is broadened due to the presence of the electric Hall field during the injection and not due to disorder. Our model is described in detail in our paper Theory of the quantum Hall effect in finite graphene devices [Phys. Rev. B, 81, 081410(R) (2010), free arxiv version] and the LDOS in a conventional semiconductor in electric and magnetic fields is given in a previous paper on electron propagation in crossed magnetic and electric fields. The tricky part is to prove the correct quantization, since the absence of any translational symmetries in the Hall potential obliterates the use of “Gedankenexperimente” relying on periodic boundary conditions or fancy loop topologies.
In order to propel the theoretical models forward, we need more experimental images of the Hall potential in a device, especially in the vicinity of the contacts. Experiments with graphene devices, where the Hall potential sits close to the surface, could help to establish the potential distribution and to settle the question which Hamiltonian is applicable for the quantum Hall effects. Is there anybody out to take up this challenge?
Trilobites revived: fragile Rydberg molecules, Coulomb Green’s function, Lambert’s theorem
The trilobite state
The trilobite Rydberg molecule can be modeled by the Coulomb Green’s function, which represents the quantized version of Lambert’s orbit determination problem.
The recent experimental realization observation of giant Rydberg molecules by Bendkowsky, Butscher, Nipper, Shaffer, Löw, Pfau [theoretically studied by Greene and coworkers, see for example Phys. Rev. Lett. 85, 2458 (2000)] shows Coulombic forces at work at large atomic distances to form a fragile molecule. The simplest approach to Rydberg molecules employs the Fermi contact potential (also called zero range potential), where the Coulomb Green’s function plays a central role. The quantum mechanical expression for the Coulomb Green’s function was derived in position space by Hostler and in momentum space by Schwinger. The quantum mechanical expression does not provide immediate insights into the peculiar nodal structure shown on the left side and thus it is time again to look for a semiclassical interpretation, which requires to translate an astronomical theorem into the Schrödinger world, one of my favorite topics.
Johann Heinrich Lambert was a true “Universalgelehrter”, exchanging letters with Kant about philosophy, devising a new color pyramid, proving that π is an irrational number, and doing physics. His career did not proceed without difficulties since he had to educate himself after working hours in his father’s tailor shop. After a long journey Lambert ended up in Berlin at the academy (and Euler choose to “escape” to St. Petersburg).
Lambert followed Kepler’s footsteps and tackled one of the most challenging problems of the time: the determination of celestial orbits from observations. In 1761 Lambert did solve the problem of orbit determination from two positions measurements. Lambert’s Theorem is a cornerstone of astronavigation (see for example the determination of Sputnik’s orbit using radar range measurements and Lambert’s theorem). Orbit determination from angular information alone (without known distances) is another problem and requires more observations.
Lambert poses the following question [Insigniores orbitae cometarum proprietates (Augsburg, 1761), p. 120, Lemma XXV, Problema XL]: Data longitudine axis maioris & situ foci F nec non situ punctorum N, M, construere ellipsin [Given the length of the semi-major axis, the location of one focal point, the points N,M, construct the two possible elliptical orbits connecting both points.]
Lambert's construction of two ellipses.
Lambert’s construction to find all possible trajectories from N to M and to map them to a ficticious 1D motion from n to m.
Lambert finds the two elliptic orbits [Fig. XXI] with an ingenious construction: he maps the rather complicated two-dimensional problem to the fictitious motion along a degenerate linear ellipse. Some physicists may know how to relate the three-dimensional Kepler problem to a four-dimensional oscillator via the Kustaanheimo–Stiefel transformation [see for example The harmonic oscillator in modern physics by Moshinsky and Smirnov]. But Lambert’s quite different procedure has its advantages for constructing the semiclassical Coulomb Green’s function, as we will see in a moment.
Shown are two ellipses with the same lengths of the semimajor axes 1/2 A1B1=1/2 A2 B2 and a common focus located at F. The centers of the two ellipses are denoted by C1 and C2. Lambert’s lemma allows to relate the motion from N to M on both ellipses to a common collinear motion on the degenerate linear ellipse Fb, where the points n and m are chosen such that the time of flight (TOF) along nm equals the TOF
along the elliptical arc NM on the first ellipse. On the second ellipse the TOF along the arc NB2M equals the TOF along nbm. The points n and m are found by marking the point G halfway between N and M. Then the major axis Fb=A1 B1=A2 B2 of the linear ellipse is drawn starting at F and running through G. On this line the point g is placed at the distance Fg=1/2(FN+FM). Finally n and m are given by the intersection points of a circle around g with radius GN=GM. This construction shows that the sum of the lengths of the shaded triangle α±=FN + FM ± NM is equal to α±=fn+ fm ± nm. The travel time depends only on the distances entering α±, and all calculations of the travel times etc. are given by one-dimensional integrations along the ficticious linear ellipse.
Lambert did find all the four possible trajectories from N to M which have the same energy (=semimajor axis a), regardless of their eccentricity (=angular momentum). The elimination of the angular momentum from Kepler’s equation is a tremendous achievement and the expression for the action is converted from Kepler’s form
• [Kepler] W(r,r‘;E)=√μ a Kc [ξ + ε sin(ξ) – ξ’ – ε sin(ξ’)], with eccentricity ε, eccentric anomaly ξ to
• [Lambert] W(r,r‘;E)=√μ a Kc[γ + sin(γ) – δ – sin(δ)], with
sin2(γ/2)=(r+r’+ |r‘-r|)/(4a) and sin2(δ/2)=(r+r’- |r‘-r|)/(4a).
The derivation is also discussed in detail in our paper [Kanellopoulos, Kleber, Kramer: Use of Lambert’s Theorem for the n-Dimensional Coulomb Problem Phys. Rev. A, 80, 012101 (2009), free arxiv version here]. The Coulomb problem of the hydrogen atom is equivalent to the gravitational Kepler problem, since both are subject to a 1/r potential. Some readers might have seen the equation for the action in Gutzwiller’s nice book Chaos in classical and quantum mechanics, eq. (1.14). It is worthwhile to point out that the series solution given by Lambert (and Gutzwiller) for the time of flight can be summed up easily and is denoted today by an inverse sine function (for hyperbolic motion a hyperbolic sine, a function later introduced by Riccati and Lambert). Again, the key-point is the introduction of the linear ficticious ellipse by Lambert which avoids integrating along elliptical arcs.
The surprising conclusion: the nodal pattern of the hydrogen atom can be viewed as resulting from a double-slit interference along two principal ellipses. The interference determines the eigenenergies and the eigenstates. Even the notorious difficult-to-calculate van Vleck-Pauli-Morette (VVPM) determinant can be expressed in short closed form with the help of Lambert’s theorem and our result works even in higher dimensions. The analytic form of the action and the VVPM determinant becomes essential for our continuation of the classical action into the forbidden region, which corresponds to a tunneling process, see the last part of our paper.
Lambert is definitely a very fascinating person. Wouldn’t it be nice to discuss with him about philosophy, life, and science?
Determining the affinities of electrons OR: seeing semiclassics in action
Electron trajectories for photodetachment in an electric field.
Negatively charged ions are an interesting species, having managed to bind one more electron than charge neutrality grants them [for a recent review see T. Andersen: Atomic negative ions: structure, dynamics and collisions, Physics Reports 394 p. 157-313 (2004)]. The precise determination of the usually small binding energy is best done by shining a laser beam of known wave length on the ions and detect at which laser frequency the electron gets detached from the atomic core.
For some ions (oxygen, sulfur, or hydrogen fluoride and many more) the most precise values given at NIST are obtained by Christophe Blondel and collaborators with an ingenious apparatus based on an idea by Demkov, Kondratovich, and Ostrovskii in Pis’ma Zh. Eksp. Teor. Fiz. 34, 425 (1981) [JETP Lett. 34, 403 (1981)]: the photodetachment microscope. Here, in addition to the laser energy, the energy of the released electron is measured via a virtual double-slit experiment. The ions are placed in an electric field, which makes the electronic wave running against the field direction turn back and interfere with the wave train emitted in the field direction. The electric-field induced double-slit leads to the build up of a circular interference pattern of millimeter size (!) on the detector shown in the left figure (the animation was kindly provided by C. Blondel, W. Chaibi, C. Delsart, C. Drag, F. Goldfarb & S. Kröger, see their orginal paper The electron affinities of O, Si, and S revisited with the photodetachment microscope, Eur. Phys. J. D 33 (2005) 335-342).
Observed time-dependent emergence of the interference pattern in an electric field. Video shown with kind permission of C. Blondel et al. (see text for full credit)
I view this experiment as one of the best illustrations of how quantum and classical mechanics are related via the classical actions along trajectories. The two possible parabolic trajectories underlying the quantum mechanical interference pattern were described by Galileo Galilei in his Discourses & Mathematical Demonstrations Concerning Two New Sciences Pertaining to Mechanics & Local Motions in proposition 8: Le ampiezze de i tiri cacciati con l’istesso impeto, e per angoli egualmente mancanti, o eccedenti l’angolo semiretto, sono eguali. Ironically the “old-fashioned” parabolic motion was removed from the latest Gymnasium curriculum in Baden-Württemberg to make space for modern quantum physics.
At the low energies of the electrons, their paths are easily deflected by the magnetic field of the Earth and thus require either excellent shielding of the field or an active compensation, which was achieved recently by
Chaibi, Peláez, Blondel, Drag, and Delsart in Eur. Phys. J. D 58, 29-37 (2010). The new paper demonstrates nicely the focusing effect of the combined electric an magnetic fields, which Christian Bracher, John Delos, Manfred Kleber, and I have analyzed in detail and where one encounters some of the seven elementary catastrophies since the magnetic field allows one to select the number of interfering paths.
We have predicted similar fringes for the case of matter waves in the gravitational field around us originating from trapped Bose-Einstein condensates (BEC), but we are not aware of an experimental observation of similar clarity as in the case of the photodetachment microscope.
Mathematically, the very same Green’s function describes both phenomena, photodetachment and atomlasers. For me this universality demonstrates nicely how mathematical physics allows us to understand phenomena within a language suitable for so many applications.
Interactions: from galaxies to the nanoscale
Microscopic model of a Hall bar
(a) Device model
(b) phenomenological potential
(c) GPU result
For a while we have explored the usage of General Purpose Graphics Processing Units (GPGPU) for electronic transport calculations in nanodevices, where we want to include all electron-electron and electron-donor interactions. The GPU allows us to drastically (250 fold !!!) boost the performance of N-body codes and we manage to propagate 10,000 particles over several million time-steps within days. While GPU methods are now rather popular within the astrophysics crowd, we haven’t seen many GPU applications for electronic transport in a nanodevice. Besides the change from astronomical units to atomic ones, gravitational forces are always attractive, whereas electrons are affected by electron-donor charges (attractive) and electron-electron repulsion. Furthermore we have a magnetic field present, leading to deflections. Last, the space where electrons can spread out is limited by the device borders. In total the force on the kth electron is given by \vec{F}_{k}=-\frac{e^2}{4\pi\epsilon_0 \epsilon}\sum_{\substack{l=1}}^{N_{\rm donor}}\frac{\vec{r}_l-\vec{r}_k}{|\vec{r}_l-\vec{r}_k|^3}+\frac{e^2}{4\pi\epsilon_0 \epsilon}\sum_{\substack{l=1\\l\ne k}}^{N_{\rm elec}}\frac{\vec{r}_l-\vec{r}_k}{|\vec{r}_l-\vec{r}_k|^3}+e \dot{\vec{r}}_k\times\vec{B}
Our recent paper in Physical Review B (also freely available on the arxiv) gives the first microscopic description of the classical Hall effect, where interactions are everything: without interactions no Hall field and no drift transport. The role and importance of the interactions is surprisingly sparsely mentioned in the literature, probably due to a lack of computational means to move beyond phenomenological models. A notable exception is the very first paper on the Hall effect by Edwin Hall, where he writes “the phenomena observed indicate that two currents, parallel and in the same direction, tend to repel each other”. Note that this repulsion works throughout the device and therefore electrons do not pile up at the upper edge, but rather a complete redistribution of the electronic density takes place, yielding the potential shown in the figure.
Another important part of our simulation of the classical Hall effect are the electron sources and sinks, the contacts at the left and right ends of the device. We have developed a feed-in and removal model of the contacts, which keeps the contact on the same (externally enforced) potential during the course of the simulation.
Mind-boggling is the fact that the very same “classical Hall potential” has also been observed in conjunction with a plateau of the integer quantum Hall effect (IQHE) [Knott et al 1995 Semicond. Sci. Technol. 10 117 (1995)]. Despite these observations, many theoretical models of the integer quantum Hall effect do not consider the interactions between the electrons. In our classical model, the Hall potential for non-interacting electrons differs dramatically from the solution shown above and transport proceeds then (and only then) along the lower and upper edges. However the edge current solution is not compatible with the contact potential model described above where an external reservoir enforces equipotentials within each contact. |
6b719b90a4405cd6 | Foreword: Conceptual Foundations of Theoretical Physics
First a foreword to the present FPU faculty in the conceptual foundations of theoretical physics, which is important you understand before considering for application. In fact, theoretical physics has nowadays found so many fields of application and has become so diversified, that there is no unique and coherent definition of what exact curricula it might entail. If you have in mind to become an applied physicist in the domains of material science, nanotechnologies, biophysics, electronics, etc., this course can greatly help too, but will not prepare you to that. What we have in mind here is something which skips the trendy mantra of forced interdisciplinarity at all costs in view of practical applications led by a managerial styled R&D. Here we are not offering you something which is supposed to project you towards a high-impact career which deals with contemporary research lines in fashion supposed to revolutionise the world. It is not about a knowledge that focuses on industrial and practical applications preparing you for the competition in a globalized world, as so many universities love to make you believe. What this course prepares to, is therefore not a domain useful for applied physics, even though lots of stuff that is studied here is a basic request for that too. If you are looking for that, please understand that most probably you are not in the right place here.
With ‘theoretical physics’ we mean the subjects that prepare you to understand and further research the deeper meaning and fundamental structure of the physical world. It is the physics which rises curiosity and the desire to know more about nature, independently from whatever that knowledge might or might not be useful for. It is about something which primarily is supposed to prepare you to tackle first and foremost with the interesting and complex philosophical questions that arise from physics. Moreover, it is intended to prepare the grounds so that you can later eventually research and contribute at a professional level to everything related to what goes beyond the standard model of particle physics (SM) in the view of a future quantum gravity theory. But it does not introduce you (at least not at this stage) to any speculative and yet not experimentally tested theory, like string theory, or canonical quantum gravity, etc., because there are good chances that these theories might not survive the test of history (however, these approaches might eventually become subjects of other courses in the future, once this introductory courses have been activated successfully).
What we are talking about is therefore first of all an understanding of the mathematical necessary basics, then classical mechanics, electromagnetism, quantum mechanics, relativity, quantum field theory (QFT), the SM, especially from the foundational theoretical perspective (but without excluding entirely the experimental aspects neither). With ‘theoretical physics’ we therefore do not mean a full-fledged course in solid state physics and even not astrophysics, cosmology or other chapters that are often cited as such, even though several aspects will be touched on and might well be included as appendix or added in one of the given courses (for instance, a brief introduction to cosmology is considered an integral part of the general relativity course).
What instead this course is going to offer you is a preparation from scratch (i.e. beginning all over again from middle school math and physics level) that will make you competent at the highest professional level to understand by your own modern approaches that go beyond the incomplete and now 40 years old SM or tackle with the philosophical issues related to quantum mechanics, relativity and QFT.
Having clarified that, let us now describe the content and structure of the proposed curriculum.
As well known, physics and mathematics have a lot in common. Not only is math absolutely essential to understand physics, but they are both subjects which have a strongly hierarchically build up foundation. You cannot begin to build a house from the roof. You can not write a novel without learning the letters of an alphabet first, the words of the vocabulary, the grammatical rules of the language. Similarly, you will never be able to play a musical instrument without having gone through a sufficiently long practice and exercise.
This seems to us quite obvious, and yet there are many who are convinced that this does not hold for physics. They firmly believe that they do not need to learn math and the basics of physics and begin immediately to imagine, speculate, fantasize and build a huge castle of misunderstandings and made of concepts on foundations that do not exist. In some sense this might indicate a strong desire to discover and inquire things as soon as possible, but the aspirant physicist of the FPU must learn to balance a good and healthy curiosity with the awareness of one’s own limits. As Newton did when he stated: "If I have seen further it is by standing on the shoulders of Giants", meaning by that, that one can discover new truths only by building on previous discoveries. In fact, while in a FPU students are even encouraged to express their interests almost from the beginning with the portfolio work which comes at the end of each course, on the other side it is made clear that the foundations of physics are not dispensable.
There is a basis and a conceptual structure that holds in place a sort of conceptual pyramid. For example, you cannot understand how atomic physics works, if you didn’t learn quantum physics. You can’t understand quantum physics if you did not learn classical mechanics first. You can’t understand classical mechanics if you didn’t learn decently calculus. You can’t understand calculus if you have no longer the basic math tools you learned in high school. The very same principle holds also inside the respective subjects. For instance, you can’t understand what a differential equation is and know how to take advantage of it, if you don’t learn what a differential of a function is. You can’t understand what a differential is if you don’t know what a derivative of a function is. You can’t understand what a derivative is if you don’t know what at all a mathematical function is in the first place. And so on...
The result of a careful selection of courses for a curriculum of a FPU faculty in the foundations of physics is illustrated in the pyramid below:
So, you must realize that almost all the concepts in physics and mathematics build up on each other. And it is dangerous to skip any steps in between, otherwise you might not be allowed to proceed further. Not because there is an authority who dictates what to do, but simply because it is inherent in the nature of things as they are.
A possible source of confusion emerges sometimes when, once you will ask experienced academicians what really is the essential minimum to learn physics, you might get very different answers. Everyone has his/her own point of view and will place an emphasis on their own interests. Some will point out a plethora of things you are supposed to learn but that you might realize (when it is too late) that they were not so relevant for your interests. For instance, nowadays physics is heavily oriented towards experimental and applied science, which means that most physicists work in a sector which, for example, needs deep knowledge of electronics, computational science or solid state physics and will try hard to convince you how essential these are even though you might be interested in entirely different aspects of physics, like the physics of black holes or the philosophy of quantum mechanics. While having a good background in these topics might be quite useful too, especially when you will have to interact with experimentalists, during the first run of your apprenticeship in the foundations of theoretical physics, these topics will not turn out to be of such paramount importance, because they are not part of the set of topics contained in a common intersection of the basics you need to acquire in order to be able to proceed later by your own. Whereas, there are other things which are part of this common intersection and will permanently present itself in the most diverse domains of theoretical physics. If you won’t learn these they will always hunt you and never allow you to fully appreciate the ideas you want to absorb. For instance, Fourier transforms, Euler-Lagrange equations, Maxwell equations or the Schrödinger equation are almost ubiquitous mathematical and physical concepts or tools you will have to learn to be accustomed with because you will find them everywhere in theoretical physics (by the way, in applied physics too). Whereas, for the theoretical physicists who is looking more to the conceptual foundation of theoretical physics, knowing how the semiconductor physics in a transistor or a diode works, is useful but not a priority. General education and an interdisciplinary culture is fine, but since it takes already many years to get acquainted with the kind of stuff we are trying to elaborate here, it is wiser to fix our attention on the necessary basics first and let eventually the student the freedom to look further (for instance, during the portfolio phase).
So, that’s enough for the courses introduction, now let us take a look at each of the courses. |
ced9a6ea38dec32d | 2019 Vol. 43, No. 9
Display Method: |
Novel theoretical constraints for color-octet scalar models
Li Cheng, Otto Eberhardt, Christopher W. Murphy
2019, 43(9): 093101. doi: 10.1088/1674-1137/43/9/093101
We study the theoretical constraints on a model whose scalar sector contains one color octet and one or two color singlet SU(2)L doublets. To ensure unitarity of the theory, we constrain the parameters of the scalar potential for the first time at the next-to-leading order in perturbation theory. Moreover, we derive new conditions guaranteeing the stability of the potential. We employ the HEPfit package to extract viable parameter regions at the electroweak scale and test the stability of the renormalization group evolution up to the multi-TeV region. Furthermore, we set upper limits on the scalar mass splittings. All results are given for both cases with and without a second scalar color singlet.
Properties of the decay ${{H\to\gamma\gamma}} $ using the approximate ${{\alpha_s^4}}$ corrections and the principle of maximum conformality
Qing Yu, Xing-Gang Wu, Sheng-Quan Wang, Xu-Dong Huang, Jian-Ming Shen, Jun Zeng
2019, 43(9): 093102. doi: 10.1088/1674-1137/43/9/093102
The decay channel \begin{document}$ H\to\gamma\gamma $\end{document} is an important channel for probing the properties of the Higgs boson. In this paper, we analyze its decay width by using the perturbative QCD corrections up to the \begin{document}$ \alpha_s^4 $\end{document} order with the help of the principle of maximum conformality (PMC). PMC has been suggested in literature for eliminating the conventional renormalization scheme-and-scale ambiguities. After applying PMC, we observe that an accurate renormalization scale independent decay width \begin{document}$ \Gamma(H\to\gamma\gamma) $\end{document} up to the N4LO level can be achieved. Taking the Higgs mass, \begin{document}$ M_{\rm H} = 125.09\pm $\end{document}\begin{document}$ 0.21\pm0.11 $\end{document} GeV, given by the ATLAS and CMS collaborations, we obtain \begin{document}$ \Gamma(H\to \gamma\gamma)|_{\rm LHC} = 9.364^{+0.076}_{-0.075} $\end{document} KeV.
Neural network study of hidden-charm pentaquark resonances
Halil Mutuk
2019, 43(9): 093103. doi: 10.1088/1674-1137/43/9/093103
Recently, the LHCb experiment announced the observation of hidden-charm pentaquark states \begin{document}$P_c(4312)$\end{document}, \begin{document}$P_c(4440)$\end{document}, and \begin{document}$P_c(4457)$\end{document} near \begin{document}$ \Sigma_c \bar{D}$\end{document} and \begin{document}$ \Sigma_c \bar{D}^\ast$\end{document} thresholds. In this present work, we studied these pentaquarks in the framework of the nonrelativistic quark model with four types of potential. We solved five-body Schrödinger equation by using the artificial neural network method and made predictions of parities for these states, which are not yet determined by experiment. The mass of another possible pentaquark state near the \begin{document}$\bar{D}^\ast \Sigma_c^\ast$\end{document} with \begin{document}$J^P=5/2^-$\end{document} is also calculated.
Study of the ${{s\to d\nu\bar{\nu}}}$ rare hyperon decays in the Standard Model and new physics
Xiao-Hui Hu, Zhen-Xing Zhao
2019, 43(9): 093104. doi: 10.1088/1674-1137/43/9/093104
FCNC processes offer important tools to test the Standard Model (SM) and to search for possible new physics. In this work, we investigate the \begin{document}$s\to d\nu\bar{\nu}$\end{document} rare hyperon decays in SM and beyond. We find that in SM the branching ratios for these rare hyperon decays range from \begin{document}$10^{-14}$\end{document} to \begin{document}$10^{-11}$\end{document} . When all the errors in the form factors are included, we find that the final branching ratios for most decay modes have an uncertainty of about 5% to 10%. After taking into account the contribution from new physics, the generalized SUSY extension of SM and the minimal 331 model, the decay widths for these channels can be enhanced by a factor of \begin{document}$2 \sim 7$\end{document}.
Activation cross-sections of titanium isotopes at neutron energies of 13.5–14.8 MeV
Fengqun Zhou, Yueli Song, Yong Li, Xiaojun Sun, Shuqing Yuan
2019, 43(9): 094001. doi: 10.1088/1674-1137/43/9/094001
The cross-sections for 46Ti(n,2n)45Ti, 46Ti(n,p)46m+gSc+47Ti(n,d*)46m+gSc, 46Ti(n,p)46m+gSc, 47Ti(n,p)47Sc+48Ti(n,d*)47Sc, 47Ti(n,p)47Sc, 48Ti(n,p)48Sc+49Ti(n,d*)48Sc,48Ti(n,p)48Sc, and 50Ti(n,α)47Ca reactions were investigated around neutron energies of 13.5–14.8 MeV by means of the activation technique. Fast neutrons were produced by the 3H(d,n)4He reaction. Neutron energies from different directions in the measurements were obtained in advance using the method of cross-section ratios for 90Zr(n,2n)89m+gZr and 93Nb(n,2n)92mNb reactions. The results obtained are analyzed and compared with the experimental data provided by the literature and verified nuclear data in the JEFF-3.3, CENDL-3.1, ENDF/B-VIII.0 libraries, as well as results calculated by Talys-1.9 code.
Tin-accompanied and true ternary fission of 242Pu
M. Zadehrafi, M. R. Pahlavani, M. -R. Ioan
2019, 43(9): 094101. doi: 10.1088/1674-1137/43/9/094101
True ternary fission and Tin-accompanied ternary fission of 242Pu are studied by using the 'Three Cluster Model'. True ternary fission is considered as a formation of heavy fragments in the region \begin{document}$ 28\leqslant Z_1,Z_2,Z_3\leqslant 38 $\end{document} with comparable masses. The possible fission channels are predicted by the potential-energy calculations. Interaction potentials, Q-values and relative yields for all possible fragmentations in equatorial and collinear configurations are calculated and compared. It is found that ternary fission with formation of a double magic nucleus like \begin{document}$ ^{132}{\rm Sn} $\end{document} is more probable than the other fragmentations. Also, the kinetic energies of the fragments for the group \begin{document}$ Z_1 = 32 $\end{document}, \begin{document}$ Z_2 = 32 $\end{document} and \begin{document}$ Z_3 = 30 $\end{document} are calculated for all combinations in the collinear geometry as a sequential decay.
Applicability of 9Be global optical potential to reactions of 7,10,11,12Be
Yong-Li Xu, Yin-Lu Han, Hai-Ying Liang, Zhen-Dong Wu, Hai-Rui Guo, Chong-Hai Cai
2019, 43(9): 094102. doi: 10.1088/1674-1137/43/9/094102
Elastic scattering angular distributions and total reaction cross-sections of 7,10,11,12Be projectiles are predicted by the systematic 9Be global phenomenological optical model potential for target mass numbers ranging from 24 to 209. These predictions provide a detailed analysis by their comparison with the available experimental data. Furthermore, these elastic scattering observables are also predicted for some targets out of the mass number range. The results are in reasonable agreement with the existing experimental data, and they are presented in this study.
Response functions of hot and dense matter in the Nambu-Jona-Lasino model
Chengfu Mu, Ziyue Wang, Lianyi He
2019, 43(9): 094103. doi: 10.1088/1674-1137/43/9/094103
We investigate current-current correlation functions, or the so-called response functions of a two-flavor Nambu-Jona-Lasino model at finite temperature and density. The linear response is investigated introducing the conjugated gauge fields as external sources within the functional path integral approach. The response functions can be obtained by expanding the generational functional in powers of the external sources. We derive the response functions parallel to two well-established approximations for equilibrium thermodynamics, namely mean-field theory and a beyond-mean-field theory, taking into account mesonic contributions. Response functions based on the mean-field theory recover the so-called quasiparticle random phase approximation. We calculate the dynamical structure factors for the density responses in various channels within the random phase approximation, showing that the dynamical structure factors in the baryon axial vector and isospin axial vector channels can be used to reveal the quark mass gap and the Mott dissociation of mesons, respectively. Noting that the mesonic contributions are not taken into account in the random phase approximation, we also derive the response functions parallel to the beyond-mean-field theory. We show that the mesonic fluctuations naturally give rise to three kinds of famous diagrammatic contributions: the Aslamazov-Lakin contribution, the self-energy or density-of-state contribution, and the Maki-Thompson contribution. Unlike the equilibrium case, in evaluating the fluctuation contributions, we need to carefully treat the linear terms in external sources and the induced perturbations. In the chiral symmetry breaking phase, we find an additional chiral order parameter induced contribution, which ensures that the temporal component of the response functions in the static and long-wavelength limit recovers the correct charge susceptibility defined using the equilibrium thermodynamic quantities. These contributions from mesonic fluctuations are expected to have significant effects on the transport properties of hot and dense matter around the chiral phase transition or crossover, where the mesonic degrees of freedom are still important.
Energy staggering parameters in nuclear magnetic rotational bands
Wu-Ji Sun, Jian Li
2019, 43(9): 094104. doi: 10.1088/1674-1137/43/9/094104
This study presents the systematics of energy staggering for magnetic rotational bands with \begin{document}$ M1 $\end{document} and \begin{document}$ E2 $\end{document} transition properties, which are strictly consistent with the features of good candidates of magnetic rotational bands in the \begin{document}$ A\sim80 $\end{document}, 110, 130, and 190 mass regions. The regularities exhibited by these bands with respect to the staggering parameter, which increases with increasing spin, are in agreement with the semiclassical description of shears mechanism. Moreover, the abnormal behaviour in the backbend regions or close to band termination has also been discussed. Taking the magnetic dipole bands with same configuration in three \begin{document}$ N = 58 $\end{document} isotones, i.e., \begin{document}$ ^{103} {\rm Rh}$\end{document}, \begin{document}$ ^{105} {\rm Ag}$\end{document}, and \begin{document}$ ^{107} {\rm In}$\end{document}, as examples, the transition from chiral to magnetic rotation with the proton number approaching \begin{document}$ Z = 50 $\end{document} is presented. Moreover, the self-consistent tilted axis and principle axis cranking relativistic mean-field theories are applied to investigate the rotational mechanism in the dipole band of \begin{document}$ ^{105} {\rm Ag}$\end{document}.
Classical model for diffusion and thermalization of heavy quarks in a hot medium: memory and out-of-equilibrium effects
Marco Ruggieri, Marco Frasca, Santosh Kumar Das
2019, 43(9): 094105. doi: 10.1088/1674-1137/43/9/094105
We consider a simple model for the diffusion of heavy quarks in a hot bath, modeling the latter by an ensemble of oscillators distributed according to either a thermal distribution or to an out-of-equilibrium distribution with a saturation scale. In this model it is easy to introduce memory effects by changing the distribution of oscillators: we model them by introducing a Gaussian distribution, \begin{document}$ {\rm d}N/{\rm d}\omega $\end{document}, which can be deformed continuously from a \begin{document}$ \delta- $\end{document}function, giving a Markov dissipation, to a broad kernel with memory. Deriving the equation of motion of the heavy quark in the bath, we remark how dissipation comes out naturally as an effect of the back-reaction of the oscillators on the bath. Moreover, the exact solution of this equation allows to define the thermalization time as the time necessary to remove any memory of the initial conditions. We find that the broadening of the dissipative kernel, while keeping the coupling fixed, lowers the thermalization time. We also derive the fluctuation-dissipation theorem for the bath, and use it to estimate the kinematic regime in which momentum diffusion of the heavy quark dominates over drift. We find that diffusion is more important as long as \begin{document}$ K_0/{\cal E} $\end{document} is small, where \begin{document}$ K_0 $\end{document} and \begin{document}$ {\cal E} $\end{document} denote the initial energy of the heavy quark and the average energy of the bath, respectively.
Anisotropic evolution of 4-brane in a 6D generalized Randall-Sundrum model
Guang-Zhen Kang, De-Sheng Zhang, Long Du, Jun Xu, Hong-Shi Zong
2019, 43(9): 095101. doi: 10.1088/1674-1137/43/9/095101
We investigate a 6D generalized Randall-Sundrum brane world scenario with a bulk cosmological constant. Each stress-energy tensor \begin{document}$ T_{ab}^{i} $\end{document} on the brane is shown to be similar to a constant vacuum energy. This is consistent with the Randall-Sundrum model, in which each 3-brane Lagrangian yielded a constant vacuum energy. By adopting an anisotropic metric ansatz, we obtain the 5D Friedmann-Robertson-Walker field equations. In a slightly later period, the expansion of the universe is proportional to the square root of time, \begin{document}$ t^{\frac{1}{2}} $\end{document}, which is similar to the period of the radiation-dominated regime. Moreover, we investigate the case with two \begin{document}$ a(t) $\end{document} and two \begin{document}$ b(t) $\end{document}. In a large range of \begin{document}$ t $\end{document}, we obtain the 3D effective cosmological constant \begin{document}$ \Lambda_{\rm eff} = -2\Omega/3>0 $\end{document}, which is independent of the integral constant. Here, the scale factor is an exponential expansion, which is consistent with our present observation of the universe. Our results demonstrate that it is possible to construct a model that solves the dark energy problem, while guaranteeing a positive brane tension.
On the possibility to determine neutrino mass hierarchy via supernova neutrinos with short-time characteristics
Junji Jia, Yaoguang Wang, Shun Zhou
2019, 43(9): 095102. doi: 10.1088/1674-1137/43/9/095102
In this paper, we investigate whether it is possible to determine the neutrino mass hierarchy via a high-statistics and real-time observation of supernova neutrinos with short-time characteristics. The essential idea is to utilize distinct times-of-flight for different neutrino mass eigenstates from a core-collapse supernova to the Earth, which may significantly change the time distribution of neutrino events in the future huge water-Cherenkov and liquid-scintillator detectors. For illustration, we consider two different scenarios. The first case is the neutronization burst of \begin{document}$ \nu^{}_e$\end{document} emitted in the first tens of milliseconds of a core-collapse supernova, while the second case is the black hole formation during the accretion phase for which neutrino signals are expected to be abruptly terminated. In the latter scenario, it turns out only when the supernova is at a distance of a few Mpc and the fiducial mass of the detector is at the level of gigaton, might we be able to discriminate between normal and inverted neutrino mass hierarchies. In the former scenario, the probability for such a discrimination is even less due to a poor statistics.
Perturbative modes and black hole entropy in f (Ricci) gravity
Chuanyi Wang, Liu Zhao
2019, 43(9): 095103. doi: 10.1088/1674-1137/43/9/095103
f (Ricci) gravity is a special kind of higher curvature gravity whose bulk Lagrangian density is the trace of a matrix-valued function of the Ricci tensor. It is shown that under some mild constraints, f (Ricci) gravity admits Einstein manifolds as exact vacuum solutions, and can be ghost-free and tachyon-free around maximally symmetric Einstein vacua. It is also shown that the entropy for spherically symmetric black holes in f (Ricci) gravity calculated via the Wald method and the boundary Noether charge approach are in good agreement.
W-hairs of the black holes in three-dimensional spacetime
Jing-Bo Wang
2019, 43(9): 095104. doi: 10.1088/1674-1137/43/9/095104
In a previous publication, we claimed that a black hole can be considered as a topological insulator. A direct consequence of this claim is that their symmetries should be related. In this paper, we give a representation of the near-horizon symmetry algebra of the BTZ black hole using the W1+∞ symmetry algebra of the topological insulator in three-dimensional spacetime. Based on the W1+∞ algebra, we count the number of the microstates of the BTZ black holes and obtain the Bekenstein-Hawking entropy. |
d9b593f250f7bf4a | AM Vol.11 No.3 , March 2020
A Mathematical Model for Redshift
Author(s) Peter Y. P. Chen
We have used model scaling so that the propagation of light through space could be studied using the well-known nonlinear Schrödinger equation. We have developed a set of numerical procedures to obtain a stable propagating wave so that it could be used to find out how wavelength could increase with distance travelled. We have found that broadening of wavelength, expressed as redshift, is proportional to distance, a fact that is in agreement with many physical observations by astronomers. There are other reasons for redshifts that could be additional to the transmission redshift, resulting in the deviation from the linear relationship as often observed. Our model shows that redshift needs not be the result of an expanding space that is a long standing view held by many astrophysicists. Any theory about the universe, if bases on an expanding space as physical fact, is open to question.
1. Introduction
Shift of spectral lines in the light spectrum from distance stars has been extensively observed and measured by researchers over many centuries. The current most acceptable model is based on Hubble’s law which was started from observations of the linear relationship between the velocity and distance of stars. If Doppler Effect due to velocity is taken into consideration, it could be shown that redshift is directly and approximately related linearly to the distance from the stars to the observers.
One of the problems with Hubble model is the large redshifts observed in quasars. Quasar ULAS J1342+0928 is known to have a redshift of 7.54, which corresponds, according to Hubble model, to a distance of approximately 29.36 billion light-years from Earth (these distances are much larger than the distance light could travel in the universe’s 13.8 billion year history). Also, based on this theory, recessional velocity of an object is proportional to the distance from observers; it means that for a distant object, its velocity could not just greater than but also hundreds or more times the speed of light. Mathematically, there is no problem to used recessional velocity, based on the assumption of an expanding space between an object and its observers; this conception is, however, difficult to be accepted physically. Many would find it difficult to understand the difference between peculiar velocity, the velocity at which an object moves through space, and recessional velocity.
There is a different theory that is much less known and much less accepted by researchers. This is known as “Tired Light” theory [1]. The idea behind this theory is that, when light is travelling through the cosmic space, it must lose energy through interaction with particles, mostly hydrogen atoms, or other minute particles. Although the space is very thinly populated by those particles, the cumulating effect through an exceedingly long cosmic distance must result in a detectable loss of energy that is manifested itself as redshifts. The proposed theory is that the loss varies exponentially from distance travelled. Although there are qualitative arguments presented on how the loss could have taken place, there is no concrete evidence from laws of physics that such an exponential relationship should exist. Therefore, this theory is closer to just being an empirical correlation between the observed redshifts and distance.
In this paper, we accept the fact that the space is not a complete vacuum. Light as a form of electromagnetic waves in their propagation through space must interact with whatsoever material present, no matter how thinly it is distributed. The transmission is therefore governed by the well-known nonlinear Schrödinger equation (NLSE). We accept the fact that we are dealing with distances not just measured in light year but it could be in billions of light years. If SI units are used, we would be solving NLSE with system parameters as small as 10−10 or less. However, these should not be any problems as we could use well-established modelling technique of scaling so that we are solving the scaled-down NSLE with much more convenient numbers. Numerical experiments could then be carried to determine the intrinsic physical properties of the system, in this case the cosmic space. In this paper, we are interested to study the fact that electromagnetic waves are known to increase in their pulse widths when propagating through a medium with anomalous dispersion (that is with a positive dispersion coefficient) [2]. We believe that this is the physical explanation of the universal observation of redshifts (or blue-shift if the coefficient is negative).
Since 1931, the linear relationship between redshift with velocity had received popular support due to Hubble’s astronomical observations on nearby stars. However, the present-day theory under the same name is involving so-called recessional velocity that is associated with a yet unproven expanding space. Although there are extensive references, discussions and reviews on this theory, we do not include any reference about them in this paper because we are presenting a completely new transmission model that is based on established physical laws and has not been studied by any researcher on this topic.
There is no dispute that NLSE is the field equation that governs how light waves are propagating through space. In recent years, NLSE is widely used in the development of optical fiber technology. But NLSE is a robust equation that allows the transmission of all sorts of waves under many different conditions. Under this scenario we consider that research works done on the solutions of NLSE are not appropriate for our purpose. We need to know precisely how wavelength changes due to the distance travelled. We do not refer to other researchers’ work because our method is unique and aimed at the needs of our model.
2. The Nonlinear Schrödinger Equation (NLSE)
The propagation of light in a medium is governed by the NLSE:
u x i 2 D ( x ) u t t i γ | u | 2 u = 0 (1)
where u is the slow varying envelope of the axial electric field, D ( x ) and γ represents the dispersion coefficient and self-phase modulation parameters, respectively, x and t is the propagation distance and time, respectively.
Introducing scaling factors, xo and to, so that
x * = x x o , t * = t t o (2)
Equation (1) becomes
u x i 2 D ( x ) u t t i | u | 2 u = 0 (3)
where the superscript * has been omitted for simplicity and
D * = D x o t 0 2 , u * = ( γ x o ) 0.5 u (4)
3. The Numerical Solution Method
We have used the Lanczos-Chebyshev pseudospectral reduction method [3] [4] to convert Equation (3) into a set of ordinary differential equations (ODE). Because the emission is a soliton pulse, we need to subdivide the computational t-domain into N divisions. Additionally, a high (M – 1)th-order power series for each sub-domain must be used in order to be able to capture the characteristics of the pulse. The resultant ODE is in the form,
A U x ( x ) i L U ( x ) = i Q ( x , U ) (5)
where U is a (M x N) vector consisting of the coefficients of the power series used. For numerical integration in the x-direction, we have used the unconditionally stable and implicit Crank-Nicholson step-wise formulation. For Equation (5) with step size ∆x,
A ( U m + 1 + U m ) i Δ x 2 [ L ( U m + 1 + U m ) ] = i Δ x 2 [ Q ( x , U m + 1 ) + Q ( x , U m ) ] (6)
Because the term Q ( x , U m + 1 ) in the LHS, Equation (6) is nonlinear, it has to be solved by an iterative procedure as described in Reference [4].
4. Stable Propagating Wave (SPW)
When applying to a given system, Equation (4) could support the stable wave propagation. Such a wave must have definite pulse energy and the correct pulse shape. The characteristics of SPW are an unchanging pulse shape and slow varying maximum amplitude when travelling along the propagation distance. Any noncompliant components present in the input wave would be dissipated and disappear progressively. However, if the input wave is too far different from a SPW, instability may occur.
Since light emitted from a cosmic object must have travelled through such a long distance, we belive that all the spectral components we received on earth are SPWs. This is feasible as we could find in multi-mode optical fibre technology that the same fiber can support multiple numbers of modes. From multiplex technology, we know that a multiple number of signals can be launched into the same fiber. Therefore, it is logical that we could choose one of the SPWs, for example, that of the hydrogen spectral line Hα, and its associated changes in wave length to find out the extent of redshift.
For the generation of SPWs, we have used numerical procedures that we had developed and used previously. We could find stationary solutions of NLSE for dispersion management in optical fibers [2]. The idea is that we could use a fiber consisting of numbers of segments, each of them has the same dispersion map, for example, half has a positive dispersion coefficient and the other half a negative one. If we take the average of the input and output waves in a segment, after adjustment for any phase change, and use it as the input to the next segment, after a small number of segments, we could obtain a stationary solution quite quickly, providing the initial input pulse is well chosen.
For our numerical model, we have chosen a dispersion map with the first half a dispersion coefficient D and the other half – D, giving 0 as the average coefficient. The reason for this choice is that the physical dispersion coefficient for the cosmic space is very close to zero. For such a dispersion map, there is a zero nett dispersion effect on the travelling wave. However, because of the presence of the nonlinear term in the NLSE, we would not get a stationary solution but a SPW.
It should be noted that in this arrangement the input to each segment is not a solution of NLSE because the input is the phase-adjusted average of the input and output of the previous segment. Therefore, it will take a short distance before the pulse evolves into a SPW.
An example of how a SPW is propagating through a segment is shown in Figure 1.
5. Numerical Investigation
For numerical solutions of Equation (3), we consider a pulse at the centre of a
Figure 1. An example of a SPW travelling through a test segment.
local time, t, window between –L to L and travelling in the x direction. We divide this time space into N subdivisions. In each subdivision, u is represented by a (M – 1)th order power series that has M coefficients. The numerical simulation of the propagation of u along x is carried out using step size, ∆x. For every 2x distance travel, the dispersion coefficient D is positive for the first x distance and –D for the next x distance. We adjust u at the end of each 2x, according to the procedures described previously. A Gaussian pulse is used as initial input with total pulse energy
E = L L | u ( t ) | 2 d t (7)
As an example, we use L = 30, N = 4, M = 20, ∆x = 0.001, x = 1, D = 1 and E =0.25. Changes to the wavelength are found from W, the pulse width at half of the maximum intensity (FWHM). Results for W found numerically are plotted in Figure 2.
Our numerical investigations reveal that the pulse requires a high order polynomial representation. There could be a loss of accuracy if the order is too high. Dividing a given t-domain into numbers of subdivisions could be a workable approach. The choice of the size of a numerical window is also important as the pulse is a narrow spike with long tails. But the pulse width is expanding along the propagation distance. There is a numerical limit on the distance as the window used could become too small for the broadened pulse involved.
For the numerical example described above, a distance of x = 40 has been found a limit. For any longer distance, the pulse needs to be re-launched into a larger window in order to cater for the larger pulse width. However, as to be described in the next section, this measure may not be required because re-scaling and calibration could be used to change x to represent any larger distance.
Figure 2. FWHM histories for 20 test segments at x = 2 each (Other parameters given in the text).
6. Calibration
As commonly used in model studies, parameters involved could be determined by calibration. We have chosen one of the transmission cycles in Figure 3 to show how we could use our results to measure redshift in star light as observed on earth. Figure 3 shows the particular SPW cycle started at x = 30. If the initial few steps are ignored we can see that W has an almost exactly linear relationship with x. As Hubble constant, Ho, is determined from physically observed data to represent the linear relationship between redshift and distance, we could calibrate our results based on Hubble’s theory:
d c = z H o (8)
where d is the distance in Mpc, c the speed of light in km·s−1, z the redshift (dimensionless) and Ho in km·s−1·Mpc−1. The usual definition for z is that
z = λ o b s λ s t λ s t (9)
where λ is the wavelength and the subscripts refer to “observed” and “starting” respectively. Since z is a dimensionless ratio, we could define it, using the assumption that λ is proportional to W:
z = W 2 W 1 W 1 (10)
where the subscript 1 and 2 refer to W measured at x1 and x2 respectively. Knowing z, Equation (8) could be used to find the distanced in unit based on what units are used in Ho. In our example as shown in Figure 3, W2 and W1 are taken at x2 = 31 and x1 = 30.5. Using Equation (10), z = 0.86. Using a value of Ho
Figure 3. A single test segment is used for calibration.
= [Ho]WMAP = 70.3 in Equation (8) d is found to be 0.01223c Mpc. Since x2 – x1 = 0.5, we can find the dimensional conversion factor fd that can be used to convert x to the unit of Mpc (assuming x is dimensionless),
( x 2 x 1 ) f d = d c (11)
It should be noted that the dimension in fd is dependent on the dimension of x. Then, assuming that in this case x is dimensionless,
f d = d c 1 x 2 x 1 = 0.01223 0.5 = 0.02446 Mpc (12)
For the local time variable, t, we can find a dimensional conversion factor ft to convert t into W, the wavelength. If we use Hα spectral line for calibration, the wavelength is 656.281 nm. From Figure 3, W1 = 2.8. Therefore, ft = 656.281 ÷ 2.8 = 234.4 nm (assuming t is dimensionless).
From the calibrations just described, it could be confirmed that our numerical results could be applied for a spectral line of any wavelength.
We could also scale up z so that the results are applicable to a larger distance. Let
z * = f z z (13)
Then, from Equation (8), it could be shown that
f z d c = z * H o (14)
and, from a scale up z * ,a scaled up distance f z d c could be found.
7. Further Applications of SPW and Discussions
Readings from Figure 2, we can see how W1 and W2 change through each computational segment as shown in Figure 4. It could be seen that both W1 and
Figure 4. The dependency of output pulse width W2 with input pulse width W1.
W2 are increasing from one segment to the next. Plotting out the corresponding redshift, z, at three selected points is shown in Figure 5 in that it could be seen that z is slightly increasing with W1. The implication is for the same system a broader the input wave will lead to a slight increase in z. We have also shown in Figure 5 that, from the same observed spectrum of a galaxy in the Hubble Deep Field, three observed redshifts for three different wavelengths, Hα, OIII and OII. There is a remarkable agreement between these two sets of data, although they are based on different units. We could see from the previous explanation that scaling in this way is quite acceptable, providing that all the data in a set come from the same system.
So far we have assumed that light waves have come from stationary sources. If they have peculiar velocities, shifts of the spectral lines due to Doppler Effect could be considered as an extra contribution to redshift. Assume all stars at a given distance from earth have randomly distributed radial velocities within a certain range. We have produced a simulated sky map, shown as Figure 6, according to the data given in Table 1.
This map could be used to explain partly why the Hubble parameter, h, is often given with a specified range.
There are many other events [5],for example, gravitation and an exploding nova that could produce spectral shifts. All could be considered as additional to what we have found.
Although we have found blue-shift in the negative diffusion coefficient segment when we generate our SPW, we do not consider this is the explanation that stars with blue-shift have been observed.
There is no reason to consider this as an exception to the astronomical principle that the universe is uniform and isotropic in every direction. In optical fibers, propagation of light-waves is affected by defects and gaps; light transmission could be abruptly disrupted by environmental conditions. Further research is needed to identify the cause of blue-shift in stars.
Table 1. Data for Sky map simulation.
Figure 5. The confirmation of redshift changes with initial pulse width.
Figure 6. A computer generated sky map.
An important area not covered by this paper is transmission loss. A loss term could be added to NLSE without introducing extra complication to the numerical procedures. On the other hand, our model could be considered to have covered small losses because our system parameters are determined by calibration with observations. For very long distance in many Giga pc, it is worthwhile to consider whether losses should be included as part of the model. The effect of pulse energy is also an area that needs further investigation.
The finding of our numerical investigations based on NLSE is that redshift is linearly proportional to distance measured from the source to earth. There is no limit to this distance. This relationship is completely independent of recessional velocity, if any exists. While our model is not limited by z, the Hubble z – d relationship, as derived originally from Hubble law is applicable only for z 1 . The fact that this relationship is used for large z is due to its empirical nature. That is Ho is determined from observed data for far distant stars and galaxies.
We acknowledge that as a mathematical tool, in order to account for redshift, it is convenient to assume that the space in which light is travelling is expanding. From observed data, it is possible to work out the relation between the recessional velocities with distance. But we are constantly been reminded that the actual physical distance does not chang. However, there are important cases that have taken the expansion to be real and physical. For example, the Big Bang theory has taken the expansion to be real so that the universe must start from a singular point. Many discussions about the size of our universe have also considered this expansion to be real.
8. Conclusions
We have shown that by using model scaling light propagating through space could be studied by using the well-known NLSE. We have devised numerical procedure to generate SPWs which could be used to show that redshifts commonly observed in light from distant objects could come from the intrinsic physical properties of space, namely dispersion coefficient and self-phase modulation parameter.
Our numerical results confirm that redshift has a linear relationship with distance between a source and our earth. This relationship is not limited by the distance. Our system once calibrated could be applicable to the real physical universe.
Our present model only considers redshift due to light travelling through space. This is known to be the major contribution to redshifts so extensively observed by astronomers. There are other less important causes that could contribute additionally to redshift.
The most important finding in our studies is that redshift needs not come from the recessional velocity of an expanding space. The implication is that any theory about the universe using an expanding space as fact is open to question.
Cite this paper
Chen, P. (2020) A Mathematical Model for Redshift. Applied Mathematics, 11, 146-156. doi: 10.4236/am.2020.113013.
[1] Sho, M.H., Wang, N. and Gao, Z.F. (2018) Tired Light Denies the Big Bang. Redefing Standard Model Cosmology-Intechopen.
[2] Chen, P.Y.P., Malomed, B.A. and Chu, P.L. (2006) Optimal Preprocessing of Pulses for Dispersion Management. Journal of the Optical Society of America B, 23, 1257-1261.
[3] Chen, P.Y.P. (2016) The Lanczos-Chebyshev Pseudospectral Method for Solution of Differential Equations. Applied Mathematics, 7, 927-938.
[4] Chen, P.Y.P. and Malomed, B.A. (2012) Lanczos-Chebyshev Pseudospectral Methods for Wave-Propagation Problems. Mathematics and Computers in Simulation, 82, 1056-1068.
[5] Carroll, A.O. and Ostlie, D.A. (2017) An Introduction to Modern Astrophysics. Cambridge University Press, United Kingdom. |
802fab4e89118538 | Biofield Science: Current Physics Perspectives
This article briefly reviews the biofield hypothesis and its scientific literature. Evidence for the existence of the biofield now exists, and current theoretical foundations are now being developed. A review of the biofield and related topics from the perspective of physical science is needed to identify a common body of knowledge and evaluate possible underlying principles of origin of the biofield. The properties of such a field could be based on electromagnetic fields, coherent states, biophotons, quantum and quantum-like processes, and ultimately the quantum vacuum. Given this evidence, we intend to inquire and discuss how the existence of the biofield challenges reductionist approaches and presents its own challenges regarding the origin and source of the biofield, the specific evidence for its existence, its relation to biology, and last but not least, how it may inform an integrated understanding of consciousness and the living universe.
Key Words: Biofield, quantum mechanics, physics
Conventional biology is based on molecular processes—ie, biochemical interactions that ultimately reduce to macromolecules such as DNA and RNA. Even organismal biology, which concerns itself with addressing organisms as wholes, still relies on the reductionist approach of understanding the whole by analyzing how the parts fit together. These approaches, although very successful in specific scientific and medical applications, fail to address phenomena that by their nature are holistic—ie, they may need to be explained from a whole organism context, crossing boundaries of scale, and thereby including quantum and conventional fields, mind, and relationship to environment. It seems that biology, despite the great successes it has achieved and the multitude of applications in theory as well as in practice, has still not undergone the types of revolutions that shook physics over the last 100 years.
Evidence for the existence of the biofield now exists, and current theoretical foundations are now being developed., The term biofield describes “a field of energy and information, both putative and subtle, that regulates the homeodynamic function of living organisms and may play a substantial role in understanding and guiding health processes.” Another definition describes it as
an organizing principle for the dynamic information flow that regulates biological function and homeostasis. Biofield interactions can organize spatiotemporal biological processes across hierarchical levels: from the subatomic, atomic, molecular, cellular, organismic, to the interpersonal and cosmic levels. As such, biofield interactions can influence a variety of biological pathways, including biochemical, neurological and cellular processes related to electromagnetism, correlated quantum information flow, and perhaps other means for modulating activity and information flow across hierarchical levels of biology.
Unified and coherent characteristics of the biofield imply a strong and perhaps unique role for quantum models. A review from the viewpoint of physical science is needed in order to identify a common body of knowledge and evaluate possible underlying principles of origin of the biofield. To that end, the review presented here surveys current models including electromagnetic processes and quantum models. We go on to speculate on processes that are not currently well understood. Central to the possible role of quantum theory, for example, we discuss quantum biology and its manifestations in such processes such as photosynthesis, avian navigation, olfactory reception, regeneration, microtubule interactions, brain dynamics, and cognition.
It has been hypothesized that biology could ultimately be built from more fundamental underlying quantum physics. This assumption is implicit in many approaches to molecular biology, genetics, and various applications in medicine and health but is often more honored in the breach. If biology truly derives from physics, then biology should be an extension of quantum physics, the most accurate and fundamental physical theory at our disposal. While quantum biology is an emerging branch of science, most practicing biologists don’t take it into account. Conventional biology and biophysics derive predominately from a biochemical and Newtonian physics standard, but biological effects that cannot be understood without reference to quantum phenomena are accumulating, as in avian magnetoreception, olfaction, and plant photosynthesis.
However, very recent work describes a theoretical foundation for biology, suggesting that biology can be put on an equal footing with physics and not simply reduced to biochemical processes. Living matter would then be seen as following basic principles and laws that are not reducible to conventional physics, though would be smoothly interwoven with quantum physical processes. In this view, we would assert that the generic science of biology is complementary to the generic science of physics (ie, the 2 are closely related but not identical). Possibly both are anchored to mutual processes through the underlying quantum vacuum.
In this regard, the evidence for the existence of the biofield holds the promise of significant growth in scientific understanding and for developing applications in medicine, health, and healing. This line of research and application of quantum physics perspective approaches living organisms through “an emergent and potentially all-encompassing biofield” that entails the existence of long-range interactions, most likely of a coherent nature. Even as experimental evidence is accumulating for the existence of precisely such a long-range, coherent biofield, theoretical understanding is still lacking. Various hurdles exist: The concept of the biofield has many aspects, the concept often means different things to different workers, and a clear language for the description of biofield interactions hasn’t been agreed upon. Further complicating the situation is that a host of relevant terms and concepts (eg, bioplasma, bioelectromagnetics, quantum vacuum) are being widely used in a variety of different contexts.
Does the theoretical understanding of biofield involve a few dominant theories? Do they depend on specific phenomena? Can such understanding be part of existing field theories (such as electromagnetism) or is new physics a necessary outcome of studies of the biofield? From the viewpoint of classical physics, another possibility that has been suggested is that the biofield consists of electromagnetic emanations from molecular transitions in living matter. This possibility is not viable due to associated short timescales. From this perspective, electromagnetic field (EMF) coherence might be an essential requirement for biofield interactions to organize biological processes. Because quantum physics underlies all electromagnetic theories and thus biochemistry and neurobiology, quantum mechanical processes, the role of the vacuum, and interpretations concerning the role of the mind itself are important aspects to consider. Also we shall discuss in greater detail below how other “quantum-like” properties of the biofield may play a key role in biofield interactions (by quantum-like, we intend macroscopic and biological correlates of quantum phenomena such as nonlocality, superposition, complementarity,, etc). If the workings of generalized, mesoscopic (molecules to mm in size) and macroscopic quantum-like processes that span both physics and biology can be demonstrated, then we will discuss in this article how the biofield itself may be an important—and perhaps to-date, crucial but ignored—missing link. In other words, if quantum-like is defined as the more general framework embracing biology and physics, then macroscopic quantum processes such as entanglement (where multiple objects exist in the same quantum state and so are linked together) and coherence (ordering of the phase angles between the components of a system in a quantum superposition) across a single organism and beyond would be crucial signposts marking what lies ahead, coherence as such being a bridge between micro- and macroscales., The recent discovery of macroscopic entanglement in 2 diamond crystals could also be pointing to the likelihood that quantum-like phenomena may, in some cases, literally be propagation of quantum level phenomena into the macroscopic scale. These recent issues will be briefly addressed in the current work.
Ultimately, for any quantum discussion, the problem of observation à la von Neumann arises. The so-called “von Neumann cut,” or the point of separation between the observer and the observed system, suggests an essential role for the observer with clear relevance to how biofield interactions may be connected to brain structure and processes. Where is the observer situated, in the brain? What is the role of mind and consciousness itself in biofield interactions? One can speculate on the many possibilities that exist with regard to the interaction of an observer with observed systems, where the cut may be (if anywhere) in biological systems, serving as a connection to the activity of the biofield. We must consider consciousness as an integral part of biofield theory and experimentation, as any discussion of quantum biology directly implicates the question of the observer and the observer requires consciousness.
The review presented here is meant as a comprehensive introduction to many aspects already known while also highlighting issues remaining and speculating upon conceptual developments that are needed to develop a theoretical framework for the copious body of data on biofield phenomena. We also refer the reader to the extensive discussion presented in the excellent compendium of relevant works in Popp and Beloussov. This book discusses in detailed chapters the idea of biophysics as being quantum biological, developmental biology and morphology and field theory, biophotonic emission studies, mitogenetic radiation as a biofield phenomena, and life and consciousness as relevant aspects to biophysics and integrative biophysics as being inclusive of this.
The concept of a biofield has been emerging steadily, with the work of several groups indicating that part of a living organism’s energy is “integrated into a sort of an all-inclusive, long range and to a certain degree coherent field. This suggests that fundamental properties like coherence, integrative function, and various long-range influences on the organism are all potentially associated with the biofield. A number of scientists have historically proposed that a biological field exists in a holistic or global organizing form. The details are different, but in general, such propositions involve coherence in electromagnetic waves, biophotons, or going beyond electromagnetism, human intention. In some suppositions, an “electromagnetic body” or “subtle body” is invoked, as related to acupuncture meridians in traditional Chinese medicine and chakras, the subtle energy centers in the Indian esoteric tradition. As Liboff notes, “Once the organism is described as an electromagnetic entity, this strongly suggests the reason for the efficacy of the various electromagnetic therapies, namely as the most direct means of restoring the body’s impacted electromagnetic field to its normal state.”
From a recent perspective, the term biofield was coined in 1994 by a panel on manual medicine modalities convened at the National Institutes of Health (NIH) to discuss complementary and alternative medicine (CAM). As result, the NIH, through the National Center for Complementary and Alternative Medicine, issued a request for applications for grant proposals to study a variety of biofield therapies, including Reiki, healing touch, qigong, and other subtle energy healing interactions. As a result of this research focus, much of the physiological evidence for the biofield has come through the application of various CAM techniques of healing.
To get at its nature in terms of fields explored in classical physics, the biofield has been defined as “the endogenous, complex dynamic EMF resulting from the superposition of component EMFs of the organism that is proposed to be involved in self-organization and bioregulation of the organisms.” A classical electromagnetic-based definition such as this one can serve as an important starting point, insofar as it involves the concept of bioinformation. However, as we will see below, any electromagnetic-based definition is limiting, since it does not encompass quantum and holistic effects. EMF theories are also themselves special cases of quantum field theories, the latter being more natural and general, and therefore able to account for the properties of coherence, nonlocality, and entanglement,, which are strikingly relevant to living organisms.
Before turning our attention to the specifics of the biofield and the underlying physics, we will examine the general role of “integrative biophysics,” a term coined by Popp and Beloussov that refers to different aspects of nonconventional biophysics and biology. Specifically, the term indicates a departure from equilibrium thermodynamics, the foundation of classical physics and chemistry on which most of biology is based. Instead, a central aspect of integrative biophysics is modeling of the organism built completely upon the field concept—this forms a common thread throughout integrative biophysics and phenomena associated with biophotons.
Quantum mechanics has established the primacy of the unseparable whole. For this reason, the basis of the new biophysics must be the insight into the fundamental interconnectedness within the organism as well as between organisms, and that of the organism with the environment. This will be an integral biophysics…. The existence of a pre-physical, unobservable domain of potentiality in quantum theory, which forms the basis of the fundamental interconnectedness and wholeness of reality and from which arise the patterns of the material world, may provide a new model for understanding the holistic features of organisms, such as morphogenesis and regeneration, and thus provide a foundation for integral biophysics.
As a starting point, evidence of bioelectromagnetic fields and the biological effects of external EMFs have historically lagged behind the successes of biochemistry, resulting in a delayed start in understanding the ubiquitous nature of biofields in living organisms. The historical emphasis on reductionist molecular biological explanations has been practical and allowed for the gains of current biomedicine. Organismal and biofield biology and their multifaceted mechanisms and forms may also offer a host of useful approaches for investigating and unlocking the mysteries of life that have been neglected.
The need for general principles in biology has been pointed out by Bizzarri, Palombo, and Cucina and by Grandpierre, Chopra, and Kafatos. Instead of looking on a more integrated approach like systems biology as merely an extension of molecular biology, these investigators strongly suggest that integrated biology and biophysics operate beyond the reductionist approach. For example, these authors are challenging genetics as being the sole discipline for explaining evolution. We hope that integrative biophysics and associated field processes, including EMFs, biophotons, and possible quantum interactions, will soon be seen as necessary, fundamental, and complementary aspects of molecular biology and biochemistry. New vistas for understanding evolution will emerge when these complementary approaches are accepted.
We now turn our attention to specific aspects of biofield, beginning with EMFs. An EMF is a physical field produced by electrically charged particles in motion. We refer to the work of Jerman, Leskovar, and Krašovec for many of the details. A widely applicable notion of the biofield is associated with endogenous EMFs of organisms., Every living cell membrane “has an electric field of very high intensity (around 10 V/m) though of a rather low voltage … one of the basic features of life.” Biomedical researchers and clinicians routinely gather meaningful data from the manifestations of endogenous EMFs through the use of skin surface measurements like electroencephalograms (EEGs) and electrocardiograms (ECGs). The human body also includes classical acoustic energy fields due, for example, to muscular contraction. Coherence is often observed in EEG, which would indicate self-organizing systems. Such coherence has been shown to increase during meditative states of settled awareness.,
Applying very–low power coherent EMFs at specific frequencies in the mm range to biological systems results in a resonance-like behavior that supports the theoretical prediction of polar coherent modes in a manner comparable to Bose condensation. Polar coherent modes are predicted to result from the high-intensity field across cell membranes, that when driven by metabolism, create coherent microwave oscillation. A Bose-Einstein condensate is a state of matter of a dilute gas of bosons cooled to temperatures very close to absolute zero. Under such conditions, macroscopic quantum phenomena become apparent. Such macroscopic quantum phenomena are hypothesized as qualities of the biofield. Moreover, according to Fröhlich, these polar coherent modes represent the basis for electromagnetic oscillations at cellular levels in the organism. The existence of endogenous EMFs at the predicted Fröhlich frequencies has not yet been proven experimentally, and their coherent nature in the body is only inferred. However, the discovery of an endogenous EMF at much lower MHz frequencies in microtubules is significant because it suggests a form of coherent electromagnetic activity that may play a role in biofield signaling, thus lending some support to the theory coherent modes of Fröhlich but at much lower frequencies than predicted theoretically.
Other indirect indications of endogenous EMFs come from biophotonics, with foundations in the pioneering work of Popp and collaborators on coherent ultraweak light emissions from cells., Bischof describes the biophoton field, summarizing 90 years of peer-reviewed published research, as follows: “All living organisms, including humans, emit a low-intensity glow that cannot be seen by the naked eye, but can be measured by photomultipliers that amplify the weak signals several million times and enable the researchers to register it in the form of a diagram. As long as they live, cells and whole organisms give off a pulsating glow with a mean intensity of several up to a few ten thousand photons per second and square centimeter,” also known as “cellular glow” or “ultraweak bioluminescence.” These biophotonic phenomena could point to long-range interactions between biological organisms. This possibility is supported by observations of intercellular signaling mediated by biophotons. via a field containing coherent states, in agreement with the pioneering conjectures of Fröhlich.
In summary, the electromagnetic basis includes the presence of at least 2 field sources: “one (static electric-transmembrane potential) that has been known for long, and the other, a high frequency oscillating and more or less coherent EMF.” The latter can be considered to have 2 further aspects manifesting in different energy or frequency ranges: (1) a microwave to MHz and lower frequency range coherence, which we can simply refer to as the Frölich field, and (2) a visible/infrared/near ultraviolet diffuse field, which we can refer to as the Popp photon field. The former has been observed but at lower frequencies than predicted; the latter is supported empirically by observations of the statistical coherence of biophotons, which produce emission spectra that are distinctly different from byproducts of biochemical reactions. This appears to be related to quantum mechanical squeezed states., Squeezed states of light belong to the class of nonclassical states of light and indicate quantum coherent states. As such, quantum mechanical effects are clearly indicated through coherence and squeezed states in both the Fröhlich and Popp fields; therefore, they constitute nonclassical fields with their own particular properties (see next section). Recently it has been suggested that the Fröhlich field and the Popp field are interconnected through strong mode coupling in living systems. An experimental and theoretical basis for defining the existence of a macroscopic coherent quantum system in living things is being developed here and extended subsequently. This has profound implications for biology and medicine.
Coherent EMFs may indeed be the organizing agent of cellular processes, which would indicate that the biophoton source is nonbiochemical. It is of course possible that these ultraweak photon fields are somehow related to biochemical processes, although concensus is that they may be guiding the entire cellular physiology. Biofield interactions could also be responsible for the organization of cellular microtubular networks and biological regulation processes that have been shown to occur via endogenous EMFs within microtubular cytoskeleton such as the following: the regulation of the dynamics of mitosis and meiosis,; chromosome packing during the mitotic phase of the cell-cycle; and interactions between ion channel activity and the phosphorylation status of binding molecules such as MAP2 and CaMKII, which act modulate cytoskeletal structure and connectivity. These experimental data are supported by theoretical prediction of classical and quantum information processing in microtubules., The coherent photon field, on the other hand, could be the dominant factor in cellular physiology, a conclusion supported by experimental observations of cell-to-cell signaling via coherent bio-photon activity.
It is of course important to also consider that neither biophotons nor biomolecular physiology are primarily causative but are instead tightly coupled processes arising codependently within biological systems. In this vein, it should be recognized that individual cellular or multicellular organisms, while temporally and spatially separate from each other when regarded from customary investigative points of view, actually have no strict and definable boundaries between themselves. In complex ways, living organisms form colonies and populations, merge with influences from the environment as they eat and breathe, behave according to shared genetic inheritance, and are inhabited by innumerable microorganisms known collectively as the microbiome, which makes even a marked visual boundary like the skin quite tenuous. It is just as important to consider the entire biosphere as a single evolving living structure comprising all seemingly separate “beings.”
Moving beyond classical EMF descriptions, the general CAM approach aims to modulate the endogenous fields. It has been suggested that this aim must include modulation of nonclassical and quantum forms of energy. Indeed, it is a logical necessity to consider that the collective biofield consists of (at least) electromagnetic, optical, acoustic, and nonclassical energy fields associated with biological entities: cells, bodies, perhaps ecosystems, and even Gaia as a whole. As stated above, the coherence of endogenous EMFs suggest, specifically that nonclassical fields are existing in biological entities., It has been proposed that the biofield may be applicable in complementary medical therapies and healing.
Potentially such therapies could be directed noninvasively at enhancing or stimulating the body’s healing process, reducing pain and anxiety, and a variety of other conditions. Many of these applications reflect the influence of mind/body interactions, suggesting that the role of the observer in quantum mechanics (QM) may be of central importance to understanding mind/body therapies and the role of mind and emotions in health and wellbeing. To what extent “mind” may also be related to the biofield lies outside the scope of this review, but we have been describing some of the basic physical biofield processes that could explain the efficacy of complementary medical therapies.
All physics, including electromagnetic theory, rests upon a nonclassical foundation. For example, the electromagnetic potential field (comprising the vector potential, A, and scalar potential, Ф, which are the sources of EMFs) mediates the classical EMFs described by Maxwell’s equations and quantum levels described by the Schrödinger equation. The electromagnetipotential acts by modulating the phase of charged particle wave functions; field interactions can occur in regions of zero electric and magnetic fields, yet nonzero A and Ф. Thus the electromagnetic potential is itself a nonclassical field functioning through a modulation of quantum phase rather than via a classical field of force. The case for other nonclassical fields has been summarized by Rein, and such fields, while not yet directly observed, are a direct consequence of both classical, relativistic, and quantum theories.
For example, because the wave equations derived from Maxwell’s equations (ie, classical electromagnetic theory) are symmetric in time, solutions exist for both the “advanced” and “retarded” electromagnetic potentials, propagating backwards and forwards in time, respectively. Other field quantities that propagate at faster-than-light speeds, such as pilot waves, follow directly from calculations in both classical and relativistic electrodynamics. In relativistic quantum theory, solutions to the Dirac equation successfully predicted the (now experimentally confirmed) existence of the positron, requiring a formulation in which the arrow of time is reversed. “Longitudinal” or “scalar” waves have also been suggested to be primary aspects of the biofield. In contrast to the transverse vector waves of classical EMF theory, such scalar waves are hypothesized to result from superposition of electromagnetic waves—eg, when 2 waves cancel each other, a transformation of energy into vacuum potentiality is thought to occur. Such scalar fields, which are not mediated by electric dipoles or electron transitions, propagate far from equilibrium and clearly don’t constitute known electromagnetic-based structures.
These connections with nonclassical fields have led several scientists to consider the body as functioning as a macroscopic quantum system.,, The existence of macroscopic biological processes linked to QM leads to quantum biology and as we will see below, to a biofield conception beyond both quanta and biological entities to the underlying vacuum and even further. In an integrated quantum description of the body, bioinformation must play a fundamental role. The implications for biomedicine are profound. Such a system would create a model for the origin and cause of broad physiological regulatory behavior that we currently lack, primary to molecular biology. Practical control of this system would lead to deep insights for healing, regeneration, morphology, disease elimination, growth, and mind/body interaction, as well as insights into the fundamental questions of what is life, what is consciousness, and what the full mechanisms underlying evolution are. It may describe a new, unique, quantum mechanical and electrically based physiological system that interfaces with both the quantum world, quantum vacuum, and biochemical world. It may be the key to integrating the science of consciousness and biology. It would certainly be an epochal paradigm shift for science.
Quantum physics provides a theoretical entry to attempt to explain the existence of the biofield and how it interacts with the body. There are qualifications to this assumption, however. Bischof indicates the fundamental sense that quantum physics has implicitly replaced the old reductionist and molecular view of science with a holistic one in which materiality forms an unbroken whole. Likewise, the most persistent paradigm in neuroscience considers the mind as an emergent property of a large and complex physical brain that mediates awareness and remembrance., In this orthodox view, “mind” appeared in the evolutionary chain because of the development of nervous systems in general, central nervous systems in particular, or only in primates and perhaps just homo sapiens.
In contrast, a view closely linked to the role of observation in quantum measurements assigns a role to subjectivity in keeping with the Copenhagen Interpretation (CI) and particularly its revision by John von Neumann, known as the orthodox quantum view. It holds that consciousness provides the individual observer with agency and freedom. As such, quantum measurement theory has yielded to what Wheeler refers to as the “participatory universe.” The conundrum of whether or not the falling tree would make a noise in the forest is irrelevant if no conscious observers were around to hear it. From this participatory viewpoint, properties of quanta and quantum systems in general are “contextual”: They don’t exist by themselves but are intrinsically tied to acts of observation.
In von Neumann’s view, nature exhibits free choice of response to an act of observation by an observer. The time evolution of a quantum system is described by the wave function, which fully characterizes such systems through the deterministically evolving Schrödinger equation. However, what value will result following an actual experimental choice is not known. Once an experiment is conducted, a single value in the probability space described by the wave function results, and this is the famous “collapse of the wave function.” Quantum theory presents us with a world following a completely different order from the world of everyday experience. In what constitutes the underlying reality, quanta are entangled in both space and time, and nonlocality is implied in quantum measurements.
By extension, a number of quantum physicists take participation to be an absolute requirement, holding that the world is primarily mental, since mental decisions implicitly play the primary role in the collapse of the wave function.,, In the CI of quantum theory, the wave function is not considered to be real. Rather, it is only a prescription of determining probabilistic potential outcomes, which are described by the square of the absolute value of the wave function, as proposed by Born., However, the variables measured must conform to macroscale classical analogues, since any apparatus in the lab would be a classical system. Thus the CI has a duality built into it. Not all physical variables of a quantum system can be simultaneously known (according to the Heisenberg Uncertainty Principle). In the CI, quantum systems behave in a complementary manner, either as particles or waves (Bohr’s Principle of Complementarity). This complementary relationship manifests in the act of observation itself. For example, the more precisely a particle’s position (particle-like aspect) is measured, the less precisely can its momentum be known (ie, wavelength or wave-like aspect). Thus the type of measurement chosen by the observer determines the outcome of experiments, suggesting a participatory role for the observer.
In von Neumann’s view, there is a universal wave function. However, as in the CI, there is also collapse through conscious observation. For von Neumann, the state transformation due to measurement (process 1) is distinct from that due to time evolution (process 2) as described by the Schrödinger time-dependent equation: Time evolution is deterministic and unitary whereas measurement is nondeterministic and nonunitary., Von Neumann’s interpretation is the gold standard against which all other interpretations must be compared. Von Neumann’s nondeterministic interpretation of measurement gives a psychological component to reality itself, casting the observer in the role of an active participant in the creation of events.
This viewpoint, that the observer’s participation plays an essential role in the outcome of events, has fundamental implications for biofield science and mind/body therapies. It has the potential for understanding how many such therapies operate. In the same breath, the issue of efficacy arises. There is a wide range of response to all medical interventions, whether in complementary or conventional scientific medicine. No 2 patients respond alike, and uncertainty is always present. Mind and body are fundamentally connected. Thus, the primary connection of the observer and the observed system, as understood in QM, has profound implications for the nature of the biofield: We cannot take the living body as an entity existing independent of the biofield to which it belongs and independent of the practitioner and the receiving subject in CAM treatments.
The primary shortcoming of molecular biology is that the “holistic” character of the physical world now recognized in quantum theory is either not acknowledged by the bioengineers or rejected as irrelevant., The world view of QM is much richer and more holistic than molecular biology would have. It is no surprise that many of the founders of QM understood the implications of wholeness in both physics and biology. For example, Planck held that wholeness must be introduced into physics as in biology. Bohr understood the significance of complementarity beyond QM and how it was paramount to biology., Schrödinger wrote an important work with the title “What is Life?” in which he approached the holistic view for both QM and life as similar. For example, primary colors are not a fundamental property of light but are related to the physiological response of the eye to light. Moreover, Heisenberg also held that mind plays a fundamental role in the universe.
Today, the evidence of macroscopic quantum effects in biology has yielded a plethora of phenomena that can be understood through the application of quantum physics. They include understandings of the role of coherence in photosynthesis,, the avian compass through which birds navigate, the sense of smell, quantum coherence in microtubules,, regeneration, and quantum processes in brain dynamics.
The application of quantum microphysics to macroscopic scales is natural and yet at the same time surprising. The naturalness is because QM is the most complete theory of physical reality that we have where classical physics is incomplete. The surprise is because most QM effects occurring in the microcosm, such as entanglement and nonlocality, don’t readily apply to everyday experience. In what follows, we refer to Kafatos as it applies to bridging the microscopic and macroscopic domains.
By quantum-like effects are meant (1) phenomena that are clearly related to QM but apply at macroscopic scales where normally they would not be expected and (2) phenomena that should be seen as extensions beyond current orthodox QM, in particular those involving life processes that cannot be accounted for by standard biochemistry, biology, or quantum theory. The Hilbert space formalism of QM, Schrödinger’s wave mechanics, and Heisenberg’s matrix mechanics don’t directly address life processes. Quantum-like processes have been theoretically invoked in a host of life processes and macroscopic physics (such as brain dynamics)., “Quantum-like” indicates that the principles of QM apply at all scales, not just the microscopic, and as such, they provide fundamental insights to phenomena in fields outside physics, such as those already touched upon—biology, neuroscience, and medicine—and potentially extending to other areas like psychology and even anomalous psi phenomena, where one might apply QM phenomena such as entanglement and nonlocality.
Reflecting on these concepts from the perspective of complexity theory, it becomes clear that many of the “peculiar” effects observed at the quantum level have biological forms: for example, biological complementarity and uncertainty., Extending QM concepts in this way leads to biological scale, quantum-like nonlocality, recursion, and entanglement. These extensions are more than analogies or metaphors. Beyond a scope usually considered as peculiar to the quantum world and not occurring in the “real world” of classical physics, we suggest that if the observable universe at its foundation is quantum mechanical, as held in standard orthodox QM,,,, then nonlocality could indeed be one of the signature aspects of an underlying mental world. This has been referred to as the “conscious universe.”, Such a universe, where consciousness is primary, would entail qualia of experience, where the qualities of the experienced world describe reality with the validity of conventional science and yet go much further by including every aspect of mind., Quantum-like can thus be understood as the (future) extension of both QM and quantum biology to account for the physical, mental, and biological realms, with the biological domain characterized by huge complexity and different levels of information rates.
In interpersonal field phenomena, the presence of nonelectromagnetic fields is indicated. These may be electromagnetic potential fields, which Aharonov and Bohm showed are very real. Tiller has suggested that these potential fields mediate between EMFs, the macroscopic quantum states of matter, and the physical vacuum. We agree with Bischof that “all the features of unbroken wholeness of reality implicit in quantum theory—non-separability, non-locality, fundamental connectedness—which are so fundamental for biological understanding, are an expression of the properties of the vacuum.” According to this view, the vacuum organizes the structure of space-time through macroscopic EMFs, and the phase-controlling property of the electromagnetic potentials plays a central role. The importance of phase-relations for complex biosystems, consisting of many oscillating fields coupled nonlinearly by their phase-relations, points to the importance of the vacuum for the biofield itself.
Relatedly, the coherence of biophoton emission has been suggested to arise from “potential information” in the organism that is virtual and nonmeasurable and a “superfluid vacuum model” has been proposed for biophoton emission of seeds and its connection to their vitality. This model characterizes the vacuum as a superfluid Bose-condensate of photons in which virtual fields in the vacuum state are involved in the manner posited by Grandpierre and Kafatos. Zeiger and Bischof make clear “that there is significantly more to the quantum vacuum than just the electromagnetic vacuum (the zero-point fluctuations),” and
the need for assuming a pre-physical dimension of potentiality for the understanding of organisms, and for the creation of the new discipline of vacuum biophysics as a basis of biophysical understanding, is postulated … The fundamental quantum mechanical nature of biological phenomena will only be fully understood if the vacuum is taken into full and explicit consideration as the essence and ground of these phenomena. The quantum vacuum may serve as a framework for a unification program in biology aimed at incorporating all relevant aspects of life into a physical picture of the organism.
In agreement with views presented above, Zeiger and Bischof also recognize the role of the observer and of consciousness itself in QM. In addition, Grandpierre and Kafatos and Grandpierre, Chopra, and Kafatos have provided arguments for the fundamental role of the quantum vacuum in biology, in the autonomy or free choice of organisms and as the driver of biological evolution.
An intriguing experimental result, known as “the phantom leaf effect,” if fully verified, may be an example of some or even all of these biofield processes. In these experiments, coronal discharge or the Kirlian photographic effect reveals a field effect in the morphological form of an intact living leaf even after part of the leaf is severed. This suggests an analogy to the subjective experience of a phantom limb reported by patients after the limb has been amputated. There might be a persisting biofield that represents the amputated limb. First described by Adamenko and reported by Tiller and by Ostrander and Schroeder, more recent validating experiments have been performed with detection methods of greater precision; these are summarized in Hubacher. In his most recent publication, Hubacher performed the experiment with highest definition photographic samples using the largest number of samples to date. Of 137 leaves severed and imaged, 96 (70%) demonstrated clear phantoms (example in the Figure).
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Example of the phantom leaf effect from Hubacher (2015).
In these experiments the phantom structure (1) appears as an integral and coherent whole, (2) is independent spatially of the organism, (3) interacts with both magnetic and electric fields and conducts current, and (4) represents the precise anatomy of the original physical leaf. Hubacher concludes that the phantom leaf, being electroconductive, may carry both information and energy and therefore possibly represents a true biofield manifestation that regulates physiological processes.
An early explanation of this effect questioned whether the phantom leaf effect might result from moisture emission from the cut portion driven into the space from which the cut section had been removed by the power of the field emission process. However, the most recent data do not support this explanation, as the precise and complex anatomical replication of the original leaf is present in minute detail.
On the other hand, it is also unclear why the effect is not seen 100% of the time (though it is more reproducible in this current cohort than it has been before). Hubacher suggests that
some parameter or group of parameters is probably needed beyond what is understood, to reliably reproduce these results. These include such things as frequency, waveform, dielectric spacing, pulse widths, and types of grounding. Other variables can include film types, gases in the electrode mechanism, humidity, power sources, times of year, plant species, [and] chemically influenced specimens, eg, perfusion with chloroform prior to photography.
Further work is clearly needed to determine the impact of these variables, but the fact remains that phantom leaves have been demonstrated using a variety of techniques. The remarkable results strongly suggest a robust effect that can arise from a very broad array of interwoven field phenomena.
In the images obtained, it is electron flux that creates the image. These data point to the existence of an intact, integral, and conductive system permeating the original leaf. Given the absence of any conductive physical structures in the severed area, the coronal discharge appears to be under the influence of a quantum-level, nonphysical field functioning below the level of EMFs, in order to support and structure those EMFs. Vacuum phantom effects have also been proposed at the molecular level for DNA., We note also that the quantum vacuum produces real measurable effects such as the Lamb shift, the Casimir effect (which occurs when charged parallel electrodes are closely adjacent), and the Bose condensation mentioned above.
The mechanisms are as yet unknown, but the various findings point to aspects that would be expected from the postulated biofield. It can be asked, then, whether a phantom structure functions like a true physiological system, as has been suggested for the biofield. A functioning system of this nature has been postulated to deliver energy and/or information systemically throughout an organism using electromagnetic signals and forces.
In this regard, it appears that the phantom leaf effect may provide an excellent model through which to explore the manifestations of a truly observable biofield (or of overlapping, interactive biofields). At the very least, the opportunity to explore biofield mechanisms at the level of EMF or below, into subtler quantum realms, is intriguing. The fact that the phantom leaf effect is highly robust in recent trials suggests that further work will identify confounding variables, which will likely uncover some of the underlying principles.
Our examination of the evidence for the biofield indicates the need for explanations to go beyond conventional classical physics and biology. In particular, one needs the consideration of holistic approaches and coherent processes. Biofields may be carried by EMFs, quantum and quantum-like processes, and other fundamental coherent states. Further research must be done on the physical origins of the biofield and how it relates to an integrated understanding of consciousness and the “living universe.” Our recommendations include new investigations that address the comprehensive issues listed below, some of which are currently speculative.
• What is the role of observation in the structure of the biofield? Does the state of the practitioner affect the structure of the biofield in medical applications, for example? Even for the same subject receiving different CAMs at different times, would the biofield depend on the person administering the treatment?
• Is the coherence seen in biofield, and particularly in biophoton emissions, indicative of the basic quantum(like) nature of life? Similarly, do nonlocality and entanglement and other quantum properties apply among different interacting organisms?
• In CAM, how is the endogenous and all-encompassing nature of the biofield in an individual tied to the biofield of the practitioner and to all biofields of living entities? For example, do biofields linking every living entity exist at all scales? How would we show this experimentally and what would the consequences be?
• If entanglements across “different” biofields are real, how might CAM modalities be developed to deliver the maximum beneficial effects to the patient?
• Can the use of CAM take advantage of the nonlocal nature of the biofield (eg, along with hands-on healing, distant healing, as in Reiki, could be equally effective)?
• Can the biofield be understood as ultimately emanating from the quantum vacuum? Would this open up new vistas for energetic healing transmission? For example, would the persistence of biofield be utilized for health benefits across space-time?
• Can we devise scientific experiments to study specific quantum-like properties of the biofield that would be useful in CAM?
• The phantom leaf effect may represent an easily performed and reproducible model system for exploring not only the primary nature of the biofield but also how CAM interventions might interact with it or even change it.
• Finally, what makes biofield research so fascinating is its immediate impact on human beings. We are living entities imbedded in the fields described by classical and quantum physics. Nature’s forces invisibly affect us every day, and science has long searched for a bridge between the quantum and classical world. If these worlds turn out to be united in a very practical way through the phenomenon of life itself, the biofield will be far more than theoretical. It will redefine what human life constitutes, where we belong in the panoply of life on the planet, and ultimately how we should live in a wider, even cosmic, context.
We would like to thank Glen Rein, David Muehsam, Beverly Rubik, and Rick Leskowitz for useful input and suggestions, including Leskowitz’s work on phantom limbs.
Disclosures The authors completed the ICMJE Form for Potential Conflicts of Interest and disclosed the following: Dr Chevalier is a consultant for Psy-Tek Laboratory outside the submitted work. Dr Chopra disclosed that he is co-owner of the Chopra Center for Wellbeing as well as payments and royalties for activities outside the submitted work. The other authors had no conflicts to disclose.
Contributor Information
Menas C. Kafatos, Chapman University, Orange, California (Dr Kafatos)
Gaétan Chevalier, The Earthing Institute and Psy-Tek Laboratory, Encinitas, California (Dr Chevalier)
Deepak Chopra, Chopra Foundation and University of California, San Diego (Dr Chopra)
John Hubacher, Pantheon Research Inc, Culver City, California (Mr Hubacher)
Subhash Kak, School of Electrical and Computer Engineering, Oklahoma State University, Stillwater (Dr Kak)
Neil D. Theise, Mount Sinai Beth Israel Medical Center, Icahn School of Medicine at Mount Sinai, New York, New York (Dr Theise)
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c5c9d7ec13c1ffc1 | Multiple Quantum Interactions in Practical Material Regulated
Materials that have controllable quantum mechanical characteristics are highly significant for the quantum computers and electronics in future. Discovery or development of practical materials having these features is very difficult.
At present, an international theory and computational team headed by Cesare Franchini at the University of Vienna has discovered that some quantum interactions can exist simultaneously in a single real material and demonstrated the way these interactions are controlled by using an electric field. The outcomes of the study have been reported in the Nature Communications journal.
The application of an electric field changes the symmetry of the crystal and drives a transition from a metal (left) to an insulator (right). (Image credit: He/Franchini)
Next-generation quantum computers and electronics are dependent on materials that reveal quantum-mechanical phenomena and related properties, which can be regulated using external stimuli, for example, by using a battery in a microelectronic circuit. For instance, quantum mechanics controls whether and at what speed electrons can travel through a material and, hence, govern whether it is a metal conducting electric current or an insulator that does not. Moreover, the interaction between the electrons and the crystal structure directs the ferroelectric properties of a material. Here, when an external electric field is applied, switching between two electric orientations becomes possible. The probability of activating several quantum-mechanical characteristics in a single material is of scientific interest, but it can also increase the spectrum of prospective applications.
A group of international researchers (headed by Professor Cesare Franchini and Dr. Jiangang He from the Quantum Materials Modelling Group - University of Vienna, in collaboration with Professor Rondinelli from Northwestern University and Professor Xing-Qiu Chen from the Chinese Academy of Science) have shown that multiple quantum interactions can exist side-by-side in a single material and that it is feasible to switch between them by applying an electric field.
This is like awakening different kinds of quantum interactions that are quietly sleeping in the same house without knowing each other.
Professor Franchini
In order to achieve this, the researchers solved the relativistic form of the Schrödinger equation, by carrying out computer simulations on the Vienna Scientific Cluster. The material selected by the researchers, Ag2BiO3, is extraordinary because, first, it is made up of bismuth; a weighty element, which enables the spin of the electron to interact with its own motion. This process is known as spin-orbit coupling - a characteristic with no analogy in classical physics. Secondly, inversion symmetry is not exhibited by its crystal structure, indicating that ferroelectricity could occur.
Harmonizing multiple quantum mechanical properties which often do not coexist together and trying to do it by design is highly complex.
Professor Rondinelli
When an electric field is applied to the Ag2BiO3 oxide, the atomic positions change and it can be determined whether the spins are coupled in pairs, developing Weyl-fermions, or separated (Rashba-splitting) and whether the material can conduct electric current.
We have found the first real case of a topological quantum transition from a ferroelectric insulator to a non-ferroelectric semi-metal.
Professor Franchini
The spin-orbit coupling is of fundamental significance as it enables the emergence of innovative quantum states of matter, and depicts one of the intriguing research fields in modern physics. Moreover, owing to prospective applications, there are propitious applications: the regulation of quantum interactions in a realistic material will allow low-power, ultrafast quantum computers, and electronics for qualitative advancements forward in data collection, processing, and exchange.
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f5c1f89b9082a3b9 | BEGIN:VCALENDAR VERSION:2.0 PRODID:-//CERN//INDICO//EN BEGIN:VEVENT SUMMARY:Resurgent Theta-functions: a Conjectured Gateway into Dimension D> 1 Quantum Mechanics DTSTART;VALUE=DATE-TIME:20190612T080000Z DTEND;VALUE=DATE-TIME:20190612T090000Z DTSTAMP;VALUE=DATE-TIME:20200811T194900Z DESCRIPTION:Speakers: André Voros (IPhT - CEA Saclay - CNRS)\nResurgent analysis of the stationary Schrödinger equation (exact-WKB method) has re mained exclusively confined to 1D systems due to its underlying linear-ODE techniques. \nHere\, building on a solvable 2D case (a Selberg trace form ula\, as analyzed with P. Cartier)\, and on a Balian--Bloch abstract quant um framework in any dimension using complex orbits\, we isolate a very spe cial generalized-heat-trace function as best candidate to start some resur gent description of quantum mechanics in general dimension.\nThe latter st atement is still quite embryonic and speculative - our main hope is to enc ourage future research.\n\n tions/3761/ LOCATION:Le Bois-Marie Centre de Conférences Marilyn et James Simons URL: END:VEVENT END:VCALENDAR |
e74ec807a10a9246 | Calling all Quantum Theorists and Cosmologists who can be patient with innumerate humanists and theists…
Hi, gentle readers. I’m moving another comment thread up onto my front page where it belongs. Please jump into the conversation — just as long as you can be very respectful to science, and very respectful to both humanism and theism, okay? All right then. Ready? Set? Go!
Thank you so much for joining in here, Maria! Welcome to the discussion.
Now Gavin, be patient here with all of us, and don’t take yield to the temptation to take these questionings that non-physicists have as simply some kind of New Age occultism, okay? Continue to be your tolerant and patient self, okay? (If you don’t, I will have to remind you that YOU lean toward many-worlds (gasp!) as the best interpretation of wave collapse — and that sounds extremely “New Agey” to most people, even though it is strictly mathematical, right?)
Maria, I am better at responding when I’ve had time to ponder, so I’ll get back to you later on, except to say that many very good minds (including scientific ones) have seen quantum indeterminacy as opening up a universe that is much more open to freedom and spontaneity than was thought in Newtonian times. And eventually, when we know more about QM, I think it may shed some brand new light on very old metaphysical questions. (BTW, I was just wondering this morning some questions along the lines of what you are saying.) So I will get back to you. Instead, I am now going to ask Gavin the quantum physicist some questions that have come to the forefront for me lately.
[Still, let me recommend to theists the epilogue to C. S. Lewis’ The Discarded Image, in which he talks about nature in a way that shows he had been talking with the physicists at Cambridge and Oxford about Copenhagen QM (quantum mechanics) and the way that many manifest processes in nature seem to depend on resources lying elsewhere. It is so strange to me that many scientists are very comfortable talking about all of this stuff — just as long as no one uses the words “God” or “spiritual,” because these words have very unwelcome connotations (and I’m not sure theists aren’t responsible for many of these bad connotations). Theists, be sure to look, too, at what Richard Dawkins, the bad man hisself, said in an interview discussed in my earlier post, “Bravo for 3 Quarks Daily…”]
In the meantime, let me ask Gavin and other physicists/cosmologists/molecular biologists and so forth, these questions that have finally crystalized for me, questions that may in the long run prove to be related to Maria’s thoughts.
1) My son took a GE course in astronomy/cosmology at Penn last spring and came home for the summer and repeated to me something I’ve heard before about quantum indeterminacy. I want to know if you agree. He said that if you throw a ball at a solid wall a billion billion times, one time it might not bounce back, but continue right on through the wall. I understand why this is said (I think) but it is up-to-date in your view?
2) Here’s another question, one that has been driving me crazy. If you shot a rifle that was not accurately lined up or had looseness in its design where it shouldn’t, isn’t it the case that you would end up with a haphazard spray of bullet holes around the center of the target, and they would be randomly distributed and you couldn’t say where each bullet would be except roughly within a certain tolerance?
Now why is it that the collapse of the wave function is so worrisome to theorists, given that the particles are bound to appear within the range of the wave function and you can even specify the probability of where any given particle might appear on the screen. Setting aside the wave/particle duality itself (if we can), why is it so problematic that we can’t say where each particle itself will land, in a strict deterministic fashion? Aren’t there many things in nature that operate this way? When water is splashing along in a stream, it doesn’t splash exactly the same way twice, but it is certainly determined within certain limits. (?)
In the humanities and social sciences, we talk about norms applying “with a certain degree of determinacy.” In other words, the manifestation is always within a certain range, but the degree to which the normative outcome applies may be very loose or somewhat loose or may apply with very little indeterminacy, but it does not have to strictly every time to be a true normative pattern. Does there really have to be a deterministic mechanism underlying everything in science? If so, what about the water splashing…. (Or Brownian motion? It is random, isn’t it? “Random” within a certain shaping description.)
3) Finally, I remember that you said that Roger Penrose was waaay off in suggesting that a new theory of quantum gravity may eventually provide an underlying mechanism for the wave/particle enigma. (And he connects this with the quantum nature of consciousness, too, which Maria has heard of and that it is in the collective psyche of our culture right now.)
So Gavin, can you clarify what is the exact nature of your disapproval of Penrose, here? Is it the gravity part of the theory that you object to, or is it the very idea that physics will discover an underlying mechanism that is currently unknown, for wave collapse? That you are convinced in sticking with the current maths like Copenhagen does and not looking for anything further?
It seems to me that the general course of scientific progress indicates that an anomaly like wave/particle collapse will eventually be resolved by a new and deeper underlying theory, as in the case of earlier anomalies like black body radiation and the Michelson-Morley experiments. Yet it seems that you like the current maths enough to invest in the “other” explanation: that the current anomaly is explained by the particle being in every possible location but in different universes. But doesn’t this interpretation mean that “we” are in all those different universes too, but each of “us” doesn’t know the others exist? That SOUNDS, at least, pretty far out. I don’t say this to provoke you, Gavin, because I understand how compelling the mathematics are said to be. But what of the “metaphysical” implications? Don’t those give you pause? And why are you so opposed to a more “traditional” way out: that a new theory will provide a new mechanism for a more deterministic explanation?
4) Oh, sorry, one more question. I’ve been reading a philosopher of science James Cushing who describes the de Broglie/Bohm theory of quantum collapse, which Bohm developed after the acceptance of the Copenhagen approach as the standard theory. Cushing says Bohm’s iis a deterministic theory, and that it explains the phenomenon as well as Copenhagen does, and he contends that it is merely historical contingency that the Copenhagen hit the scene first and became orthodoxy. Do you have any thoughts on this? I would expect that physicists would opt for a deterministic theory if it really held water for them, even if another theory got there first….?
Sorry to load all of this questions onto poor Gavin. Any other physicists or others out there who care to comment on any of these questions? Or innumerate humanists and/or theists?
25 thoughts on “Calling all Quantum Theorists and Cosmologists who can be patient with innumerate humanists and theists…
1. Gavin
I can’t play ask-a-physicist without setting some limits, or I won’t have time for anything else in my life. Here are some rules:
1. I don’t discuss theories outside the main-steam of science. Science is a huge search. Vast regions of possible truth have been searched and have produced nothing. Small regions are proving fertile, and we are concentrating our attention there. I cannot go over every barren region again with newcomers. Faith healing, the realm of Platonic forms, a spiritual plane, Penrose’s link between quantum gravity and wave function collapse, and Bohm’s theory are all in the vast barren region. Sorry, we’ve moved on.
I do make some exception for widely held or forcefully promoted ideas: creationism and intelligent design, and quantum woo. I will not, however, be polite. There’s nothing useful to be said about these concepts while remaining polite.
2. First priority is always going to the issues that are directly related to the faith issues that we eventually want to reach. The quantum nature of the universe is at the wrong energy scale for addressing questions of God or a soul. The issues surrounding John McCain are far more relevant.
Now for Janet’s remaining questions.
1) No, the ball will not go through the wall. I know what they are trying to say, but this is the wrong way to say it.
2) This question contained seven questions, so I’m going to pick one: “Now why is it that the collapse of the wave function is so worrisome to theorists[?]” The problem is that quantum mechanics obeys one rule between measurements, and another when a measurement occurs. This would be fine if somebody could tell me what a measurement is. So, there are two different rules and no sure way to know which one should be used. That is the problem.
You seem to think the problem is that we lost determinism. It is not. We don’t like randomness, and we don’t like many-worlds either, but we understand that nature doesn’t care what we like. We can cope with randomness, if we have clear rules for when to use it. We know how to use it for water splashing and rifle shots.
2. Uh huh. Okay!
Yeah, I get your rules. I’m very sympthetic to the time constraint, which I feel when trying to explain semiotic theory. So maybe others will help out. Even very short answers help me a lot with the issues that come up in my own field that I can’t explain to the physicists exactly.
After all, aren’t we learning that we can’t become experts in one another’s fields. So we have to be willing to talk at each other in this hafl-frustrating and half-very-valuable manner? Those who simply say, study the books are giving up on the conversation. Because I am studying the textbooks and I still have questions that only those much more informed than I am can clarify. And those clarifications are more to help me think from my own vantage point and training than to think as a physicst…(and vice-versa).
But I must say that I’m surprised Roger Penrose is in “the vast barren region” with all those others Gavin names. There’s no indication he is either thinking about God (a hidden agenda, so to speak) or that he’s in the “quantum woo” camp that uses Bohm a lot, is there? So Gavin just reports that quantum gravity is unfruitful? Or is it the brain-and-QM connection in general that seems unfruitful? (Gavin, isn’t that question inevitable given the centrality of defining measurement?)
Moving on. Gavin says: “We can cope with randomness, if we have clear rules for when to use it. We know how to use it for water splashing and rifle shots.” And he says: “The problem is that quantum mechanics obeys one rule between measurements, and another when a measurement occurs.”
Okay, that helps me much.
This helps me too. But, then, why don’t you expect an underlying mechanism to be found that will explain both behaviors in one theory? Or is it just that every avenue for finding that has proven fruitless, so you are sticking with the current enigma as being more true to the data?
Gavin, are there measurements made in nature without any intentional observer or measurer or a measuring machine made by such, or is that precisely what we cannot know because we’d have to measure to find out? Like a photon hitting an eyeball. Is that a measurement? (The electromagnetic wave acts like a particle?)
Finally, this is really fascinating! Gavin says: “The quantum nature of the universe is at the wrong energy scale for addressing questions of God or a soul. The issues surrounding John McCain are far more relevant.”
What??? This is so surprising to me. In the West, starting with the Greeks, the divine has always been sought at the most fundamental level of the material universe. We look to the most underlying element(s) to find the “causes” of the abundant orders and the coherent emergences we see all around us. But that is “causes” in the explanatory sense, not necessarily the mechanical sense that science has focused on from the beginning.
Can you say why the meaning structures surrounding “John McCain” seem more fruitful for you? Is it simply that you think that for God to exist, God must have a physical kernel of reference, like the physical body of John McCain? But that is absolutely ruled out by the very definition of “God” in the West.
(This btw is why the notion of the Incarnation is so entirely scandalous. To localize the non-local in a physical body? To assume finiteness and vulnerability (and even death) by what is infinite and omnipotent? These are supposed to be utterly mind-blowing and very offensive contradictions. Something that is “a foolishness to the Greeks, and to the Jews, a stumbling block.”)
The Schrodinger equation gives you probabilities. Why is it worrisome to physicists that each wave can only collapse in ONE of those probable locations. Isn’t that what happens in every statistically probable future? What ACTUALLY happens is only one of the several likely results that WOULD or COULD happen?
I know a lot of you are following this conversation (thank heavens for blog stats!) so someone else should try to relieve Gavin once in awhile. On the other hand, sometimes it takes a lot of prior conversation to get the point where you have covered enough common ground to communicate across these barriers of disciplinary background….
3. HI
Poor Gavin, indeed. I will try to comment on Janet’s questions 3) and 4), even though I don’t have credentials of a working physicist.
3) Regarding Penrose:
It was a long time ago when I read “The Emperor’s New Mind” by Roger Penrose. It didn’t make much sense to me even when I tried to read it carefully then and it is even more difficult to follow his argument as I try to skim through it now. So, I cannot give too detailed comments but will give you my impressions from reading it a long time ago.
As I understand, Penrose is trying to somehow connect three poorly understood subjects, quantum gravity, quantum measurement and consciousness. Since we don’t understand much about any of these, we cannot say outright that Penrose is wrong. When we are ignorant, there is more room for speculations, as I think Gavin wrote before. But that doesn’t mean that there is high probability that any such speculation is right. And Penrose didn’t make a very convincing case that his particular speculation is right. In fact, I think many people have reasons to think that Penrose is likely to be wrong on this.
Do we need to take gravity into account in order to understand quantum measurement? We can ignore gravity to understand most quantum phenomena, because the effect of gravity is negligibly weak. So, it doesn’t make much sense that when we think about the measurement, quantum gravity suddenly becomes fundamental. Even if he is right, how helpful is his speculation when there is no successful theory of quantum gravity yet? Penrose may be thought-provoking, but he is not providing any thing very substantial, unlike EPR paradox and Bell’s theorem that lead to better understanding of quantum measurement. I think that the right way to progress is to try better understanding of each subject. If there indeed is a fundamental connection between these subjects of the kind Penrose proposes, it is bound to be found. But there is not a convincing reason to believe in such connection now.
Is quantum effects important for consciousness? Again, it seems to me that quantum effects don’t play significant role in most of basic neurobiological processes, such as firing of neurons and synaptic transmissions. And while neuroscientists may have yet to explain consciousness, they have learned great deal about how the neurons and our brains work. Penrose, as brilliant man as he is, is not an expert of neuroscience. Don’t you think it is a bit arrogant of him to claim that he knows better than the neuroscientists, especially when he is not making a good argument about the connection between quantum effects and neurobiological phenomena? You have to realize that the revolutions of quantum mechanics and relativity are very exceptional events in the history of science. The problem is rarely that we don’t have adequate fundamental laws. Often the difficult part is to understand the more complex higher order phenomena from the simple laws and the basic processes. Classical physics is sufficient to predict the weather, but it remain difficult to forecast the weather. Taking quantum mechanics and relativity into account won’t help it. Suppose that the string theorists are successful in coming up with the so-called “theory of everything” that unite quantum mechanics and general relativity. It is not going to affect biologists, chemists, nor even condensed-matter physicists. You don’t even need quarks to explain the structure of DNA, or chemical bonds, or superconductivity.
I might add that I don’t find it very fruitful to connect quantum measurement to consciousness (like Wigner did, for example). (And it is interesting that Penrose doesn’t like Wigner’s interpretation, either.) It is difficult enough to quantum mechanically describe a measuring apparatus. Consciousness is even more poorly defined. To think that consciousness somehow does something magical seems like baseless speculation and wishful thinking to me.
4) Regarding Bohm’s theory:
I have never studied Bohm’s theory, so all I know is based on what other people wrote about it.
My understanding is that Bohm’s theory is a variation of hidden variables theories. These are deterministic theories of quantum mechanics and the probabilistic nature is explained as a consequence of our ignorance on hidden variables. Most straightforward hidden variables theories are ruled out. Apparently Bohm’s theory is not ruled out unlike other hidden variables theories, but it comes with a great cost. It includes non-local interactions that are weird and complicated. It may be deterministic alright, but it looks too contrived and not appealing to many physicists. They’d rather take probabilistic quantum mechanics that may not be entirely satisfactory, but is simpler.
4. Gavin
I said there are two rules in quantum mechanics, one for use between measurements and one for measurements. The one to use between measurements is the Schrödinger equation, which is deterministic. The Schrödinger equation does not give probabilities. You tell it what wave function or density operator you have at the start and it tells you exactly what wave function or density operator you will have at the end. There’s nothing random about it; it’s totally deterministic.
The rule for measurements is wave function collapse. You tell it what wave function or density operator you have and it gives you the probabilities for a bunch of different wave functions or density operators that you might get out, with their associated measurement results. This is a random, non-deterministic process, but the probabilities are predictable.
Janet asks “But, then, why don’t you expect an underlying mechanism to be found that will explain both behaviors in one theory?” In fact, the underlying mechanism has already been found. There is one theory that explains both the Schrödinger equation and wave function collapse. That theory is the Schrödinger equation. If you just use the Schrödinger equation all the time, even when you do a measurement, then you can predict all of the features of wave function collapse. What should we call this new theory? We call it quantum mechanics, which is exactly what we called the old theory, so I can understand why people get confused. The process of wave function collapse is called “decoherence.”
Note that it doesn’t work the other way. You can’t use wave function collapse to explain the Schrödinger equation. It just doesn’t work. So, the reason we got rid of the non-deterministic random aspect of quantum mechanics isn’t because we don’t like randomness (although we don’t like randomness, so we’re pretty happy about this), it is because wave function collapse was a useless and ill defined part of the theory.
All of this is well understood and accepted by the experts in the field. We can use the Schrödinger equation to understand measurement in great detail. In particular, we can answer with confidence all of the questions you ask about measurement:
1) “Gavin, are there measurements made in nature without any intentional observer or measurer or a measuring machine made by such…?” Yes, all the time. In fact, natural measurements happen at an absolutely staggering rate. This is the main reason that the macroscopic world does not look quantum mechanical (and why the ball won’t go through the wall).
2) “Like a photon hitting an eyeball. Is that a measurement?” Yes. The photon hitting a tree or a rock is a measurement too. A photon hitting a mirror is not.
3) “The electromagnetic wave acts like a particle?” Yes, mostly. It retains enough of its wave characteristic for us to see color (approximately).
Not only can we understand what is an isn’t a measurement, we can perform partial measurements. This isn’t just a crazy theory, partial measurements are used in modern atomic clocks. The state of excited atoms is partially measured in one stage, and the measurement is finished in another much later stage. The long duration of the measurement allows a much more accurate measurement of the frequency of the atoms’ oscillations, in accordance with the Heisenberg uncertainty principle. Also, The field of quantum computing is base on understanding the details of measurements, including partial measurements.
Now, I said that all of this is well understood and accepted by the experts. That is not a terribly large population, and it certainly doesn’t include all physicists. I can back up everything I’ve said, but the math is difficult, even by physicist’s standards. Furthermore, the some of the implications are startling and not fully understood, causing some concern. None the less, the people who actually have to earn a living doing quantum measurements are on board because it is the only approach that makes sense and works.
Roger Penrose does not have to earn a living doing quantum measurements. Quantum gravity is a worthy pursuit (it is what I do), Prof. Penrose is a good physicist, and wave function collapse is an interesting (solved) problem. Prof. Penrose’s link between quantum gravity and wave function collapse didn’t pan out. The quantum woo community’s link between consciousness and wave function collapse also failed. (Penrose promoted this connection in The Emperor’s New Mind, so his record on wave function collapse is not good. He’s very good at relativity.) Decoherence is the winner.
Perhaps I jumped the gun with my comment about God and souls. If you could tell me one thing that God or a soul does, that would be very helpful.
Roger Penrose is not “the mathematician” that Maria mentions. I think she is talking about compact extra dimensions, which are common in string theory and are explained in several popular books about string theory, including Lisa Randal’s ,Warped Passages.
5. Maria Kirby
God gives life. God creates. God predicts the future. Souls live eternally. Souls might be considered the life force of bodies. -Not particularly testable actions.
I was speaking of Lisa Randal and Raman Sundrum, whose theories elaborated in their papers RS1 and RS2 will hopefully be tested next year at CERN.
I do think its very interesting that we create a reality or form like Hamlet or the Saint Paul Cathedral (before it was constructed) and then proceed to create an instance of the form as an actual building. (Some persons take on the role or character of a literary form, thus giving them an instance, maybe for the duration of the play, maybe for a life time. An actor who does this brings the character to ‘life’ for the audience.
It seems like mathematics/physics does a similar process, albeit some times in reverse. We observe certain phenomenon or instances. We then try to create a form (mathematical equations) which we can use to reproduce an equivalent instance to the one observed. The form is validated when it can not only predict an equivalent instance but can be used to predict other observable instances. The laws of motion work for dropping balls from the tower of Pisa as well as the movement of the planets.
It seems to me that when it comes to spiritual things we’re kind of like blind people trying to understand color. I think its very interesting that churches report more miracles in locales where persons are more prone to believe in demons and magic. While I think psychology is an important factor, I don’t think it explains everything. It seems that there is some connection between what we believe/think (its form) and what is observed to occur in the physical realm (its instance). And I don’t necessarily think that quantum mechanics can explain the phenomenon.
(I personally think that a number of people have hijacked quantum theories to make them support certain philosophical ideas, instead of letting quantum theories speak for themselves. But there does seem to be implied corrolaries, I’ve heard the laws of motion used in connection with human interaction.)
If mathematicians/physicists can prove that there are more that three dimensions, then it seems that the obvious next question is what is in those dimensions? Does what is observed in three dimensions project at all into any of those other dimensions? If something like gravity can project from a fourth dimension into our three, then can the strong, weak, or electro-magnetic force project into the forth? And what would that look like?
I may be naive, but I still believe in angels. It seems to me that angels are spiritual beings, who at times, have a physical presence. (Unlike ourselves who are physical beings with a spiritual presence.) Why would it be so unreasonable to say that what we attribute to spirit is not a force of another dimension?
6. I’ll respond to Hi, then Gavin, and then Maria.
Thanks so much! I’m really interested to hear that Wigner tried to make a QM-consciousness connection, too, and your judicious comments about Penrose and the different areas of his work were very helpful and clarifying too. (I want to look at Wigner — do you happen to know where he published this?) Eugene Wigner, remember folks, wrote the elegant essay on “The Unreasonable Success of Mathematics in Physics” that we discussed earlier under Part 4 of my lit theory lectures….
On Bohm and non-locality, I’ve been wanting to prod Hi and Gavin to say something about EPR and Bell’s theorum…. Gavin, would you rule out from a physics standpoint any connection between non-locality in QM and the idea that deep reality might be non-local? (At one point I read Bell’s entire book, Speakable and Unspeakable in QM, so I have a right to ask this, I think! But it’s not right for H & G to have to answer it. But how can you science-guys blame us innumerate humanists for getting stirred up by this stuff. It is downright IN-EV-IT-A-BLE.)
As for Gavin, Gavin, I don’t think you realized how new-style you are (as opposed to old-style scientific thought). You grew up with paradigm shift and an indefinite future for physics to evolve into and so you always are surprised that I am concerned with determinism or old-style locality in space and time. But speaking HISTORICALLY, it was precisely that emphasis of Galileo, Descartes, and Newton on “reality” being a natural world of empirical time and space (and Newton did try for an absolute time and space against which to measure relativistic space and motion) that led to the general incapacity of modern Westerners to imagine as “real” anything that is not something you could rest your coffee cup on (or kick).
Yet, the thoughtful or philosophical among Christians have always been uncomfortable with the modern notion that God or the soul are supposed to be “immaterial” entities that are divorced from the natural world. Yes, modern theists often do think of them this way, but this is because Descartes cut the mind or soul out of the material world and made it into a non-material thing, and created the “ghost” in the machine.
Now Maria, you are brave and straight-forward and obviously doing some reading and thinking here, and I’m glad you joined us. Knowing Gavin and Hi, though, as I do, do you mind if I echo your remarks in a somewhat different manner?
Also, may I say in passing — and this may be only my view and not Maria’s — that I think we theists would get further if we made it clear that in saying that “God creates” or “the soul is immortal” are not meant as naturalistic knowledge-claims. Theists bear witness to what they have come to believe on the basis of other disciplines and practices. These are things we say we “know” in the sense of having intimately experienced and/or of being committed to as a grounding hope. (The soul’s immortality, for example, is to me one of the most speculative of religious beliefs. The Jews in OT times generally didn’t have an afterlife in view. They worshipped Yahweh but in THIS life. It didn’t make them any less theistic. But the VALUE and PRECIOUSNESS of the soul is not speculative, because it is not about the unknowable future. It is experienced as a present reality, and as an ethical and esthetic commitment that is imposed upon anyone who desires to “imitate” God.)
Okay, so Gavin asks, “name something God or a soul DOES.” But you mean, “Name something God or the soul does that I cannot account for in other ways, scientifically.”
Remember, Gavin, that a scientific account is just that. It is a naturalistic account of something physically detectable in the world, according to the standards and methods of science.” You want physical causes or physical effects. But there are other causes and other accounts, depending on the discipline or the way of knowing you are working in.
I want to say that God “does” everything that happens in nature and physics and chemistry and biology, but not as a physical cause-and-effect. That would make God either a mere part of the physical world, and else an absolute determiner of the physical world, so that it would have no independent life or being. Instead, God “does” it all, in the sense of giving to the world the capacities and potencies to unfold as it therefore can, and to do all the kinds of things that it therefore can do.
And as it says in Genesis, or as Plato and Aristotle realized, the higher living creatures and especially this strange “speaking” and “thinking” creature that we are is “most like” that underlying immanence called potentiality or capacity, because we are capable of recognizing and thinking about and naming and re-enacting those unfolding laws and principles and kinds of things.
In other words, I am thinking of the world as something that can move into the future on its own, based on inherent potentialities that shape what is possible but do not absolutely determine it. God is the name we use to refer to the nature of those potentialities as potentialities, and therefore the name of their source and their direction, even if those are internal to universe itself.
Perhaps I should simply say that theists and pre-scientific thinkers all tend to see the universe as dynamic, not inert. And whatever it is in the cosmos and in living things and in history that keeps the patterned changes going and the developments developing and the processes processing and the “inert” elements being formed and the exploitations of every possibility for higher-order complexities exploited and evolution evolving, that is the indwelling divinity of the universe. And yet this divinity comes to us somehow as itself and not merely as the sum of all the separate processes. This is the fundamental human reaction to nature, and even the non-theistic scientists share it, as long as it isn’t called “God.”
By the way, no matter how “personal” a God one worships, I think in our day something is missing if one does not have the god of the philosophers included in the notion of God. (Perhaps this is why the early and medieval Christian church was so much more profound in its thought than the modern churches tend to be.)
So when Maria answers, “God creates.” And “a soul is the life-force of the body.” Then I want to say….
… that Maria doesn’t mean that God creates in the sense of a physical cause or a physical law or mechanism “creating” a certain physical state. It is not a push-pull cause and effect like in Newtonian mechanics. It is something far more philosophical, and yet felt by people on a daily level. Sometimes it is said that God is the “condition of possibility” for these natural mechanisms to exist and operate, and that God is the “reason” that there is something rather than nothing. Science is getting waaay beyond its sphere of expertise when anyone in science claims that science speaks to these questions or is able to speak to them.
Suppose that a mechanism for the Big Bang is found and we come to know that it occurred because of a whole series of other and preceding factors or suppose we come to find that it makes no sense to talk of a “before” the Big Bang (which was Augustine’s position) or some other scientific insight is arrived at 10 years or 1000 years from now. This will not do away with the basic philosophical questions and their cogency for human beings. It is wishful thinking for scientists to try to say that science will or science does do away with these questions. They are rational and inevitable. And I’m not going to be here in 1000 years and maybe not in 10. So I have to go with looking at all of the arts and sciences and all of my life journey and doing the best I honestly can to arrive at a worldview that I can keep faith with and that accords with my deepest knowing.
Now part of my deepest knowing is the model of the scientific, and science is so beautiful to me that I don’t want any of those laws to be abridged or changed by any interventions. (I hate the idea of miracles, if you want to know the truth!!) But what I cannot get away from is the way those beautiful coherencies and those intricate emergences of higher-order complexities depend upon potentialities that lie within the natural world itself. What is im-manent — what “abides in” those empirical things — and at the same time “lies beyond” those things, is the most fundamental meaning of the term “God” for me, and in our Western tradition of thought.
In other words, our cosmos has had the potentiality within itself from the beginning to bring forth all that it has brought forth and will bring forth, and that potentiality ITSELF is exactly what Aristotle meant by the word “form.” The potentiality is something separate from every single instance of it. There is something in this universe of which the universe is an instance and yet that is not the same as this universe itself. There is something unfolded in the history of this universe of which this universe’s history is only an instance, and that is not the same as the history of this universe.
Folks, this is highly philosophical, what I just said. Only human beings (as far as we know) of living creatures in our universe can observe the “existence” of what I what just speaking about. I think this is why Heidegger said that we are the kind of being that “raises the question of the being of beings,” and by so doing, identify ourselves as the kind of being that we uniquely are.
So I love Maria’s the soul is the life-force of the body. And the life-force of the body is clearly something different from the “stuff” that is left as the “body” without its life, because we have all seen the life leave the body, and the body is no longer a real and living body without it. (That doesn’t mean that the life-force exists forever, by the way, or even that the life can be life without its body. It is interesting that before Descartes, Westerners believed that angels, like all created beings, HAD to have bodies, even if the bodies were made of “ether” or some other more perfect composition. And all self-directing “bodies” had to have an indwelling formal element that held them together.)
Now I am one of those who thinks that every single thing that happens in our consciousness has to be related to brain chemistry. The soul or mind or personality for me is not something detached from the brain. It is instead an emergent phenomenon that is entirely based on chemistry and physics but that has a complex “being” and an organization on its own level of being.
One hundred years ago (Gavin) to say that everything in our minds was based on brain chemistry would have been to say that our minds are strictly determined by rigid laws of cause-and-effect. Scientific determinism raised those questions of free will that so occupied people in the Newtonian period. But now, as Dennett says, we see that, scientifically and naturalisticly speaking, “freedom evolves.” The more highly developed the consciousness of the kind of creature turns out to be, the more room for freedom has evolved in its mental determinations, beginning with moving away from danger and twoard prey and so forth.
Maria, I think the notion of “dimensions” is quite different for a mathematician or a physicist than for most of the rest of us. Extra dimensions beyond the ones we normally perceive do not necessarily imply a mysteriously “other” world of being in nature. Perhaps I should apply this advice to myself too, as regards EPR non-locality. But I don’t want to find a mysterious other realm of being. I just want to be able to say that much of our cosmos is alive, and that what is not alive is nonetheless potently capable of exploiting the conditions for life. It may take 4 billion years, but the amino acids will get it together! And the hydrogen molecules had to have already condensed.
I don’t see how science threatens the overwhelming reality of a universe that contains within itself potentialities such as we have seen and such as we have resulted from. To reduce this universe to a purely mechanistic model will not work any more, even if it seemed to (for some) in the 18th and 19th centuries. This universe has had direction from the beginning in the “form” of certain inherent potentialities, and it has evolved not only life but freedom and conscience. The anthropic principle cannot prove or disprove a creator God, or that our universe is a purposeful universe in a strictly religious sense. But it illustrates that we can no longer view this universe as an inert machine (as Dawkins knows).
So like good little liberal arts students we move back-and-forth between these new-style physical sciences to the other disciplines for a renewed conversation between all of them about the most basic metaphysical issues. I think we are verging toward naming an indwelling determinacy that is neither a law of strict necessity nor a chaos of pure chance, but that leaps into a future from a presence that came out of the past. Aristotle called it a coherent wholeness, one that is based on “that which is possible, according either to probability or necessity.”
Yes. In my own work (off-line) I am trying to formalize what Maria is focusing on here in a way that works for all the ways of knowing. We often don’t see the importance of this form-al mediation because we tend to reduce it to our modern notion of “abstraction.” (Very 18th century!) In an earlier post, I tried talking about this instead under the name of “rehearsal.” (As soon as I get the software working, I’m going back to explaining the semiotic codes or normative principles we encounter with language and structures of language, by the way.)
Hume changed everything in the West when he questioned “induction.” He pointed out that no matter how many instances we encounter, we cannot be SURE and CERTAIN that the next instance won’t be a counter-instance and destroy the general principle. So Hume realized that when we go from empirical instances to form-ality we are moving away from Descartes’ ABSOLUTE certainty. This required Kant to go to work to save induction and cause-and-effect, the other big formalization that Hume demolished.
But look at how this Humean thinking is based on the requirement of absolute certainty. The very definition of “Knowledge” became after Hume “what we can know with absolute certainty.” None of our science today would survive this requirement, because we realize today that our knowing is always “open to the future.” In the future, we may revise or re-understand what we know today. (notice that the empirical is always something that is slipping away into the past. We are left thinking about its significance for the future!
But Plato & Aristotle rightly thought that induction was a dynamic part of everyday life and every human learning, beginning with language. We wouldn’t know what a word meant if we had to be absolutely certain or if the word was tied down to a static one-to-one relationship to a “closed” meaning. Instead we develop a theoretical construct open to the future, for every word we learn. This is why poststructuralists are always talking about how problematic imposed closures (associated with the absolutism of the 18th century) are for the well-being of our knowing and being.
Aistotle thought, contra Hume, that on occasion, even ONE instance could be enough for a form-al interpretation. (like the way we judge the other person on a date?) And most of the time, and in relation to most of the things we really, really need to know, we have to go with likelihood and probability and the hopes of acheiving a high degree of confidence, but not “absolute knowledge.” So why not just go back to the notion of ike (techne or episteme) of the Greeks, where an ike is an attempt to come to know better the formal characteristics of a kind of thing (or kind of process)? It will have exactly as much likelihood as the kind of thing itself allows, but it will still be a valid discipline of that kind of thing. (MAJOR assumption of Western education before the laws of motion installed absolute certainty as the norm for a real “science.”)
And let’s particularly notice the role of time — and “the future” — here. When the Newtonian laws of motion became the BIG cultural paradigm for knowledge, it appeared that the future could be predicted absolutely by laws, and hence was determined. This didn’t hold up, even for the physical sciences, and Hi talks about this above. Atmospheric science uses deterministic laws and you cannot predict the weather with more than certain varying degrees of probability.
For the Newtonian worldview (which of course is not the same as Newton), it looked like the future was only the current “actuality” all over again, repeating itself. But Aristotle — esp. in his literary theory but everywhere else as well — looked at it in this way. First, the past is no longer open. Once it has happened, it’s determined in the sense that it can’t be changed.
But only a part of what happened (actuality) was because of ordered principles of causation. A lot of it was accidental or contingent “stuff” because various causations happened to intersect in a random manner. So Aristotle thought of predicting the future as taking what was coherently causal in the past and formalizing it and then projecting that formality into the possible future, knowing that we can’t be certain because of all the different causal processes and their random interferences. And because some causal processes are simply less deterministic than others by nature.
THIS IS WHY EPISTEME IS FUNDAMENTAL TO KNOWING. Each kind of causation needs its own disciplinary community. We cannot know the whole world directly. We need to find coherent parts of the world (kinds of things) to formalize and we need to learn about formalization itself, first.
But where do we step back and put all the ikes together and think about the whole of life and the whole of where everything is going? (It’s called “First Philosophy” or metaphysics and there is a discipline for it. Theologies are also inherently a kind of first philosophy.)
There’s an existential core to each of our lives and we have a drive to achieve an integrated worldview and make some sense of things and also determine what kind of person we should be and how we should act (and how we get so we can act like that when we know we want to — the biggest problem of all).
There’s no absolute certainty is THESE areas, the ones that finally matter the most. And you don’t make any progress by simply accepting a ready-made worldview or ideology or religion either, especially if it’s Christianity, because this faith is so counter-intuitive and demands so much thought work and so much willingness to ALWAYS scrutinize and overturn one’s assumptions. (You can try to resist this, but it always gets done for you, anyway.)
Being a Christian is being called to a continuous inward revolution and requires the activity of the full mind and the whole person. Christianity is based upon the paradox that there is a fullness of truth toward which we aim our passion and we do experience it from time to time and try to chart our course by it, but we can never have in our finite and limited selves an adequate conception of the truth or what it means. The more we try to cling to and insist upon certainty, the more likely that the shells of our certainties will be overthrown to get us to a deeper truth.
8. Let’s get back to the questions of the “existence” of:
Hamlet (the character — but consider the play)
John McCain
the electron
Note that with regard to “electron” the difference between “an electron” and “the electron.” Here’s that Form-al awareness Maria was talking about, entering into the picture. (Bertrand Russell had to spend 100+ pages on the meaning of “the” in his Principia Mathematica!)
“An electron” usually refers to a particular instance of the electron, whereas “the electron” refers to the formal mode of being of the electron, as a theoretical construct (what Plato & Aristotle called a “logos,” a formal definition or account), the electron as a topic or a subject matter for formal inquiry. (The “eidetic,” as I am calling it in my off-line work, after Plato’s eidos or “Form” or Idea.)
Sooo…. Let’s not dismiss Plato’s Forms too quickly to the barren wilderness or the realm of quantum woo…. Too often, they have been interpreted to suggest an otherworldly realm of pure Ideality, but in practice, in the dialogues Plato wrote 2400 years ago, they emerge as tentative or provisional idea(l)s of the topic, and then the Form is used, paradoxically (or dialectically), to critique or to call into question all of the current (received) ideas about the topic. (Experimental and reflective testing is built in to the notion of the eidetic or the Form-al. The naming of the kind of thing within a philosophical inquiry opens the space of inquiry by opposing the Form as the ideal reality to the theory so far, or to whatever we unreflectively may have supposed.)
So, I’d like to say that the Form or the eidos is “The Putative Reality, As It Might Get To Be Known in the Future”! It is the practical and servicable goal of our quest, though we never reach it. It is the Ideal Answer that we strive toward but do not yet have in its entirety. And there’s no sense, with Plato or Aristotle, that our disciplinary knowing is useless unless or until we do arrive at ultimate knowledge. The search is substantial and makes progress, and that gives us the experiential contact with reality that we need as human beings. It seasons us and makes us committed to the search for truth.
By the way, guess where the word “future” comes from? It comes from the Greek word physis, from which we get “physics.” It’s that active ending -sis added to the verb phuo that means to grow, bring forth, or give rise to. So again we see that what any discipline does is attend to what can be observed to have already happened and to be happening, in regard to a certain formal kind of thing and its process of coming-to-be, and then weed out the irrelevant noise and accidents and incidentals, and then formalize the potentialities that might have been in action there. Then, we will have the kind of episteme that enables us to make predictions about the future that are better than those of persons who do not have the episteme.
It’s not the predicting itself that matters here, though it is fundamental to scientific method. Episteme is not so that we gain “control” of the future per se. Knowing, instead, is about assimilating the know-how or expertise or deeper understanding that gives the member of the ike the “power to know” — the power to know “how to do” certain things, and that involves being able to gauge what most likely might or will or would happen. The important thing here is that the knower is trying to follow something that has produced a pattern in the past — and follow it into the future.
Remember Paul on how hard it is to dream up experiments to test new ideas? Harder than coming up with the ideas themselves? Science is inventive and creative, working along lines already laid down, and projecting them into a “future” that we MIGHT get to FROM HERE. This means that the Present must be viewed as being structured by formal organizations that can be hypothetically discerned from the past and projected into the future along the same principles. We are trying to move from the past into the future by assuming something that operated in the past (we think) and in the present (we think) will continue to operate (more or less, apart from accidental interferences and incidental complexities) in the future. We project our knowing as an expectation about the future that comes FROM the formal principles upon which we’ve come to think some of the stuff in the present and the past were based.
All of this requires the use of what we moderns tend to call “imagination,” but Aristotle called “poietike,” a kind of “making” of a “fictive future” of what “would” happen, in the sense of what “might” happen IF, as we suppose, our analysis in (of) the past has indeed been moving in a fruitful direction. The Possible, or The Possible-Probable, of Aristotle is not confined to the worlds of art. (P.S. The imagination is a Romantic concept only 200 years old and a bit too free-wheeling to pull together science and the arts as ways of knowing, in my judgment.)
Now, I want to remind us all that I insisted on adding to Gavin’s list of “things that exist” a couple more items (with a view to eventually discussing God and faith issues, as well as the liberal arts).
So let’s add:
John McCain
the market (as in “the market sets the prices)
“Summer” differs from the first three “things” because, unlike Hamlet, it is based more im-mediately upon empirical or physical observations and sensations and measurements of “it,” like “John McCain,” but we can’t just point to a “summer” sitting there as an entity in the empirical world. So it is like an “electron,” in that we have a theoretical construct to define something we have detected empirically, but it is unlike an electron in that it doesn’t have the same coherent wholeness or entity-ship.
“Summer” has edges that are blurry, and different cultures may divide up the seasons somewhat differently, so it may be a local construct. On the other hand, there is certainly in nature a fairly regular recurrence and patterning in the swinging around and repetition of the seasons. Yet you cannot simply identify “summer” with its empirical measurements, as I’ll show, in part because what constitutes a summer (actual temperatures, weather) may be different in Alaska than in Maylasia, and yet we still speak of a summer in both cases. (This is exactly like the identity of phonemes or morphemes in language.)
We CAN identify summer MOST coherently IN DIFFERENTIAL RELATIONSHIP TO THE OTHER SEASONS. “Seasons” is the fruitful category here, like a genus, and then we need the differentia that make the seasons differ from one another in each case….
So finally “summer” as a “thing” is a theoretical construct that “exists” for us because we have defined it in relationship to other closely related things within a certain coherent context (the cycle of seasons). But is the existence of this “summer” out there in the actual empirical world? If there is no one to observe the patterns in the weather and compare and contrast them from year to year and name them in the common language so little toddlers begin to learn about “summer” and “winter” as theoretical constructs, then does “summer” exist empirically in the natural world? This is NOT a yes/no question!
We can even say things like: “This was a very cool summer, hardly like a summer at all. More like late winter.” We are talking about and interacting with the natural world in these sentences, and we are also using the culturally prevalent constructions of all of that empirical data into the particular units or wholes that in our language and culture enable us to talk about the data on this more powerful formal level in meaningful terms.
But when we say that the cool summer was not really a “summer” at all, what exactly do we mean by summer? The cool summer is an actual instance. The summer that it is not, is our idealized or typical summer in our minds (Plato’s Form), against which we measure each actual occurrence.
So why don’t we just call this summer a winter if it “more like” a winter than a summer. You know why. We have a whole theoretically precise set of constructs in place, and as a result, just because the specific manifestations of this particular summer don’t resemble the formal identity of summer, it is still a summer. For us, in terms of interpreting the data…
The identity of many “things” does not depend on their physical make-up so much as on the normative structures (based on physical instances) that we bring to evaluating them. (John McCain is a man, a senator, a POW….) With regard to linguistic units, this is so much the case that Saussure compares it to a game of chess, in which the formal rules remain and make the various “pieces” what they are. So you can replace a pawn or a rook with anything you want, a coin, say, and it is still a pawn or a rook so long as it differs from the other pieces enough for us to keep its identity straight.
The “being a Pawn” — or the mode of being called a pawn — does not depend on any physical substance the pawn is made out of. But this is NOT saying that the identities of pawns or of summers are merely socially constructed. It is simply the case that we aren’t done defining them if we designate a piece of polished wood of a certain shape or a set of temperature ranges and weather patterns. Physical structures are involved at every level, but the identities are formal and relational (differential) identities. As every structuralist knows, a relationship is always also a contrast and an identity is also always a difference, because identities as regonized by human knowers are always defined within a coherence context and with reference to one another.
Then, of course, Shakespeare’s Richard III says, “This is the winter of our discontent, made glorious summer by this son of York….” These are metaphors, not references to a “real” summer or “winter” at all, it would seem, and yet of course they are references to real summers! We wouldn’t even understand the metaphors if we didn’t have a form-al notion of “summer” based on many actual summers, in contrast to many actual winters, experienced and named by our speech community.
For the Greeks, that hypothetical or normative Idea is the “Real” and the actual summer is merely one actual instance of that real thing…. It’s a very, very helpful contrast for us, this contrast between the empirically actual, which is always gone (into the past), and the Real-ity of the formal theoretical constructs which we human knowers come up with, to use as we seek a deeper understanding.
But how in the world are we going to talk about the existence of “the market”? Where is it? (Like the Internet. It’s in our heads, and it’s Real, and it’s actual.) Here we have to start talking about invisible codes of “behavior” that connect all of the members of the economic community and “information” and “market forces” and these are not occult. They exist, if we can rely on observations, but the mode of their existence? I heard Alan Greenspan’s replacement say that if we could only figure out what causes “confidence” we could predict the market absolutely, but we can’t….
These “names” are technical vocabulary and refer to things going on in the world. Their existence is clearly in the mode of the “Real” or Form-al or ideal “things” we’ve talked about, like “being a pawn,” and not the merely actual or physical objects, only these constructs are removed from the first-hand data by more layers of theoretical construct. (We don’t even know what the data we want is until we have some kind of theory going.)
The big question in American academia the last 30 years or so has been, are the theoretical constructs in our heads also out there in the physical world? This is such a naive question. Only English speakers with our own tradition of reductive empiricism, from our “scientific” philosophy that valiantly struggled to model itself “logically” upon geometry, would think that if a thing is a construct that cannot be simply equated with a physical object, then “the construct” is “just socially constructed.”
All human knowing is “constructed” knowing, esepcially in the sciences, with those constructions always, always based upon constant interaction with the world. The reason we don’t see this as self-evident is that we have forgotten that there is a difference between an actual “thing” and a “kind of thing,” even though we never ever perceive and know any actual thing without the theory of the kind of thing and the theory of its difference from and relationship to other kinds of things mediating our knowing of it.
The very words in the lexicon of our language that we learn as we emerge as human persons in early childhood refer to the formal kindness of things, as we have learned to name them in the past (langue), and because they are formal constructs of that sort, therefore we can in the future use those form-al words to make specific references to instances of those kinds of things in the world, and in memory, in dream, in literature…. The formality of their identity enables us to transpose them into various realms that are realms of projected formal being….
9. Gavin
I’m just going to pick one thing as an example. You say:
I am inclined to say, “Fine.” I personally don’t see any reason to personify those things, but if want to, that’s great.
However, I run into trouble. My friend, Brent agrees with you but adds that God thinks homosexuality is an abomination. Then there’s Andrey who agrees with you and Brent, but also thinks that God has asked him to intimidate gays with physical violence, to the point of death.
How can I be respectful of non-empirical knowledge in some cases, and then oppose these other ways of knowing in other cases. As an elder in the Presbyterian church I spent considerable time watching men and women argue about what God wanted us to be doing in bed. It was typically a He said He said debate with everyone quoting and interpreting passages from the bible. It had no connection, that I could see, to the world because it wasn’t based on anything empirical, and they got nowhere. If we had decided to work with empirical evidence, then the issue would have been rather easily resolved.
Asking everyone to stick to empirical evidence seems to be the best way to make progress in debates about practically anything, which make me reluctant to say “fine” if you claim to have personal knowledge of some deity whose every action is undetectable.
10. HI
Janet wrote:
But don’t you see exactly where non-theists have a problem with? Here you are talking as if God only means such “potentialities.” But of course that is not all what the God is to you and most theists. Don’t Christians use the words such as loving and caring to describe their god? And didn’t you confess how real that kind of God is to you? Why would you worship “potentialities” anyway? But the problem is that it is not self-evident that God that is loving and caring and God of “potentialities” or “the condition of possibility” are one and the same. (And I suspect that God of “potentialities” is not the primary motivation for the faith of most theists.)
John McCain is a senator and a former POW at the same time. But a senator is not necessarily a former POW. We only know that John McCain is both a senator and a former POW, because we know that John McCain is a senator and we know that John McCain was a POW and we know that John McCain the senator and John McCain the former POW is the same person. Can you make a similar connection for God? It is more convenient to just to talk about the more philosophical concept of God, but that is not going to be enough.
And even if we forget about your personal God of love and focus on the philosophical God, there still remains a question of how meaningful such a concept of God is. Read what Sean Carroll wrote.
(Also, in a different thread on Cosmicvariance, someone called Ali made a following comment. I’m not sure if this is the same Ali who also comments on the thread above.
This sounds similar to what you are attempting. Do you care to comment on that?)
Regarding Wigner, I really don’t know much anything beyond what was written in popular science books or what you can find on internet. Among other things, Wigner proposed a thought experiment called “Wigner’s friend”, which is a variation of Schroedinger’s cat that essentially replaced the cat with a human (and the human doesn’t have to die unlike Schroedinger’s cat). It was supposed to illustrate the importance of a conscious observer in the measurement, but it seems to illustrate the flaw in his thinking to me.
11. Maria Kirby
Isn’t that exactly what Christians are claiming? The soul is immortal because we have empirical evidence that Jesus rose from the dead, in a new eternal body?
And we also know the Form of God because we know the Jesus who is the Word become flesh, the Form became an empirical experience?
12. All of you are keeping ME on my toes!
Maria, you point out something I badly need to clarify, and it is connected with all of our impasses about empirical and semiotic and so forth.
I’m drafting a reply to everyone. Thanks!
13. By the way, Hi, those links aren’t working for me. To Sean Carrol and Ali. Can you offer them again or name the posts to see at cosmicvariance? Thanks!
Gavin, your links are truly horrifying. Thanks for alerting us.
14. I just read Sean’s post and I am surprised at him — I think he is being incredibly reductive and narrow-minded. (His review of Dawkins was much better, imho.) And Ali, there is so much more to the discussion of “existence” than what this “religious scholar” refers to.
If you scientific folks think that MY grasp of QM is not adequate, and I’ve done a lot of work there, then I have to say that to me these accounts of the theological, philosophical, and logical issues at stake surrounding “God” are not even kindergarten-level. And yet they don’t seem to recognize that they don’t know anything about what they are dismissing with their knock-down arguments; that there are intellectual worlds there of which they have no knowledge whatsoever, not to mention cultural, ethical, and daily worlds of which they apparently know nothing. It is as though they are color-blind or tone-deaf. Only what their own way of knowing illuminates, can “exist” or make a difference. Anything that other people might perceive or treasure simply doesn’t exist, because their little elite group has the only way of knowing, and anything else would complicate things too much. They are bound and determined in advance to know about nothing except what they want to know about.
And I don’t buy this demonizing of the Christian rank-and-file as having no theological or philosophical sophistication. You can’t have a genuine experience of God without having those profound philosophical ramifications entering into your new experience of life. (It should go without saying that there are of course “Christians” who have had no genuine experience of God, and non-Christians who have had. The Bible is full of this — remember the Pharisees?)
I will try again and reread that post tomorrow, but I am very sad. Sean speaks of a single world with a single way of knowing what’s what, and you guys agree with him? So what have we been talking about all this time? The very idea that so many people seem to think it is okay for Dawkins or anyone else to dismiss the very question of God as stupidity, without knowing any theology, makes me want to weep. If all YOU happen to know is the straw man that Dawkins attacks, then you simply are as uninformed as he is. It doesn’t mean it’s okay to reduce the whole thing to what you’ve encountered. This is prejudice and bigotry. Fanaticism always works like this.
How is this insistence that there’s no content to faith in God any different from narrow-minded and fanatical Christians saying that evolution is wrong and blasphemous, when they don’t know the science or understand or credit the scientific method on its own terms?
This kind of reductive and fanatical insistence that one way of knowing is the single obvious monolithic truth and that it speaks for itself and that everyone else is just dead wrong is just appalling. Asking other ways of knowing to justify themselves by the standards of your own field is deadly to thought and to any prospects for human peace and advancement.
You cannot base an argument on ignorance. You may decide you don’t like religion and that YOU don’t WANT to know anything about it, but you can’t then dismiss it and claim to be able to close it off and dispose of it in advance as empty and void of truth or reality for anyone. That is simply fanaticism, and Dawkins is in this respect as simple-minded and fanatical as they come. He is blindingly ignorant of what it is that he is dismissing. He even says that his atheism is “a victimless crime” — that it hurts no one. (Not being an atheist, but his militant attacks on all religions and religious people.) He is living in a dream world. He is fomenting hatred and aggression against what many human beings hold to be their most precious possession. That isn’t hurting anyone? Militant attacks always hurt people. The militants themselves above all.
Look, no one can tell me anything about the evils of religion that I don’t know first hand. But if you don’t know anything about its treasures and its depth and its meaningfulness and the daily goodness it has also supported, then how can you begin to make an evaluation of it? I don’t mind Dawkins disliking religion and he is entitled to his opinions. It is his claim that an entire rich dimension of human exploration and experience is worthless and empty and can be known to be worthless from outside, that qualifies him as a bigot. You cannot ignore the voices of human beings with other backgrounds and think that you don’t need to value their experiences and their insights — just because you know better than they do, in advance.
What if we asked artists to “name one thing that art has done that makes a difference.” Or, “How would the universe be different without art?” Or music?
What if we made it harder and asked, how would the universe be any different without government and politics? Just look at all the terrible things governments have done. Look at how destructive political fanaticism or ideologies have been? Let’s stop believing in it and it will go away.
Everyone is trying to make the universe much simpler than it is. For a person to dismiss as nonsense something like the “condition of possibility” is simply ignorance. It’s pitiful, to anyone in those fields. Dawkins’ arguments do make you cringe, just like Terry Eagleton says, they are so sophomoric. It’s exactly like a Fundamentalist getting an easy laugh from the audience by ridiculing the idea that humans descended from apes. It is a pitiful spectacle to watch supposedly liberally educated people indulge themselves in demeaning and demonizing whole segments of the human race instead of attempting to understand them and hear them on their own terms.
Depth experiences of God occur in all cultures, and in the biblical faiths, the experience of a “personal” God is simultaneous with the experience of an ultimate reality and with “the ground of existence” and the condition of possibility. These aren’t empty phrases pointing to nothing at all. Does me not knowing and understanding advanced elements of a field of science make that science empty and meaningless? Only if I think I can dismiss the work of other human beings in an arduous common enterprise, just because I haven’t been drawn to it or trained in it.
Does that mean we accept everything that claims a religious basis (like gay-bashing) or a scientific basis (like experiments that cause unconscionable suffering to helpless dogs and cats and other higher animals) just because they claim to be religious or scientific? No. We have to keep on struggling to interpet and distinguish. It’s never easy.
Gavin says we’re better off to simply “stick with the empirical,” and then we could settle things more easily, without the ambiguities of religion. It’s a nice hope, but I think that everything including science is pretty ambiguous ethically, and we are stuck in the middle of the whole mess having to struggle constantly with interpretations and decisions, individual and collective. We’re all in this together and demonizing each other isn’t helping. (Talk about “dishonest.” All these knee-jerk reactions and wholesale dismissals and sweeping assumptions that what is self-evident to me is therefore universally applicable to everyone, without even checking with the others first?)
I hate it when Christians are self-righteous and reductive and judgmental, but it isn’t really any nicer to see it in atheists, either.
If you “believe in God,” it is either because you have accepted a form of religion passed down to you, or because God has become unmistakably manifest to you, or both. In the latter cases, you don’t add up the “arguments” pro and con. You try to integrate the continuing reality of God with everything else you know, and that usually means finding a tradition that is capable of helping you to grow in your relationship with God.
Because you feel incredible gratitude to God, and a profound sense of the sacredness and goodness of the sacred dimension in your life, religion can become a powerful force for good or evil, and it is just as liable to become distorted and destructive as a marriage or a family or a community or any other human institution is. One feels of course that God is on the side of health and fruitfulness in all of these cases. But for us to know the good, and then to do the good? That is always the problem. But we have an evolving tradition that is very rich and profound to guide us.
I think that one of the differences that God makes is deeply inward. Genuine experience of God moves one into a journey of discovery in which you are just as foolish and intolerant as anyone else, but you aren’t left to your own resources. There’s an inexorable pressure to see through your own excuses eventually and become more humane. And there’s knowing and loving this incredibly suffering and loving presence…. I could say so much more, but I would have to do it by speaking of my own tradition and not so generally about the religious dimension in general.
What does God see when God looks at the world — imagine this, as a thought experiment if you will. The Christian tradition says that God sees the spiritual suffering and struggle of the world, because God values that above all else. (The Jewish tradition, also.) And that the spiritual is not separated from the yearnings of the natural and the animal world as well. God looks inwardly and sees the inward heart of things and God values even the smallest increase in the kingdom of love. And God is broken by every violence that breaks any one of us. God suffers with us and in us and for us. There’s nothing easy here. Nothing snappy. Just something unbearably relevant and real.
15. Gavin says: “How can I be respectful of non-empirical knowledge in some cases, and then oppose these other ways of knowing in other cases.”
But you have to. You have to try to distinguish genuine ways of knowing from ones you cannot accept as genuine. You have to distinguish the Christian tradition from gay-bashing, for instance. You have to distinguish scientific knowing from the hideous torture of animals. You have to do the best you can, as thoughtfully as you can, and take your stand as best you can, but you can’t just throw out whole ways of knowing because they change, disagree, and sponsor terrible things.
And here you are saying “non-empirical” again…. Anything that requires human observation over time is no longer strictly empirical, but a weaving together of empirical observations at different times into a construct that is both empirically based and that “exists” in human consciousness, language, and history. You are talking about a way of knowing that has as its ultimate arbiter the conformity of these constructions to experimental testing.
Such a way of knowing cannot do ethics and perform in many other vital areas. It can help inform ethical decision-making, but it cannot make the decisions, because it isn’t designed to do that. How would empirical considerations settle the Presbyterian elders’ debates about what people should do in bed? It could inform the debate, but you would still need to make larger ethical arguments for how to interpret the scientific data in an ethical framework.
The science on homosexuality did settle my own stance as a Christian on homosexuality, but that is only because I have a larger context of religious and ethical theory, i.e. that the Cross shows that nothing trumps divine love. Therefore, if people are born with different sexual orientations, and have no choice in the matter, as we now believe based on science, then I don’t believe Christ would condemn non-heterosexual persons to live without intimacy and physical love. But I’m not allowed to condemn and hate Christians who cannot, in good faith, come to this view of the matter. (I know some of them for whom this is tearing them apart. Great suffering here on all sides.) To me, we are in another period of historical change. We’ve gone through this with abolition of slavery, with Christians on both sides, and then again with women in ministry, and now we are going through it again with homosexuality. But I do have to oppose any hating and persecuting of other persons because of homosexuality (or for any other reason).
The whole church will come around on this as it has on the other issues. We’re a species that is now evolving culturally as well as (or more than) genetically, but we still resist at every step the manifestations of a transcendent love and compassion that we also prize and adore above all else.
16. Gavin,
I know many, many of the Greek Orthodox here in Seattle quite personally, because my Episcopal parish shared our building with a Greek Orthodox mission congregation until they were big enough to have their own building, but we are still very close, and I went to their larger gatherings, and more gentle and loving people you would never hope to meet. They took care of their elderly and adored their children and reached out to everyone — they were on fire with love. I can’t put it any other way. Their children would come home crying from second grade because little Evangelical children had told them they “weren’t really Christians.” When their tradition goes straight back to the early church. There isn’t much limit to our human pettiness and iniquity. And there isn’t much limit to those people’s love and the good that they do and are.
Aren’t the sociological reasons for those Russian men’s looking for scapegoats pretty obvious? Sean Carroll’s review of Richard Dawkins’ The God Delusion is a sophisticated discussion of why we can’t attribute all evil by religious persons to religious factors.
Also, folks, I’d like to add to the list of questions we’ve been building.
What difference does innocence make?
What difference does forgiveness make?
What difference does vicarious sacrifice make?
What does the figure of God on the Cross mean to the mothers of those who’ve been “disappeared” in South America, and does it make a difference for them that God’s son too was put to death as a criminal?
This narrow rationalism is as thin as water. God takes on our flesh and our blood and speaks to us in our deepest sufferings and rebukes all our iniquities by taking them all on personally. (But I can also see how people who have been suffocated by the perversions of religion can find in science freedom and space and fresh air, while for me the scientific attitude was the source of great harm. This is where we need semiotic theory. Things take on their identity in large part from the surrounding system of associations and the rules we have in place, like summer from the other seasons and a pawn in chess. The “Christ-event” for one person might be a forest and for another romantic love and for another science itself. Maybe we should read Dante together here….)
Don’t let the distortions fool you. The most powerful goods can be turned into the most powerful evils quite easily and it happens all the time. Humans are deeply irrational creatures, as well as deeply rational creatures, and we are in desperate need interventions on all levels of our being. Wow. This is turning into a lead-in to discussing Shusako Endo’s _Silence_! Starts tomorrow… on All Saints Day, in fact, as it happens.
17. Gavin
As I said before, I agree with Sean except for his use of the word “dishonest.” You respond:
This passage stands out for its clarity, but mischaracterizations and insults continue throughout. I will not participate in a conversation like this.
Good luck,
18. I am very sad. You have been a wonderful conversation partner, Gavin.
I was fresh from the lacerations I had just received from reading Sean’s piece and some of the comment thread. I should have waited until I was calmer.
I wasn’t talking about you and Sean personally, Gavin. I was talking about this militant attack on religion as a way of thinking and viewing the world. I still believe it is as tragically narrow as the fundamentalist biblical literalists who are crusading against Darwin is.
What about all of us in the middle? I hope you reconsider, but in any case, I’ll always treasure the conversation — and reread the QM parts!
(Looks like I need to learn some spiritual lessons in humility from Shusaku Endo.)
19. Maria Kirby
I would like to go back to your example of the reality of Hamlet as an example of semiotic knowledge and empirical knowledge. Hamlet as a play, as words written down, expresses certain ideas and concepts embedded in the character of Hamlet. When an actor performs Hamlet, he converts the semiotic knowledge into empirical knowledge. To the extent that the actor’s representation or characterization reflects accurately the semiotic knowledge of Hamlet, that semiotic knowledge becomes empirical to the actor and his audience.
I believe the same is true for religious concepts, particularly our knowing God through Jesus. To the extent that we understand the semiotic knowledge expressed in the Bible about Jesus, to the extent that our semiotic knowledge is developed through philosophy, nature, or other means, we can convert that knowledge into empirical knowledge through how we behave towards others, through how we embody Christ, or Love, or forgiveness.
It seems to me that one of the major themes in the Bible is that of transformation. God transforms evil into good. Forgiveness transforms enemies into friends. God’s love transforms us from dying or dead into living and alive. The resurrection transforms death into life. I see a similar phenomenon occurring in biological systems where DNA is torn apart, replicated, and restored. And in the process a new set of DNA is created and life is duplicated. Evil and death tear apart the present life. Forgiveness and love restores life, but its not a restoration to the previous conditions, its a restoration to new life, eternal life as seen (witnessed, empirically experienced) in the resurrection of Jesus.
Because the new life that Jesus lived after the resurrection had a physical form, an empirical form, and because when we forgive each other we are converting the semiotic form that Jesus represents into an empirical form of earthly experience, I would like to think that we are also creating an eternal empirical form of an eternal life. It seems that many passages in the NT indicate that eternal life is something that we receive not only because God forgives us, but because we forgive others.
20. Thanks, Maria. I have been working on a response to your emails which I’ll be able to post soon.
And I’m posting on Shusaku Endo later today. Thanks everyone.
21. That image of the DNA being torn apart and re-united with another “torn” DNA is a powerful image.
“Except a seed fall into the ground and die,” right?
Semiotics, word theory, is filled with deaths and rebirths within words, sustaining them. It is like Heidegger’s unconcealment (truth) and re-concealment being dialectically related.
I think from a semiotics standpoint, I want to comment on the nature of both what you call semiotic knowledge and what you call empirical knowledge, though I certainly see what you mean. The two are much more inter-related than usually appears on the surface. It’s fascinating to remember that the “word” sustains even what we think of as empirical being. More on this soon. (I keep saying soon, but truly….)
As for forgiveness, it’s forgiving oneself that is often most difficult, isn’t it? The intricate interrelationship between the present and the future is something I’ll be hitting on too, I hope. Thanks. More soon.
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15bfa179173d9026 | In silico chemistry: Pursuit of chemical accuracy
Kirk A. Peterson from the Department of Chemistry, Washington State University discusses the fundamentals of in silico chemistry
In silico chemistry simply refers to carrying out investigations of chemical processes entirely by computational methods. Over the last few decades, computational chemistry has been an invaluable tool in understanding chemical reactivity, structure, and thermodynamics. This is particularly true for short-lived species such as free radicals and reaction intermediates, as well as novel species that have yet to be observed by experiment. Computer modelling can also provide for the study of a chemical system in a pristine, well-defined environment without some of the additional complexities occurring in an experiment that might complicate the interpretation of a fundamental process.
With the increasing power of modern computing resources, in silico chemistry has seen significant success in the prediction of thermochemical properties of molecules in the gas phase, e.g., bond enthalpies, heats of formation, ionization potentials, etc. The benchmark standard has long been the so-called “chemical accuracy” threshold, loosely defined as an accuracy of 1 kcal/mol (~4 kJ/mol).
For molecules consisting solely of atoms ranging from hydrogen to chlorine, this threshold can now almost routinely be met, and for relatively small molecules (perhaps not more than 5 non-hydrogen atoms) accuracies on the order of 0.25 kcal/mol (1 kJ/mol) are possible. The latter certainly then becomes competitive with or even exceeds the accuracy attainable with many experimental approaches to these quantities.
However, as the numbers of electrons increase and the electronic structure of the elements become more complicated, e.g., transition metals and heavy elements such as lanthanides and actinides, achieving 1 kcal/mol accuracy becomes much more difficult for purely first principle or ab initio methods. It does seem clear from current research, however, that accuracies of ~3 kcal/mol are possible even in these instances. So, what exactly is required to attain such a high level of accuracy that is reliable enough to perhaps even replace experiment in some cases?
Schrödinger’s equation
Just as in calculating the physics of everyday macroscopic particles where Newton’s equation, F=ma, must be solved, the relevant equation of quantum mechanics that describes the properties of atoms, electrons, and hence molecules, is the Schrödinger equation (SE), HΨ=EΨ. This modest-looking equation yields the possible energy states (E) of the system, as well as the wavefunction (Ψ), which is related to the probability of finding the quantum mechanical particles at some location in space.
For a given molecule (or collection of molecules), this equation describes the motion of the individual nuclei together with all their associated electrons, which, unfortunately except for the very simplest of molecules, makes this equation impossible to exactly solve, and intractable to even obtain approximate solutions.
Fortunately, the Born-Oppenheimer approximation, which recognises that nuclei and the much lighter electrons move at very different speeds, allows their motion to be separated. This leads to two separate Schrödinger equations, one for the nuclei and one for the electrons moving in the presence of the nuclei fixed in space. One then solves the latter electronic SE at different positions of the nuclei (bond lengths, angles, etc.) and the resulting potential energy function is used in the nuclear SE to obtain energies of molecular rotation, vibration, etc.
Relevant to the present discussion, thermodynamic properties can also be extracted from these calculations – with the main limitation to the final accuracy coming from solutions of the electronic SE. Unfortunately, even this cannot be exactly solved for any molecule larger than H2+, and approximate solutions yielding the desired accuracy can be very computationally demanding. In particular, this cost very steeply increases with the size of the chemical system, in terms of both the number of electrons and nuclei.
The way forward for accurate ab initio thermochemistry is via so-called composite methods [1, 2]. In these calculations, the results of a series of smaller, tractable calculations are combined to approximate the results of a single large target calculation that would presumably be impossible or impractical to carry out. In order to achieve chemical or sub-chemical accuracy, all appreciable sources of error in a calculation must be accounted for in a systematic way. The two central ones are related to how the wavefunction Ψ is approximated in the solution of the SE, and they are strongly coupled: (a) how is Ψ represented in terms of the underlying atomic orbitals and (b) how are these orbitals numerically represented.
The first is generally referred to as the quantum mechanical method, while the second refers to what is called the basis set, generally consisting of Gaussian-type functions. A major breakthrough for quantitative in silico chemistry was made more than 25 years ago when Dunning [3] introduced the family of correlation consistent Gaussian basis sets, which had the unique property of providing systematic convergence towards the complete basis set (CBS) limit, i.e., a limiting result that corresponds to the exact solution of the chosen quantum mechanical method. This effectively eliminates one of the major sources of error in a very systematic way. With contributions from our research group at Washington State University, correlation consistent families of basis sets are now available for nearly the entire periodic table [4, 5].
Hence a composite thermochemistry calculation with a goal of chemical accuracy begins with the use of an accurate, but not exact, quantum mechanical method with a sequence of correlation consistent basis sets of increasing size. These individual solutions to the electronic SE are then extrapolated to the CBS limit to remove basis set errors. Smaller contributions are then accounted for which may be chosen based on their appropriateness for the chemical system under study. Generally, these always include the effects of special relativity and molecular vibrational effects, but could also involve more esoteric contributions such as Born-Oppenheimer breakdown terms (when hydrogen atoms are involved) or quantum electrodynamics (QED). The resulting accuracy is then nearly completely dictated by the initial choice of quantum mechanical method. Coupled cluster methods are often the best choice since they can in principle be extended towards the exact solution, albeit with high computational cost.
The key to chemically accurate ab initio thermochemistry is clear – a systematic approach that in principle leads towards the exact solution of a relativistic SE is mandatory, and fortuitous error compensation must be avoided at all costs. This is what leads to truly predictive in silico chemistry.
1 Peterson, K. A.; Feller, D.; Dixon, D. A. Chemical accuracy in ab initio thermochemistry and spectroscopy: current strategies and future challenges. Theoretical Chemistry Accounts 2012, 131, 1079.
2 Dixon, D. A.; Feller, D.; Peterson, K. A. A practical guide to reliable first principles computational thermochemistry predictions across the periodic table. In Annual Reports in Computational Chemistry; Elsevier, 2012; Vol. 8, pp 1-28.
3 Dunning, T. H., Jr. Gaussian-basis sets for use in correlated
molecular calculations. I. The atoms boron through neon and
hydrogen. The Journal of Chemical Physics 1989, 90, 1007.
4 Feng, R.; Peterson, K. A. Correlation consistent basis sets for
actinides. II. The atoms Ac and Np–Lr. The Journal of Chemical Physics 2017, 147, 084108.
5 Figgen, D.; Peterson, K. A.; Dolg, M.; Stoll, H. Energy-consistent pseudopotentials and correlation consistent basis sets for the
5d elements Hf-Pt. The Journal of Chemical Physics 2009, 130, 164108.
Please note: this is a commercial profile
Kirk A Peterson
Edward R Meyer
Professor of Chemistry
Department of Chemistry
Washington State University
Tel: +1 509 335 7867
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25ba312462a4ae2f | Tuesday, September 20, 2005
Faster than light or not
I don't know about the rest of the world but here in Germany Prof. Günter Nimtz is (in)famous about his display experiments that he claims show that quantum mechanical tunneling happens instantaneously rather than according to Einstein causality. In the past, he got a lot of publicity for that and according to Heise online he has at least a new press release.
All these experiments are similar: First of all, he is not doing any quantum mechanical experiments but uses the fact that the Schrödinger equation and the wave equation share similarities. And as we know, in vacuum, Maxwell's equations imply the wave eqaution, so he uses (classical) microwaves as they are much easier to produce than matter waves of quantum mechanics.
So what he does is to send a pulse these microwaves through a region where "classically" the waves are forbidden meaning that they do not oscillate but decay exponentially. Typically this is a waveguide with diameter smaller than the wavelength.
Then he measures what comes out at the other side of the wave guide. This is another pulse of microwave which is of course much weaker so needs to be amplified. Then he measures the time difference between the maximum of the weaker pulse and the maximum of the full pulse when the obstruction is removed. What he finds is that the weak pulse has its maximum earlier than the unobstructed pulse and he interprets that as that the pulse has travelled through the obstruction at a speed greater than the speed of light.
Anybody with a decent education will of course immediately object that the microwaves propagate (even in the waveguide) according to Maxwell's equations which have special relativity build in. Thus, unless you show that Maxwell's equations do not hold anymore (which Nimtz of course does not claim) you will never be able to violate Einstein causality.
For people who are less susceptible to such formal arguments, I have written a little programm that demonstrates what is going on. The result of this programm is this little movie.
The programm simulates the free 2+1 dimensional scalar field (of course again obeying the wave equation) with Dirichlet boundary conditions in a certain box that is similar to the waveguide: At first, the field is zero everywhere in the strip-like domain. Then the field on the upper boundary starts to oscillate with a sine wave and indeed the field propagates into the strip. The frequency is chosen such that that wave can in fact propagate in the strip.
(These are frames 10, 100, and 130 of the movie, further down are 170, 210, and 290.) About in the middle the strip narrows like in the waveguide. You can see the blob of field in fact enters the narrower region but dies down pretty quickly. In order to see anything, in the display (like for Nimtz) in the lower half of the picture I amplify the field by a factor of 1000. After the obstruction ends, the field again propagates as in the upper bit.
What this movie definitely shows is that the front of the wave (and this is what you would use to transmit any information) everywhere travels at the same speed (that if light). All what happens is that the narrow bit acts like a high pass filter: What comes out undisturbed is in fact just the first bit of the pulse that more or less by accident has the same shape as a scaled down version of the original pulse. So if you are comparing the timing of the maxima you are comparing different things.
Rather, the proper thing to compare would be the timing when the field first gets above a certain level, one that is actually reached by the weakend pulse. Then you would find that the speed of propagation is the same independant of the obstruction being there or not.
Update: Links updated DAMTP-->IUB
Friday, September 16, 2005
Negative votes and conflicting criteria
Yesterday, Matthijs Bogaards and Dierk Schleicher ran a session on the electoral system for the upcoming general election we are going to have on Sunday in Germany. I had thought I I know how it works but I was proven wrong. Before I was aware that there is something like Arrow's impossibility theorm which states that there is a certain list of criteria your electoral system is supposed to fulfill but which cannot hold all at the same time for any implementation. What typically happens are cyclic preferences (there is a majority for A over B and one for B over C and one for C over A) but I thought all this is mostly academic and does not apply to real elections. I was proven wrong and there is a real chance that there is a paradoxical situation coming up.
Before explaining the actual problem, I should explain some of the background. The system in Germany is quite complicated because it tries to accomodate a number of principles: First, after the war, the British made sure the system contains some component of constituency vote: Each local constituency (electoral district for you Americans) should send one candidate to parliament that is in principle directly responsible to the voters in that district so voters have something like "their representative". Second, proportional vote, that is the number of seats for a party should reflect the percentage of votes for that party in the popular vote. Third, Germany is a federal republic, so the sixteen federal states should each send their own representatives. Finally, there are some practical considerations like the number of seats in parliament should be roughly 600 and you shouldn't need a PhD in math and political science to understand your ballot.
So this is how it works. Actually, it's slightly more complicated but that shall not bother us here. And I am not going into the problem of how to deal with rounding errors (you can of course only have integer seats) which brings with it its own paradoxes. What I am going to cover is how to deal with the fact, that the number of seats has to be non-negative:
The ballot has two columns: In the first, you vote for a candidate from your constituency (which is nominated by its party). In the second, you vote for a party for the proportional vote. Each voter makes one cross in each column, one for a candidate from the constituency and one for a party in the proportional vote. There are half as many constituencies as there are seats in parliament and these are filled immediately according to majority vote of the first column.
The second step is to count the votes in the second column. If a party neither gets more than five percent of those nor wins three or more constituencies their votes are dropped. The rest is used to work out how many of the total of 600 seats each of the parties gets.
Now comes the federal component: Let's consider party A and assume the popular vote says they should get 100 seats. We have to determine how these 100 seats are distributed between the federal states. This is again done proportionally: Party A in federal state (i) gets that percentage of the 100 seats that reflects the percentage of the votes for party from state (i) of the total votes for party A in all of Germany. Let's say this is 10. Further assume that A has won 6 constituencies in federal state (i). Then, in addition to these 6 candiates from the constituencies, the top four candidates from party A's list for state (i) are send to Berlin.
So far, everything is great: Each constituency has "their representative" and the total number of seats for each party is proportional to its share of the popular vote.
Still, there is a problem: The two votes in the two columns are independent. And as the constituencies are determined by majority vote, except in a few special cases (Berlin Kreuzberg where I used to live before moving to Cambridge being one with the only constituency winner from the green party) it does not make much sense to vote for a constituency candidate that is not nominated by one of the two big parties. Any other vote would likely be irrelevant and effectively your only choice is between the candidate of SPD or CDU.
Because of this, it can (and in fact often does for the two big parties) happen that a party wins more constituencies in a federal state than it is entitled to for that state according to the popular vote. In that case (because there are no negative numbers of candidates from the list to balance this) the rule is that all the constituency winners go to parliament and none from the list of that party. The parliament is enlarged for these "excess mandates". So that party gets more seats than their proportion of the popular vote.
This obviously violates the principle of proportional elections but it gets worse: If that happens in a federal state for party A you can hurt this party by voting for it: Take the same numbers as above but assume A has won 11 constituencies in (i). If there are no further excess mandates, in the end, A gets 101 seats in the enlarged parliament of 601 seats. Now, assume A gets an additional proportional vote. It is not impossible that this does not increase A's total share of 100 votes for all of Germany but increases to proportional share for the A's candidates in federal state (i) from 10 to 11. This does not change anything for the represenatives from (i), still the 11 constituency candidates go to Berlin but there is no excess mandate anymore. Thus, overall, A sends only 100 representatives to a parliament of 600, one less than with the additional vote!
As a result, in that situation the vote for A has a negative weight: It decreases A's share in the parliament. Usually, this is not so much of a problem, because the weights of votes depend on what other people have voted (which you do not know when you fill out your ballot) and chances are much higher that your vote has positive weight. So it is still save to vote for your favourite party.
However, this year, there is one constituency in Dresden in the federal state of Saxony where one of the candidates died two weeks before election day. To ensure equal chances in campaining, the election in that constituency has been postponed for two weeks. This means, voters there will know the result from the rest of the country. Now, Saxony is known to be quite conservative so it is not unlikely that the CDU will have excess mandates there. And this might just yield the above situation: Voters from Dresden might hurt the CDU by voting for them in the popular vote and they would know if that were the case. It would still be democratic in a sense, it's just that if voters there prefer CDU or FDP they should vote for FDP and if they prefer SPD or the Greens they should vote for CDU. Still, it's not clear if you can explain that to voters in less then two weeks... I find this quite scary, especially since all polls predict this election to be extremely close and two very different outcomes are withing one standard deviation.
If you are interested in alternative voting systems, Wikipedia is a good starting point. There are many different ones and because of the above mentioned theorem they all have at least one drawback.
Yesterday, there was also a brief discussion of whether one should have a system that allows fewer or more of the small parties in parliament. There are of course the usual arguments of stability versus better representation of minorities. But there is another argument against a stable two party system that is not mentioned often: This is due to the fact that parties can actually change their policies to please more voters. If you assume, political orientation is well represented by a one dimensional scale (usually called left-right), then the situation of icecream salesmen on a beach could occur: There is a beach of 4km with two competing people selling icecream. Where will they stand? For the customers it would be best if they are each 1km from the two ends of the beach so nobody would have to walk more than 1km to buy an icecream and the average walking distance is half a km. However, this is an unstable situation as there is an incentive for each salesman to move further to the middle of the beach to increase the number of customers to which he is closer
than his competitor.
So, in the end, both will meet in the middle of the beach and customers have to walk up to 2km with an average distance of 1km. Plus if that happens with two parties in the political spectrum they will end up with indistinguishable political programs and as a voter you don't have a real choice anymore. You could argue that this has already taken place in the USA or Switzerland (there for other reasons) but that would be unfair to the Democrats.
I should have had many more entries here about politics and the election like my role models on the other side of the Atlantic. I don't know why these never materialised (vitualised?). So, I have to be brief: If you can vote on Sunday, think of where the different parties actually have different plans (concrete, rather than abstract "less unemployment" or "more sunshine") and what the current government has done and if you would like to keep it that way (I just mention the war in Iraq and foreign policy, nuclear power, organic food as a mass market, immigration policy, tax on waste of energy, gay marriage, student fees, reform of academic jobs, renewable energy) your vote should be obvious. Mine is.
The election is over and everybody is even more confused than before. As the obvious choices for coalitions do not have a majority one has to look for the several colourful alternatives and the next few weeks will show us which of the several impossibilities will actually happen. What will definitely happen is that in Dresden votes for the CDU will have negative weight (linked page in German with an excel sheet for your own speculations). So, Dresdeners, vote for CDU if you want to hurt them (and you cannot convince 90% of the inhabitants to vote for the SPD).
Wednesday, September 14, 2005
Natural scales
When I talk to non-specialists and mention that the Planck scale is where quantum gravity is likely to become relevant sometimes people get suspicious about this type of argument. If I have time, I explain that to probe smaller length details I would need so much CM energy that I create a black home and thus still cannot resolve it. However, if I have less time, I just say: Look, it's relativistic, gravity and quantum, so it's likely that c, G and h play a role. Turn those into a length scale and there is the Planck scale.
If they do not believe this gives a good estimate I ask them to guess the size of an atom: Those are quantum objects, so h is likely to appear, the binding is electromagnetic, so e (in SI units in the combination e^2/4 pi epsilon_0) has to play a role and it comes out of the dynamics of electrons, so m, the electron mass, is likely to feature. Turn this into a length and you get the Bohr radius.
Of course, as all short arguments, this has a flaw: there is a dimensionless quantity around that could spoil dimension arguments: alpha, the fine-structure constant. So you also need to say, that the atom is non-relativistic, so c is not allowed to appear.
You could similarly ask for a scale that is independant of the electric charge, and there it is: Multiply the Bohr radius by alpha and you get the electron Compton wavelength h/mc.
You could as well ask for a classical scale which should be independent of h: Just multiply another power of alpha and you get the classical electron radius e^2/4 pi epsilon_0 m c^2. At the moment, however, I cannot think of a real physical problem where this is the characteristic scale (NB alpha is roughly 1/137, so each scale is two orders of magnitude smaller than the previous).
Update: Searching Google for "classical electron radius" points to scienceworld and wikipedia, both calling it the "Compton radius". Still, there is a difference of an alpha between the Compton wavelength and the Compton radius.
Thursday, September 08, 2005
Reading through the arxive's old news items I became aware of hep-th/9203227 for which the abstract reads
\Paper: 9203227
From: harvey@witten.uchicago.edu (J. B. Harvey)
Date: Wed 1 Apr 1992 00:25 CST 1992
A solvable string theory in four dimensions,
by J. Harvey, G. Moore, N. Seiberg, and A. Strominger, 30 pp
\We construct a new class of exactly solvable string theories by generalizing
the heterotic construction to connect a left-moving non-compact Lorentzian
coset algebra with a right-moving supersymmetric Euclidean coset algebra. These
theories have no spacetime supersymmetry, and a generalized set of anomaly
constraints allows only a model with four spacetime dimensions, low energy
gauge groups SU(3) and spontaneously broken SU(2)xU(1), and three families
of quarks and leptons. The model has a complex dilaton whose radial mode
is automatically eaten in a Higgs-like solution to the cosmological
constant problem, while its angular mode survives to solve the strong CP
problem at low energy. By adroit use of the theory of parabolic cylinder
functions, we calculate the mass spectrum of this model to all orders in
the string loop expansion. The results are within 5% of measured values,
with the discrepancy attributable to experimental error. We predict a top
quark mass of $176 \pm 5$ GeV, and no physical Higgs particle in the spectrum.
It's quite old and there are some technical problems downloading it.
Tuesday, September 06, 2005
Local pancake and axis of evil
This would then be an explanation of this axis of evil.
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a4b83247def189aa | Take the 2-minute tour ×
It's well-known that hydrogen atom described by time-independent Schrödinger equation (neglecting any relativistic effects) is completely solvable analytically.
But are any initial value problems for time-dependent Schrödinger equation for hydrogen solvable analytically - maybe with infinite nuclear mass approximation, if it simplifies anything? For example, an evolution of some electron wave packet in nuclear electrostatic field.
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What do you mean by "analytically"? You probably don't mean the math definition, which is that the function converges to its Taylor series. If you mean "involving simple functions" the you should know there's no qualitative difference between numerical integration and special functions. In fact many common special functions are evaluated by your computer via the differential equation they satisfy. – Chris White Feb 23 '14 at 22:44
@ChrisWhite I mean explicit solution, in terms of such functions, which don't require to set up a dense spatial grid and propagate the solution in small temporal steps to find the value at a given point in spacetime with required precision. – Ruslan Feb 24 '14 at 4:15
3 Answers 3
What you do have available is an explicit knowledge of the eigenvalues and eigenvectors (also for the continuous spectrum). By expanding your initial wavepacket in terms of the eigenvectors you then obtain its value for later times as a sum (or integral for continuous spectrum) with added weight factors exp[-i$\lambda$t], where $\lambda$ is the eigenvalue associated with the corresponding eigenvector.
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The problem with this approach is that one needs to do lots of numeric 3D integrations to find projections of initial state on eigenstates. And if the initial state has too high localization so that much of continuous spectrum states have contribution to initial state, then one would have to do even more integrals. This approach doesn't seem as a very good one in general, this is why I asked about direct solution of time-dependent equation. – Ruslan Jan 28 '14 at 12:34
Yes, I see your problem. Did you consider a Green's function approach? – Urgje Jan 29 '14 at 11:31
I don't know much about Green's functions. Could you elaborate on how to apply them to this problem (and what to read to understand your explanation better)? – Ruslan Jan 29 '14 at 11:38
The Green's function of the time-independent case is known, both in coordinate and momentum representation. Whether or not that helps in your case I do not know. – Urgje Jan 30 '14 at 16:08
Urgje gave you the answer. In its basic form (Schrödinger), the Hamiltonian is time-independent, therefore the general theory will tell you how to write the general solution of the Schrödinger equation as the sum/integral of the solutions of the spectral equation weighed by time-dependent exponentials.
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Why repeat already existing answer, especially if comments under it say why it's not acceptable? – Ruslan Feb 23 '14 at 16:58
Solution of an initial value problem can be written as integral of the initial function $\psi_0$ multiplied by the propagator of the Schr. equation. Depending on the function $\psi_0$, the integral may or may not be calculable in terms of simple functions. I do not know of any initial function $\psi_0$ and potential $A(t)$ that would admit simple exact solution; the equation with time-dependent term is difficult to solve. More rewarding way seems to be to find the solution with a computer. The real problem is I think elsewhere - how do we find appropriate function $\psi_0$ to describe real atoms? Often the first eigenfunction of the Hamiltonian is used, but I do not think this is particularly well motivated.
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My question was "are there any initial value problems", not "what is the definite solution". Of course, by initial conditions I mean not such trivial ones as superposition of finite eigenfunctions, but some form of wave packets. – Ruslan Mar 25 '14 at 19:06
OK, I've edited my answer accordingly. Sorry I can't be of more help. – Ján Lalinský Mar 26 '14 at 21:07
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a5faac878358def2 | Course Description
From 15th ISeminar 2011/12 Operator Semigroups for Numerical Analysis
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The course concentrates on the numerical solution of initial value problems of the type
u′(t) = Au(t) + f(t), \quad t \ge 0,
u(0) = u_0 \in D(A),
where A is a linear operator with dense domain of definition D(A) in a Banach space X, and u0 is the initial value. A model example is the Laplace operator A = Δ with appropriate domain in the Hilbert space L2(Ω). In this case the above partial differential equation describes heat conduction inside Ω. One way of finding a solution to this initial value problem is to imitate the way in which one solves linear ordinary differential equations with constant coefficients: First define the exponential etA in suitable way. Then the solution of the homogeneous problem is given by this fundamental operator applied to the initial value u0 , i.e., u(t) = etAu0. This is where operator semigroup theory enters the game: the fundamental operators T(t): = etA form a so-called strongly continuous semigroup of bounded linear operators on the Banach space X. That is to say the functional equation T(t + s) = T(t)T(s) and T(0) = I holds together with the continuity of the orbits t\mapsto T(t)u_0. If such a semigroup exists, we say that the initial value problem is well-posed. Once existence and uniqueness of solutions are guaranteed, the following numerical aspects appear.
• In most cases the operator A is complicated and numerically impossible to work with, so one approximates it via a sequence of (simple) operators Am hoping that the corresponding solutions \mathrm{e}^{tA_m} (expected to be easily computable) converge to the solution of the original problem etA in some sense. This procedure is called space discretisation. This discretisation may indeed come from a spatial mesh (e.g., for a finite difference method) or from some not so space-related discretisations, e.g., from Fourier-Galerkin methods.
• Equally hard is the computation of the exponential of an operator A. One idea is to approximate the exponential function z\mapsto\mathrm{e}^z by functions r that are easier to handle. A typical example, known also from basic calculus courses, is the backward Euler scheme r(z) = (1 − z) − 1. In this case the approximation means r(0) = r'(0) = e0, i.e., the first two Taylor coefficients of r and of the exponential function coincide. This leads to the following idea. If r(tA) is approximately the same as etA for small values of t (up to an error of magnitude t2), we may take the nth power of it. To compensate for the growing error, we take decreasing time steps as n grows and obtain
\left[r(\tfrac{t}{n}A)\right]^n \approx \big[\mathrm{e}^{\tfrac{t}{n}A}\big]^n=\mathrm{e}^{tA}
by the semigroup property. This procedure is called temporal discretisation.
• Due to numerical reasons, one is usually forced to combine the above two methods and add further spice to the stew: operator splitting. This is usually done when the operator A has a complicated structure, but decomposes into a finite number of parts that are easier to handle.
In semigroup theory the above methods culminate in the famous Lax Equivalence Theorem and Chernoff’s Theorem, describing precisely the situation when these methods work. In this course we shall develop the basic tools from operator semigroup theory needed for such an abstract treatment of discretisation procedures.
Topics to be covered include:
• initial value problems and operator semigroups,
• spatial discretisations, Trotter–Kato theorems, finite element and finite difference approximations,
• fractional powers, interpolation spaces, analytic semigroups,
• the Lax Equivalence Theorem and Chernoff’s Theorem, error estimates, order of convergence, stability issues,
• temporal discretisations, rational approximations, Runge–Kutta methods, operator splitting procedures,
• applications to various differential equations, like inhomogeneous problems, non-autonomous equations, semilinear equations, Schrödinger equations, delay differential equations, Volterra equations,
• exponential integrators.
Some of these topics will be elaborated on in Phase 2, where the students will have the possibility to work on projects which are related to active research.
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I have no background in physics. This isn't for homework, just for interest.
In quantum physics, it's described that a particle can act as both a particle and a wave.
Quoted from HowStuffWorks "Wave-Particle Duality"
I have trouble visualizing a particle transforming into a wave and vice-versa. The quote says that light travels away from a source as an electromagnetic wave. What does that even look like? How can I visualize "a wave"? Is that supposed to look like some thin wall of advancing light? And then, the quote says, at the moment of impact, the wave disappears and a photon appears. So, a ball of light appears? Something that resembles a sphere? How does a sphere become something like an ocean wave? What does that look like?
My (completely uneducated) guess is, by a particle becoming a wave, does that mean that this expansive wave is filled with tons of ghost copies of itself, like the one electron exists everywhere in this expansive area of the wave, and then when it hits the wall, that property suddenly disappears and you're left with just one particle. So, this "wave", is really tons of identical copies of the same photon in the shape and form and with the same properties of, a wave? My guess comes from reading about how shooting just one photon still passes through two slits in the double-slit experiment. So the photon actually duplicated itself?
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Possible duplicate: – Qmechanic Nov 6 '12 at 22:13
The wave nature is what evolves with time and the particle nature is what is observed. This link may help:… – centralcharge Jun 18 '13 at 9:41
What we observe in nature exists in several scales. From the distances of stars and galaxies and clusters of galaxies to the sizes of atoms and elementary particles.
Now we have to define "observe".
Observing in human size scale means what our ears hear, what our eyes see, what our hands feel, our nose smells , our mouth tastes. That was the first classification and the level of "proxy", i.e. intermediate between fact and our understanding and classification, which is biological. (the term proxy is widely used in climate researches)
A second level of observing comes when we use proxies, like meters, thermometers, telescopes and microscopes etc. which register on our biological proxies and we accumulate knowledge. At this level we can overcome the limits of the human scale and find and study the enormous scales of the galaxies and the tiny scales of the bacteria and microbes. A level of microns and milimeters. We observe waves in liquids with such size wavelengths
Visible light is of the order of Angstroms, 10^-10 meters. As science progressed the idea of light being corpuscles ( Newton) became overcome by the observation of interference phenomena which definitely said "waves".
Then came the quantum revolution, the photoelectric effect (Particle), the double slit experiments( wave) that showed light had aspects of a corpuscle and aspects of a wave. We our now in a final level of use of proxy, called mathematics
The wave particle duality was understood in the theory of quantum mechanics. In this theory depending on the observation a particle will either react as a "particle" i.e. have a momentum and location defined , or as a wave, i.e. have a frequency/wavelength and geometry defining its presence BUT, and it is a huge but, this wavelength is not in the matter/energy itself that is defining the particle , but in the probability of finding that particle in a specific (x,y,z,t) location. If there is no experiment looking for the particle at specific locations its form is unknown and bounded by the Heisenberg Uncertainty Principle.
What is described with words in the last paragraph is rigorously set out in mathematical equations and it is not possible to understand really what is going on if one does not acquire the mathematical tools, as a native on a primitive island could not understand airplanes. Mathematics is the ultimate proxy for understanding quantum phenomena.
Now light is special in the sense that collectively it displays the wave properties macroscopically, and the specialness comes from the Maxwell Equations which work as well in both systems, the classical and the quantum mechanical, but this also needs mathematics to be comprehended.
So a visualization is misleading in the sense that the mathematical wave function coming from the quantum mechanical equations is like a "statistical" tool whose square gives us the probability of observing the particle at (x,y,z,t). Suppose that I have a statistical probability function for you, that you may be in New York on 17/10/2012 and probabilities spread all over the east coast of the US. Does that mean that you are nowhere? does that mean that you are everywhere? Equally with the photons and the elementary particles. It is just a mathematical probability coming out of the inherent quantum mechanical nature of the cosmos.
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Thanks for the really detailed post and the concept of proxies. Surely mathematics isn't "the final" proxy to observe and learn? There may be others? – Jason Oct 17 '12 at 7:38
Mathematics itself has several levels used in physics, and those are continually expanding as research progresses. The question becomes esoteric to mathematics, imo. – anna v Oct 17 '12 at 10:35
Any source that is content to describe scientific theories in terms of black magic is worse than useless.
How can I visualize "a wave"?
You can visualize it as y=sin(x). The wave's strength oscillates both over time (if you stand in one place and watch it pass), and space (if you "freeze" it in time). Light is more complex than more familiar waves (eg, waves in water), in that it's made up of oscillating electrical and magnetic waves.
So the photon actually duplicated itself?
No, it hasn't duplicated itself, just spread itself out so that it can pass through two slits simultaneously (just like a wave in water would do). The "collapsing" occurs due to the quantization of light, which is evident when the light gets absorbed by matter.
Realize that trying to visualize is a very limited vehicle for understanding quantum-scale physics. Since such tiny scales are out the domain of our ordinary sense perception, all we have available is hypothesis based on experiments. So, to understand a theory is to understand the paradoxes and experiments that gave rise to it; in the case of the wave-particle duality, that would be (among others) the double slit experiment, as you mentioned. On the quantization of light, a good class of experiments to ponder is those of emission/absorption spectra of elements.
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Visualization is difficult since, for example, the 'waves' that we're talking about a probability amplitude density waves. (In fact, we (teachers) should probably discourage initiates into the subject from trying to do this outright.)
One thing that has always helped when I describe this to folks is something I got from Paul Tipler's undergrad book (!) a long time ago when I was a teaching assistant. He makes a very useful distinction: when an electron (as a canonical example of a quantum 'particle') propagates it behaves in a wavelike fashion; when it exchanges energy with other systems, it does so discretely, like a particle.
In this sense then the 'duality' of quantum mechanics is less paradoxical and perhaps less seemingly contradictory. Electrons behave as waves and particles but never 'at the same time.'
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Particles in quantum mechanics are always particles and act as particles. E.g. an electron or a photon are always defined as particles according to the Standard Model and Wigner representation. E.g. an electron is defined as a particle with mass $m_e$, spin 1/2 and charge e and always behaves as a particle, never as a wave. As emphasized in CERN website "everything in the Universe is found to be made from twelve basic building blocks called fundamental particles".
There is not need for a wave-particle duality in modern interpretations of quantum mechanics (in fact quantum mechanics can be formulated without wavefunctions) and the wave-particle duality term is often considered a "myth", Klein prefers the term "misnomer". The historical roots of the wave-particle duality myth are explained in Ballentine's celebrated textbook Quantum Mechanics: A Modern Mechanics Development:
As stated by Klein:
"The miraculous "wave-particle duality" continues to flourish in popular texts and elementary text books. However, the rate of appearance of this term in scientific works has been decreasing in recent years."
Look Akira Tonomura’s video clip .wmv .mpeg for a beautiful demonstration of the appearance of statistical wave pattern on a double-slit interference experiment when a large number of independent single particles (electrons) impact the detector.
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In advanced formulations of QM, wavefunctions are substituted by kets, density matrices, Wigner distributions... Have you heard of some ket-particle, matrix-particle, or distribution-particle duality? No, because there is none. Moreover, wavefunctions are not waves, are functions. – juanrga Oct 21 '12 at 11:07
Thank you! It was -4 some days ago. I would like to know two things: (i) why do they appeal to a hypothetical wave-particle duality, when $\Psi$ is not a physical wave but a mathematical function? and (ii) what term would they use in those advanced formulations of QM, where there is no wavefunction $\Psi$? E.g. in the Wigner-Moyal formulation of QM the state of the system is given by the Wigner distribution W and the evolution equation is not the Schrödinger equation but the Moyal equation $\dot{W} = \{H, W\}_{M}$. – juanrga Oct 22 '12 at 14:59
I really think, this answer is spot on. – Iota Mar 12 '14 at 21:04
POV. Many physicists would agree with you. Particle physicists, for example. And Feynman. But there is no universal consensus on this, it is merely a POV. Axiomatic QFT people, like doing Streater--Wightman stuff, believe in fields, and would not agree with you. Oh, and string-theorists. Of course I have to admit I myself do not believe in strings, but you should not pretend that no physicists believe that strings, not particles, are the fundamental building blocks. CERN is a resort for particle physicists: obviously a particle physicist will believe in particles. – joseph f. johnson Nov 26 '15 at 18:14
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in the Born Oppenheimer Approximation, one expands the molecular wavefunction $\Psi(x,X)$ in terms of the electronic wavefunctions $\phi(x;X)$: $$\Psi(x,X)= \sum_k(c(X)_k\phi(x;X)_k)$$ ($x$ are the electronic coordinates and $X$ are the nucleonic coordinates)
Now, since the electronic wavefunctions are eigenstates of the electronic Hamiltonian, the constitute a complete basis of the electronic space. Thus any electronic wavefunction can be expanded in terms of the eigenfunctions. But, how can we be sure that any molecular wavefunction can be expanded in terms of the electronic wavefunctions? How can we be sure that the molecular Hilbert space is not larger than the space which is spanned by the eigenstates of the electronic Hamiltonian?
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I thought the Born Approximation was the latest Matt Damon movie. Thanks for the clarification! – twistor59 Feb 4 '13 at 7:24
The key word here is approximation. Its not the that the hilbert space of the full molecule is not larger, its that the velocities of the much more massive nucleons can be neglected and only their positions need to be considered. In fact, the BO approximation breaks down in some cases when non-adiabatic effects are not negligible (e.g. loss or gain of energy due to changes in electronic orbits). An important point that Born and Oppenheimer make in their paper is their approximation is possible because earlier classically based approximations overstated the nuclear contribution because it neglected spin, which is purely a quantum phenomenon with no classical analog. This causes the nuclear contribution to be of the fourth order instead of the second order. This is another excellent case to show where classical mechanics breaks down in the description of the universe.
I guess to answer your question though, the hilbert space of the full system is actually larger then the electronic states, its just that under certain conditions some nuclear degrees of freedom can be ignored.
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I know this question was asked a long time ago, but since I thought very hard about the same question today and didn't find the other answer very helpful, I decided to write my own. The problem is that OP is not really asking about why the product form used in the BO approximation is valid, but how the given expansion (which is claimed to be exact, see e.g. the scan of a book chapter provided in this post: Non Adiabatic Coupling Term in Born Oppenheimer Approximation) is justified.
Even though this problem confused me a lot, it is actually quite simple, and I understood it best using some mathematical language. I call the space of electronic coordinates $A$ (so $x \in A$) and $\Psi(x, X)$ is the exact solution of the complete (electronic and nuclear) Schrödinger equation. Then for each set of nuclear coordinates $X$, we can define a function \begin{equation} \Psi_X: A \rightarrow \mathbb{C}, \quad \Psi_X(x) := \Psi(x, X). \end{equation} We know that for all $X$, the set of eigenfunctions of the electronic Hamiltonian, {$\phi(x;X)_k$}, is a complete basis for electronic wave functions. We now expand each electronic wave function $\Psi_X$ in a different set of basis functions, namely \begin{equation} \Psi_X(x) = \sum_k c(X)_k \phi(x;X)_k, \end{equation} where $c(X)_k$ is the expansion coefficient belonging to the $k$th basis function associated with the electronic wave function parametrized by $X$. But since $\Psi_X(x) = \Psi(x, X)$ we already obtained what OP asked for.
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5b9364ffed3c3c29 | Ab Initio Molecular Orbital Theory
To make a quantum mechanical model of the electronic structure of a molecule, we must solve the Schrödinger equation.
Solving this equation is a very difficult problem and cannot be done without making approximations. We have covered some of these approximations in the Semiempirical MO Theory handout. In this handout we focus on ab initio methods of solving the equation, in which no integrals are neglected in the course of the calculation.
The Born-Oppenheimer Approximation
The first approximation is known as the Born-Oppenheimer approximation, in which we take the positions of the nuclei to be fixed so that the internuclear distances are constant. Because nuclei are very heavy in comparison with electrons, to a good approximation we can think of the electrons moving in the field of fixed nuclei. We first choose a geometry (with fixed internuclear distances) for a molecule and solve the Schrödinger equation for that geometry. We then change the geometry slightly and solve the equation again. This continues until we find an optimum geometry with the lowest energy.
The Independent Electron Approximation
When more than one electron is present, the Schrödinger equation is impossible to solve because of the interelectron terms in the Hamiltonian. Consider, for instance, the Hamiltonian for the hydrogen molecule in the Born-Oppenheimer approximation.
The first two terms are due to the kinetic energy of the electrons. The last six terms express the potential energy of the system of four particles. The potential energy term due to the repulsion of the electrons makes the Schrödinger equation impossible to solve.
To produce a solvable Schrödinger equation we assume that the Hamiltonian is a sum of one-electron functions, fi, with an approximate potential energy that takes the average interaction of the electrons into account. This leads to a set of one-electron equations, called the Hartree-Fock equations, where is a one-electron wavefunction.
The total wavefunction that is a solution to the total Schrödinger equation, , is approximated as the product of the solutions to the one-electron equations.
This product must be adjusted to satisfy the Pauli Exclusion principle, but we won't get into that here. If you are familiar with determinants, it involves writing the wavefunction as a determinant.
The Hartree-Fock Self-Consistent Field (SCF) Approximation
The question remains about the approximate potential energy in the one-electron functions that take the average interaction of the electrons into account. What is the form of the functions fi in the Hartree-Fock equations? The most common way of handling this is to define
where vi is an average potential energy due to the interaction of one electron with all the other electrons and nuclei in the molecule. The average potential depends on the orbitals, , of the other electrons, which means we must solve the Hartree-Fock equations iteratively.
The iterative solution of the Hartree-Fock equation is as follows.
1. Guess reasonable one-electron orbitals (wavefunctions), , and calculate the average potential energies, vi.
2. Using the variation principle, solve the Hartree-Fock equations,
to give new one-electron orbitals, . Use these new orbitals to calculate new and improved average potential energies, vi. Because the solution of the Hartree-Fock equations depends on the variation principle, the Hartree-Fock energy should be higher than the true energy.
3. Repeat the second step until the one-electron orbitals and potential energies don't change (are self-consistent).
Restricted Hartree-Fock Calculations
To take the Pauli Principle into account, we must include electron spin in our wavefunctions. The orbitals that are calculated by the Hartree-Fock method actually are spin orbitals that are a product of a spatial wavefunction and a spin function.
In a spin orbital, is the spatial wavefunction describing the probability of finding the electron in space and or are spin wavefunctions.
For a closed shell system, in which all of the electrons are paired, during the solution of the self-consistent field equations, we can restrict the solution so that the spatial wavefunctions for paired electrons are the same. This is called a restricted Hartree-Fock (RHF) calculation and generally is used for molecules in which all the electrons are paired. When the spin functions are removed, we are left with a set of spatial orbitals, each occupied by two electrons.
An example would be the restricted Hartree-Fock solution to the Schrödinger equation for the hydrogen molecule, H2. This would lead to two spatial orbitals, one occupied by the pair of electrons and one unoccupied. The orbitals holding electrons are called occupied orbitals and the unoccupied orbitals are called virtual orbitals.
Unrestricted Hartree-Fock Calculations
For open shell systems that contain unpaired electrons, the assumption made in the restricted Hartree-Fock method obviously won't work. There is more than one way of handling this type of problem. One way is to not constrain pairs of electrons to occupy the same spatial orbital - the unrestricted Hartree-Fock (UHF) method. In this method there are two sets of spatial orbitals - those with spin up () electrons and those with spin down
() electrons. This leads to two sets of orbitals as pictured at the right and to a lower energy than if the restricted method were used.
Basis Sets
For molecular calculations, the Hartree-Fock SCF equations
still cannot be solved without one further approximation. To solve the equations, each SCF orbital,, is written as a linear combination of atomic orbitals. For instance, for the H2 molecule, the simplest approximation is to write each spatial SCF orbital as a combination of 1s atomic orbitals, each centered on one of the protons.
This reduces the problem to solving for the coefficients, c1 and c2, since the atomic orbitals do not change.
The set of atomic orbitals that is chosen to represent the SCF orbitals is called a basis set. The {1sA, 1sB} basis set shown above is a minimal basis set - the smallest set of orbitals possible that describe an SCF orbital. Usually, the quality of a basis set depends on its size. For instance, a larger basis set, such as {1sA, 1sB, 2sA, 2sB}would do a better job approximating the SCF orbital than {1sA, 1sB}.
For many-electron atoms, we don't know the actual mathematical functions for the atomic orbitals, so substitutes are used - usually either Slater-type orbitals (STO) or Gaussian-type orbitals (GTO). We won't concern ourselves with the exact form of STO and GTO. Suffice it to say that they are chosen to behave mathematically like the actual atomic orbitals: s-type, p-type, d-type, and f-type, for instance. A few commonly used basis sets are listed below. The symbol of the basis set is given in the left column and the characteristics of the basis set in the center. At the right is the basis set that would be used to represent methane. For instance, the STO-3G basis set for methane would be {1sH, 1sH, 1sH, 1sH, 1sC, 2sC, 2pxC, 2pyC, 2pzC}.
Basis Sets1
Basis Set Example (CH4)
A minimal basis set (although not the smallest possible) using three GTOs to approximate each STO. This basis set should only be used for qualitative results on very large systems
Each H: 1s
C: 1s, 2s, 2px, 2py, 2pz
Inner shell basis functions made of three GTOs. Valence s- and p-orbitals each represented by two basis functions (one made of two GTOs, the other of a single GTO). Use for very large molecules for which 6-31G is too expensive.
Each H: 1s, 1s'
C: 1s, 2s, 2px, 2py, 2pz, 2s', 2px', 2py', 2pz'
Inner shell basis functions made of six GTOs. Valence s- and p- orbitals each represented by two basis functions (one made of three GTOs, the other of a single GTO). Adds six d-type basis functions to non-hydrogen atoms. This is a popular basis set that often is used for medium and large systems.
Each H: 1s, 2s
C: 1s, 2s, 2px, 2py, 2pz, 2s', 2px', 2py', 2pz', 3dx2, 3dy2, 3dz2, 3dxy, 3dxz, 3dyz
Like 6-31G(d) except p-type functions also are added for hydrogen atoms. Use when hydrogens are of interest and for final, accurate energy calculations.
Each H: 1s, 2s, 2px, 2py, 2pz
Generally, the larger the basis set the more accurate the calculation (within limits) and the more computer time that is required. As an example, consider the calculation of the bond length of H-F using different basis sets, as shown below. 1
Basis Set
Bond Length (Å)
| Error (Å) |
You might notice that although the large basis set, 6-311++G(d,p), predicts the correct answer to within 0.001 Å, several others are correct to within 0.01 Å (well within the criteria of chemical accuracy). Although a larger basis set usually gives better results, you often have diminishing returns as you choose larger sets. A point may be reached beyond which the additional computer time is not worth it.
Post-SCF Calculations
Even with a very large basis set calculation, Hartree-Fock results are not exact because they rely on the independent electron approximation. Hartree-Fock SCF Theory is a good base-level theory that is reasonably good at computing the structures and vibrational frequencies of stable molecules and some transition states2. Electrons are not independent, though. We say that they are correlated with each other and that the Hartree-Fock method neglects electron correlation. This means that Hartree-Fock calculations do not do a good job modeling the energetics of reactions or bond dissociation. There are several ways of correcting SCF results to take electron correlation into account.
One method of taking electron correlation into account is Møller-Plesset many-body perturbation theory, which is used after a RHF or UHF calculation has been made. It is assumed that the relationship between the exact and Hartree-Fock Hamiltonians is expressed by an additional term, H(1), so that H = fi + H(1). Calculations based on this assumption lead to corrections that can improve SCF results. Various levels of perturbation theory can be applied to the problem. They are called MP2, MP3, MP4, etc. MP2 calculations are not time-consuming and usually give quite accurate geometries and about one-half of the correlation energy. Because perturbation theory is not based on the variation principle, the energy predicted by MP calculations can fall below the actual energy.
Another important method of correcting for the correlation energy is configuration interaction (CI). Conceptually we can think of CI calculations as using the variation principle to combine various SCF excited states with the SCF ground state, which lowers its energy. We won't use CI calculations in our exercises at this level.
SCF Molecular Orbitals
When calculating molecular orbitals, you should remember that molecular orbitals are not real physical quantities. Orbitals are a mathematical convenience that help us think about bonding and reactivity, but they are not physical observables. In fact, several different sets of molecular orbitals can lead to the same energy. Nevertheless, they are quite useful. We will use ethylene as an example to illustrate MO concepts.
The basis functions in SCF molecular orbitals are like atomic orbitals. A RHF/6-31G(d) calculation on ethylene uses 38 basis functions (15 for each carbon and 2 for each hydrogen). Since the molecular orbital wavefunction is expanded in terms of the all the basis functions,
it might seem that constructing a picture of the orbital would be difficult. Luckily, most of the coefficients are zero, so the molecular orbitals are easy to picture. Consider, for instance, the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) of ethylene.
The HOMO is a bonding-orbital.
The LUMO is an antibonding -orbital.
Scaling Vibrational Frequencies
In the last part of the job output from a frequency calculation you will find the predicted vibrational frequencies (cm-1) of the normal modes of the molecule. Also supplied are the predicted intensities of the IR and Raman bands corresponding to these normal modes.
1 2 3
B1 B2 A1
Frequencies -- 1335.5948 1383.4094 1679.4157
4 5 6
A1 A1 B2
Frequencies -- 2027.8231 3160.8817 3232.9970
Computational results usually have systematic errors. In the case of Hartree-Fock level calculations, for instance, it is known that calculated frequency values are almost always too high by 10% - 12%. To compensate for this systematic error, it is usual to multiply frequencies predicted at the HF/6-31G(d) level by an empirical factor of 0.893. Similarly, frequencies calculated at the MP2/6-31G(d) level are scaled by 0.943. 1
The predicted frequencies after applying the 0.893 scale factor are listed below.
1 2 3
B1 B2 A1
Scaled Frequencies -- 1193 1235 1450
4 5 6
A1 A1 B2
Scaled Frequencies -- 1811 2822 2887
1J. B. Foresman and Æ. Frisch, Exploring Chemistry with Electronic Structure Methods, Gaussian, Pittsburgh, 1995-96, p 102.
2J. B. Foresman and Æ. Frisch, Exploring Chemistry with Electronic Structure Methods, Gaussian, Pittsburgh, 1996, p 115.
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Are there analytic solutions to the time-Dependent Schrödinger equation, or is the equation too non-linear to solve non-numerically?
Specifically - are there solutions to time-Dependent Schrödinger wave function for an Infinite Potential Step, both time dependent and time inpendent cases?
I have looked, but everyone seems to focus on the time-Independent Schrödinger equation.
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The time dependent equation for a time independent potential is solved by superpositions of the solutions to the time-independent problem, with the coefficients varying as exp(iEt). The equation is completely linear. – Ron Maimon Nov 30 '11 at 4:20
@RonMaimon: Not so simple. If this were the case, there'd be no reason to ever make a rotating wave approximation. Time dependence is particularly troublesome when there is no interaction picture which removes it. In an example case of a sinusoidal drive on a low dimensional system, there is no rotating frame without time dependence, and in general no analytic solution. Having said that, even in the time independent case you are fortunate if your ODE has an attainable analytic solution beyond a hand-wavey "it's just a matrix exponential". – qubyte Nov 30 '11 at 17:09
@Mark: Yes it is so simple, your examples are completely irrelevant. No need muddy this simple question--- it's just asking about the time-dependent equation for a time independent potential. A sinusoidal drive is not time independent. Rotating frames produce time independent forces in the rotating frame, a global centrifugal potential and a coriolis magnetic-type force, so they can be made time independent. Lots and lots of problems have analytic solutions if you start with an known ground state and vary coefficients. – Ron Maimon Nov 30 '11 at 18:08
@RonMaimon: Calm down there. I misread the start of your comment. In any case I'm not "muddying the question" since the comment was clearly directed at you. Anyway, in my experience it's only the special cases that can be solved. Throw a few small systems together and you've got a many body problem (unless you have a lot of symmetry). As we rarely deal with just a few two level systems, that pretty much describes everything unless you're willing to approximate. But I digress... – qubyte Nov 30 '11 at 18:22
@Everitt: I agree. I didn't mean to sound abrasive, just haven't had my coffee yet. – Ron Maimon Nov 30 '11 at 18:35
3 Answers 3
The complete solution for the time dependent equation with an infinite potential step is found by the method of images. Given any initial wavefunction
$$ \psi_0(x) $$
for x<0, you write down the antisymmetric extension of the wavefunction
$$ \psi_0(x) = \psi_0(x) - \psi_0(-x) $$
And you solve the free Schrodinger equation. So any solution of the free Schrodinger equation gives a solution for the infinite potential step. This is not completely trivial to make, because the solutions do not vanish in any region. But, for example, the spreading delta-function
$$ \psi(x,t) = {1\over \sqrt{2\pi it}} e^{-(x-x_0)^2\over it } $$
Turns into the spreading, reflecting, delta function
$$ \psi(x,t) = {1\over \sqrt{2\pi it}} e^{-(x-x_0)^2\over it } - {1\over \sqrt{2\pi it}} e^{-(x+x_0)^2\over it } $$
You can do the same thing with the spreading Gaussian wavepacket, just subtract the solution translated to +x from the solution translated to -x. In this case, normalizing the wavefunction is hard when the wavefunction start out close to the reflection wall.
Time independent infinite potential wall
The solution to the time independent problem of the infinite potential wall are all wavefunctions of the form
$$ \sin(kx) $$
for all k>0. Superposing these solutions gives all antisymmetric functions on the real line.
To find this solution, note that the time independent problem (eigenvalue problem) for the Schrodinger equation is solved by sinusoidal waves of the form $e^{ikx}$, and you need to superposes these so that they are zero at the origin, to obey the reflection condition. This requires that you add two k-waves up with opposite signs of k and opposite sign coefficients.
The opposite sign of k just means that the wave bounces off the wall (so that k changes sign), while the opposite sign of the coefficient means that the phase is opposite upon reflection, so that the wave at the wall cancels.
General solution
The time dependent problem for a time independent potnetial is just the sum of the solutions to the time independent problem with coefficients that vary in time sinusoidally.
If the eigenfunctions $\psi_n$ are known, and their energies $E_n$ are known, and the potential doesn't change in time, then the,
$$ \psi(t) = \sum_n C_n e^{-iE_n t} \psi_n(x) $$
is the general solution of the time dependent problem. This is so well known that generally people don't bother saying they solved the time-dependent problem once they have solved the eigenvalue problem.
The general solution of the time-dependent Schrodinger equation for time dependent potentials doesn't reduce to an eigenvalue problem, so it is a different sort of thing. this is generally what people understand when you say solving the time-dependent equation, and this reflects the other answers you are getting. I don't think this was the intent of your question, you just wanted to know how to solve the time dependent equation for a time independent potential, in particular, for an infinite reflecting potential wall. This is just the bouncing solution described above.
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Full marks Ron, this was just what I was trying to understand. Ironic that its so trivial no one mentions it. Can you recommend a text or source for this? – metzgeer Dec 1 '11 at 1:09
I don't think there is any more to say about it that what I said above, also, look here: physics.stackexchange.com/questions/12611/… . The grandaddy of all exact solution for a time independent problem is the Gaussian wavepacket, which you can work out most easily from knowing the stochastic version. I worked it out here: en.wikipedia.org/wiki/User:Likebox/Schrodinger , it used to be on Wikipedia, before that project degenrated. I can't recommend any other literature on elementary stuff, unfortunately. – Ron Maimon Dec 1 '11 at 1:41
@metzgeer: It doesn't contain this problem specifically (I think, it may since have been updated), but I highly recommend Introduction to Quantum Mechanics by Griffiths. It contains some introductory examples and problems to work though which are very helpful for solidifying the methods used in quantum mechanics at this level. It is also in a conversational tone that will probably appeal to you if you enjoy using SE. :) – qubyte Dec 1 '11 at 4:15
@MarkS.Everitt: Oh thank Saint Albertus Magnus the patron saint of scientists, for physics books at a conversational tone - or I would never make it :) – metzgeer Dec 1 '11 at 10:22
@metzgeer: Eek, sorry I hope I didn't cause any offence. I recommend it because it's my favourite! – qubyte Dec 1 '11 at 10:32
The equation is analytically solveable if you allow the potential to vary in magnitude but keep the borders fixed. Then, you can assume the form $\sum_{n} A_{n}(t) \sin\left(\frac{\pi n x}{L}\right)$ for the wavefunction.
Substituting this form of the wavefunction into $\frac{\hbar}{i}\frac{d\psi}{dt}=-\frac{\hbar^{2}}{2m}\frac{d^{2}\psi}{dx^{2}}+V(t)\psi$ results in:
$$0=\sum_{n} \sin\left(\frac{\pi n x}{L}\right)\left[ \frac{\hbar \dot A_{n}}{i} - \frac{\hbar^{2}A_{n}\pi^{2}n^{2}}{2mL} +V A_{n} \right]$$
Since each sine term has independent nodes and antinodes, each of the enclosed factors must be independently zero. The solution integrates to:
$$A_{n}=a_{n}\exp\left(\frac{\pi^{2}\hbar n^{2} i t}{2mL^{2}}\right)\exp\left(\frac{i}{\hbar}\int V(t) dt\right)$$
Where the $a_{n}$ are constants. Note that this differs from the standard infinite square well solution only by the second factor involving the integral of the potential energy. Also, note that there is nothing that depends on $n$ in this term, so this solution can be pulled out of the sum entirely, and thus simply multiplies the old wavefunction by an overall phase, and generates a physically identical wavefunction to our old, constant-$V(t)$ solution.
I'm almost certain that exact anlaytic solutions exist for less trivial solutions, but varying the potential on the infinite square well doesn't do much, in the end.
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This is for a infinite potential well isn't it, I'm trying to understand an infinite potential step, do I simply increase the width of the well to let L approach infinity? – metzgeer Nov 30 '11 at 4:08
@metzgeer: I don't understand what an infinite potential step is--do you mean a potential that is $V=V(t)$ for $x > 0$, and $\infty$ otherwise? – Jerry Schirmer Nov 30 '11 at 16:46
you're quite right jerry. I was thinking of a time varying $\psi(x,t)$ variable, not in terms of V(x,t) I should have said V(x) = $\infty$ for all time. Mea culpa. – metzgeer Dec 1 '11 at 1:00
@metzgeer: in that case, the time-independent schrodinger solutions, with each basis function multiplied by $\exp(iEt)$ gives you the correct time-dependent wavefunction. This will be true for all potentials that don't depend on the time. – Jerry Schirmer Dec 1 '11 at 12:18
The universe has a sense of humour, so I cannot resist supplementing a question
which has the word « analytic » in its title... in a minute, you will see why I would have preferred you to use the phrase « closed-form », with the following observation--answer.
For any time-independent potential $V(x)$, let $H = -{\partial^2\over\partial x^2} + V(x)$. (What I am about to say works for any time-independent $H$, and if a system is isolated, the Hamiltonian is always time-independent even though Schroedinger's equation is time-dependent.) Suppose the system starts in the initial state $\psi_o$, a wave function of $x$ of course. Then the following analytic function of time gives the solution to the time-dependent Schroedinger equation:
$$\psi(x,t) = e^{itH}\cdot\psi_o.$$
This is similar to the last formula given by Mr. Maimon which told you, as he explained, how to get the solution to the time-dependent Schroedinger equation once you have solved the time-independent equation for all its eigenvalues and eigenstates. The difference is that in that formula, $E$ was one of the many eigenvalues of the operator $H$, but here we can simply plug in $H$ as an operator into the holomorphic (analytic) function $e^z$ (one way is by using the power series of this analytic function).
Hence the universes' pun between analytic as a synonym for closed-form expression but also as a synonym for holomorphic which means it can be expressed as a convergent power series and extended as a function to the entire complex number plane, which is done in some approaches to Quantum Field Theory, e.g., by Streater and Wightman, and in some approaches to path integrals.
This approach is less practical for your specific situation than the answer by Mr. Maimon which is well-adapted to your specific problem...but it gives a closed-form formula, generalises well even to potentials $V$ with singularities, infinities, etc., and sometimes can help you think about the physics of the problem without getting lost in the gory details of calculating the answer.
I have often wondered whether it can be extended to time-varying potentials...I suspect it could be...
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It can only be generalized as $\psi(x,t) = \exp (-i \int H(t') dt' ) \psi(x,0)$ if the Hamiltonian commutes with itself at different times, $[H(t),H(t')] = 0$. – perplexity Jan 19 '12 at 13:56
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3194b3b12b78d209 | Friday, September 12, 2008
Aether and Lorentz invariance
The concept of Lorentz invariance is basic postulate of special relativity theory and one of most deeply misunderstood concepts of Aether theory, being considered incompatible with relativity in general. The true is, the light speed invariance can be derived easily from Maxwell's Aether theory of light, based on transversal wave spreading. As I demonstrated already, the ability of luminiferous Aether to spread the light of whatever energy density effectively implies the very dense environment and the transversal character of energy spreading (which is required for casual spreading of information) in it, because the sparse Aether cannot spread the EM waves of whatever energy density. Therefore the famous Michelson-Morley experiment (MMX) shouldn't be used for disapproval of Aether theory - but for effective confirmation of it, instead. It shoud be noted, with contrary to widespread belief the negative result of MMX cannot serve as a confirmation of relativity, because the light speed invariance isn't theorem, but a postulate (i.e. sort of axiomatic tautology) in special relativity theory.
Another general source of Aether misunderstanding is the common belief, the concept of particle environment isn't compatible with light speed invariance and the relativistic physics in general. This is nonsense, because the common interpretation of Galileo transform isn't compatible with relativistic Lorentz transform. The spreading of sound wave in air cannot be considered as analogy of light spreading in vacuum, until we consider the sound wave as the only source of information including the time and distance intervals measurement, i.e. by the same way, like during light spreading in vacuum. The common understanding of wave spreading in particle medium usually involves at least TWO kind of waves (the light wave, used for time/distance measurement and the studied/observed wave itself), while during light spreading in vacuum the only kind of energy spreading can be considered (the light wave serves here both as the subject of observation, both the mean of observation). This general inconsistency in experiment interpretation leads to the (false) conclusion, the Newtonian mechanics and the invariance of energy wave speed in particle environment is incompatible with the light speed invariance (and the relativity theory in general). As we can see, it's just a result of fundamental inconsistency of experimental arrangement, instead. Therefore MMX cannot give a positive result, simply because it’s virtually impossible to detect every environment just by it’s waves. If some particle is serving for wave spreading, it cannot be observed just by this wave and nothing very strange is about. No object can serve as a mean of it’s own observation and the inner and outer perspectives cannot be mixed.
The similar mistake consists in widespread belief, the absence of reference frame excludes the existence of luminiferous Aether, the particle environment in general. In fact, no particle in such environment cannot serve as an subject of observation and the mean of observation at the same time, therefore the absence of reference frame is the natural consequence of energy wave spreading inside of such environment, if we make sure, the same kind of wave is serving as object and mean of observation, i.e. by the same way, like during light spreading in vacuum. As the waves in particle environment are mixture of longitudinal and transversal waves in general, we can follow the above rule and the absence of reference frame most efficiently at the cases, when only transversal wave spreading prevails - for example at the case of capillary waves spreading along water surface, which is driven by surface tension (nearly) completely. With respect of these waves the water surface is behaving like thin elastic membrane with (nearly) no underwater (motion/reference frame) at all - so we can see clearly, the transversal wave spreading in particle environment is really background independent and no additional postulates are required to consider here.
By analogous way, we cannot observe the water surface by using of water waves and nothing very strange is about. The water surface will always appear as a void, empty space from surface waves perspective, because it just serves as an environment for these waves. The common observation of water waves by light waves cannot serve as a direct analogy of observation of light waves by using waves in vacuum, simply because in vacuum only one kind of waves can be always involved in experiment - the waves of light. So here’s nothing strange about different results of "classical physics" experiments, which were made in different arrangement(s). This doesn’t mean of course, the classical mechanics differs from reality conceptually - it just means, we aren’t observing wave phenomena by the same way, like during experiments in vacuum - that’s all. The Lorentz invariance (symmetry) of Aether is valid as long the transversal character of wave spreading is retained. Because the transversal wave spreading is the only causal way of information spreading considered for human creatures, the Lorentz invariance follows automatically from unitary time arrow and vice-versa: the quantum uncertainty related to multiplicity of time arrows and longitudinal energy wave spreading is equivalent to Lorentz symmetry violation.
Note that the transversal wave is the case, where the energy spreads by the slowest speed through such environment, i.e. here's a minimum of the celerity / wavelength dependence. This makes the environment as large, as possible from internal observer perspective - so we can say, the Universe appears so large for us just because of transversal character of light spreading. It's somewhat surprising, these fundamental connections were revealed after nearly four hundred years after postulation of particle luminiferous Aether concept by R. Descartes (1644) and Ch. Huygens (1678) on behalf of positivistic, ad-hoced (i.e. belief based) consideration of relativistic postulates.
"All our attempts to make ether real failed. It revealed neither its mechanical construction nor absolute motion. Nothing remained of all the properties of the ether except that for which it was invented, i.e., its ability to transmit electromagnetic waves. Our attempts to discover the properties of the ether led to difficulties and contradictions. After such bad experiences, this is the moment to forget the ether completely and to try never to mention its name."
(The Evolution of Physics Einstein 1938)
Tuesday, September 09, 2008
Sacred geometry and Aether concept
By AWT all structures inside of our Universe are formed as a "jammed structures" of another structures, recursively. The high degree of nested compactification is the source of complexity of observable reality. One of most remarkable features of AWT is its close connection to sacred geometry, the geometry of mutually cicPlatonic solids in the theory of five elements in particular - which is closely related to the heterosis of Aether foam by gradual compactification /condensation of foam gradients (membranes). During shaking of soap foam, the newly created density gradients are formed in the corners of existing ones and this process is completely reversible, if the foam bubbles are filled just by their own vapor:
The odd/atemporal/male (bosonic) symmetry alternates the even/temporal/female (fermionic) one during mutual heterosis. The most symmetric level of particle compactification possible leads to the solid dodecahedral structure of foam, assigned to Prana in Vedantic philosophy ("Aether" or vacuum). the foam bubbles can be approximated by platonic solids, where the dodecahedron is the most complex one in 3D space. The dark matter foam structure and E8 Lie group exhibits this symmetry too. The five-fold (A5) rotational symmetry of icosahedron serves as a symbol of water in sacred geometry of five elements in accordance to icosahedron symmetry of fluids, the glass and water clusters in particular.
The dodecahedron foam is the most regular lattice, which we can met inside of our 3D Universe generation and the number of condensation steps required for its formation is quite limited. Therefore the geometry of real foam driven by principle of least action remain close to dodecahedron structure. It still doesn't fit the 3D space completely, though - which is the reason, why M-theory operates in 10-dimensional space. The another condensation inside of dodecahedron will lead to cubic structure again and we can achieve the same structure by topological inversion of this structure, which follows to AdS5/CFT4 correspondence. The temperature of CMB (i.e. the interior of Universe) corresponds the Hawking radiation of black hole, whose lifespan corresponds the age of our Universe generation and the mass density of which corresponds energy density of vacuum (i.e. the 3rd power of Planck constant for 3D space perspective).
Logo of Aether Wave Theory
Aether and the definition of time
By AWT everything, what we can observe from reality are just a "changes", i.e. the Aether density gradients ("gradient driven" reality). This follows from analogy of particle gas or fluid, where only density gradients/fluctuation can be observed directly. This is because every density gradient is behaving like place, where particles of environment are moving in circles like particles bouncing at the undulating water surface. It "undulates at place" in compactified (hidden) dimensions, thus making "permanent changes", so it can be perceived as a atemporal, persistent entity/piece of reality. We can see, the phase transition is nothing else, then the compactification of underlying space-time. Note that the space dimension compactified becomes a time dimension in the space-time, which is formed by compactification of previous generation of "hyperspace" ("false vacuum"). This corresponds the relativistic perspective of the matter wave motion through space-time along geodesics, thus fulfilling Fermat's theorem. Note that the Fermat's theorem is consequence of Huyghens principle, which is the special case of principle of least action, which leads to the description of matter motion along geodesics as a Hamiltonian flow. By such way, the AWT explains clearly, what the time is and how it is related to the spatial dimensions. Ironically, it was Heidegger in his anti-philosophical work (Being and Time), who concretised time as the essence of "existence", whereas Einstein's relativistic stance might imply time's abstract nature.
In AWT the existence of space-time follows from asymmetry between spatial and time dimensions, which was created during Universe inflation. We can expect, from sufficiently distant perspective this asymmetry will be replaced by another one, because Universe is arranged randomly. By AWT the time is asymmetric, because it’s formed by density gradient of Aether. In analogy to local space-time with water surface, the space dimensions are the directions parallel with water surface, while the time is the direction normal (perpendicular) to this surface gradient - as such it’s always oriented from past to future (it exhibit's an "arrow").
The above animation illustrates, how the same phase transition occurs in more dimensions. We can see, the space-time formation is nothing else, then the condensation of matter into density gradients, forming this space-time. Note that the life on particularly stable/atemporal space-time (mem)brane implies the existence of pair conjugated time dimensions (1, 2, 3, 4), because the membranes of foam are consisting of pairs of surface gradients. The existence of more time dimensions can be derived by many ways. The backward time arrow is connected with negative (i.e. repulsing) gravity action and negative rest mass, for example. The particle of antimatter are living in backward time arrow partially: they're dissolve into radiation, while the particles of normal matter are condensing by their gravity. The existence of multiple time dimensions is related to the longitudinal energy spreading at higher of lower energy density. We can met with its analogy at the water surface, where the very small or large waves are of pronounced longitudinal character due the dispersion (compare the celerity/wavelength dependence for surface water waves).
The uncertainty of quantum mechanics follows from pronounced longitudinal wave spreading of energy between fluctuations of vacuum and it's manifestation of many time arrows of space-time convoluted at Planck scale (compare the Feynman's many path integral formalism). The vacuum foam tends to be formed by spherical bubbles after introducing of energy or near gravitating objects . The separation of surfaces forming the Aether foam (mem)branes and the path splitting of chiral bosons (light cones) at GUT energy scales (10+14 GeV) manifest itself like so called Faraday/Kerr and birefringence effect in vacuum. At high values of electrostatic or magnetic field near magnetars or inside of strong gravitational field of rotating black holes such birefringence leads to the formation of multiple event horizons / space-time branes.
In my understanding, inside our generation of observable Universe a two reciprocal time dimensions are dominant, related by 1:10{+500} duality at 1.27 cm scale. In AWT object is moving in time dimension, when it expands (to past) or collapses (to past) above 1.27 cm scale. For objects smaller then 1.27 cm the direction of both time arrow becomes reversed. For example diffusion, evaporation and/or condensation is a travel accros time dimension(s), as it requires transfer of energy (usually in form of inertial acelleration) without general change of spatial location.
Albert Einstein: "The only reason for time is so that everything doesn't happen at once."
Monday, September 08, 2008
E8 Lie group and Aether theory
Mr. Garett’s E8 group model can be understood on background of Aether particle theory easily. This is because Lie E8 group is not just some void geometrical structure. It’s root vector system is describing the tightest structure of kissing scale invariant hyperspheres ("unparticles"), where the kissing points of spheres are sitting at the centers of another hyperspheres, recursively. The Aether Wave Theory proposes at least two dual ways, how to interpret such structure:
The cosmological one is maybe easier to realize: it considers, the current Universe generation is formed by interior of giant dense collapsar, which is behaving like black hole from outer perspective. This collapse was followed by phase transition, which proceeded like crystallization from over-saturated solution by avalanche-like mechanism. During this, the approximately spherical zones of condensing false vacuum have intersect mutually, and from these places the another vacuum condensation has started (a sort of nucleation effect). We can observe the residuum of these zones as a dark matter streaks. The dodecahedron structure of these zones should corresponds the E8 group geometry, as being observed from inside (i.e. from past perspective due the Universe "expansion").
The second interpretation of E8 is relevant for Planck scale, i.e. for outer perspective (the future). The dense interior of black hole is forming the physical vacuum, which is filled by spongy system of density fluctuations, similar to nested foam. Such structure has even a behavior of soap foam, because it gets more dense after introducing of energy by the same way, like soap foam shaken inside of closed vessel. Such behavior leads to the quantum behavior of vacuum and particle-wave duality. Every energy wave, exchanged between pair of particles (i.e. density fluctuations of foam) is behaving like less or more dense blob of foam, i.e. like gauge boson particle. Every boson can exchange its energy with another particles, including other gauge bosons, thus forming the another generation of intercalated particles.
Therefore the E8 Lie group solves the trivial question: "Which structure should have the tightest lattice of particles, exchanged/formed by another particles?". And such question has perfect meaning even from classical physics point of view! Such question has a perfect meaning in theory, describing the most dense structure of inertial particles, which we can ever imagine, i.e. the interior of black hole. AWT inteprets a rotation of Lie group in general reference frame, which leads to another particle generation as a Penrose-Terrell effect, formalized in Wick rotation approach.
Correspondence of AWT and other theories
The Aether concept as a "zero dimensional particle theory" is quite fundamental TOE and it cannot be omitted from physics due the Occam's razor and Anderson's "More is different" principle. It serves as a conceptual Newtonian mechanics based glue between relativity theory and quantum mechanics both on cosmic energy/distance scale, both on Planck scale. But the Aether multi-particle concept goes even deeper - it can help us to redefine the whole observable reality on the background of probability calculus. It's not still the ultimate approach, as it cannot explain, which the Aether is composed from and why reality exists. But if we consider, the reality exists and its composed of infinitely many pieces/units, it enables us to predict the appearance of this reality on the background of scale invariant fluctuations of such system.
The Aether Wave Theory is closely related to Constructal theory, Process Physics, Unparticle Physics and Emergence Theory - it can serve as the conceptual base of all these theories. The concept of Aether foam covers the quantum foam or spin network of LQG theory, the protosimplex lattice of Heim theory or the recent string net liquid concept. And the non-linear properties of Aether foam can reconcile the dual aspects of general relativity and quantum mechanics and quantum field theories (the free fermion models of string field theories in particular). As a particle theory the AWT is closely related to Garett's E8 group theory, because the Lie E8 group describes the tightest structure of hypersphere particles, exchanged by another particles, recursively.
If all these theories are relevant each other - as the correspondence principle requires - here must exist some common connecting point/concept and we are forced to find it, because it's unacceptable to have so many theories, which are incompatible each other. Therefore the AWT isn't limited to realm of physics, as it describes the general connections/interactions inside of all multicomponent systems, including the biology and sociology and theories of information spreading.
Sunday, September 07, 2008
Mass of photon
By AWT the problem of rest mass of photon must be separated from luminal speed of light wave, as expected by special relativity. Light wave isn't photon and special relativity doesn't care about photon existence at all - it just considers fully harmonic light wave, which is atemporal and of unlimited range by its very nature. Another correction - just negative one - brings the presence of cosmic microwave background (CMB). Due the random character of vacuum at presence of CMB photons, real empty spacetime isn't completelly flat, so that every light wave can be considered a dynamic mixture of photons and tachyons of negative rest mass. As a whole, this mixture has a zero rest mass just at the CMB scale, which is indeed not the case of photons itself. Therefore special relativity can still have its portion of truth - but real photons would undergo a subtle dispersion in CMB field, which decreases their speed a bit, because no isolated object can remain in complete rest with respect to this field.
By AWT every artifact with positive curvature should have a positive (i.e. nonezero) rest mass and the photon - being an isolated particle - is no exception. The particle like character of photons can be observed easily during spreading of gamma rays in spark chamber or by scintillator in spinthariscope, where they're behaving like distinst well defined particles ("scintilla" means "spark" in Greek). Therefore it's nothing strange, if photon increases mass of resonator, whenever it gets trapped into it - as we can observe by mass spectrometer during excitation of atom nuclei, for example.
The theoretical rest mass of photon can be extrapolated as a dynamic mass of photon, when the (wavelength of) photon becomes so large, it will fit the whole observable Universe, so that the photon cannot move and it stay at rest in it. This value is incredibly low, though and it can be estimated by using of E=hν formula to some 10E-61 kg. Albeit low, it can result into observable violation of Compton law at Planck scale (pair formation) and into light speed invariance violation at cosmic scale (for example by polarization of microwaves by vacuum and by dispersion of gamma rays, as observed by GZK limit or by MAGIC telescope during Mkn 501 flash).
The effective rest mass of photon could become even higher (~10E-17 kg), if we consider, the photons, whose wavelength is longer then human scale would dissapear in the noise of cosmic microwave backround (CMB) radiation, where only entangled light waves can spread effectivelly. In adition, photons of wavelength larger the human / CMB scale (~1.7 cm) are behaving rather like weak holes in the ocean of CMB photons, so they should be expelled by them in gravity field, instead.
The general problem in misunderstanding of special relativity consist in mixing of light and photon concepts. Light wave can be local, but the photon isn't never quite local thing, it has a finite (albeit typically quite small) size. It means, only light wave can move by speed of light, but not photon. For wavelength comparable to CMB radiation the light can consist only from waves, but not photons, because these size of photons are comparable to CMB noise size, so they cannot be distinguished from it. For longer wavelengths, then those of CMB the negative rest mass photons can be postulated, and the speed of such "negative curvature" photons becomes superluminal - the character of such waves will converge to longitudinal gravitational waves, which are inherently superluminal. The superluminal portion of microwave light enables to escape it from black hole as a Hawking radiation for example, which makes the whole concept testable.
but he is mistaken." (Albert Einstein, 1954)
Aether and quantum mechanics
Richard Feynman: "It is safe to say that nobody understands quantum mechanics".
John Wheeler: "If you are not completely confused by quantum mechanics, you do not understand it."
Roger Penrose: "Quantum mechanics makes absolutely no sense." (via
OK. The interpretation of quantum mechanics by Aether Wave Theory is easy and it's based on the foam behavior of Aether fluctuations, which are getting more dense after introduction of some energy by the same way, like soap foam during shaking inside of evacuated vessel. This behavior can be modeled easily by common computer and you can play with it by using of interactive Java applet. Note the similarity of foam behavior and dynamic mesh approach, used in numerical simulations.
As the result, every wave propagates through vacuum like so called quantum wave packet ("particle"), where the mass density of vacuum (so called the probability function, denoted by blue line on the picture bellow) is proportional the the actual energy density in each space and time interval (red color), thus fulfilling the Schrödinger equation - a fundamental equation of quantum mechanics.
The foam model illustrates the particle-wave duality, i.e. the fact, every isolated wave (soliton) propagates through vacuum like less or more pronounced density blob (wave packet), the density of which is proportional the wave frequency, which keeps the energy of wave packet quantized. This leads to interesting phenomena during wave packets collision, when the gradient forming blob becomes so large, so that the wave will bounce from internal walls of wave packet like wave inside of glass sphere or similar resonator and it will change itself into isolated standing wave of particle, undulating at place (a process known as a materialization of radiation).
Structure of observable reality
By AWT, the Aether structures are given by probability laws inside the inertial chaos, composed of many states (virtual particles). These less or more deterministic fluctuations of chaos density (i.e. the chaos density gradients) are having a structure of scale invariant Perlin noise, which we can perceive as a foam from local perspective. This structure can be derived from number theory as well, if we realize, the natural numbers are representing countable objects (colliding particles) and repeating sequences in random numbers are the less frequent, the more deterministic states (i.e. the similar numbers, the linearly increasing serii, etc.) they contain.
This is because, the density fluctuations are everything, what we can see from inertial chaos and the density fluctuations is the only way, how the energy/information can propagate at distance. So when the density of system increases, the foamy character of Perlin noise will become a clearly pronounced, so we can approximate all Aether structures (including time and space) by nested foam. This mechanism is analogous to formation of foamy density fluctuations inside of condensing supercritical liquid - so we can say, the whole observable reality has a structure of nested foam or exaggerated density fluctuations of heavily compressed particle gas or fluid: i.e. the interior of black hole.
Inside of large chaotic field the number of states, which are observable at the same moment is always quite limited because of limited speed of energy/information spreading. We can say, we are seeing something from such chaos just because we cannot see everything from it at the single moment. Therefore such system can never appear completely chaotic for us by the same
way, like the color patterns, formed by limited number of color states inside of random field of colored dots. Note that these patterns are scale invariant, they're always appear the same, despite the number of entities involved - they're forming so called unparticles.
When the density/scale of system increases, the foamy character of Perlin noise will become a clearly pronounced, so we can approximate all Aether structures (including time and space) by nested foam, which we can observe both in large scales as a streaks of dark matter, both in Planck scale as a "quantum foam" or as a "fabric of space".
It means, the Universe "as such" is completely random and it has no apparent structure or laws. But the observable portion of Universe cannot be completely random, or it wouldn't observable for us at all. Because we are highly ordered creatures, we have tendency to consider just a well organized pieces of Universe as a reality, by the same way, like we cannot see the chaotic
portion of condensing supercritical fluid - only the gradient driven portion of it. From the above follows, the observable reality is gradient driven, because we are forced to see it so. We can see exactly the same things, we could see inside of superdense particle fluid.
Aether and Maxwell's Equations
Well, the vacuum is probably a dense fluid. But which fluid? A fluid composed of its own scale invariant vortices as a boson condensate. This concept explains just a hydrodynamic properties of vacuum, its vorticity in particular, which can be described by tensor fields. Therefore it belongs into realm of relativity theory, the LQG and twistor theory in particular.
The connection between fluid vorticity and electromagnetism is known for years. Whole the Maxwell's theory was based on inertial fluid concept, which Maxwell has used for explanation of his displacement current concept. No wonder, Maxwell's equations are all isomorphous with Navier-Stokes equations. The most pronounced analogy we can met at the case of hydrodynamic analogy of Biot-Savart law:
Richard Cunningham Patterson Jr.: "If something looks like a duck, walks like a duck, and quacks like a duck, it's probably a duck."
The magnetic field transforms the vacuum into field of many tiny vortices, through which the charged particle with spin is moving along curved path, being dragged by vortex field. The analogy of Faraday-Lentz force and Newton-Magnus-Robbins force by AWT follows from the picture bellow:
For explanation of quantum mechanics properties of vacuum we are forced to adhere on foam model of vacuum, which follows from AWT as well. Only one real-life system covers both aspects of vacuum by analogy: its a condensing supercritical fluid, which can be described both fluid, both foam at the same moment. And this is where the AWT has started after one hundred years, when it was left abandoned by Sir Oliver J. Lodge, who had proposed it in 1904.
History of Aether Wave Theory
How the AWT affects expert's thinking in "quiet"..
Before two years a former Harvard professor Lubos Motl well known in biosphere was firmly convinced proponent of anti-aether lobby - but now he's promoting an Aether concept openly, albeit he's working in physics for years and he censored it in his blog comments obstinately.
This example just demonstrates clearly, how scrambled many people (even those most formally "qualified" ones..) can be concerning the trivial Aether concept. From AWT follows, the character of new ideas spreading corresponds a common phase transition inside of multiparticle system as a result of symmetry breaking, for example the character of boiling near water surface.
At the very beginning the new ideas are propagating like tiny isolated islands through society. Most of ideas will not survive negativistic stance of surrounding environment and they will collapse again like bubbles near boiling surface. Their proponents don't understand their common points, so they're repelled by surface tension mutually like tiny bubbles and they're even fighting mutually due energy competetion. Gradually, the number of people understanding new ideas increases and the mainstream community is starting to integrate/steal them into system of existing theories (for example, string and quantum foam, fractal or gradient reality, emergence or unparticle concept of Aether theory adopted by mainstream theories as an example).
At certain moment, an inverse population is reached and the intersubjective thinking will suddenly switch into new conceptual paradigm from distant outer perspective, so such transition appears sharp like surface of black hole event horizon. However, from internal observer perspective such transition often appears seamlessly continuous, because their proponents didn't realize change of intersubjective thinking, being isolated from reality in their ivory towers like tiny isolated black holes or elementary particles due their strong surface gradient of information density (compare the "fuzzball" concept of event horizon). These proponents of old paradigm will become isolated in their stance gradually, so they play a role of rare antiparticles persisting in diaspora inside of new conceptual continuum. And whole evolution can repeat again.
We can observe many other analogies to material world here. For example, active proponents of ideas are often attracted by super-symmetrical particles, which are playing role of opposition by the same way, like antiparticle clouds of dark matter occurs at the presence of massive objects as a result of strong gradient of gravity field. The short-seeing proponents of ideas are often behaving like black holes due total reflection, so they lose ability to exchange their ideas with the rest of society at all. We are saying, such person anticipate their time in relation to omni-directional space-time expansion.
As the result, behavior of biological systems or society and propagation of enthropy density and memes can learn us a lot about energy and matter spreading through Aether - and vice-versa.
And that's the memo. ;-) |
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For a time dependent wavefunction, are the instantaneous probability densities meaningful? (The question applies for instances or more generally short lengths of time that are not multiples of the period.)
What experiment could demonstrate the existence of a time dependent probability density?
Can an isolated system be described by a time dependent wavefunction? How would this not violate conservation of energy?
I see the meaning of the time averaged probability density. Is the time dependence just a statistical construct?
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Hello Praxeo, Welcome to Physics.SE. Please try to ask questions only related to the topic (of yours) or consider a revision to focus on your question. The problem (for now) is, you're asking lot of questions... – Waffle's Crazy Peanut Nov 15 '12 at 16:38
up vote 1 down vote accepted
1) Why do you believe that instantaneous probability densities are not meaningful?
2) Essentially any non-stationary state for which you need to compute time-dependent wavefunctions: e.g. chemical reaction dynamics, particle scattering, etc.
3) Yes, the time dependant Schrödinger equation applies to isolated systems.
4) By definition energy is conserved in an isolated system. Moreover, the Schrödinger equation conserves energy because the generator of time translations is the Hamiltonian and this commutes with itself $[H,H]=0$, i.e. energy is conserved. For isolated systems, the Hamiltonian is time-independent (explicitly) and the time-dependent wavefunction $\Psi$ has the well-known form $\Psi = \Phi e^{-iEt/\hbar}$, with $E$ the energy of the isolated system.
5) I do not understand the question.
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In (4), one needs a further condition that the Hamiltonian is itself time-independent, $\frac{\partial H}{\partial t} = 0$. – Stan Liou Nov 16 '12 at 9:34
As well, one has to distinguish the energy certainty from the energy conservation. – Vladimir Kalitvianski Nov 16 '12 at 15:17
@StanLiou: Conservation, by definition, implies zero production $d_iH/dt=0$. If the Hamiltonian has explicit time dependence then the equation of motion contains a 'flow' term $d_eH/dt$ but the production term continues being zero. – juanrga Nov 16 '12 at 18:28
@VladimirKalitvianski: Not sure what do you mean, but the conservation law $[H,H]=0$ is independent of the kind of quantum state. – juanrga Nov 16 '12 at 18:32
It's certainly correct and tautologous to say that energy is conserved in an isolated system. But if your last sentence were correct, all quantum systems whatever would conserve energy, because $[H,H] = 0$ is an exact identity and $H$ is always the generator of time translation. Hence, I expected the point of $[H,H] = 0$ to be a reference to $\frac{dA}{dt} = \frac{\partial A}{\partial t} + \frac{1}{i\hbar}[A,H]$ in the Heisenberg picture or analogous expectations in the Schrödinger picture. In the Lagrangian formalism, energy through Noether's theorem also needs no explicit time dependence. – Stan Liou Nov 16 '12 at 19:13
Yes, $|\psi(t)|^2$ is an instantaneous probability density.
Passage of a wave packet can be experimentally observed.
An isolated system can be in a superposition of different energy eigenfunctions. It does not violate the energy conservation law because initially the system is not in an eigenstate - it has some energy uncertainty at $t=0$. This uncertainty evolves as any other uncertainty.
EDIT: Let us make a superposition of two states: $$\psi(t)=c_1\psi_1(x)e^{-iE_1 t}+c_2\psi_2(x)e^{-iE_2 t}.$$ It means that we can find in experiment the system in state 1 with probability $|c_1|^2$ and in state 2 with probability $|c_2|^2$. The system is free and this is due to coefficients $c_1$ and $c_2$ being constant in time (occupation numbers do not depend on time).
Measuring the system energy will give sometimes $E_1$ and sometimes $E_2$, with the same probabilities. So initially and later on the system does not have a certain energy. The state $H\psi$ depends on time as $$H\psi=c_1 E_1 \psi_1(x)e^{-iE_1 t}+c_2 E_2 \psi_2(x)e^{-iE_2 t}.$$ It is not an eigenstate of the Hamiltonian, so the time derivative $\partial\psi/\partial t$ is not proportional to $\psi$.
The Hamiltonian expectation value, however, does not depend on time: $$\langle\psi|H|\psi\rangle = |c_1|^2 E_1 + |c_2|^2 E_2 = const.$$ In other words, it is the energy expectation value that conserves, not the energy. The latter is undefined, uncertain in this free state.
You invoke the "energy conservation law" $dH/dt=0$ which is an operator relationship. If the system has a certain energy $E_n$ in the initial state, this value remains the system energy in later moments, so your "conservation law" may be cast in a form $dE(t)/dt=0$ that means $E=const=E(0)=E_n$.
But if the system does not have a certain energy at the initial state $\psi(0)$, then there is no $E(0)$ to conserve and your operator relationship turns into conservation of the expectation value.
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4ad8e41917276c90 | Complex number
From Wikipedia, the free encyclopedia
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A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. "Re" is the real axis, "Im" is the imaginary axis, and i is the imaginary unit which satisfies i2 = −1.
A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, that satisfies the equation i2 = −1.[1] In this expression, a is the real part and b is the imaginary part of the complex number.
Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane. A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the complex numbers contain the ordinary real numbers while extending them in order to solve problems that cannot be solved with real numbers alone.
As well as their use within mathematics, complex numbers have practical applications in many fields, including physics, chemistry, biology, economics, electrical engineering, and statistics. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers. He called them "fictitious" during his attempts to find solutions to cubic equations in the 16th century.[2]
Complex numbers allow for solutions to certain equations that have no solutions in real numbers. For example, the equation
(x+1)^2 = -9 \,
has no real solution, since the square of a real number cannot be negative. Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the imaginary unit i where i2 = −1, so that solutions to equations like the preceding one can be found. In this case the solutions are −1 + 3i and −1 − 3i, as can be verified using the fact that i2 = −1:
((-1+3i)+1)^2 = (3i)^2 = (3^2)(i^2) = 9(-1) = -9,
((-1-3i)+1)^2 = (-3i)^2 = (-3)^2(i^2) = 9(-1) = -9.
According to the fundamental theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers.
An illustration of the complex plane. The real part of a complex number z = x + iy is x, and its imaginary part is y.
A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying i2 = −1. For example, −3.5 + 2i is a complex number.
The real number a is called the real part of the complex number a + bi; the real number b is called the imaginary part of a + bi. By this convention the imaginary part does not include the imaginary unit: hence b, not bi, is the imaginary part.[3][4] The real part of a complex number z is denoted by Re(z) or ℜ(z); the imaginary part of a complex number z is denoted by Im(z) or ℑ(z). For example,
\operatorname{Re}(-3.5 + 2i) &= -3.5 \\
\operatorname{Im}(-3.5 + 2i) &= 2.
Hence, in terms of its real and imaginary parts, a complex number z is equal to \operatorname{Re}(z) + \operatorname{Im}(z) \cdot i . This expression is sometimes known as the Cartesian form of z.
A real number a can be regarded as a complex number a + 0i whose imaginary part is 0. A purely imaginary number bi is a complex number 0 + bi whose real part is zero. It is common to write a for a + 0i and bi for 0 + bi. Moreover, when the imaginary part is negative, it is common to write abi with b > 0 instead of a + (−b)i, for example 3 − 4i instead of 3 + (−4)i.
The set of all complex numbers is denoted by , \mathbf{C} or \mathbb{C}.
Some authors[5] write a + ib instead of a + bi, particularly when b is a radical. In some disciplines, in particular electromagnetism and electrical engineering, j is used instead of i,[6] since i is frequently used for electric current. In these cases complex numbers are written as a + bj or a + jb.
Complex plane[edit]
Main article: Complex plane
Figure 1: A complex number plotted as a point (red) and position vector (blue) on an Argand diagram; a+bi is the rectangular expression of the point.
A complex number can be viewed as a point or position vector in a two-dimensional Cartesian coordinate system called the complex plane or Argand diagram (see Pedoe 1988 and Solomentsev 2001), named after Jean-Robert Argand. The numbers are conventionally plotted using the real part as the horizontal component, and imaginary part as vertical (see Figure 1). These two values used to identify a given complex number are therefore called its Cartesian, rectangular, or algebraic form.
A position vector may also be defined in terms of its magnitude and direction relative to the origin. These are emphasized in a complex number's polar form. Using the polar form of the complex number in calculations may lead to a more intuitive interpretation of mathematical results. Notably, the operations of addition and multiplication take on a very natural geometric character when complex numbers are viewed as position vectors: addition corresponds to vector addition while multiplication corresponds to multiplying their magnitudes and adding their arguments (i.e. the angles they make with the x axis). Viewed in this way the multiplication of a complex number by i corresponds to rotating the position vector counterclockwise by a quarter turn (90°) about the origin: (a+bi)i = ai+bi2 = -b+ai.
History in brief[edit]
Main section: History
The solution in radicals (without trigonometric functions) of a general cubic equation contains the square roots of negative numbers when all three roots are real numbers, a situation that cannot be rectified by factoring aided by the rational root test if the cubic is irreducible (the so-called casus irreducibilis). This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545, though his understanding was rudimentary.
Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.
Many mathematicians contributed to the full development of complex numbers. The rules for addition, subtraction, multiplication, and division of complex numbers were developed by the Italian mathematician Rafael Bombelli.[7] A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.
z_{1} = z_{2} \, \, \leftrightarrow \, \, ( \operatorname{Re}(z_{1}) = \operatorname{Re}(z_{2}) \, \and \, \operatorname{Im} (z_{1}) = \operatorname{Im} (z_{2})).
Because complex numbers are naturally thought of as existing on a two-dimensional plane, there is no natural linear ordering on the set of complex numbers.[8]
There is no linear ordering on the complex numbers that is compatible with addition and multiplication. Formally, we say that the complex numbers cannot have the structure of an ordered field. This is because any square in an ordered field is at least 0, but i2 = −1.
Elementary operations[edit]
Main article: Complex conjugate
Geometric representation of z and its conjugate \bar{z} in the complex plane
The complex conjugate of the complex number z = x + yi is defined to be xyi. It is denoted \bar{z} or z*.
Formally, for any complex number z:
\bar{z} = \operatorname{Re}(z) - \operatorname{Im}(z) \cdot i .
Geometrically, \bar{z} is the "reflection" of z about the real axis. Conjugating twice gives the original complex number: \bar{\bar{z}}=z.
The real and imaginary parts of a complex number z can be extracted using the conjugate:
\operatorname{Re}\,(z) = \tfrac{1}{2}(z+\bar{z}), \,
\operatorname{Im}\,(z) = \tfrac{1}{2i}(z-\bar{z}). \,
Moreover, a complex number is real if and only if it equals its conjugate.
Conjugation distributes over the standard arithmetic operations:
\overline{z+w} = \bar{z} + \bar{w}, \,
\overline{z-w} = \bar{z} - \bar{w}, \,
\overline{z w} = \bar{z} \bar{w}, \,
\overline{(z/w)} = \bar{z}/\bar{w}. \,
The reciprocal of a nonzero complex number z = x + yi is given by
\frac{1}{z}=\frac{\bar{z}}{z \bar{z}}=\frac{\bar{z}}{x^2+y^2}.
This formula can be used to compute the multiplicative inverse of a complex number if it is given in rectangular coordinates. Inversive geometry, a branch of geometry studying reflections more general than ones about a line, can also be expressed in terms of complex numbers. In the network analysis of electrical circuits, the complex conjugate is used in finding the equivalent impedance when the maximum power transfer theorem is used.
Addition and subtraction[edit]
Complex numbers are added by adding the real and imaginary parts of the summands. That is to say:
Similarly, subtraction is defined by
(a+bi) - (c+di) = (a-c) + (b-d)i.\
Multiplication and division[edit]
The multiplication of two complex numbers is defined by the following formula:
In particular, the square of the imaginary unit is −1:
i^2 = i \times i = -1.\
The preceding definition of multiplication of general complex numbers follows naturally from this fundamental property of the imaginary unit. Indeed, if i is treated as a number so that di means d times i, the above multiplication rule is identical to the usual rule for multiplying two sums of two terms.
(a+bi) (c+di) = ac + bci + adi + bidi (distributive law)
= ac + bidi + bci + adi (commutative law of addition—the order of the summands can be changed)
= ac + bdi^2 + (bc+ad)i (commutative and distributive laws)
= (ac-bd) + (bc + ad)i (fundamental property of the imaginary unit).
The division of two complex numbers is defined in terms of complex multiplication, which is described above, and real division. When at least one of c and d is non-zero, we have
\,\frac{a + bi}{c + di} = \left({ac + bd \over c^2 + d^2}\right) + \left( {bc - ad \over c^2 + d^2} \right)i.
Division can be defined in this way because of the following observation:
\,\frac{a + bi}{c + di} = \frac{\left(a + bi\right) \cdot \left(c - di\right)}{\left (c + di\right) \cdot \left (c - di\right)} = \left({ac + bd \over c^2 + d^2}\right) + \left( {bc - ad \over c^2 + d^2} \right)i.
As shown earlier, cdi is the complex conjugate of the denominator c + di. At least one of the real part c and the imaginary part d of the denominator must be nonzero for division to be defined. This is called "rationalization" of the denominator (although the denominator in the final expression might be an irrational real number).
Square root[edit]
The square roots of a + bi (with b ≠ 0) are \pm (\gamma + \delta i), where
\gamma = \sqrt{\frac{a + \sqrt{a^2 + b^2}}{2}}
\delta = \sgn (b) \sqrt{\frac{-a + \sqrt{a^2 + b^2}}{2}},
where sgn is the signum function. This can be seen by squaring \pm (\gamma + \delta i) to obtain a + bi.[9][10] Here \sqrt{a^2 + b^2} is called the modulus of a + bi, and the square root sign indicates the square root with non-negative real part, called the principal square root; also \sqrt{a^2 + b^2}= \sqrt{z\bar{z}}, where z = a + bi .[11]
Polar form[edit]
Figure 2: The argument φ and modulus r locate a point on an Argand diagram; r(\cos \varphi + i \sin \varphi) or r e^{i\varphi} are polar expressions of the point.
Absolute value and argument[edit]
An alternative way of defining a point P in the complex plane, other than using the x- and y-coordinates, is to use the distance of the point from O, the point whose coordinates are (0, 0) (the origin), together with the angle subtended between the positive real axis and the line segment OP in a counterclockwise direction. This idea leads to the polar form of complex numbers.
The absolute value (or modulus or magnitude) of a complex number z = x + yi is
\textstyle r=|z|=\sqrt{x^2+y^2}.\,
If z is a real number (i.e., y = 0), then r = | x |. In general, by Pythagoras' theorem, r is the distance of the point P representing the complex number z to the origin. The square of the absolute value is
\textstyle |z|^2=z\bar{z}=x^2+y^2.\,
where \bar{z} is the complex conjugate of z.
The argument of z (in many applications referred to as the "phase") is the angle of the radius OP with the positive real axis, and is written as \arg(z). As with the modulus, the argument can be found from the rectangular form x+yi:[12]
\varphi = \arg(z) =
\arctan(\frac{y}{x}) & \mbox{if } x > 0 \\
\arctan(\frac{y}{x}) + \pi & \mbox{if } x < 0 \mbox{ and } y \ge 0\\
\arctan(\frac{y}{x}) - \pi & \mbox{if } x < 0 \mbox{ and } y < 0\\
\frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y > 0\\
-\frac{\pi}{2} & \mbox{if } x = 0 \mbox{ and } y < 0\\
\mbox{indeterminate } & \mbox{if } x = 0 \mbox{ and } y = 0.
The value of φ is expressed in radians in this article. It can increase by any integer multiple of and still give the same angle. Hence, the arg function is sometimes considered as multivalued. Normally, as given above, the principal value in the interval (−π,π] is chosen. Values in the range [0,2π) are obtained by adding if the value is negative. The polar angle for the complex number 0 is indeterminate, but arbitrary choice of the angle 0 is common.
The value of φ equals the result of atan2: \varphi = \mbox{atan2}(\mbox{imaginary}, \mbox{real}).
Together, r and φ give another way of representing complex numbers, the polar form, as the combination of modulus and argument fully specify the position of a point on the plane. Recovering the original rectangular co-ordinates from the polar form is done by the formula called trigonometric form
z = r(\cos \varphi + i\sin \varphi ).\,
Using Euler's formula this can be written as
z = r e^{i \varphi}.\,
Using the cis function, this is sometimes abbreviated to
z = r \operatorname{cis} \varphi. \,
In angle notation, often used in electronics to represent a phasor with amplitude r and phase φ, it is written as[13]
z = r \ang \varphi . \,
Multiplication and division in polar form[edit]
Multiplication of 2 + i (blue triangle) and 3 + i (red triangle). The red triangle is rotated to match the vertex of the blue one and stretched by 5, the length of the hypotenuse of the blue triangle.
Formulas for multiplication, division and exponentiation are simpler in polar form than the corresponding formulas in Cartesian coordinates. Given two complex numbers z1 = r1(cos φ1 + i sin φ1) and z2 = r2(cos φ2 + i sin φ2), because of the well-known trigonometric identities
\cos(a)\cos(b) - \sin(a)\sin(b) = \cos(a + b)
\cos(a)\sin(b) + \sin(a)\cos(b) = \sin(a + b)
we may derive
z_1 z_2 = r_1 r_2 (\cos(\varphi_1 + \varphi_2) + i \sin(\varphi_1 + \varphi_2)).\,
In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. For example, multiplying by i corresponds to a quarter-turn counter-clockwise, which gives back i2 = −1. The picture at the right illustrates the multiplication of
(2+i)(3+i)=5+5i. \,
Since the real and imaginary part of 5 + 5i are equal, the argument of that number is 45 degrees, or π/4 (in radian). On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan(1/3) and arctan(1/2), respectively. Thus, the formula
holds. As the arctan function can be approximated highly efficiently, formulas like this—known as Machin-like formulas—are used for high-precision approximations of π.
Similarly, division is given by
\frac{z_1}{ z_2} = \frac{r_1}{ r_2} \left(\cos(\varphi_1 - \varphi_2) + i \sin(\varphi_1 - \varphi_2)\right).
Euler's formula[edit]
Euler's formula states that, for any real number x,
where e is the base of the natural logarithm. This can be proved through induction by observing that
i^0 &{}= 1, \quad &
i^1 &{}= i, \quad &
i^3 &{}= -i, \\
i^4 &={} 1, \quad &
i^5 &={} i, \quad &
i^7 &{}= -i,
and so on, and by considering the Taylor series expansions of eix, cos(x) and sin(x):
Natural logarithm[edit]
Euler's formula allows us to observe that, for any complex number
where r is a non-negative real number, one possible value for z's natural logarithm is
\ln (z)= \ln(r) + \varphi i
Because cos and sin are periodic functions, the natural logarithm may be considered a multi-valued function, with:
\ln(z) = \left\{ \ln(r) + (\varphi + 2\pi k)i \;|\; k \in \mathbb{Z}\right\}
Integer and fractional exponents[edit]
We may use the identity
\ln(a^{b}) = b \ln(a)
to define complex exponentiation, which is likewise multi-valued:
\ln (z^n)=\ln((r(\cos \varphi + i\sin \varphi ))^{n})
= n \ln(r(\cos \varphi + i\sin \varphi))
= \{ n (\ln(r) + (\varphi + k2\pi) i) | k \in \mathbb{Z} \}
= \{ n \ln(r) + n \varphi i + nk2\pi i | k \in \mathbb{Z} \}.
When n is an integer, this simplifies to de Moivre's formula:
z^{n}=(r(\cos \varphi + i\sin \varphi ))^{n} = r^n\,(\cos n\varphi + i \sin n \varphi).
The nth roots of z are given by
\sqrt[n]{z} = \sqrt[n]r \left( \cos \left(\frac{\varphi+2k\pi}{n}\right) + i \sin \left(\frac{\varphi+2k\pi}{n}\right)\right)
for any integer k satisfying 0 ≤ kn − 1. Here nr is the usual (positive) nth root of the positive real number r. While the nth root of a positive real number r is chosen to be the positive real number c satisfying cn = x there is no natural way of distinguishing one particular complex nth root of a complex number. Therefore, the nth root of z is considered as a multivalued function (in z), as opposed to a usual function f, for which f(z) is a uniquely defined number. Formulas such as
\sqrt[n]{z^n} = z
(which holds for positive real numbers), do in general not hold for complex numbers.
Field structure[edit]
The set C of complex numbers is a field. Briefly, this means that the following facts hold: first, any two complex numbers can be added and multiplied to yield another complex number. Second, for any complex number z, its additive inverse z is also a complex number; and third, every nonzero complex number has a reciprocal complex number. Moreover, these operations satisfy a number of laws, for example the law of commutativity of addition and multiplication for any two complex numbers z1 and z2:
z_1+ z_2 = z_2 + z_1,
z_1 z_2 = z_2 z_1.
These two laws and the other requirements on a field can be proven by the formulas given above, using the fact that the real numbers themselves form a field.
Unlike the reals, C is not an ordered field, that is to say, it is not possible to define a relation z1 < z2 that is compatible with the addition and multiplication. In fact, in any ordered field, the square of any element is necessarily positive, so i2 = −1 precludes the existence of an ordering on C.
When the underlying field for a mathematical topic or construct is the field of complex numbers, the topic's name is usually modified to reflect that fact. For example: complex analysis, complex matrix, complex polynomial, and complex Lie algebra.
Solutions of polynomial equations[edit]
Given any complex numbers (called coefficients) a0, …, an, the equation
a_n z^n + \dotsb + a_1 z + a_0 = 0
has at least one complex solution z, provided that at least one of the higher coefficients a1, …, an is nonzero. This is the statement of the fundamental theorem of algebra. Because of this fact, C is called an algebraically closed field. This property does not hold for the field of rational numbers Q (the polynomial x2 − 2 does not have a rational root, since 2 is not a rational number) nor the real numbers R (the polynomial x2 + a does not have a real root for a > 0, since the square of x is positive for any real number x).
There are various proofs of this theorem, either by analytic methods such as Liouville's theorem, or topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of odd degree has at least one real root.
Because of this fact, theorems that hold for any algebraically closed field, apply to C. For example, any non-empty complex square matrix has at least one (complex) eigenvalue.
Algebraic characterization[edit]
The field C has the following three properties: first, it has characteristic 0. This means that 1 + 1 + ⋯ + 1 ≠ 0 for any number of summands (all of which equal one). Second, its transcendence degree over Q, the prime field of C, is the cardinality of the continuum. Third, it is algebraically closed (see above). It can be shown that any field having these properties is isomorphic (as a field) to C. For example, the algebraic closure of Qp also satisfies these three properties, so these two fields are isomorphic. Also, C is isomorphic to the field of complex Puiseux series. However, specifying an isomorphism requires the axiom of choice. Another consequence of this algebraic characterization is that C contains many proper subfields that are isomorphic to C.
Characterization as a topological field[edit]
The preceding characterization of C describes only the algebraic aspects of C. That is to say, the properties of nearness and continuity, which matter in areas such as analysis and topology, are not dealt with. The following description of C as a topological field (that is, a field that is equipped with a topology, which allows the notion of convergence) does take into account the topological properties. C contains a subset P (namely the set of positive real numbers) of nonzero elements satisfying the following three conditions:
• P is closed under addition, multiplication and taking inverses.
• If x and y are distinct elements of P, then either xy or yx is in P.
• If S is any nonempty subset of P, then S + P = x + P for some x in C.
Moreover, C has a nontrivial involutive automorphism xx* (namely the complex conjugation), such that x x* is in P for any nonzero x in C.
Any field F with these properties can be endowed with a topology by taking the sets B(x, p) = { y | p − (yx)(yx)* ∈ P } as a base, where x ranges over the field and p ranges over P. With this topology F is isomorphic as a topological field to C.
The only connected locally compact topological fields are R and C. This gives another characterization of C as a topological field, since C can be distinguished from R because the nonzero complex numbers are connected, while the nonzero real numbers are not.
Formal construction[edit]
Formal development[edit]
Above, complex numbers have been defined by introducing i, the imaginary unit, as a symbol. More rigorously, the set C of complex numbers can be defined as the set R2 of ordered pairs (a, b) of real numbers. In this notation, the above formulas for addition and multiplication read
(a, b) + (c, d) &= (a + c, b + d)\\
(a, b) \cdot (c, d) &= (ac - bd, bc + ad).
It is then just a matter of notation to express (a, b) as a + bi.
Though this low-level construction does accurately describe the structure of the complex numbers, the following equivalent definition reveals the algebraic nature of C more immediately. This characterization relies on the notion of fields and polynomials. A field is a set endowed with addition, subtraction, multiplication and division operations that behave as is familiar from, say, rational numbers. For example, the distributive law
(x+y) z = xz + yz
must hold for any three elements x, y and z of a field. The set R of real numbers does form a field. A polynomial p(X) with real coefficients is an expression of the form
where the a0, ..., an are real numbers. The usual addition and multiplication of polynomials endows the set R[X] of all such polynomials with a ring structure. This ring is called polynomial ring.
The quotient ring R[X]/(X 2 + 1) can be shown to be a field. This extension field contains two square roots of −1, namely (the cosets of) X and X, respectively. (The cosets of) 1 and X form a basis of R[X]/(X 2 + 1) as a real vector space, which means that each element of the extension field can be uniquely written as a linear combination in these two elements. Equivalently, elements of the extension field can be written as ordered pairs (a, b) of real numbers. Moreover, the above formulas for addition etc. correspond to the ones yielded by this abstract algebraic approach—the two definitions of the field C are said to be isomorphic (as fields). Together with the above-mentioned fact that C is algebraically closed, this also shows that C is an algebraic closure of R.
Matrix representation of complex numbers[edit]
Complex numbers a + bi can also be represented by 2 × 2 matrices that have the following form:
a & -b \\
b & \;\; a
Here the entries a and b are real numbers. The sum and product of two such matrices is again of this form, and the sum and product of complex numbers corresponds to the sum and product of such matrices. The geometric description of the multiplication of complex numbers can also be expressed in terms of rotation matrices by using this correspondence between complex numbers and such matrices. Moreover, the square of the absolute value of a complex number expressed as a matrix is equal to the determinant of that matrix:
|z|^2 =
a & -b \\
b & a
= (a^2) - ((-b)(b)) = a^2 + b^2.
The conjugate \overline z corresponds to the transpose of the matrix.
Though this representation of complex numbers with matrices is the most common, many other representations arise from matrices other than \bigl(\begin{smallmatrix}0 & -1 \\1 & 0 \end{smallmatrix}\bigr) that square to the negative of the identity matrix. See the article on 2 × 2 real matrices for other representations of complex numbers.
Complex analysis[edit]
Color wheel graph of sin(1/z). Black parts inside refer to numbers having large absolute values.
Main article: Complex analysis
The study of functions of a complex variable is known as complex analysis and has enormous practical use in applied mathematics as well as in other branches of mathematics. Often, the most natural proofs for statements in real analysis or even number theory employ techniques from complex analysis (see prime number theorem for an example). Unlike real functions, which are commonly represented as two-dimensional graphs, complex functions have four-dimensional graphs and may usefully be illustrated by color-coding a three-dimensional graph to suggest four dimensions, or by animating the complex function's dynamic transformation of the complex plane.
Complex exponential and related functions[edit]
The notions of convergent series and continuous functions in (real) analysis have natural analogs in complex analysis. A sequence of complex numbers is said to converge if and only if its real and imaginary parts do. This is equivalent to the (ε, δ)-definition of limits, where the absolute value of real numbers is replaced by the one of complex numbers. From a more abstract point of view, C, endowed with the metric
\operatorname{d}(z_1, z_2) = |z_1 - z_2| \,
is a complete metric space, which notably includes the triangle inequality
|z_1 + z_2| \le |z_1| + |z_2|
for any two complex numbers z1 and z2.
Like in real analysis, this notion of convergence is used to construct a number of elementary functions: the exponential function exp(z), also written ez, is defined as the infinite series
\exp(z):= 1+z+\frac{z^2}{2\cdot 1}+\frac{z^3}{3\cdot 2\cdot 1}+\cdots = \sum_{n=0}^{\infty} \frac{z^n}{n!}. \,
and the series defining the real trigonometric functions sine and cosine, as well as hyperbolic functions such as sinh also carry over to complex arguments without change. Euler's identity states:
\exp(i\varphi) = \cos(\varphi) + i\sin(\varphi) \,
for any real number φ, in particular
\exp(i \pi) = -1 \,
Unlike in the situation of real numbers, there is an infinitude of complex solutions z of the equation
\exp(z) = w \,
for any complex number w ≠ 0. It can be shown that any such solution z—called complex logarithm of a—satisfies
\log(x+iy)=\ln|w| + i\arg(w), \,
where arg is the argument defined above, and ln the (real) natural logarithm. As arg is a multivalued function, unique only up to a multiple of 2π, log is also multivalued. The principal value of log is often taken by restricting the imaginary part to the interval (−π,π].
Complex exponentiation zω is defined as
z^\omega = \exp(\omega \log z). \,
Consequently, they are in general multi-valued. For ω = 1 / n, for some natural number n, this recovers the non-uniqueness of nth roots mentioned above.
Complex numbers, unlike real numbers, do not in general satisfy the unmodified power and logarithm identities, particularly when naïvely treated as single-valued functions; see failure of power and logarithm identities. For example, they do not satisfy
\,a^{bc} = (a^b)^c.
Both sides of the equation are multivalued by the definition of complex exponentiation given here, and the values on the left are a subset of those on the right.
Holomorphic functions[edit]
A function f : CC is called holomorphic if it satisfies the Cauchy–Riemann equations. For example, any R-linear map CC can be written in the form
with complex coefficients a and b. This map is holomorphic if and only if b = 0. The second summand b \overline z is real-differentiable, but does not satisfy the Cauchy–Riemann equations.
Complex analysis shows some features not apparent in real analysis. For example, any two holomorphic functions f and g that agree on an arbitrarily small open subset of C necessarily agree everywhere. Meromorphic functions, functions that can locally be written as f(z)/(zz0)n with a holomorphic function f, still share some of the features of holomorphic functions. Other functions have essential singularities, such as sin(1/z) at z = 0.
Complex numbers have essential concrete applications in a variety of scientific and related areas such as signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, and vibration analysis. Some applications of complex numbers are:
Control theory[edit]
In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The system's poles and zeros are then analyzed in the complex plane. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane.
In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes, i.e. have real part greater than or less than zero. If a linear, time-invariant (LTI) system has poles that are
If a system has zeros in the right half plane, it is a nonminimum phase system.
Improper integrals[edit]
In applied fields, complex numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this; see methods of contour integration.
Fluid dynamics[edit]
In fluid dynamics, complex functions are used to describe potential flow in two dimensions.
Dynamic equations[edit]
In differential equations, it is common to first find all complex roots r of the characteristic equation of a linear differential equation or equation system and then attempt to solve the system in terms of base functions of the form f(t) = ert. Likewise, in difference equations, the complex roots r of the characteristic equation of the difference equation system are used, to attempt to solve the system in terms of base functions of the form f(t) = rt.
Electromagnetism and electrical engineering[edit]
Main article: Alternating current
In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. This approach is called phasor calculus.
In electrical engineering, the imaginary unit is denoted by j, to avoid confusion with I, which is generally in use to denote electric current, or, more particularly, i, which is generally in use to denote instantaneous electric current.
Since the voltage in an AC circuit is oscillating, it can be represented as
V(t) = V_0 e^{j \omega t} = V_0 \left (\cos \omega t + j \sin\omega t \right ),
To obtain the measurable quantity, the real part is taken:
v(t) = \mathrm{Re}(V) = \mathrm{Re}\left [ V_0 e^{j \omega t} \right ] = V_0 \cos \omega t.
The complex-valued signal V(t) is called the analytic representation of the real-valued, measurable signal v(t). [14]
Signal analysis[edit]
Complex numbers are used in signal analysis and other fields for a convenient description for periodically varying signals. For given real functions representing actual physical quantities, often in terms of sines and cosines, corresponding complex functions are considered of which the real parts are the original quantities. For a sine wave of a given frequency, the absolute value | z | of the corresponding z is the amplitude and the argument arg(z) is the phase.
If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as complex valued functions of the form
x(t) = Re \{X( t ) \} \,
X( t ) = A e^{i\omega t} = a e^{ i \phi } e^{i\omega t} = a e^{i (\omega t + \phi) } \,
where ω represents the angular frequency and the complex number A encodes the phase and amplitude as explained above.
This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis (and wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, and video signals.
Another example, relevant to the two side bands of amplitude modulation of AM radio, is:
\cos((\omega+\alpha)t)+\cos\left((\omega-\alpha)t\right) & = \operatorname{Re}\left(e^{i(\omega+\alpha)t} + e^{i(\omega-\alpha)t}\right) \\
& = \operatorname{Re}\left((e^{i\alpha t} + e^{-i\alpha t})\cdot e^{i\omega t}\right) \\
& = \operatorname{Re}\left(2\cos(\alpha t) \cdot e^{i\omega t}\right) \\
& = 2 \cos(\alpha t) \cdot \operatorname{Re}\left(e^{i\omega t}\right) \\
& = 2 \cos(\alpha t)\cdot \cos\left(\omega t\right)\,.
Quantum mechanics[edit]
The complex number field is intrinsic to the mathematical formulations of quantum mechanics, where complex Hilbert spaces provide the context for one such formulation that is convenient and perhaps most standard. The original foundation formulas of quantum mechanics—the Schrödinger equation and Heisenberg's matrix mechanics—make use of complex numbers.
In special and general relativity, some formulas for the metric on spacetime become simpler if one takes the time component of the spacetime continuum to be imaginary. (This approach is no longer standard in classical relativity, but is used in an essential way in quantum field theory.) Complex numbers are essential to spinors, which are a generalization of the tensors used in relativity.
Certain fractals are plotted in the complex plane, e.g. the Mandelbrot set and Julia sets.
Every triangle has a unique Steiner inellipse—an ellipse inside the triangle and tangent to the midpoints of the three sides of the triangle. The foci of a triangle's Steiner inellipse can be found as follows, according to Marden's theorem:[15][16] Denote the triangle's vertices in the complex plane as a = xA + yAi, b = xB + yBi, and c = xC + yCi. Write the cubic equation \scriptstyle (x-a)(x-b)(x-c)=0, take its derivative, and equate the (quadratic) derivative to zero. Marden's Theorem says that the solutions of this equation are the complex numbers denoting the locations of the two foci of the Steiner inellipse.
Algebraic number theory[edit]
Construction of a regular pentagon using straightedge and compass.
As mentioned above, any nonconstant polynomial equation (in complex coefficients) has a solution in C. A fortiori, the same is true if the equation has rational coefficients. The roots of such equations are called algebraic numbers – they are a principal object of study in algebraic number theory. Compared to Q, the algebraic closure of Q, which also contains all algebraic numbers, C has the advantage of being easily understandable in geometric terms. In this way, algebraic methods can be used to study geometric questions and vice versa. With algebraic methods, more specifically applying the machinery of field theory to the number field containing roots of unity, it can be shown that it is not possible to construct a regular nonagon using only compass and straightedge – a purely geometric problem.
Another example are Gaussian integers, that is, numbers of the form x + iy, where x and y are integers, which can be used to classify sums of squares.
Analytic number theory[edit]
Analytic number theory studies numbers, often integers or rationals, by taking advantage of the fact that they can be regarded as complex numbers, in which analytic methods can be used. This is done by encoding number-theoretic information in complex-valued functions. For example, the Riemann zeta function ζ(s) is related to the distribution of prime numbers.
The earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of the Greek mathematician Hero of Alexandria in the 1st century AD, where in his Stereometrica he considers, apparently in error, the volume of an impossible frustum of a pyramid to arrive at the term \scriptstyle \sqrt{81 - 144} = 3i\sqrt{7} in his calculations, although negative quantities were not conceived of in Hellenistic mathematics and Heron merely replaced it by its positive (\scriptstyle \sqrt{144 - 81} = 3\sqrt{7}).[17]
The impetus to study complex numbers proper first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians (see Niccolò Fontana Tartaglia, Gerolamo Cardano). It was soon realized that these formulas, even if one was only interested in real solutions, sometimes required the manipulation of square roots of negative numbers. As an example, Tartaglia's formula for a cubic equation of the form \scriptstyle x^3 = px + q[18] gives the solution to the equation x3 = x as
At first glance this looks like nonsense. However formal calculations with complex numbers show that the equation z3 = i has solutions i, {\scriptstyle\frac{\sqrt{3}}{2}}+{\scriptstyle\frac{1}{2}}i and {\scriptstyle\frac{-\sqrt{3}}{2}}+{\scriptstyle\frac{1}{2}}i. Substituting these in turn for {\scriptstyle\sqrt{-1}^{1/3}} in Tartaglia's cubic formula and simplifying, one gets 0, 1 and −1 as the solutions of x3x = 0. Of course this particular equation can be solved at sight but it does illustrate that when general formulas are used to solve cubic equations with real roots then, as later mathematicians showed rigorously, the use of complex numbers is unavoidable. Rafael Bombelli was the first to explicitly address these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic trying to resolve these issues.
The term "imaginary" for these quantities was coined by René Descartes in 1637, although he was at pains to stress their imaginary nature[19]
[...] sometimes only imaginary, that is one can imagine as many as I said in each equation, but sometimes there exists no quantity that matches that which we imagine.
([...] quelquefois seulement imaginaires c’est-à-dire que l’on peut toujours en imaginer autant que j'ai dit en chaque équation, mais qu’il n’y a quelquefois aucune quantité qui corresponde à celle qu’on imagine.)
A further source of confusion was that the equation \scriptstyle \sqrt{-1}^2=\sqrt{-1}\sqrt{-1}=-1 seemed to be capriciously inconsistent with the algebraic identity \scriptstyle \sqrt{a}\sqrt{b}=\sqrt{ab}, which is valid for non-negative real numbers a and b, and which was also used in complex number calculations with one of a, b positive and the other negative. The incorrect use of this identity (and the related identity \scriptstyle \frac{1}{\sqrt{a}}=\sqrt{\frac{1}{a}}) in the case when both a and b are negative even bedeviled Euler. This difficulty eventually led to the convention of using the special symbol i in place of −1 to guard against this mistake.[citation needed] Even so, Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.
In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the complicated identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be simply re-expressed by the following well-known formula which bears his name, de Moivre's formula:
(\cos \theta + i\sin \theta)^{n} = \cos n \theta + i\sin n \theta. \,
In 1748 Leonhard Euler went further and obtained Euler's formula of complex analysis:
\cos \theta + i\sin \theta = e ^{i\theta } \,
by formally manipulating complex power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.
The idea of a complex number as a point in the complex plane (above) was first described by Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's De Algebra tractatus.
Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra. Gauss had earlier published an essentially topological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1". It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology. The English mathematician G. H. Hardy remarked that Gauss was the first mathematician to use complex numbers in 'a really confident and scientific way' although mathematicians such as Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.[20] Augustin Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of complex analysis to a high state of completion, commencing around 1825 in Cauchy's case.
The common terms used in the theory are chiefly due to the founders. Argand called \scriptstyle \cos \phi + i\sin \phi the direction factor, and \scriptstyle r = \sqrt{a^2+b^2} the modulus; Cauchy (1828) called \cos \phi + i\sin \phi the reduced form (l'expression réduite) and apparently introduced the term argument; Gauss used i for \scriptstyle \sqrt{-1}, introduced the term complex number for a + bi, and called a2 + b2 the norm. The expression direction coefficient, often used for \cos \phi + i\sin \phi, is due to Hankel (1867), and absolute value, for modulus, is due to Weierstrass.
Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others.
Generalizations and related notions[edit]
The process of extending the field R of reals to C is known as Cayley–Dickson construction. It can be carried further to higher dimensions, yielding the quaternions H and octonions O which (as a real vector space) are of dimension 4 and 8, respectively.
However, just as applying the construction to reals loses the property of ordering, more properties familiar from real and complex numbers vanish with increasing dimension. The quaternions are only a skew field, i.e. for some x, y: x·yy·x for two quaternions, the multiplication of octonions fails (in addition to not being commutative) to be associative: for some x, y, z: (x·yzx·(y·z).
Reals, complex numbers, quaternions and octonions are all normed division algebras over R. However, by Hurwitz's theorem they are the only ones. The next step in the Cayley–Dickson construction, the sedenions, in fact fails to have this structure.
The Cayley–Dickson construction is closely related to the regular representation of C, thought of as an R-algebra (an R-vector space with a multiplication), with respect to the basis (1, i). This means the following: the R-linear map
\mathbb{C} \rightarrow \mathbb{C}, z \mapsto wz
for some fixed complex number w can be represented by a 2 × 2 matrix (once a basis has been chosen). With respect to the basis (1, i), this matrix is
\operatorname{Re}(w) & -\operatorname{Im}(w) \\
\operatorname{Im}(w) & \;\; \operatorname{Re}(w)
i.e., the one mentioned in the section on matrix representation of complex numbers above. While this is a linear representation of C in the 2 × 2 real matrices, it is not the only one. Any matrix
J = \begin{pmatrix}p & q \\ r & -p \end{pmatrix}, \quad p^2 + qr + 1 = 0
has the property that its square is the negative of the identity matrix: J2 = −I. Then
\{ z = a I + b J : a,b \in R \}
is also isomorphic to the field C, and gives an alternative complex structure on R2. This is generalized by the notion of a linear complex structure.
Hypercomplex numbers also generalize R, C, H, and O. For example, this notion contains the split-complex numbers, which are elements of the ring R[x]/(x2 − 1) (as opposed to R[x]/(x2 + 1)). In this ring, the equation a2 = 1 has four solutions.
The field R is the completion of Q, the field of rational numbers, with respect to the usual absolute value metric. Other choices of metrics on Q lead to the fields Qp of p-adic numbers (for any prime number p), which are thereby analogous to R. There are no other nontrivial ways of completing Q than R and Qp, by Ostrowski's theorem. The algebraic closure \overline {\mathbf{Q}_p} of Qp still carry a norm, but (unlike C) are not complete with respect to it. The completion \mathbf{C}_p of \overline {\mathbf{Q}_p} turns out to be algebraically closed. This field is called p-adic complex numbers by analogy.
The fields R and Qp and their finite field extensions, including C, are local fields.
See also[edit]
1. ^ Charles P. McKeague (2011), Elementary Algebra, Brooks/Cole, p. 524, ISBN 978-0-8400-6421-9
2. ^ Burton (1995, p. 294)
3. ^ Complex Variables (2nd Edition), M.R. Spiegel, S. Lipschutz, J.J. Schiller, D. Spellman, Schaum's Outline Series, Mc Graw Hill (USA), ISBN 978-0-07-161569-3
4. ^ Aufmann, Richard N.; Barker, Vernon C.; Nation, Richard D. (2007), "Chapter P", College Algebra and Trigonometry (6 ed.), Cengage Learning, p. 66, ISBN 0-618-82515-0
5. ^ For example Ahlfors (1979).
6. ^ Brown, James Ward; Churchill, Ruel V. (1996), Complex variables and applications (6th ed.), New York: McGraw-Hill, p. 2, ISBN 0-07-912147-0, In electrical engineering, the letter j is used instead of i.
7. ^ Katz (2004, §9.1.4)
8. ^
9. ^ Abramowitz, Milton; Stegun, Irene A. (1964), Handbook of mathematical functions with formulas, graphs, and mathematical tables, Courier Dover Publications, p. 17, ISBN 0-486-61272-4 , Section 3.7.26, p. 17
10. ^ Cooke, Roger (2008), Classical algebra: its nature, origins, and uses, John Wiley and Sons, p. 59, ISBN 0-470-25952-3 , Extract: page 59
11. ^ Ahlfors (1979, p. 3)
12. ^ Kasana, H.S. (2005), "Chapter 1", Complex Variables: Theory And Applications (2nd ed.), PHI Learning Pvt. Ltd, p. 14, ISBN 81-203-2641-5
13. ^ Nilsson, James William; Riedel, Susan A. (2008), "Chapter 9", Electric circuits (8th ed.), Prentice Hall, p. 338, ISBN 0-13-198925-1
14. ^ Electromagnetism (2nd edition), I.S. Grant, W.R. Phillips, Manchester Physics Series, 2008 ISBN 0-471-92712-0
15. ^ Kalman, Dan (2008a), "An Elementary Proof of Marden's Theorem", The American Mathematical Monthly 115: 330–38, ISSN 0002-9890
16. ^ Kalman, Dan (2008b), "The Most Marvelous Theorem in Mathematics", Journal of Online Mathematics and its Applications External link in |work= (help)
17. ^ Nahin, Paul J. (2007), An Imaginary Tale: The Story of −1, Princeton University Press, ISBN 978-0-691-12798-9, retrieved 20 April 2011
18. ^ In modern notation, Tartaglia's solution is based on expanding the cube of the sum of two cube roots: \scriptstyle \left(\sqrt[3]{u} + \sqrt[3]{v}\right)^3 = 3 \sqrt[3]{uv} \left(\sqrt[3]{u} + \sqrt[3]{v}\right) + u + v With \scriptstyle x = \sqrt[3]{u} + \sqrt[3]{v}, \scriptstyle p = 3 \sqrt[3]{uv}, \scriptstyle q = u + v, u and v can be expressed in terms of p and q as \scriptstyle u = q/2 + \sqrt{(q/2)^2-(p/3)^3} and \scriptstyle v = q/2 - \sqrt{(q/2)^2-(p/3)^3}, respectively. Therefore, \scriptstyle x = \sqrt[3]{q/2 + \sqrt{(q/2)^2-(p/3)^3}} + \sqrt[3]{q/2 - \sqrt{(q/2)^2-(p/3)^3}}. When \scriptstyle (q/2)^2-(p/3)^3 is negative (casus irreducibilis), the second cube root should be regarded as the complex conjugate of the first one.
19. ^ Descartes, René (1954) [1637], La Géométrie | The Geometry of René Descartes with a facsimile of the first edition, Dover Publications, ISBN 0-486-60068-8, retrieved 20 April 2011
20. ^ Hardy, G. H.; Wright, E. M. (2000) [1938], An Introduction to the Theory of Numbers, OUP Oxford, p. 189 (fourth edition), ISBN 0-19-921986-9
Mathematical references[edit]
Historical references[edit]
• Burton, David M. (1995), The History of Mathematics (3rd ed.), New York: McGraw-Hill, ISBN 978-0-07-009465-9
• Katz, Victor J. (2004), A History of Mathematics, Brief Version, Addison-Wesley, ISBN 978-0-321-16193-2
• Nahin, Paul J. (1998), An Imaginary Tale: The Story of \scriptstyle\sqrt{-1} (hardcover edition ed.), Princeton University Press, ISBN 0-691-02795-1
A gentle introduction to the history of complex numbers and the beginnings of complex analysis.
• H.D. Ebbinghaus; H. Hermes; F. Hirzebruch; M. Koecher; K. Mainzer; J. Neukirch; A. Prestel; R. Remmert (1991), Numbers (hardcover ed.), Springer, ISBN 0-387-97497-0
An advanced perspective on the historical development of the concept of number.
Further reading[edit]
• The Road to Reality: A Complete Guide to the Laws of the Universe, by Roger Penrose; Alfred A. Knopf, 2005; ISBN 0-679-45443-8. Chapters 4–7 in particular deal extensively (and enthusiastically) with complex numbers.
• Unknown Quantity: A Real and Imaginary History of Algebra, by John Derbyshire; Joseph Henry Press; ISBN 0-309-09657-X (hardcover 2006). A very readable history with emphasis on solving polynomial equations and the structures of modern algebra.
• Visual Complex Analysis, by Tristan Needham; Clarendon Press; ISBN 0-19-853447-7 (hardcover, 1997). History of complex numbers and complex analysis with compelling and useful visual interpretations.
• Conway, John B., Functions of One Complex Variable I (Graduate Texts in Mathematics), Springer; 2 edition (12 September 2005). ISBN 0-387-90328-3.
External links[edit] |
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Explores the growth of twentieth-century physics, including theoretical developments like relativity, quantum mechanics, and symmetry-based thinking, and the rise of new subdisciplines such as atomic physics, condensed-matter physics, nuclear physics, and particle physics. Some attention is given to the societal context of physics, the institutions of the discipline, and the relations between 'pure' and 'applied' physics. Students who have taken or are concurrently taking any physics course numbered over 100 will not receive credit for this course. Familiarity with standard secondary school mathematics is required.
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1ec89a1de47dc323 | Do all quantum trails inevitably lead to Everett?
I’ve been thinking lately about quantum physics, a topic that seems to attract all sorts of crazy speculation and intense controversy, which seems inevitable. Quantum mechanics challenges our deepest held most cherished beliefs about how reality works. If you study the quantum world and you don’t come away deeply unsettled, then you simply haven’t properly engaged with it. (I originally wrote “understood” in the previous sentence instead of “engaged”, but the ghost of Richard Feymann reminded me that if you think you understand quantum mechanics, you don’t understand quantum mechanics.)
At the heart of the issue are facts such as that quantum particles operate as waves until someone “looks” at them, or more precisely, “measures” them, then they instantly begin behaving like particles with definite positions. There are other quantum properties, such as spin, which show similar dualities. Quantum objects in their pre-measurement states are referred to as being in a superposition. That superposition appears to instantly disappear when the measurement happens, with the object “choosing” a particular path, position, or state.
How do we know that the quantum objects are in this superposition before we look at them? Because in their superposition states, the spread out parts interfere with each other. This is evident in the famous double slit experiment, where single particles shot through the slits one at a time, interfere with themselves to produce the interference pattern that waves normally produce. If you’re not familiar with this experiment and its crazy implications, check out this video:
So, what’s going on here? What happens when the superposition disappears? The mathematics of quantum theory are reportedly rock solid. From a straight calculation standpoint, physicists know what to do. Which leads many of them to decry any attempt to further explain what’s happening. The phrase, “shut up and calculate,” is often exclaimed to pesky students who want to understand what is happening. This seems to be the oldest and most widely accepted attitude toward quantum mechanics in physics.
From what I understand, the original Copenhagen Interpretation was very much an instrumental view of quantum physics. It decried any attempt to explore beyond the observations and mathematics as hopeless speculation. (I say “original” because there are a plethora of views under the Copenhagen label, and many of them make ontological assertions that the original formulation seemed to avoid, such as insisting that there is no other reality than what is described.)
Under this view, the wave of the quantum object evolves under the wave function, a mathematical construct. When a measurement is attempted, the wave function “collapses”, which is just a fancy way of saying it disappears. The superposition becomes a definite state.
What exactly causes the collapse? What does “measurement” or “observation” mean in this context? It isn’t interaction with just another quantum object. Molecules have been held in quantum superposition, including, as a new recent experiment demonstrates, ones with thousands of atoms. For a molecule to hold together, chemical bonds have to form, and for the individual atoms to hold together, the components have to exchange bosons (photons, gluons, etc) with each other. All this happens and apparently fails to cause a collapse in otherwise isolated systems.
One proposal thrown out decades ago, which has long been a favorite of New Age spiritualists and similarly minded people, is that maybe consciousness causes the collapse. In other words, maybe it doesn’t happen until we look at it. However, most physicists don’t give this notion much weight. And the difficulties of engineering a quantum computer, which require that a superposition be maintained to get their processing benefits, seems to show (to the great annoyance of engineers) that systems with no interaction with consciousness still experience collapse.
What appears to cause the collapse is interaction with the environment. But what exactly is “the environment”? For an atom in a molecule, the environment would be the rest of the molecule, but an isolated molecule seems capable of maintaining its superposition. How complex or vast does the interacting system need to be to cause the collapse? The Copenhagen Interpretation merely says a macroscopic object, such as a measuring apparatus, but that’s an imprecise term. At what point do we leave the microscopic realm and enter the classical macroscopic realm? Experiments that succeed at isolating ever larger macromolecules seem able to preserve the quantum superposition.
If we move beyond the Copenhagen Interpretation, we encounter propositions that maybe the collapse doesn’t really happen. The oldest of these is the deBroglie-Bohm Interpretation. In it, there is always a particle that is guided by a pilot wave. The pilot wave appears to disappear on measurement, but what’s really happening is that the wave decoheres, loses its coherence into the environment, causing the particle to behave like a freestanding particle.
The problem is that this interpretation is explicitly non-local in that destroying any part of the wave causes the whole thing to cease any effect on the particle. Non-locality, essentially action at a distance, is considered anathema in physics. (Although it’s often asserted that quantum entanglement makes it unavoidable.)
The most controversial proposition is that maybe the collapse never happens and that the superposition continues, spreading to other systems. The elegance of this interpretation is that it essentially allows the system to continue evolving according to the Schrödinger equation, the central equation in the mathematics of quantum mechanics. From an Occam’s razor standpoint, this looks promising.
Well, except for a pesky detail. We don’t observe the surrounding environment going into a superposition. After a measurement, the measuring apparatus and lab setup seem just as singular as they always have. But this is sloppy thinking. Under this proposition, the measuring apparatus and lab have gone into superposition. We don’t observe it because we ourselves have gone into superposition.
In other words, there’s a version of the measuring apparatus that measures the particle going one way, and a version that measures it going the other way. There’s a version of the scientist that sees the measurement one way, and another version of the scientist that sees it the other way. When they call their colleague to tell them about the results, the colleague goes into superposition. When they publish their results, the journal goes into superposition. When we read the paper, we go into superposition. The superposition spreads ever farther out into spacetime.
We don’t see interference between the branches of superpositions because the waves have decohered, lost their phase with each other. Brian Greene in The Hidden Reality points out that it may be possible in principle to measure some remnant interference from the decohered waves, but it would be extremely difficult. Another physicist compared it to trying to measure the effects of Jupiter’s gravity on a satellite orbiting the Earth: possible in principle but beyond the precision of our current instruments.
Until that becomes possible, we have to consider each path as its own separate causal framework. Each quantum event expands the overall wave function of the universe, making each one its own separate branch of causality, in essence, its own separate universe or world, which is why this proposition is generally known as the Many Worlds Interpretation.
Which interpretation is reality? Obviously there’s a lot more of them than I mentioned here, so this post is unavoidably narrow in its consideration. To me, the (instrumental) Copenhagen Interpretation has the benefit of being epistemically humble. Years ago, I was attracted to the deBroglie-Bohm Interpretation, but it has a lot of problems and is not well regarded by most physicists.
The Many Worlds Interpretation seems absurd, but we need to remember that the interpretation itself isn’t so much absurd, but its implications. Criticizing the interpretation because of those implications, as this Quanta Magazine piece does, seems unproductive, akin to criticizing general relativity because we don’t like the relativity of simultaneity, or evolution because we don’t like what it says about humanity’s place in nature.
With every experiment that increases the maximally observed size of quantum objects, the more likely it seems to me that the whole universe is essentially quantum, and the more inevitable this interpretation seems.
Now, it may be possible that Hugh Everett III, the originator of this interpretation, was right that the wave function never collapses, but that some other factor prevents the unseen parts of the post-measurement wave from actually being real. Referred to as the unreal version of the interpretation, this seems to be the position of a lot of physicists. Since we have no present way of testing the proposition as Brian Greene suggested, we can’t know.
From a scientific perspective then, it seems like the most responsible position is agnosticism. But from an emotional perspective, I have to admit that the elegance of spreading superpositions are appealing to me, even if I’m very aware that there’s no way to test the implications.
What do you think? Am I missing anything? Are there actual physics problems with the Many Worlds Interpretation that should disqualify it? Or other interpretations that we should be considering?
56 thoughts on “Do all quantum trails inevitably lead to Everett?
1. There is no such thing as objectivity. Not all physicist dismiss some interaction with consciousness. Some very prominent physicists at least think its a plausible possibility. I love how folks want to dismiss any quantum connections to consciousnesses as mysticism. One has to wonder what would ever be enough evidence to at least get those who don’t want to give credence to at least say its plausible. They so easily accept Hawking’s many worlds and parallel universe hypotheses even though thus far no real method to test them exists either. But suggests that perhaps consciousness is more than the body and your a mystic. I like Penrose’s Biocentrism ideas. But I like others to include perhaps its an illusion or we live in a real matrix. But I certainly don’t consider myself a mystic.
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1. Well Im not as learned as you so I would say what is the evidence for it not causing the collapse? Testable verifiable evidence? Just like several hypotheses in quantum Physics testable, repeatable and verifiable evidence is often out of reach. I’m not saying anything is absolutely true but to dismiss a hypothesis out of hand when your preferred solution is just as untestable and unverifiable is not objective and not fair. In the end all of it might be wrong. If consciousness is merely something that fades after the death of the organism so be it, nothing we can do its natural law in that case. But until we know, if we can know, we should at least aknowledge some very smart Physicist do indeed think consciousness may play a role. Roger Penrose is no crazy person nor is Stuart Hammerhoff an uneducated loon. Those are just two people who have a wide variety of hypotheses that say its plausible that consciousness interacts with exotic quantum particles. Many point out the double slit Experiment among other things as an example of what might be. Nobody knows what is as of yet but to call legitimate scientists mystics for saying maybe is just unfair. In the end you and scientists like you might indeed be right but until its proven please give all legitimate scientists the same respect you gave Hawking when he proposed String Theory and all the craziness, parallel worlds, many copies of me on parallel worlds, and all the other things I watched on The Scifi Channel, that come with it. Now some scientists are saying consciousness is an illusion and that’s funny really. When you cant solve it say it doesn’t exist solves the problem only it doesn’t because it does exist only its nature is a mystery
2. To the very limited extent that I understand the mathematics of decoherence, it does seem to make Everett the most natural interpretation. Why should orthogonal states just vanish when their effect on us diminishes? “Us” meaning the states of observers whose device registered a particle going through the left slit, for example, and “orthogonal ” meaning approximately orthogonal, to within some rounding error.
The fact that decoherence is in principle a smooth process, albeit a fast one, takes a lot of the sting out of the Many Worlds label. It’s kind of a misnomer. It would be equally fair to say there’s one world in Everett, but many superposed states that have extremely weak interactions.
A good resource is the wiki article on decoherence. Another is David Wallace, The Emergent Multiverse .
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1. Thanks for the references. I agree on the wiki article. I’ll check out the Wallace one.
Good point about the label. The main reason I described MWI the way I did was to downplay the new universes thing. Dewitt reportedly used it as a selling tool, but I think it makes too many people dismiss it as outlandish without understanding what’s actually being proposed.
3. Nobody knows the source or nature of consciousness. There is evidence you remain conscious after the heart stops and blood flow to the brain ceases. For how long is still being examined. Previously this was not thought possible. Now some adjust there position saying activity continues till clinical brain death. No one as of yet can provide evidence consciousness is not affecting quantum particles or the double slit Experiment because nobody knows the nature, origin, components or make up of consciousness. Hell some just give up altogether and say its not real anyway, its an illusion. So all human beings are, what they have acomplished over millions of years of evolution is an illusion. Anyone who matter of factly claims they can prove consciousnesses is not affecting the quantum relm or vis versa know there wrong. Nobody even knows what consciousness is composed of let alone its origins so they can’t say for sure one way or another. They can dismiss it as woo or mysticism, they can belittle those who at least say maybe but, just like those who subjectively hope consciousness doesn’t die, they cant prove anything one way or the other. I wouldn’t be so harsh if people disparage brilliant scientists like Penrose and others by calling it mysticism. No better way to disparage a scientist than to call his or her hypothesis mysticism. Nobody called Hawking a mystic when ge hypothesized String Theory which is a parallel worlds theory with absolutely no direct evidence of it being true. Honestly parallel universes with my double in them sounds pretty darn mystical to me.
4. Don’t confuse the scientific method with the actual scientists. Scientists are people, human beings, and like all human beings they are almost incapable of objectivity on there own. If you can pick it up, put it in a beaker, and test it using the Scientific Method thats objective. Supposedly if the math works that is a good sign it could be true but even if tte math works it still can be wrong. If you can’t pick it up and test it it could be wrong. Quantum Physics reaches out into a largely untestable area of science. In fact many well known scientists ponder aloud that maybe we have reached or soon will reach all we are capable of knowing leaving infinite amounts of questions unanswered and unknowable.
1. Hi Matthew,
“…I would say what is the evidence for it not causing the collapse? Testable verifiable evidence? ”
I alluded to some in the post: the difficulty in constructing a quantum computer. Quantum computing’s unique value is being able to process possible paths in parallel, which requires maintaining a superposition as long as possible. However, long before any conscious entity becomes aware of what’s happening, the superposition decoheres. This is a serious challenge for QC. If it could be overcome simply by keeping conscious systems from seeing it, it likely would have been solved decades ago. As it is, many QC processors have to operate at near 0 Kelvin to minimize interaction with the environment and even that only keeps the qubit circuits in superposition for a very brief time.
“Nobody knows the source or nature of consciousness.”
I think neuroscience is making steady progress in understanding it. (See the posts in my Mind and AI category for why.) Of course, many people don’t like what’s being found, so the assertion that science is utterly helpless in this area remains a popular one.
“Don’t confuse the scientific method with the actual scientists.”
A crucial part of scientific methods (there isn’t just one) is guarding against human bias. It’s why results must be repeatable, transparent, and subject to peer review. In my experience, the ones that pass this test don’t affirm expansive conceptions of consciousness.
But as you note, there is no unique evidence for any one interpretation of quantum physics. It’s why I said that the responsible position is agnosticism on them. For now.
5. Maybe a little beside the point … Please forgive me.
As someone who could not even bother with elementary school and for several years has not been able to master English … he claims that scientists do not understand the basic processes of the universe.
Well, it can be said, it’s just a stupid Pole.
But I will not be giving hundreds of examples of scientific indolence. Only one.
Just what to think of the state of the scientific mind, when one of the most prominent minds, carries out such a thought experiment … whether it was just a joke or just a word of despair
Throw a book into the black hole. The book carries information. Perhaps that information is about physics, perhaps that information is the plot of a romance novel – it could be any kind of information. But as far as anyone knows, the outgoing Hawking radiation is the same no matter what went into the black hole. The information is apparently lost – where did it go?
Do we see one of the greatest idiocys of quantum physics?
Do we see how beautiful minds are stupidity?
Maybe just a stupid pole is dumber than it would seem?
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1. Stan,
From what I understand, information lost to a black hole remains a problem that hasn’t been solved. I’ve read some speculation that maybe it’s smeared across the event horizon as a sort of hologram, which sounds like it could conceivably affect Hawking radiation, but it all sounds highly speculative.
One of the problems with physics today is that too much of the theoretical work happens far outside of testable conditions. On the one hand, this should be fine since we never know when such exploration might turn up something testable. But until it does, we have to be stringent in remembering that it’s informed speculation.
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6. Mike.
Only this is not a problem with the information that carries the object that falls into a black hole.
This applies to the information that the object carries about itself.
Is known that information is the basis of the quantum universe.
1. Throw two stones into a black hole. On one we paint the flag US and the second flag of Poland.
Does such information mean something.
2. Now we will fire two cannonballs towards the black hole.
A stone ball from Poland and a ball of uranium from the US.
Is this the sense of information for quantum physics?
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7. Mike.
If I didn’t believe in your wonderful reasoning … after all, I read your wise statements.
If something is to blame, it is my tragic English.
Besides, the scientists themselves, although they are so wonderful in quantum physics, admit that they absolutely have no idea why this works.
so I disappear… but not on twitter.
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8. My problem with MWI is the same one many have: where do all those new realities come from? What does it suggest about matter and energy?
Tegmarkians can talk about how the square root of 4 is both +2 and -2, and no one worries about where the extra answer came from. But I don’t believe we live in a Tegmarkian universe.
There is also, to me, an issue of reality explosion: Wear a pair of polarizing sunglasses, and each photon that hits them has a chance of passing through or not. So each photon seems to be creating new realities. Billions and billions of new realities. Every instant.
MWI fans have said this doesn’t happen, but I’m not clear on why not.
I have played with the idea that what happens is that the standing wave of the universe becomes more complex with each possible branch such that all possible paths that could have been taken are part of that wave. But there’s only one actual reality that emerges from that wave.
I’ve never found the waveform collapse all that mysterious. A particle in flight is a vibration in the relevant particle field, the energy of that quanta is spread out in the wave. But for that energy to interact with, say, an electron in the wall it hits, that single spread out quanta “drains” into the contact point.
The mystery, if I understand it, has to do with what “selects” that contact point, and how does the energy of the wave “drain” into that point? We have no maths for that.
I suspect the contact point gets selected per the same mechanism that “selects” which atom of a radioactive sample decays next. Or as how the first bird of a flock decides to take to the air. Maybe it is literally random (which it seems to be).
I sure wish someone would discover something new. QFT and GR have been at loggerheads far too long.
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1. I have to admit that I wonder about the energy aspect of this as well. If every part of the wave becomes a full particle in its own branch of the superposition, then how is the energy of that wave, and every other wave, not effectively magnified? My understanding is that we still don’t understand at a fundamental level how mass is generated. (The Higgs supposedly only explains a subset of it.) If the non-visible parts of the post-measurement wave aren’t real, then maybe that has something to do with it.
What’s interesting about the explosion of superpositions, is virtually all quantum events average out until the macroscopic deterministic world emerges. To me, that implies that most of the “universes” being generated are virtually identical. (There would have been far more divergence in the early instances of the big bang when quantum events generated patterns that later grew into voids and galactic superclusters.) Today, it seems like it would only be the rare case of quantum indeterminancy “bleeding” through that would lead to divergences. It might be that most of the exploding superpositions end up converging back to one reality, or only a few of them. (I have no idea if the mathematics lend any credence whatsoever to just conjecture.) And I’ve read some variances of the interpretation that, instead of proliferating universes, it’s really just interacting ones.
That actually isn’t my understanding of what happens. As I understand it, the entire wave instantly disappears, replaced by the particle, even if the wave has been spread around and fragmented over vast distances, that there’s no timeline for it to drain. (Which admittedly also makes “collapse” a questionable word for the phenomenon.) That said, decoherence isn’t supposed to be instantaneous either, just very fast, so who knows.
Totally agreed that it would be good to see progress somewhere. I remember many physicists hoping the LHC would provide something, anything, unexpected so they’d have something to work with, but other than failing to confirm supersymmetry, most of what they’ve gotten just seemed to reaffirm the Standard Model.
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Yeah, the mass of protons and neutrons, for example, comes mainly from the energy of the quark and gluon interactions, which means most of the mass from matter isn’t due to the Higgs.
Which is why I find it easier to think about in terms of energy, although I usually see mass and energy as two faces of the same thing.
“To me, that implies that most of the “universes” being generated are virtually identical.”
Which I think is how MWI fans respond to the question about sunglasses and photons. My question in return is how identical is “virtually” identical?
Remember Bradbury’s famous short story, The Sound of Thunder? Do worldlines converge and merge, or do even quantum differences ultimately diverge and result in separate realities?
A lot of MWI fans think Occam and parsimony support their position, but I (so far) see it the opposite. MWI doesn’t sound like the simple explanation, and the explosion problem defies parsimony.
But then I’m not sure I truly understand MWI, and I’ve gotten the impression a lot of its fans don’t really understand it, either. Plus, there seem to be multiple versions of the theory since Everett.
Greg Egan has a short story, The Infinite Assassin, in his collection, Axiomatic. It’s about an illegal drug that allows users to interact with parallel universes, which turns out to be a Very Bad Thing. What I really liked about the story was the sense of continuum Egan gives to parallel worlds.
One can’t help but wonder what makes them distinct.
Sean Carroll gave a talk about MWI (which I found unconvincing), and he had an experiment set up remotely that did a photon-half-silver-mirror thing with two detectors. Through a phone app he was able to trigger the experiment and get a (random) result which he used to determine if he should jump to the left or to the right. (The right, in this case, IIRC.)
The claim was that this generated two realities accommodating his jumping both ways. Which generated two different audiences (and sets of video viewers) who remember him jumping both ways. Which led to this comment where I recall him jumping right. Presumably the alternate me remembers it differently.
But I keep wondering about those sunglasses and all the quantum interactions happening all the time. I’ve just never heard anything from MWI that gets me past this key objection.
Yes, agreed. (That’s why I quoted “drains” — best word I could think of but hardly adequate.) I think we’re on the same page here, I’m just trying to imagine an ontology that makes sense of “waveform collapse.”
I’ve been thinking about this a bit as I try to wrap my head around some of the strange variations of the two-slit thing. (Have you see the three-slit experiment? Mind-blowing!)
In a single photon event, the laser emits a “photon” with no location but a wave (with momentum) that expands from the laser into the surrounding environment. It’s a single quanta of energy causing a vibration in the EM field.
Now that energy has to go somewhere, and what we see happening is that waveform somehow interacting with some electron in some atom such that the electron is raised to a new energy level. At that point, the photon does have a location (and presumably we can no longer talk about its momentum).
That interaction requires the full energy of the quanta, so the energy in the field “goes” (or “drains” or some better word) into that interaction.
But this is just me pondering the “waveform collapse” issue and WAG-ing at an ontology.
“I remember many physicists hoping the LHC would provide something, anything,”
Yeah, and now it’s shut down for two years for an upgrade. You’d think not finding SUSY at all would take the wind out of certain sails, but they just keep redefining the target. Part of the problem is that String Theory seems to need it, so no SUSY threatens ST.
There’s also that chart you’ve probably seen showing how the three forces unify at very high energies? Those curves intersect at the same point only if SUSY is true. Without SUSY, they don’t.
So it’s a dream that’s hard to kill.
There was some hope of seeing something new in very esoteric sectors involving (IIRC) weak decay. I can’t recall what it was exactly, and no one is jumping up and down, so whatever they saw may have not survived more analysis. They were seeing bumps in both CMS and ATLAS, I think, and combining the two bumps gave them a nice sigma, but the data weren’t compatible so combining them didn’t really say anything.
Or something like that.
Merry Christmas!
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1. “My question in return is how identical is “virtually” identical?”
My conception is that normal events, such as all the deterministic events we see in nature where the quantum events average out, don’t create deviations. It’s only when we tie a macroscopic event to a specific quantum outcome, that a notable divergence happens. As you note, even a minor “meaningless” macroscopic event (such as which way Carroll jumped) might eventually butterfly into major changes.
Of course, we can’t rule out the possibility that quantum indeterminacy doesn’t “bleed” into the macroscopic world outside the precision of our instruments and butterfly all on its own, so the idea of similar universes may not be tenable.
There are definitely lots of versions in the Everettian family of interpretations. One I recently heard about on the Rationally Speaking podcast was relational quantum mechanics, which posits that whether a wave has decohered is relative to an observer. In other words, like the relativity of simultaneity in Einstein’s theories, this holds that where you are in the sequence of events determines when you see the collapse. Schrodinger’s cat sees the collapse as soon as the detection device is triggered, but Schrodinger himself doesn’t see it until he opens the box. However, the relational interpretation is reportedly agnostic about the reality of the other outcomes. (It doesn’t seem agnostic to me, but I probably don’t grasp the full idea.)
I need to look up that Egan story. It sounds interesting.
Ah, ok, I missed the quotes on “drain.” Thanks for the description of the photon. Part of what I find interesting about this is that the electrons are presumably constantly exchanging photons with each other and the nucleus, but despite that exhibit quantum waveness to those of us outside the relationship, which makes me think of the relational interpretation again.
I don’t think I knew that uniting all three forces required SUSY. Interesting. I know the weak and electomagnetic one were already shown to be the same. (Which strikes me as an odd pair.)
All in all, I think I’m happy I’m not a physicist right now.
Merry Christmas!
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1. “It’s only when we tie a macroscopic event to a specific quantum outcome, that a notable divergence happens.”
That matches what I’ve heard from MWI fans, but it seems to suffer the same micro/macro issues as many quantum things do. What is a “notable divergence” and what happens? Reality doesn’t diverge at all (why not?), or the diverged lines merge into one (again, why?).
That Egan story is good at pointing out how, if we take MWI at face value, our own reality is a fuzzy continuum of indistinguishable nearby realities. At what point am “I” no longer really me?
Chaos theory suggests (to me) that even minute differences may result in large changes down the road. What if, butterfly fashion, a photon that did pass through my sunglasses accounts for some minute change that ultimately destroys Saturn?
I’ve long wanted to sit down with a working theoretical physisict who’s really into, has really studied, MWI, because I’d like to understand how people like Sean Carroll identify MWI as their preferred interpretation. Some even say it’s the mostly glaringly obvious interpretation!
Doesn’t part of that thinking also come up in Copenhagen? The idea that the cat isn’t superposed to itself, but is to the scientist who hasn’t opened the box. Likewise, the science writer standing outside the lab is superposed until the scientist informs them of the result. And millions of readers are superposed until they read the writer’s article. (And everyone in Andromeda remains superposed probably forever.)
I’m not sure I believe in the idea of macro objects being superposed. What does it mean to suggest I’m superposed? Can experiments demonstrate it? Or is it just that I lack knowledge?
Ugh. We really need some advances in HE physics. We’re just grasping in the dark here.
I think at least some of that is accounted for in the difference between virtual photons and actual photons. I’ve seen some physics videos recently emphasizing the difference between them and how you can’t treat virtual photons as real — they’re almost an accounting device, although obviously something physical is going on. Lamb shift and so forth.
Same here! Electro-weak theory. (And the weak force is the one many books hand-wave on that “has something to do with radioactive decay” … yeah, and making the sun work, too!)
It sure made it seem like unification was a thing though, didn’t it. If two things as seemingly different as EM and weak force are unified, why not the strong force?
Again, we need more information! We don’t even really know if gravity is a force!
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2. “At what point am “I” no longer really me?”
Michael and I discussed this as well somewhere else on this thread. It seems like reality likes ruining our clean little categories, such as what is life or non-life (see prions or viriods), what is the border between species (some members of species A can mate with species B, but others can’t), what is computation, or what is a planet. It won’t surprise me too much if it scrambles our ideas of the self.
I told you to stop playing with those glasses Wyrd! Now look at what you’ve done. Who’s going to clean up this mess? We’ve got Saturn all over everything! 🙂
I recently went back and read Sean Carroll’s blog post on the MWI. I’m not sure his instincts on explaining it are the best. He tends to emphasize the multiple universes thing, which I think is a mistake.
Paul Torek above recommended David Wallace’s ‘The Emergent Multiverse’, which I’m thinking about picking up. It looks pretty good in the preview. My only pause is it’s pricey. Of course I’ve often spent more on neuroscience books. I just have to decide if I’m interested enough and willing to invest the work it would require.
I can see why people say the MWI is the most straightforward interpretation though. It does explain a lot. I see it as a candidate for reality. The only question is whether the implications of it in any way falsify it. But as I commented on Carroll’s post, that’s the problem with these interpretations. None of them are uniquely testable.
“— they’re almost an accounting device, although obviously something physical is going on. ”
Didn’t quantum physics start with Max Planck introducing a quanta purely as an accounting device? There was a similar disclaimer on Copernicus’ book. It seems like a lot of physics starts with someone saying, “Don’t worry, this is only for calculating convenience. It’s not it’s real or anything.”
“Again, we need more information! We don’t even really know if gravity is a force!”
Totally agreed on needing more information. Although wouldn’t you say we know gravity is a force? Or did you mean if it’s a force like the others in the Standard Model, with bosons (gravitons) and the like?
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3. “It won’t surprise me too much if it scrambles our ideas of the self.”
Yeah. The more I learn and think about “the self” the more complex and puzzling it seems.
“[MWI] does explain a lot.”
That I do realize. I’m confounded by the whole multiple universes thing; that’s pretty much the entirety craw stick.
I vaguely remember reading that Sean Carroll post. Think I’ll go back and re-read it this evening.
The Wallace book sounds kinda interesting… once I read about it. The title put me off, because while I’m open-minded-but-skeptical on MWI, I’m disbelieving (and disinterested) in multiverse theories. I found an online review of the Wallace book that sounds like another read for this evening.
“Didn’t quantum physics start with Max Planck introducing a quanta purely as an accounting device?”
Ha, yes, good point!
“Or did you mean if [gravity is] a force like the others in the Standard Model, with bosons (gravitons) and the like?”
Exactly. I want GR to be essentially correct with some minor correction to accommodate quantum, and I want QFT to turn out to be essentially epicyles — a theory that matches our instruments but is seriously wrong in some key regard.
We know matter/energy is quantized, but the jury is out on time/space. I want them to be smooth (providing yet another duality to reality). And that gravity is due to warped spacetime and there is no such thing as a graviton.
My spacetime wishlist. 😀
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4. Wow, that review is 19 pages long. I thought I might sneak a quick read before responding, but I think I’ll just add it to my queue too. Thanks for linking to it!
On GR and QM, I don’t really have preferences on which one wins (assuming they both don’t eventually have to be heavily modified). If spacetime does appear to be smooth, I wonder if we could ever be sure it wasn’t quantized at a size below the level of precision of whatever we were using to measure it.
And an infinitely divisible spacetime seems like it would come with its own potential multiverses. If the space between elementary particles is infinitely divisible, it allows patterns to exist there below our notice, such as entire micro-universes. And entire other universes could have been born, existed, and died in the Planck time at the beginning of the big bang. For that matter, an infinity of universes might have existed during the time you read this reply. (Don’t hit me.)
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5. I gave up (for now) on that review once I got to the discussion section. They were a little too glowing in their assessment for me to trust, and there was already a bit of a “yelling at the screen” thing going on here on the material they mentioned to that point.
The book does sound interesting, though. I found myself wondering if Wallace explains some of the stuff that was making me yell.
Continuous spacetime does seem to have the same weird issues the real numbers have. Maybe matter/energy being quantized saves the day?
While space might be infinitely small, matter isn’t, so no micro-galaxies hiding in the dust motes. Quantum limits on energy might also affect the minimum time it takes anything to happen (like c limits causality).
The question might be whether we can trust scale. Atoms have sizes due to their properties, so maybe certain things can only happen on certain scales. (And we use atomic vibrations to define the second.)
Or maybe they’ll find a graviton (or a chronon), and that will end the matter. But until then… well, just say that I look at GR and think, yes, that makes sense, but look at QFT and think, wait, what?!
Obviously the universe is under no obligation to fulfill my sense of how it ought to behave (oh, if only). 🙂
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6. “While space might be infinitely small, matter isn’t, so no micro-galaxies hiding in the dust motes.”
I actually wasn’t thinking the micro-universe patterns would be made of any matter/energy as we understand it, but something else, something we never see because it exists too far below the scales we can detect. Call it Mini-Me matter which could have it own smaller Mini-me quanta sizes. Of course, between Mini-Me matter might be Mini-mini-Me matter, and so forth and so on. Turtles all the way down.
Or if in fact there is only the matter/energy we’re familiar with, that means an infinite emptiness between every occurrence of it, which would itself be profound.
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7. Yes, as profound as the next real number after zero!
Talk about macro objects in superposition… I’m totally superposed on the real numbers being, in fact, real or, as sure seems sometimes, a fabrication of our imagination.
The thing is: how real is a circle, its diameter, and their ratio? If they are real, so is pi.
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9. I don’t get the whole ‘measuring changes quantum particles behavior’ thing. And by ‘not get’ it seems like it doesn’t work or is a simplification that lost important details on the way. For example if ‘measuring’ changes the quantum particles, then at what distance can you measure them? Any distance? If so wow, you’ve invented an instantaneous communication device that’s…faster than light. Nice. Or if the distance actually matters, then ‘measure’ is a term that is a heuristic and lacks the actual details like what distances are involved and where does the effect run out?
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1. You’re totally right not to get it. “Measurement” or “observation” is a maddeningly vague aspect of this. It reflects the lived experiences of scientists running experiments on quantum phenomena. Niels Bohr reportedly insisted that the description of this be limited to “ordinary” language, presumably because any attempt at a more precise description would imply knowledge we don’t really have.
It’s called “the measurement problem,” and it’s at the heart of the absurd nature of quantum mechanics. Attempts to solve it have led people down all kinds of bizarre paths.
I sometimes think QM represents the limits of our reality, where that reality emerges from some other underlying meta-reality. It might be that any “interpretation” is simply a vain attempt to map that meta-reality back into our little parochial reality. As patterns in and of the parochial reality, we simply may not be equipped to understand the wider meta-reality.
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1. FWIW, I see “measurement” as anything that resolves superposition. For me, the cat was always (obviously) either alive or dead, because the detector monitoring the radioactive sample is the measurement. There is no superposition; there is only a lack of knowledge about the cat.
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10. Excellent post, Mike. I enjoy mulling these quantum conundrums around. I am left feeling like an extremely poor sommelier of ideas–I get hints of different flavors but… really I have no idea what I’m tasting. It’s just really, really complex and intriguing. My own opinion is that we just don’t really know what we’re studying, and that at some point there will be a breakthrough in our conception of what reality actually is that will assist us in fitting the pieces of the puzzle we’ve found so far into a more insightful framework. As an example, I think our notions of physical and non-physical have pretty much broken down, and we have only vague ideas as to what consciousness might be, most of them extremely myopic, so that we’re in the position of using pretty poor tools for the job.
Just as one example, in that Quanta article to which you linked, Brian Greene suggests that each copy of you in the MWI is really you, and that the true you is the sum total of these you’s. Something like that. When a scientist says that a “self” might be a superposition of conscious selves occupying subtly related windows of reality, it’s an interesting idea to some folks and frowned upon by others–while when the classic New Age book Seth Speaks posits the same notion it is deemed woo woo foo foo to that crowd, but accepted by the other. This is, in a sense, what I mean about once clear concepts and divisions breaking down. So my own feeling is everyone’s a little bit right, and the answer is somehow a superposition of a great many ideas out there… 🙂
I don’t suspect a ton of physicists are lining up to endorse Brian Greene’s idea of the self. I have no idea, actually. But it’s always interesting to me when these parallels emerge. I think it’s safe to say whatever “models” or “conceptual frameworks” we use to try and organize our phenomenal observations are all wanting right now. What I dislike about the Copenhagen Interpretation is that it seems like a consequential moment in defining the purpose of science–which accepts setting aside questions about what the universe really is, and accepting as complete descriptions of what it does. For me, science is much less interesting when only one of the two questions remains in play…
Happy Holidays, Mike!
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1. Thanks Michael, and great hearing from you! Your comments are always thought provoking.
On Brian Greene’s notion of the self spanning multiple copies, I think, much like the notion of additional selves that originate from the idea of mind uploading, it’s a matter of philosophy, in other words, not a fact of the matter, but a personal choice. In both cases, the issue gets blurred as the copies get farther and farther away from the original.
For example, is someone born with my exact genetics, but due to an early quantum branching, lived a radically different life, still me? What about someone who branched away from me before I became a skeptic? Or even before I became interested in science? Or someone who branched away before I broke up with one of my old girlfriends, but instead married her and proceeded to have a large family?
My attitude is that these would all be a sort of sibling, albeit in the case of recent copies, far closer to me than any brother or sister. The only way I might be tempted to ever consider them to be me is if we could somehow share memories, but even then I’d expect difference to arise based on the order in which the various copies received the different memories.
On the Copenhagen Interpretation, I can understand not liking its inherent instrumentalism. I totally agree it’s a lot more inspirational to think of science as the pursuit of truth. The pursuit of models that accurately predict future observations…just doesn’t have the same inspirational resonance.
On the other hand, maybe the idea that the pursuit of truth is anything other than the pursuit of predictive models is an illusion. The real dividing line is whether we want to get into models that make predictions we can’t test. The Copenhagen Interpretation (apparently heavily influenced by the logical positivism in vogue during its formulation), labels that as undesirable.
I think by calling these models that go beyond the mathematics of quantum mechanics “interpretations”, physics has found a way to have its cake and eat it too. It allows us to label the predictive aspects of QM as settled science, but keep trying to figure out what it means.
Although as I’ve noted to you before, and as I did to Callan above, I sometimes wonder if quantum phenomena isn’t right at the edge of the reality we, as a subset of that reality, have any ability to make sense of. It might be a hole we can navigate around mathematically, but can never enter. (Although I hope we never stop trying.)
Happy Holidays to you too Michael!
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11. I have some strong opinions about this issue, and have been meaning to bring this up with Sabine Hossenfelder over at So far I’ve been too shy however. This is a woman who I absolutely love! She’d like to help “fix” a physics community that seems to have gotten “lost in the math”. Similarly I’d like to help a science community that attempts to function without generally accepted principles of metaphysics, epistemology, and axiology (or the three elements of “philosophy”). Perhaps if I feel that I’m able to develop my QM ideas here well enough, then I’ll become confident enough to speak with her about this over there some time? Well maybe.
Rather than get caught up in all sorts of higher speculation initially, I like to begin with QM basics. We humans perceive matter in terms of “particles” and in terms of “waves”. Are such perceptions good enough? Apparently they are not. When we try to pin down the exact state of a particle we’re confounded with wave like characteristics. Then when we try to pin down the exact state of a wave we’re confounded by particle like characteristics. So it should instead be better to consider matter to function as both. But apparently we can’t measure matter as some kind of hybrid of the two. Therefore it makes sense to me that we’d witness fundamental uncertainty as expressed by Heisenberg’s uncertainty principle, or an inequality that references Planck’s constant.
So to me there isn’t too much to worry about here. If we must measure particles in one way and waves in another way, though matter ultimately functions as neither but both, then we should expect to be confounded by more exacting measurements in either regard. Given the circumstances, is this not logical?
For example, let’s say that we find a material that’s similar to both rock and wood. So if we assess it as a kind of rock then the harder we look at it from this perspective, the more confounding this stuff should seem to us. Or the same could be said if we assess it as a kind of wood. So that’s essentially what I’m saying is happening with our assessments of matter. If it’s effectively “particle-wave”, though we can only provide measurements in one way or the other, then we should naturally fail as our measurements become more precise. Thus I’m good with quantum mechanics as I understand it. Apparently we’re too stupid or whatever to understand what’s going on.
The controversy however seems to be that most physicists (unlike Einstein) haven’t been content settling for such human epistemic failure. So apparently they’ve decided that no, it’s not that we’re trying to measure something as particle or wave that’s neither. Instead it must be that the uncertainty associated with either variety of measurement reflects an ontological uncertainty which exists in nature itself! So the argument is not that we’re stupid, but rather that nature itself functions outside the bounds of causality, or thus nature functions “stupidly”.
It could be that this view is entirely correct, but what irks me here it is that these physicists also refuse to admit that they thus forfeit their naturalism. Apparently they want to call themselves naturalists, but interpreting QM such that nature functions without causality — well that ain’t natural!
It’s the borderlands of science, such as here, brain study, and so on, that seem most in need of effective principle of philosophy. For this issue I offer my single principle of metaphysics. It reads:
To the extent that causality fails, there’s nothing to figure out anyway.
Unless I’m missing something this “Many worlds” interpretation appears in violation. I interpret it as physicists deciding that reality functions without causality (or “magically”), and then attempt to make sense of this anyway by theorizing “many worlds”. The more that we leave the bounds of causality behind, or thus introduce magical function, explanations should grow obsolete. From here reality should just be what it is. So I consider these sorts of interpretations of quantum mechanics to illustrate category error.
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1. A lot of your criticism seems aimed at the more ontological versions of the Copenhagen Interpretation, the ones that say that not only are we faced with an epistemic limit, but that there’s nothing else there, that reality isn’t set until the measurement. That’s usually the version of the CI that critics inveigh against, and I agree with that criticism. The ontological versions of the CI seem excessively pessimistic.
I think Neil Bohr’s version of the CI was closer to your sentiment. Here are the observations, and here are mathematics that can make predictions about those observations, with limitations, but within those limitations predictions are accurate enough to build technologies on top of them, so, “shut up and calculate!” I’ve grown to respect this view more as I’ve continued to learn about quantum physics. It’s not satisfying, but it’s at least epistemically humble.
But I think an MWI enthusiast would respond to you that their interpretation does restore determinism. Unfortunately, it’s determinism for reality overall, not a determinism we can observe. Which of course raises the question, if something is deterministic but not deterministic from any observer’s perspective, is that really deterministic? Who is it deterministic for?
One question I’d have for you is, how do you define naturalism? Is that definition mutable on new evidence? Myself, if I encounter phenomena that doesn’t meet my understanding of naturalism, I would still want to understand the phenomena as much as I could. But naturalism for me is just a set of working assumptions, ones subject to being adjusted as I learn more.
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2. If I may interrupt, two quick thoughts:
Firstly, I’m also a big fan of Sabine’s blog, been reading it for years. I highly recommend it. (Peter Woit also has a good blog.)
Secondly, just as (and I very much agree) physicists benefit from philosophy, philosophers can benefit from looking into some of the math involved. Quantum physics is highly mathematical, and the wave-particle duality confusion is, at least in part, a failure of language. At the math level, the confusion essentially goes away.
The way it’s usually put is that matter (as in particles) is something outside our direct experience that has wave-like properties and particle-like properties depending on what aspect of the particle one tests.
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3. Wyrd,
I was hoping to hear from you most of all! Perhaps on some level I mentioned Sabine because I recall you mentioning her another time? Anyway it was late 2015 that I became interested in her. Massimo Pigliucci had blogged about her position from a Munich physics conference that he attended.
On philosophers benefiting from math and physics, I certainly agree. I was initially most interested in philosophy as a university student, but didn’t want to become acclimated to accept no generally accepted agreements in the field. And beyond questions what could they teach me without generally accepted positions? Mental and behavioral sciences were next, though I found them far too speculative for comfort. So I looked for a field that could teach me how to learn. Yes physics! But alas, my own mind would not get me through upper division courses. I eventually earned a degree in economics, which I chose somewhat because it corresponded with my own amoral theory of value.
I didn’t mean to imply that modern physicists would improve if they were to become versed in modern philosophy. I actually believe that the field has tremendous problems, though needs improvement in order to better found science.
Regarding language, that’s one of my own main themes. So QM interpretations work pretty well mathematically? But I suppose that natural language explanations are needed most. Mathematics is many orders less descriptive than English. Notice that there’s nothing in mathematics which can’t be described in English, and yet much in English can’t be described in mathematics. Still the English interpretation of the mathematical QM interpretation that you’ve provided seems pretty close to mine.
It’s good to hear that you oppose the ontological version of the Copenhagen Interpretation. Actually I was under the impression that Bohr’s interpretation was more ontological, though perhaps not. Did he ever support Einstein’s “I, at any rate, am convinced that He [God] does not throw dice.”? (Though in practice I support Einstein about that, my own metaphysics is a bit more pragmatic. It’s more like “To the extent that God throws dice, nothing exists to figure out anyway!”)
If Many World enthusiasts are truly causal determinists, then tell me this. Do you think their position holds that all of these worlds actually exist? As in ontologically exist? As a solipsist I can stomach all sorts of crazy notions from a supernatural premise. But in a causal sense that position seems utterly ridiculous. Conversely if these many worlders are simply going epistemological with their position, as in “It can be helpful for us to think about QM this way…” then I could give their position some reasonable consideration.
Yep Mike, it’s deterministic. Who for? All that exists. Once again, I’m a solipsist. Reality is reality regardless of the human’s various idiotic notions.
I define naturalism as a belief that reality functions causally in the end. This definition is a definition, and therefore isn’t mutable to new evidence. Even if I ultimately decide that reality does not function causally, I should still consider this to be a useful definition. Here I’d either be a supernaturalist, or a hypocrite that changes my definition in order to call myself a naturalist.
I understand the desire to understand. This seems quite human and adaptive. Even the most faithful god fearing person should need to use reason in his or her life in order to get along. But to the extent that causality fails, as in ontological interpretations of the uncertainty associated with Heisenberg’s principle, things should not exist to figure out anyway.
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1. Eric,
Bohr very much did not support Einstein in his statement about God not playing dice. His response was along the lines of, “Einstein, don’t tell God what to do.” Honestly, while I think his and Heisenberg’s initial strategy was more epistemic, more instrumental, I do get the impression that they crossed the line in later debates. But it’s the instrumental version that I think remains useful.
“Do you think their position holds that all of these worlds actually exist? As in ontologically exist?”
It depends on which ones you talk to. Some are agnostic about whether the other wave function branches continue to exist. Others feel they don’t. But the most vocal proponents tend to think they do exist.
As I mentioned to Wyrd, it’s an old trick in physics to introduce something but then say, “Don’t panic, this is just a useful accounting gimmick. It’s not like this crazy thing is real or anything.” This has been particularly true for quantum mechanics. Max Planck originally introduced quanta purely to make his calculations work. I suspect some Everettians take this tack to side step the ontological debates. The thing is, many things that are mathematically convenient go on to become ontological necessity.
“Reality is reality regardless of the human’s various idiotic notions.”
That may be true, but how do we know whether we know reality? I think the only answer is whether our predictions are accurate. Of course, QM can’t predict a single quantum event, only the probabilities of certain outcomes. But as the numbers of events climb, those probabilities average out to solid predictions.
Given the above, whatever QM is, it has to be isomorphic with the reality in some way, otherwise those predictions would fail. As Wyrd mentioned, this may only be in the sense that epicycles were useful in Ptolemaic cosmology. (Interestingly, epicycles today remain as a useful perspective observational concept, despite the fact that we know they’re an illusion.)
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4. Mike,
If it’s the case that Bohr and Heisenberg began with a responsible epistemological position for their Copenhagen Interpretation, then why would they escalate it to ontology? Might I suggest a bit of jealousy? Even then Einstein was “the great one”. How wonderful it would feel to up him! But perhaps Einstein should mainly be blamed for selfishly not realizing that a responsible epistemological position had actually been presented, and so he chose to interprete their interpretation ontologically? Notice that “God doesn’t play dice” is an ontological claim. If he used this to counter the CI then he effectively should have goaded them into an irresponsible ontological position. And apparently they not only accepted, but used it to kick his ass! Today in popular media, and even among physicists, it’s thought that Einstein really blew it regarding QM.
I account for this incidence through a far larger structural problem. Notice that we’re asking physicists to do physics, though without provide them with any effective rules of metaphysics or epistemology to work from. Thus we should need a community of professionals armed with generally accepted rules from which to guide the function of science. Notice that the field of philosophy today has the flavor of “art and culture” rather than “science” to it. I’m not saying that this needs to change however. I’m saying that a new community of professionals must emerge that has a single mission — to straighten out science by means of its own accepted principles of metaphysics, epistemology, and axiology.
And what specifically do I propose to fix this particular mess? I’d mandate that the authors of any given position clearly state whether their proposal is theorized to just be “useful” (epistemology), or to also be “real” (ontology). Then as for those ambitions theorists that insist upon proposing an ontology regarding QM, there would be my single principle of metaphysics to contend with. Theorizing that any given bit of reality is not causally determined to occur exactly as it does occur, takes the theorist beyond the bounds of naturalism. Here there can be nothing to explain because without causal dynamics, no explanation will thus exist. This is the realm of magic. And I’m not saying that this doesn’t effectively occur. I’m saying that the position of Einstein and I, conversely, happens to be “natural”.
Well yes today, though once we have a community of professionals that’s able to effectively regulate the function of science through proven principles, there should only be “epistemic necessity”.
The only reality that I “know” exists, is that I exist in some form or other. If you’re conscious then you could say the same about yourself. And I consider it quite special to be able to truly know even that. Conversely my computer shouldn’t know that it exists (if it does exist), let along anything else.
I consider quantum mechanics to mark an incredible human achievement, though epistemologically rather than ontologically. And I do believe that it’s isomorphic with reality. But if any associated dynamic is not causally determined to occur exactly as it does occur, or “ontological uncertainty”, then the theory should effectively describe the function of magic.
But wait a minute, as I define it no explanation can exist to describe non-causal function, or magic. Right… So the effectiveness of QM theory suggests that all associated dynamics must be causally determined to occur exactly as they do occur. You’re not going to like that bit of circularity! I’ll remind you however that we’re measuring particles and waves here, though apparently matter functions as something associated but different.
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1. Eric,
I don’t know if you remember, but I actually think the distinction between instrumentalism and scientific-realism is a false dichotomy. We never have access to reality. We only ever have theories, predictive models about that reality. The “real” is only another more primally felt model. In the end, all we have are the models.
(This actually includes our model of self, as counter-intuitive as that sounds. Psychology has shown that access to our own mind is subject to just as many limitations as the information we get from the outside world.)
The only real distinction is between predictions that are testable and those that aren’t. The ones that are testable, and which have been demonstrated to have some level of accuracy, are “right” to whatever level they meet. But predictions that haven’t or can’t be tested should be regarded as speculative to varying degrees.
An untested or untestable prediction which is tightly bound to a tested prediction has a higher chance of eventually being shown to be accurate. But the more steps beyond observation to get to the prediction, the shakier the ground it rests on.
Under this guideline, the successfully tested predictions we have are the evolution of the wave function according to the Schrodinger equation, until information about it leaks into the environment, then we have the more definite state (position of the particle), etc. This is the instrumental Copenhagen Interpretation.
Everything else: assertions that the Copenhagen Interpretation is the only reality, pilot waves, spreading superpositions continuing under the Schrodinger equation, etc, have to be viewed as speculation, at least until someone can figure out some way to test them.
Still, speculation is fun, and should be fine as long as we acknowledge what we’re doing.
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5. Mike,
Well it sounds like we’re generally on the same page with that, though I wouldn’t refer to the distinction between instrumentalism and scientific–realism as a false dichotomy. Even if science only ever has models, we of course need words such as “real” which reference what actually exists beyond our models. And if some of these MWI’ers have decided that the lack of certainty in our measurements mandate “many worlds” in truth rather than simply as an accounting heuristic, then this would seem to be a wonderful example of “scientific realism”. This also strikes me as “the tail wagging the dog”.
Furthermore I don’t mind going ontological myself in some ways. I happen to believe that “God doesn’t throw dice”, which is to say I believe in absolute causality regardless of what we humans are able to figure out. Perhaps a reasonable name for this position would be “extreme naturalist”? So then what shall a person be called who makes the ontological claim that some things under a QM framework aren’t causality determined to occur exactly as they do occur? “Super-naturalist” seems over the top, and even quasi-naturalist”. So I’ll just go with straight “naturalist”, but in addition note that from this distinction “spooky stuff” does ontologically occur in some capacity.
Then there is my logical proposition from last time. My metaphysics holds that if something functions without causality, then nothing exists here to even theoretically figure out. Why? Because it’s the causality that would found any ontological explanation for any given event. The causality would be the vital element regardless of any potential understanding — nothing would otherwise exist to even look for.
I’m fine with how the QM probability distribution produces a macroscopic world which seems to function causally. But how can it be possible for something that is not perfectly caused to do whatever it does, to in the end become a causal constituent for a causal realm? I see that as a contradiction. Non-causal function, where by definition nothing exists to potentially figure out, should have no potential to produce causal function. (I suspect that there’s a simple way for this to be illustrated mathematically.) Thus if we notice that quantum function does produce causal function, then from here it must only be possible that all elements of quantum function occur causally in the end, and even if things continue to seem random to us humans.
Yes speculation is fun! Furthermore once science has better rules from which to work, it should also become more productive than today. (I see you’ve now put up a post on Sean Carrol. Sweet!)
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1. Interesting observation Steve! I’ve noticed a couple of interpretations for the Law of Large Numbers. One is that with enough trials, all sorts of implausible things eventually occur. The other seems more relevant however. It’s that the more times that you run a given experiment, the more statistically verified a given result will be. It’s essentially that all of these “random” results end up building a stronger and stronger case for a given figure. Is that what you meant?
I can see how it seems appropriate to apply this principle to quantum mechanics given that we’re discussing probability distributions for matter rather than exact states of being. But then again, my sense is that the LLN was set up to address every day causal events rather than quantum events that are theorized to not function causally. Does it address quantum strangeness as well? Have you found an infinitely better challenge to Einstein than the utterly pathetic “Don’t tell God what to do”? Is this a true answer, as in “God’s dice create order”? This deserves some academic consideration!
I’d be surprised if something fully beyond causality in an ontological sense is able to then go on to construct the causal function observed in nature. Causality is kind of my thing. But I’d love for this theory to get out there as a challenge to us causalists.
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1. While it’s true that a large number of random events will yield some rare outliers as part of the ensemble, when taken as a whole, it leads to highly predictable results. It’s the basis of statistical mechanics. Even in classical statistical mechanics, individual particles are assumed to behave randomly, but when the ensemble contains 10^23 particles, the values of pressure, temperature, etc are entirely deterministic. My statistical mechanics lecturer at university joked that when very large numbers are involved, “it is better to gamble than to count.”
Causality may be an illusion, as well as ontological fact.
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2. I agree entirely with your former professor’s observations Steve, and indeed, the Law of Large Numbers as I believe it’s traditionally been used. This is to say that if you do a single experiment a large number of times, it will continue to validate the same point in the end. And I also agree with that other interpretation. Even though a psychic may get a given prediction right, the LLN shall demonstrate the truth or falsity of this person’s powers over time.
And why does the LLN remain solid? Because of causality itself. Without an ordered world where cause leads to associated effect and the converse, it might be that the exact same experiment would not generally continue to provide the same sort of result. Or it might be that a human could indeed gain psychic powers and all sorts of “spooky” stuff. Causal order is required in order for the LLN to remain valid. Otherwise we’d need to count rather than to gamble.
I suppose that this is why advocates of ontological voids in causality haven’t yet tried to use the LLN to argue their case. Thus we instead get pedigreed snake oil carnival hawkers like Sean Carrol. Apparently people love hearing this sort of thing.
(I haven’t yet found a mathematical proof that causality can’t emerge from non-causality, but perhaps I will.)
I prefer “emergent” to “illusion”, but it’s the same concept.
If it’s true that there is a fundamental uncertainty to QM function, then yes, the causality that we observe must emerge from non-causality. Or it could be that there is a causality which we don’t grasp here given that we erroneously perceive existence in terms of particles and waves.
Causality may not be the fundamental thing we take it to be.
Right. But a better way to say this might be that causality may or may not be absolute. Somehow to me your statement implies that we’d still call something “causal” even if it isn’t. Or perhaps I’m being pedantic? You wouldn’t term something “causal” if it weren’t causally mandated to occur in the exact manner that it does would you?
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1. Eric,
If causality is emergent, that is, real but a composite process made up of lower level processes which are not themselves causal, then I would use it in the same manner I use “temperature”, “weather”, or “molecule”. Each of these things objectively exist, but are composed of things which are not that thing, in other words, they are composite phenomena.
The idea that causality is a composite phenomena is very counter-intuitive, but then so are many things in science.
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3. All true Mike, so apparently I was being pedantic there. If causality emerges from non-causality then it isn’t the fundamental thing that we take it for, similar to “molecule” and all the rest. But given our flawed perspectives I do still suspect that it’s fundamental in the end.
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12. Great post, and a clear summary of the position. I (like most people) have problems with all the proposed solutions, and that is as it should be, since none of them are entirely persuasive. The most unconvincing commentators are those who argue passionately for one particular interpretation.
My gut feeling is that we are still missing a fundamental insight, and I hope this will emerge either through some new observation, or else a new theory. My instinct is that entanglement holds the key to unlocking the answer. Disclaimer – it may be that this is wrong, and that it is just me who is lacking the fundamental insight 🙂
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1. Thanks Steve!
In recent decades, decoherence has become the preferred description of what happens when the wave appears to become a particle. Under that description, what actually happens is the wave becomes “entangled” with the environment. So your gut may be on to something!
It feels like all physicists can keep doing is testing the boundaries of this stuff until something unexpected comes up. After all, it was the necessity of dealing with bizarre observations that initially forced them to their current understanding of QM, such as it is. The answer probably lies in continuing to pile up those observations until something new emerges from the data, but that might take decades or centuries.
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6893511c9b5233be | The Schrödinger Equation
Today I’d like to start doing some physics. I wish I could say that we were going to derive the Schrödinger equation — which is basically the master equation of quantum mechanics — but it doesn’t follow from a simple examination of mathematical principles. Its justification comes from the fact that it seems to accurately describe the physical world. So I’m going to walk through some of its history, and the experiments and physical facts leading up to it, and will end up with an equation and, more importantly, an explanation of what the quantities being solved for actually mean.
A bit of history
Our story begins in 1900. At this time, our understanding of physics wasn’t quite complete — there was still some argument over whether “atoms” had any physical reality or were simply a useful calculational tool for chemistry, and we were still trying to sort out just how we moved relative to the æther — but we were confident enough in our understanding that Lord Kelvin could comfortably say that “There is nothing new to be discovered in physics now; all that remains is more and more precise measurement.”
One of the few matters still not well-understood was blackbody radiation. It was already well-known that an object, when heated, emitted a spectrum of light which was a combination of two components: an “emission spectrum,” which was a unique fingerprint of the chemical composition of the material (which fact had revolutionized analytical chemistry) and a “blackbody spectrum,” which depended only on the temperature of the object. Unfortunately, the best theoretical models of blackbody radiation, coming from application of the laws of radiation and thermodynamics, predicted that the intensity of blackbody radiation should vary with the frequency of emitted light as I(\nu) \sim \nu^3. Since we are not, in fact, instantaneously annihilated by an infinite amount of X-rays every time we strike a match, there was clearly something in this model which needed improvement.
In a seminal paper, Max Planck noted that, if one assumed that light energy was not simply a wave in the electromagnetic field, but that instead it came in discrete packets (“quanta”), each packet having an energy proportional to its frequency, and the total energy and intensity of the light depending on the number of such quanta, that by a fairly straightforward stat mech calculation1 you could derive an intensity-frequency relationship which exactly matched experimental data. The constant of proportionality between energy and frequency was a hitherto-unknown constant which he labelled h, and which is thus known as Planck’s constant:
E = h\nu;\ h \approx 6.26068 \cdot 10^{-34} J\cdot sec.
He did not, however, have any good explanation for why this should be the case; as far as he was concerned, this was a mathematical hack whose main virtue was that it happened to work. Only five years later, though, Einstein found a fascinating physical confirmation of the result in his paper on the photoelectric effect, the basis of modern solar panels.2 It was known that when light shines on an object, it releases electrons whenever the frequency of the light was above a certain threshold frequency characteristic of the material. Above this threshold, the current (the number of electrons released) was proportional to the intensity of light; below this threshold, no electrons were released regardless of the intensity. Einstein pointed out that this would be perfectly explained if light, indeed, came in quanta — he called them “photons” — whose energy was given by Planck’s formula, and that the criterion for ejecting an electron from the material was for the energy of a single photon to exceed the binding energy of the electron.
So at this point, the idea of the existence of quanta of light was becoming fairly well-established, much though it mystified everyone who had become quite used to thinking of light as a wave. In parallel, some mysteries were developing in the nascent theory of the atom. Thomson had demonstrated the existence of electrons in 1897, and showed that matter contained them; he proposed a “plum-pudding” model of the atom, consisting of “a number of negatively-electrified corpuscules enclosed in a sphere of uniform positive electrification.” But this did not hold up to experiment; in 1911, Rutherford gave a talk at the Manchester Literary and Philosophical Society detailing the results of his work with Geiger and Marsden, demonstrating — by means of a brilliant experiment — the existence of an atomic nucleus, positively charged, and so small as to be almost pointlike.3
It wasn’t hard for people to come up with a physical model for this; the Coulomb force is an inverse-square law, after all, and so one could imagine electrons orbiting a nucleus like planets around a small star. There was only one problem; an electron moving in an elliptical orbit would be continuously accelerating, and accelerating charged particles radiate; this “synchrotron radiation” (as it later came to be called) would burn up all of the electron’s energy within about 10-24 sec. Furthermore, it gave no good explanation for the emission spectrum of gases, something which was increasingly of interest to physicists to explain.
Niels Bohr gave the first good explanation for this in 1913, based on a bit of radical ad-hockery: suppose, he said, that there were only some discrete orbits allowed for the electron: what if the angular momentum were required to be an integer multiple of \hbar? Then synchrotron radiation would somehow be impossible, since it would lead to a continuous decrease in angular momentum. Instead, the electrons could only “jump” between these levels. He showed that if the energy released by such a change were emitted as light, then the energy differences between these levels matched the measured emission spectrum for atomic Hydrogen!
The effect of this paper was revolutionary. The next twelve years were spent working out this new “quantum theory” in detail. And I am going to skip explaining it, because it was based entirely on this sort of ad-hockery. Quantum mechanics didn’t spring full-formed from the mind of Zeus; it was the product of several decades of brilliant people banging their heads against something seemingly incomprehensible, trying various things until they worked. So rather than walking through all of that, I’m going to skip to the results which finally led out of the maze, and to how they were ultimately interpreted by Schrödinger to lead to the modern quantum theory.4
Two Slits, and all that
The first bit of useful progress, although it wasn’t immediately recognized as such, was when de Broglie made a radical proposition that, just as photons are somewhat “particle-like” in this new quantum theory, we should also regard all matter as being somewhat “wave-like,” and we should consider a particle moving with momentum p to have an effective “wavelength”
\lambda = \frac{h}{p}.
He noted that this formula, together with requiring that the wave be a standing wave (i.e., periodic boundary conditions around an orbit) was enough to derive Bohr’s angular momentum quantization rule, and put the whole model on a firmer footing.5
Interruption to clean up notation: I’m going to dispense with the freshman-physics quantities ν and λ at this point, and instead use the angular frequency \omega = 2\pi\nu, and the wave number k = 2\pi/\lambda. These clean up the equations considerably; Planck’s formula is now
E = \hbar \omega,
de Broglie’s is
p = \hbar k,
and the equation of an ordinary wave (à la classical wave mechanics) is e^{i(k\cdot x - \omega t)}. The units are more useful, as well; angular frequencies are measured in inverse seconds (as opposed to Hertz, which are cycles per second, and contrary to any rumors you may have heard are not the same as inverse seconds; if you multiply a raw frequency by a time and take the sine of it, you will get nonsense, not trigonometry), and wave numbers in inverse meters. This is also the last time you will see the “unreduced” Planck’s constant h; all the annoying factors of 2π will go away now.
The key experiment which started to make matters clearer was Young’s two-slit experiment, which attempted to directly test this hypothesis. In standard optics, if a light source is shined through two parallel thin slits, the distance between the slits being comparable to the wavelength of light, then the image projected through those slits onto a wall is a distinctive interference pattern. (If you aren’t familiar with this experiment in optics, I suggest scanning through the Wikipedia article linked above) The result is very distinctive and had a well-understood origin in the simple interference of two waves; you probably derived the pattern in freshman physics, simply by adding up two sine waves originating at each of the two respective slits. Since the distance of any point on the screen from each of the two slits is different, the light from each of those slits will hit that point on the screen out-of-phase, and the combination of sines gives you bumps.
What Young showed was that the same could be done with electrons; the relevant frequency and wavelength matched those predicted by the Planck and de Broglie relations. Furthermore, if you slowed the rate of electrons going through the system so that they went through one at a time, something even more curious happened: each electron would go through the slits and produce a single, clear “dot” on the screen behind it. But as more and more electrons came through, the overall distribution of the electrons followed the shape of an interference pattern!
From this, we can conclude a number of rather startling things.
1. There appears to be some sort of wave equation governing the motion of matter particles. Call the function which obeys this equation — the “wave function” — \Psi(x, t).
2. This is a linear wave equation, as evidenced by the nice, linear-superposition interference pattern. When particles move along different paths, their wave functions add.
3. At least for a free particle in space, the wave function appears to be that of a free wave, \Psi \sim e^{i(k\cdot x - \omega t)}.
4. The energy and momentum of the particle are related to the wave number and angular frequency of the wave function by E = \hbar\omega and \vec{p} = \hbar\vec{k}.
5. The wave function seems, in some way, to describe the “probability” that the particle will be found at (x, t).
6. The wave function itself, however, is not the probability; otherwise a single particle flying through no screens would show a sinusoidal distribution of positions, which it (evidently) doesn’t.
Schrödinger (and many others) puzzled over these properties a great deal. One key idea was about how to deal with (6). The simplest approach was to imagine that Ψ was a complex number, and the probability was given by some function of the magnitude of Ψ — say, |\Psi|^2. One could also imagine adding more internal structure to Ψ, such as having it be vector-valued. It turned out that the simplest approach worked well; having Ψ be the square root of a probability, a.k.a. a probability amplitude, indeed matched all experiments.6 (And we will end up adding a great many internal degrees of freedom to Ψ as we go along in QM; for now, though, the simplest wave functions are just complex functions of position)
The most interesting idea, though, was for the simultaneous handling of (3) and (4).7 We know that Ψ is going to be subject to some kind of linear differential equation, by (2). Linear differential equations can be written in the language of linear algebra; after all, the set of functions forms a vector space, and if D is some arbitrary combination of x’s and \partial‘s, then D(a\Psi_1 + b \Psi_2) = a D\Psi_1 + b D\Psi_2, so such a D is a linear operator. Its eigenvectors (commonly referred to as eigenfunctions, in this case) must form a basis for the set of all functions, and so on.
Now, if we look at the waveform in (3), there seems to be an obvious set of linear operators which would correspond to those very values of energy and momentum:
P = -i\hbar\frac{\partial}{\partial x}
H = +i\hbar\frac{\partial}{\partial t}
(We use H for the operator corresponding to total energy, H being short for the Hamiltonian operator which defines the total energy in classical mechanics; we will use lowercase p for the numeric [scalar] value of momentum, but a capital E for the scalar value of energy. This will be our one exception to the rule of using capital letters for operators, and the analogous lowercase letter to describe their eigenvalues) If we act with these operators on Ψ, we see
P\Psi = -i\hbar\frac{\partial}{\partial x}e^{i(k x - \omega t)} = \hbar k \Psi
H\Psi = i\hbar\frac{\partial}{\partial t} e^{i(k x - \omega t)} = \hbar\omega \Psi
If we are treating |\Psi|^2 as a probability weight, we should be able to talk about the mean value of physical quantities by taking the expectation value; but using these equations as a guide, we write out this general relationship in the slightly more cautious form
\left<A\right> = \int {\rm d} x \Psi^\star(x) A \Psi(x),
where A is any one of these linear operators – X, P, H, and so on. Thus, for example, for a free wave
\left<P\right> = \int {\rm d} x e^{-ipx/\hbar} (-i\hbar \partial_x) e^{ipx/\hbar} = p \left<1\right> = p.
Note that we must also, therefore, require that Ψ be normalizable —
\left<1\right> = \int_{-\infty}^\infty|\Psi(x)|^2{\rm d}x=1.
This actually seems to rule out the free wave solution from being a valid wave function. In practice, on those occasions where we do have to deal with it — and it will show up, when studying free particles — we deal with it by letting space have some finite extent L, multiplying Ψ by an appropriate normalization factor, and at the very end taking L to infinity.
Given this relationship between waves, operators, and physically observable quantities, the main question which remains to be answered is what the wave equation actually is. Well, we know a natural relationship between energy and momentum; so did Schrödinger, and so armed with these ideas he immediately wrote down the famous Klein-Gordon Equation:
(H^2 - P^2 c^2) \Psi = -\hbar^2\left(\partial_t^2 - c^2 \nabla^2\right)\Psi = m^2 c^4 \Psi
…. oh.
You were expecting the Schrödinger equation?
Well, so was he. The problem was that Schrödinger wanted to write down a nice, relativistic equation. Unfortunately, the above equation has serious problems, which basically boil down to the fact that it’s second-order in time derivatives. This means that solutions come out in pairs with energies that are equal and of opposite sign — so you end up with a hierarchy of solutions of arbitrarily negative energy, and you can’t define decent physics at all. Schrödinger came up with this equation in 1921, and spent three years bashing his head against it until he heard that Heisenberg was getting close to publishing. Spurred into sudden action, he ditched any attempt at relativistic nicety, and instead published a paper based on the nonrelativistic equation that now bears his name:
\boxed{H \Psi = \left(\frac{P^2}{2m}+V\right) \Psi\ ,}
or in terms of functions and derivatives,
\boxed{-\frac{\hbar^2}{2m}\nabla^2\Psi+V\Psi=i\hbar \frac{\partial}{\partial t}\Psi\ .}
The first equation is written in terms of abstract operators, and is just the energy-momentum relationship of classical mechanics with an arbitrary potential energy, V. The second is rewritten in terms of derivatives of functions, and is the familiar form of the Schrödinger equation.
We’re going to spend much of the rest of this course looking at solutions of this equation for various potentials, and learning about the laws of physics from them. It will turn out that pretty much everything in nonrelativistic quantum mechanics comes down to understanding this one equation and the things one can measure it. But first, we’re going to plunge a bit further into understanding the equation “in its own right” — looking at it a bit more in the language of linear algebra, and finding some remarkable conclusions.
Next time: The Uncertainty Principle!
1 I wish I could go into it here, but the calculation requires basic stat mech, which is beyond the scope of this class. However, you can find it in any intro to statistical mechanics; e.g., Kittel & Kroemer, Thermal Physics, chapter 4.
2 This was one of Einstein’s three major papers of 1905; the other two were his paper on Brownian Motion, which was the final smoking-gun proof of the existence of atoms, and the special theory of relativity.
“It’s people like this who make you realize how little you’ve accomplished. It is a sobering thought, for example, that when Mozart was my age, he had been dead for three years.” — Tom Lehrer
3 The history of this, and the details of the experiment, are fascinating. If you’re interested, I suggest reading the first few chapters of Richard Rhodes’ The Making of the Atomic Bomb; they give a wonderful overview of the development of atomic and nuclear physics in the first half of the 20th century, and are remarkably readable.
4 If you read an older QM textbook such as Pauling & Wilson, you’ll see the text divided into discussing the “old quantum theory” and the “new quantum theory.” This “old” theory was basically the mass of pre-Schrödinger work; it’s somewhat fascinating to see how far people got by doing things as hard-to-justify as Bohr’s angular momentum rule. If you look at the very end of P&W, you’ll find a description of this radical new “matrix mechanics” of Dirac’s; it was very complicated for them to explain, because by and large physicists had never heard of matrices or linear algebra at the time; they were considered an obscure tool of mathematicians. It was only after Dirac’s major cleanup of QM using them in 1934, for which he basically derived the entire theory of linear algebra from scratch, that some mathematicians came by and said “Hey, you know that there’s a whole branch of math for this…” One other side effect of this is that, if you look at older texts and especially original papers, the derivations are a lot more complicated than the ones here.
5 And his theory was regarded as nearly insane. He proposed this in his Ph.D. thesis, and his degree was nearly not granted; it was only the personal intercession of Einstein, who thought the idea might have some merit, with his committee which saved him.
6 You may ask what justifies the formula P = |\Psi|^2, as opposed to, say, |\Psi|^4 or something. The short answer is “because it matches experiment;” we’ll also see in the next lecture that it is particularly clean mathematically. But there has been research on this subject; oddly enough, it turns out that if P \sim |\Psi|^n for n not an even integer, one could construct a “postselection machine” which could not only solve NP-complete problems in constant time, but also allow you to travel back in time with a special (and consistent) way of dealing with the grandfather paradox. (The first paper actually proves only poly time; the second paper improves it to a constant, as a side effect of being able to build time machines) So if you do manage to detect any deviation from the |\Psi|^2 behavior of probabilities, by all means, patent it! I suggest U. S. Patent No. 1.
7 What I’m about to give is a considerably simpler explanation than what Schrödinger originally did, in no small part because I’m going to use what we know about linear algebra quite freely. Schrödinger’s original approach started from Poisson brackets, and was quite rigorous, but as a result was also extremely complicated to explain; there’s a reason that QM was considered an advanced graduate subject for quite some time afterwards. I will shamelessly take advantage of nearly a century of improved mathematical techniques to give this the easy way.
Published in: on August 4, 2010 at 10:10 Comments (6)
1. Hey Yonni – I’m really enjoying following your course!
(It has the added benefit that reading your posts makes me feel like I’m at least doing something productive while I avoid actual work.)
If only my insane ideas happened to match those of de Broglie… sigh.
• It’s a virtuous cycle; writing them makes me feel like I’m doing something productive while I avoid actual work.
2. Add me to the list of those enjoying these posts.
Undergrad level QM became much easier for me when I realized the entire course (well, all the problems given anyway) was about boundary conditions. Unfortunately, this realization came after I took it.
Also, there’s an interesting discussion going on at Chad Orzel’s blog about teaching Stat. Mech. that matches well with my experience.
3. tiny nit: it’s Niels Bohr, not Neils.
Some more background of what a “free wave” is would have been nice.
• Whoops — typo. Fixed.
A free wave is a solution to the wave equation in the absence of any background forces or potentials; it’s the equation which describes a wave propagating in a medium without any forces on it. It’s the same equation for, e.g., sound waves, light waves, and so on, and its solution (in an infinite, empty space, so that there aren’t any complicated boundary conditions) is e^{i(kx-\omega t)}. That’s a sine wave with wave number k, with the wave peaks moving along at a velocity \omega/k.
All of the usual rules of optics and so on follow from this; e.g., you can get the formula for the interference pattern you see when you push light through slits, gratings, and so on by adding up a bunch of free waves of the relevant sort. (For light, the wave velocity is just the speed of light)
Does that answer your question?
• Ah, yes indeed, that answers it. I wasn’t sure if the term was referring to something specifically in QM, for a particular medium, etc. (There are plenty of equations in QM that look simple and innocuous, after all.)
I’d also note that this is the first time I’d seen “capital letters, as a rule, are operators” called out explicitly. Everything else I’ve seen seemed to assume you’d pick it up eventually – which I guess is true, but it’s quite nice to see the context made explicit.
Comments are closed.
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7f40f5df4134db18 | @inproceedings{11799, abstract = {We study the problem of matching bidders to items where each bidder i has general, strictly monotonic utility functions u i,j (p j ) expressing her utility of being matched to item j at price p j . For this setting we prove that a bidder optimal outcome always exists, even when the utility functions are non-linear and non-continuous. Furthermore, we give an algorithm to find such a solution. Although the running time of this algorithm is exponential in the number of items, it is polynomial in the number of bidders.}, author = {Dütting, Paul and Henzinger, Monika H and Weber, Ingmar}, booktitle = {5th International Workshop on Internet and Network Economics}, isbn = {978-364210840-2}, issn = {1611-3349}, location = {Rome, Italy}, pages = {575--582}, publisher = {Springer Nature}, title = {{Bidder optimal assignments for general utilities}}, doi = {10.1007/978-3-642-10841-9_58}, volume = {5929}, year = {2009}, } @misc{11905, abstract = {Given only the URL of a web page, can we identify its topic? This is the question that we examine in this paper. Usually, web pages are classified using their content, but a URL-only classifier is preferable, (i) when speed is crucial, (ii) to enable content filtering before an (objection-able) web page is downloaded, (iii) when a page's content is hidden in images, (iv) to annotate hyperlinks in a personalized web browser, without fetching the target page, and (v) when a focused crawler wants to infer the topic of a target page before devoting bandwidth to download it. We apply a machine learning approach to the topic identification task and evaluate its performance in extensive experiments on categorized web pages from the Open Directory Project (ODP). When training separate binary classifiers for each topic, we achieve typical F-measure values between 80 and 85, and a typical precision of around 85. We also ran experiments on a small data set of university web pages. For the task of classifying these pages into faculty, student, course and project pages, our methods improve over previous approaches by 13.8 points of F-measure.}, author = {Baykan, Eda and Henzinger, Monika H and Marian, Ludmila and Weber, Ingmar}, booktitle = {18th International World Wide Web Conference}, isbn = {9781605584874}, location = {New York, NY, United States}, pages = {1109--1110}, publisher = {Association for Computing Machinery}, title = {{Purely URL-based topic classification}}, doi = {10.1145/1526709.1526880}, year = {2009}, } @inproceedings{11906, abstract = {In the origin detection problem an algorithm is given a set S of documents, ordered by creation time, and a query document D. It needs to output for every consecutive sequence of k alphanumeric terms in D the earliest document in $S$ in which the sequence appeared (if such a document exists). Algorithms for the origin detection problem can, for example, be used to detect the "origin" of text segments in D and thus to detect novel content in D. They can also find the document from which the author of D has copied the most (or show that D is mostly original.) We concentrate on solutions that use only a fixed amount of memory. We propose novel algorithms for this problem and evaluate them together with a large number of previously published algorithms. Our results show that (1) detecting the origin of text segments efficiently can be done with very high accuracy even when the space used is less than 1% of the size of the documents in $S$, (2) the precision degrades smoothly with the amount of available space, (3) various estimation techniques can be used to increase the performance of the algorithms.}, author = {Abdel Hamid, Ossama and Behzadi, Behshad and Christoph, Stefan and Henzinger, Monika H}, booktitle = {18th International World Wide Web Conference}, isbn = {9781605584874}, location = {Madrid, Spain}, pages = {61--70}, publisher = {Association for Computing Machinery}, title = {{Detecting the origin of text segments efficiently}}, doi = {10.1145/1526709.1526719}, year = {2009}, } @article{2795, abstract = {The collapse of turbulence, observable in shear flows at low Reynolds numbers, raises the question if turbulence is generically of a transient nature or becomes sustained at some critical point. Recent data have led to conflicting views with the majority of studies supporting the model of turbulence turning into an attracting state. Here we present lifetime measurements of turbulence in pipe flow spanning 8 orders of magnitude in time, drastically extending all previous investigations. We show that no critical point exists in this regime and that in contrast to the prevailing view the turbulent state remains transient. To our knowledge this is the first observation of superexponential transients in turbulence, confirming a conjecture derived from low-dimensional systems.}, author = {Björn Hof and de Lózar, Alberto and Kuik, Dirk J and Westerweel, Jerry}, journal = {Physical Review Letters}, number = {21}, publisher = {American Physical Society}, title = {{Repeller or attractor? Selecting the dynamical model for the onset of turbulence in pipe flow}}, doi = {10.1103/PhysRevLett.101.214501}, volume = {101}, year = {2008}, } @article{2892, author = {Azevedo, Ricardo B and Lohaus, Rolf and Tiago Paixao}, journal = {Evolution & Development}, number = {5}, pages = {514 -- 515}, publisher = {Wiley-Blackwell}, title = {{Networking networks}}, doi = {10.1111/j.1525-142X.2008.00265.x}, volume = {10}, year = {2008}, } @article{3030, abstract = {Telomeres in many eukaryotes are maintained by telomerase in whose absence telomere shortening occurs. However, telomerase-deficient Arabidopsis thaliana mutants (Attert -/-) show extremely low rates of telomere shortening per plant generation (250-500 bp), which does not correspond to the expected outcome of replicative telomere shortening resulting from ca. 1,000 meristem cell divisions per seed-to-seed generation. To investigate the influence of the number of cell divisions per seed-to-seed generation, Attert -/- mutant plants were propagated from seeds coming either from the lower-most or the upper-most siliques (L- and U-plants) and the length of their telomeres were followed over several generations. The rate of telomere shortening was faster in U-plants, than in L-plants, as would be expected from their higher number of cell divisions per generation. However, this trend was observed only in telomeres whose initial length is relatively high and the differences decreased with progressive general telomere shortening over generations. But in generation 4, the L-plants frequently show a net telomere elongation, while the U-plants fail to do so. We propose that this is due to the activation of alternative telomere lengthening (ALT), a process which is activated in early embryonic development in both U- and L-plants, but is overridden in U-plants due to their higher number of cell divisions per generation. These data demonstrate what so far has only been speculated, that in the absence of telomerase, the number of cell divisions within one generation influences the control of telomere lengths. These results also reveal a fast and efficient activation of ALT mechanism(s) in response to the loss of telomerase activity and imply that ALT is probably involved also in normal plant development.}, author = {Růčková, Eva and Jirí Friml and Procházková Schrumpfová, Petra and Fajkus, Jiří}, journal = {Plant Molecular Biology}, number = {6}, pages = {637 -- 646}, publisher = {Springer}, title = {{Role of alternative telomere lengthening unmasked in telomerase knock-out mutant plants}}, doi = {10.1007/s11103-008-9295-7}, volume = {66}, year = {2008}, } @article{3031, abstract = {Many aspects of plant development, including patterning and tropisms, are largely dependent on the asymmetric distribution of the plant signaling molecule auxin. Auxin transport inhibitors (ATIs), which interfere with directional auxin transport, have been essential tools in formulating this concept. However, despite the use of ATIs in plant research for many decades, the mechanism of ATI action has remained largely elusive. Using real-time live-cell microscopy, we show here that prominent ATIs such as 2,3,5-triiodobenzoic acid (TIBA) and 2-(1-pyrenoyl) benzoic acid (PBA) inhibit vesicle trafficking in plant, yeast, and mammalian cells. Effects on micropinocytosis, rab5-labeled endosomal motility at the periphery of HeLa cells and on fibroblast mobility indicate that ATIs influence actin cytoskeleton. Visualization of actin cytoskeleton dynamics in plants, yeast, and mammalian cells show that ATIs stabilize actin. Conversely, stabilizing actin by chemical or genetic means interferes with endocytosis, vesicle motility, auxin transport, and plant development, including auxin transport-dependent processes. Our results show that a class of ATIs act as actin stabilizers and advocate that actin-dependent trafficking of auxin transport components participates in the mechanism of auxin transport. These studies also provide an example of how the common eukaryotic process of actin-based vesicle motility can fulfill a plant-specific physiological role.}, author = {Dhonukshe, Pankaj and Grigoriev, Ilya S and Fischer, Rainer and Tominaga, Motoki and Robinson, David G and Hašek, Jiří and Paciorek, Tomasz and Petrášek, Jan and Seifertová, Daniela and Tejos, Ricardo and Meisel, Lee A and Zažímalová, Eva and Gadella, Theodorus W and Stierhof, York-Dieter and Ueda, Takashi and Oiwa, Kazuhiro and Akhmanova, Anna and Brock, Roland and Spang, Anne and Jirí Friml}, journal = {PNAS}, number = {11}, pages = {4489 -- 4494}, publisher = {National Academy of Sciences}, title = {{Auxin transport inhibitors impair vesicle motility and actin cytoskeleton dynamics in diverse eukaryotes}}, doi = {10.1073/pnas.0711414105}, volume = {105}, year = {2008}, } @article{3032, abstract = { Cell polarity manifested by the polar cargo delivery to different plasma-membrane domains is a fundamental feature of multicellular organisms. Pathways for polar delivery have been identified in animals; prominent among them is transcytosis, which involves cargo movement between different sides of the cell [1]. PIN transporters are prominent polar cargoes in plants, whose polar subcellular localization determines the directional flow of the signaling molecule auxin [2, 3]. In this study, we address the cellular mechanisms of PIN polar targeting and dynamic polarity changes. We show that apical and basal PIN targeting pathways are interconnected but molecularly distinct by means of ARF GEF vesicle-trafficking regulators. Pharmacological or genetic interference with the Arabidopsis ARF GEF GNOM leads specifically to apicalization of basal cargoes such as PIN1. We visualize the translocation of PIN proteins between the opposite sides of polarized cells in vivo and show that this PIN transcytosis occurs by endocytic recycling and alternative recruitment of the same cargo molecules by apical and basal targeting machineries. Our data suggest that an ARF GEF-dependent transcytosis-like mechanism is operational in plants and provides a plausible mechanism to trigger changes in PIN polarity and hence auxin fluxes during embryogenesis and organogenesis.}, author = {Kleine-Vehn, Jürgen and Dhonukshe, Pankaj and Sauer, Michael and Brewer, Philip B and Wiśniewska, Justyna and Paciorek, Tomasz and Eva Benková and Jirí Friml}, journal = {Current Biology}, number = {7}, pages = {526 -- 531}, publisher = {Cell Press}, title = {{ARF GEF dependent transcytosis and polar delivery of PIN auxin carriers in Arabidopsis}}, doi = {10.1016/j.cub.2008.03.021}, volume = {18}, year = {2008}, } @inbook{3033, abstract = { Embryogenesis in Arabidopsis thaliana depends on the proper establishment and maintenance of local auxin accumulation. In the course of elucidating the connections between developmental progress and auxin distribution, several techniques have been developed to investigate spatial and temporal distribution of auxin response or accumulation in Arabidopsis embryos. This chapter reviews and describes two independent methods, the detection of the activity of auxin responsive transgenes and immunolocalization of auxin itself.}, author = {Sauer, Michael and Jirí Friml}, booktitle = {Plant Embryogenesis}, editor = {Suárez, María F and Bozhkov, Peter V}, pages = {137 -- 144}, publisher = {Humana Press}, title = {{Visualization of auxin gradients in embryogenesis }}, doi = {10.1007/978-1-59745-273-1_11}, volume = {427}, year = {2008}, } @article{3034, abstract = {They can't move away from shade, so plants resort to a molecular solution to find a place in the sun. The action they take is quite radical, and involves a reprogramming of their development. }, author = {Friml, Jirí and Sauer, Michael}, journal = {Nature}, number = {7193}, pages = {298 -- 299}, publisher = {Nature Publishing Group}, title = {{Plant biology: In their neighbour's shadow}}, doi = {10.1038/453298a}, volume = {453}, year = {2008}, } @inbook{3035, abstract = {Embryogenesis of Arabidopsis thaliana follows a nearly invariant cell division pattern and provides an ideal system for studies of early plant development. However, experimental manipulation with embryogenesis is difficult, as the embryo develops deeply inside maternal tissues. Here, we present a method to culture zygotic Arabidopsis embryos in vitro. It enables culturing for prolonged periods of time from the first developmental stages on. The technique omits excision of the embryo by culturing the entire ovule, which facilitates the manual procedure. It allows pharmacological manipulation of embryo development and does not interfere with standard techniques for localizing gene expression and protein localization in the cultivated embryos.}, author = {Sauer, Michael and Jirí Friml}, booktitle = {Plant Embryogenesis}, editor = {Suárez, María F and Bozhkov, Peter V}, pages = {71 -- 76}, publisher = {Humana Press}, title = {{In vitro culture of Arabidopsis embryos }}, doi = {10.1007/978-1-59745-273-1_5}, volume = {427}, year = {2008}, } @article{3036, abstract = {Plants exhibit an exceptional adaptability to different environmental conditions. To a large extent, this adaptability depends on their ability to initiate and form new organs throughout their entire postembryonic life. Plant shoot and root systems unceasingly branch and form axillary shoots or lateral roots, respectively. The first event in the formation of a new organ is specification of founder cells. Several plant hormones, prominent among them auxin, have been implicated in the acquisition of founder cell identity by differentiated cells, but the mechanisms underlying this process are largely elusive. Here, we show that auxin and its local accumulation in root pericycle cells is a necessary and sufficient signal to respecify these cells into lateral root founder cells. Analysis of the alf4-1 mutant suggests that specification of founder cells and the subsequent activation of cell division leading to primordium formation represent two genetically separable events. Time-lapse experiments show that the activation of an auxin response is the earliest detectable event in founder cell specification. Accordingly, local activation of auxin response correlates absolutely with the acquisition of founder cell identity and precedes the actual formation of a lateral root primordium through patterned cell division. Local production and subsequent accumulation of auxin in single pericycle cells induced by Cre-Lox-based activation of auxin synthesis converts them into founder cells. Thus, auxin is the local instructive signal that is sufficient for acquisition of founder cell identity and can be considered a morphogenetic trigger in postembryonic plant organogenesis.}, author = {Dubrovsky, Joseph G and Sauer, Michael and Napsucialy-Mendivil, Selene and Ivanchenko, Maria G and Jirí Friml and Shishkova, Svetlana and Celenza, John and Eva Benková}, journal = {PNAS}, number = {25}, pages = {8790 -- 8794}, publisher = {National Academy of Sciences}, title = {{Auxin acts as a local morphogenetic trigger to specify lateral root founder cells}}, doi = {10.1073/pnas.0712307105}, volume = {105}, year = {2008}, } @article{3037, author = {Feraru, Elena and Friml, Jirí}, journal = {Plant Physiology}, number = {4}, pages = {1553 -- 1559}, publisher = {American Society of Plant Biologists}, title = {{PIN polar targeting}}, doi = {10.1104/pp.108.121756}, volume = {147}, year = {2008}, } @article{3038, abstract = {Lateral roots originate deep within the parental root from a small number of founder cells at the periphery of vascular tissues and must emerge through intervening layers of tissues. We describe how the hormone auxin, which originates from the developing lateral root, acts as a local inductive signal which re-programmes adjacent cells. Auxin induces the expression of a previously uncharacterized auxin influx carrier LAX3 in cortical and epidermal cells directly overlaying new primordia. Increased LAX3 activity reinforces the auxin-dependent induction of a selection of cell-wall-remodelling enzymes, which are likely to promote cell separation in advance of developing lateral root primordia.}, author = {Swarup, Kamal and Eva Benková and Swarup, Ranjan and Casimiro, Ilda and Péret, Benjamin and Yang, Yaodong and Parry, Geraint and Nielsen, Erik and De Smet, Ive and Vanneste, Steffen and Levesque, Mitchell P and Carrier, David and James, Nicholas and Calvo, Vanessa and Ljung, Karin and Kramer, Eric and Roberts, Rebecca and Graham, Neil and Marillonnet, Sylvestre and Patel, Kanu and Jones, Jonathan D and Taylor, Christopher G and Schachtman, Daniel P and May, Sean and Sandberg, Göran and Benfey, Philip N and Jirí Friml and Kerr, Ian and Beeckman, Tom and Laplaze, Laurent and Bennett, Malcolm J}, journal = {Nature Cell Biology}, number = {8}, pages = {946 -- 954}, publisher = {Nature Publishing Group}, title = {{The auxin influx carrier LAX3 promotes lateral root emergence}}, doi = {10.1038/ncb1754}, volume = {10}, year = {2008}, } @article{3039, abstract = {During the development of multicellular organisms, organogenesis and pattern formation depend on formative divisions to specify and maintain pools of stem cells. In higher plants, these activities are essential to shape the final root architecture because the functioning of root apical meristems and the de novo formation of lateral roots entirely rely on it. We used transcript profiling on sorted pericycle cells undergoing lateral root initiation to identify the receptor-like kinase ACR4 of Arabidopsis as a key factor both in promoting formative cell divisions in the pericycle and in constraining the number of these divisions once organogenesis has been started. In the root tip meristem, ACR4 shows a similar action by controlling cell proliferation activity in the columella cell lineage. Thus, ACR4 function reveals a common mechanism of formative cell division control in the main root tip meristem and during lateral root initiation.}, author = {De Smet, Ive and Vassileva, Valya and De Rybel, Bert and Levesque, Mitchell P and Grunewald, Wim and Van Damme, Daniël and Van Noorden, Giel and Naudts, Mirande and Van Isterdael, Gert and De Clercq, Rebecca and Wang, Jean Y and Meuli, Nicholas and Vanneste, Steffen and Jirí Friml and Hilson, Pierre and Jürgens, Gerd and Ingram, Gwyneth C and Inzé, Dirk and Benfey, Philip N and Beeckman, Tom}, journal = {Science}, number = {5901}, pages = {594 -- 597}, publisher = {American Association for the Advancement of Science}, title = {{Receptor-like kinase ACR4 restricts formative cell divisions in the Arabidopsis root}}, doi = {10.1126/science.1160158}, volume = {322}, year = {2008}, } @article{3040, abstract = {The polar, sub-cellular localization of PIN auxin efflux carriers determines the direction of intercellular auxin flow, thus defining the spatial aspect of auxin signalling. Dynamic, transcytosis-like relocalizations of PIN proteins occur in response to external and internal signals, integrating these signals into changes in auxin distribution. Here, we examine the cellular and molecular mechanisms of polar PIN delivery and transcytosis. The mechanisms of the ARF-GEF-dependent polar targeting and transcytosis are well conserved and show little variations among diverse Arabidopsis ecotypes consistent with their fundamental importance in regulating plant development. At the cellular level, we refine previous findings on the role of the actin cytoskeleton in apical and basal PIN targeting, and identify a previously unknown role for microtubules, specifically in basal targeting. PIN protein delivery to different sides of the cell is mediated by ARF-dependent trafficking with a previously unknown complex level of distinct ARF-GEF vesicle trafficking regulators. Our data suggest that alternative recruitment of PIN proteins by these distinct pathways can account for cell type- and cargo-specific aspects of polar targeting, as well as for polarity changes in response to different signals. The resulting dynamic PIN positioning to different sides of cells defines a three-dimensional pattern of auxin fluxes within plant tissues.}, author = {Kleine-Vehn, Jürgen and Łangowski, Łukasz and Wiśniewska, Justyna and Dhonukshe, Pankaj and Brewer, Philip B and Jirí Friml}, journal = {Molecular Plant}, number = {6}, pages = {1056 -- 1066}, publisher = {Oxford University Press}, title = {{Cellular and molecular requirements for polar PIN targeting and transcytosis in plants}}, doi = {10.1093/mp/ssn062}, volume = {1}, year = {2008}, } @article{3041, abstract = {The rate, polarity, and symmetry of the flow of the plant hormone auxin are determined by the polar cellular localization of PIN-FORMED (PIN) auxin efflux carriers. Flavonoids, a class of secondary plant metabolites, have been suspected to modulate auxin transport and tropic responses. Nevertheless, the identity of specific flavonoid compounds involved and their molecular function and targets in vivo are essentially unknown. Here we show that the root elongation zone of agravitropic pin2/eir1/wav6/agr1 has an altered pattern and amount of flavonol glycosides. Application of nanomolar concentrations of flavonols to pin2 roots is sufficient to partially restore root gravitropism. By employing a quantitative cell biological approach, we demonstrate that flavonoids partially restore the formation of lateral auxin gradients in the absence of PIN2. Chemical complementation by flavonoids correlates with an asymmetric distribution of the PIN1 protein. pin2 complementation probably does not result from inhibition of auxin efflux, as supply of the auxin transport inhibitor N-1-naphthylphthalamic acid failed to restore pin2 gravitropism. We propose that flavonoids promote asymmetric PIN shifts during gravity stimulation, thus redirecting basipetal auxin streams necessary for root bending. © 2008 by The American Society for Biochemistry and Molecular Biology, Inc.}, author = {Santelia, Diana and Henrichs, Sina and Vincenzetti, Vincent and Sauer, Michael and Bigler, Laurent and Klein, Markus B and Bailly, Aurélien and Lee, Yuree and Jirí Friml and Geisler, Markus and Martinoia, Enrico}, journal = {Journal of Biological Chemistry}, number = {45}, pages = {31218 -- 31226}, publisher = {American Society for Biochemistry and Molecular Biology}, title = {{Flavonoids redirect PIN mediated polar auxin fluxes during root gravitropic responses}}, doi = { 10.1074/jbc.M710122200}, volume = {283}, year = {2008}, } @article{3042, abstract = {All eukaryotic cells present at the cell surface a specific set of plasma membrane proteins that modulate responses to internal and external cues and whose activity is also regulated by protein degradation. We characterized the lytic vacuole-dependent degradation of membrane proteins in Arabidopsis thaliana by means of in vivo visualization of vacuolar targeting combined with quantitative protein analysis. We show that the vacuolar targeting pathway is used by multiple cargos including PIN-FORMED (PIN) efflux carriers for the phytohormone auxin. In vivo visualization of PIN2 vacuolar targeting revealed its differential degradation in response to environmental signals, such as gravity. In contrast to polar PIN delivery to the basal plasma membrane, which depends on the vesicle trafficking regulator ARF-GEF GNOM, PIN sorting to the lytic vacuolar pathway requires additional brefeldin A-sensitive ARF-GEF activity. Furthermore, we identified putative retromer components SORTING NEXIN1 (SNX1) and VACUOLAR PROTEIN SORTING29 (VPS29) as important factors in this pathway and propose that the retromer complex acts to retrieve PIN proteins from a late/pre-vacuolar compartment back to the recycling pathways. Our data suggest that ARF GEF- and retromer-dependent processes regulate PIN sorting to the vacuole in an antagonistic manner and illustrate instrumentalization of this mechanism for fine-tuning the auxin fluxes during gravitropic response.}, author = {Kleine-Vehn, Jürgen and Leitner, Johannes and Zwiewka, Marta and Sauer, Michael and Abas, Lindy and Luschnig, Christian and Jirí Friml}, journal = {PNAS}, number = {46}, pages = {17812 -- 17817}, publisher = {National Academy of Sciences}, title = {{Differential degradation of PIN2 auxin efflux carrier by retromer dependent vacuolar targeting}}, doi = {10.1073/pnas.0808073105}, volume = {105}, year = {2008}, } @article{3043, abstract = {Plant development is characterized by a profound phenotypic plasticity that often involves redefining of the developmental fate and polarity of cells within differentiated tissues. The plant hormone auxin and its directional intercellular transport play a major role in these processes because they provide positional information and link cell polarity with tissue patterning. This plant-specific mechanism of transport-dependent auxin gradients depends on subcellular dynamics of auxin transport components, in particular on endocytic recycling and polar targeting. Recent insights into these cellular processes in plants have revealed important parallels to yeast and animal systems, including clathrin-dependent endocytosis, retromer function, and transcytosis, but have also emphasized unique features of plant cells such as diversity of polar targeting pathways; integration of environmental signals into subcellular trafficking; and the link between endocytosis, cell polarity, and cell fate specification. We review these advances and focus on the translation of the subcellular dynamics to the regulation of whole-plant development.}, author = {Kleine Vehn, Jürgen and Friml, Jirí}, journal = {Annual Review of Cell and Developmental Biology}, pages = {447 -- 473}, publisher = {Annual Reviews}, title = {{Polar targeting and endocytic recycling in auxin-dependent plant development}}, doi = {10.1146/annurev.cellbio.24.110707.175254}, volume = {24}, year = {2008}, } @article{3044, abstract = {The signalling molecule auxin controls plant morphogenesis via its activity gradients, which are produced by intercellular auxin transport. Cellular auxin efflux is the rate-limiting step in this process and depends on PIN and phosphoglycoprotein (PGP) auxin transporters. Mutual roles for these proteins in auxin transport are unclear, as is the significance of their interactions for plant development. Here, we have analysed the importance of the functional interaction between PIN- and PGP-dependent auxin transport in development. We show by analysis of inducible overexpression lines that PINs and PGPs define distinct auxin transport mechanisms: both mediate auxin efflux but they play diverse developmental roles. Components of both systems are expressed during embryogenesis, organogenesis and tropisms, and they interact genetically in both synergistic and antagonistic fashions. A concerted action of PIN- and PGP-dependent efflux systems is required for asymmetric auxin distribution during these processes. We propose a model in which PGP-mediated efflux controls auxin levels in auxin channel-forming cells and, thus, auxin availability for PIN-dependent vectorial auxin movement.}, author = {Mravec, Jozef and Kubeš, Martin and Bielach, Agnieszka and Gaykova, Vassilena and Petrášek, Jan and Skůpa, Petr and Chand, Suresh and Eva Benková and Zažímalová, Eva and Jirí Friml}, journal = {Development}, number = {20}, pages = {3345 -- 3354}, publisher = {Company of Biologists}, title = {{Interaction of PIN and PGP transport mechanisms in auxin distribution-dependent development}}, doi = {10.1242/dev.021071}, volume = {135}, year = {2008}, } @article{3045, abstract = {Dynamically polarized membrane proteins define different cell boundaries and have an important role in intercellular communication - a vital feature of multicellular development. Efflux carriers for the signalling molecule auxin from the PIN family are landmarks of cell polarity in plants and have a crucial involvement in auxin distribution-dependent development including embryo patterning, organogenesis and tropisms. Polar PIN localization determines the direction of intercellular auxin flow, yet the mechanisms generating PIN polarity remain unclear. Here we identify an endocytosis-dependent mechanism of PIN polarity generation and analyse its developmental implications. Real-time PIN tracking showed that after synthesis, PINs are initially delivered to the plasma membrane in a non-polar manner and their polarity is established by subsequent endocytic recycling. Interference with PIN endocytosis either by auxin or by manipulation of the Arabidopsis Rab5 GTPase pathway prevents PIN polarization. Failure of PIN polarization transiently alters asymmetric auxin distribution during embryogenesis and increases the local auxin response in apical embryo regions. This results in ectopic expression of auxin pathway-associated root-forming master regulators in embryonic leaves and promotes homeotic transformation of leaves to roots. Our results indicate a two-step mechanism for the generation of PIN polar localization and the essential role of endocytosis in this process. It also highlights the link between endocytosis-dependent polarity of individual cells and auxin distribution-dependent cell fate establishment for multicellular patterning.}, author = {Dhonukshe, Pankaj and Tanaka, Hirokazu and Goh, Tatsuaki and Ebine, Kazuo and Mähönen, Ari Pekka and Prasad, Kalika and Blilou, Ikram and Geldner, Niko and Xu, Jian and Uemura, Tomohiro and Chory, Joanne and Ueda, Takashi and Nakano, Akihiko and Scheres, Ben and Jirí Friml}, journal = {Nature}, number = {7224}, pages = {962 -- 966}, publisher = {Nature Publishing Group}, title = {{Generation of cell polarity in plants links endocytosis auxin distribution and cell fate decisions}}, doi = {10.1038/nature07409}, volume = {456}, year = {2008}, } @inproceedings{3194, abstract = {We consider the problem of optimizing multilabel MRFs, which is in general NP-hard and ubiquitous in low-level computer vision. One approach for its solution is to formulate it as an integer linear programming and relax the integrality constraints. The approach we consider in this paper is to first convert the multi-label MRF into an equivalent binary-label MRF and then to relax it. The resulting relaxation can be efficiently solved using a maximum flow algorithm. Its solution provides us with a partially optimal labelling of the binary variables. This partial labelling is then easily transferred to the multi-label problem. We study the theoretical properties of the new relaxation and compare it with the standard one. Specifically, we compare tightness, and characterize a subclass of problems where the two relaxations coincide. We propose several combined algorithms based on the technique and demonstrate their performance on challenging computer vision problems.}, author = {Kohli, Pushmeet and Shekhovtsov, Alexander and Rother, Carsten and Vladimir Kolmogorov and Torr, Philip H}, pages = {480 -- 487}, publisher = {Omnipress}, title = {{On partial optimality in multi label MRFs}}, doi = {10.1145/1390156.1390217}, year = {2008}, } @inproceedings{3195, abstract = {Graph cut is a popular technique for interactive image segmentation. However, it has certain shortcomings. In particular, graph cut has problems with segmenting thin elongated objects due to the ldquoshrinking biasrdquo. To overcome this problem, we propose to impose an additional connectivity prior, which is a very natural assumption about objects. We formulate several versions of the connectivity constraint and show that the corresponding optimization problems are all NP-hard. For some of these versions we propose two optimization algorithms: (i) a practical heuristic technique which we call DijkstraGC, and (ii) a slow method based on problem decomposition which provides a lower bound on the problem. We use the second technique to verify that for some practical examples DijkstraGC is able to find the global minimum.}, author = {Vicente, Sara and Vladimir Kolmogorov and Rother, Carsten}, publisher = {IEEE}, title = {{Graph cut based image segmentation with connectivity priors}}, doi = {10.1109/CVPR.2008.4587440}, year = {2008}, } @article{3196, abstract = {Among the most exciting advances in early vision has been the development of efficient energy minimization algorithms for pixel-labeling tasks such as depth or texture computation. It has been known for decades that such problems can be elegantly expressed as Markov random fields, yet the resulting energy minimization problems have been widely viewed as intractable. Algorithms such as graph cuts and loopy belief propagation (LBP) have proven to be very powerful: For example, such methods form the basis for almost all the top-performing stereo methods. However, the trade-offs among different energy minimization algorithms are still not well understood. In this paper, we describe a set of energy minimization benchmarks and use them to compare the solution quality and runtime of several common energy minimization algorithms. We investigate three promising methods-graph cuts, LBP, and tree-reweighted message passing-in addition to the well-known older iterated conditional mode (ICM) algorithm. Our benchmark problems are drawn from published energy functions used for stereo, image stitching, interactive segmentation, and denoising. We also provide a general-purpose software interface that allows vision researchers to easily switch between optimization methods. The benchmarks, code, images, and results are available at http://vision.middlebury.edu/MRF/.}, author = {Szeliski, Richard S and Zabih, Ramin and Scharstein, Daniel and Veksler, Olga and Vladimir Kolmogorov and Agarwala, Aseem and Tappen, Marshall F and Rother, Carsten}, journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence}, number = {6}, pages = {1068 -- 1080}, publisher = {IEEE}, title = {{A comparative study of energy minimization methods for Markov random fields with smoothness-based priors}}, doi = {10.1109/TPAMI.2007.70844}, volume = {30}, year = {2008}, } @inproceedings{3198, abstract = {In this paper we present a new approach for establishing correspondences between sparse image features related by an unknown non-rigid mapping and corrupted by clutter and occlusion, such as points extracted from a pair of images containing a human figure in distinct poses. We formulate this matching task as an energy minimization problem by defining a complex objective function of the appearance and the spatial arrangement of the features. Optimization of this energy is an instance of graph matching, which is in general a NP-hard problem. We describe a novel graph matching optimization technique, which we refer to as dual decomposition (DD), and demonstrate on a variety of examples that this method outperforms existing graph matching algorithms. In the majority of our examples DD is able to find the global minimum within a minute. The ability to globally optimize the objective allows us to accurately learn the parameters of our matching model from training examples. We show on several matching tasks that our learned model yields results superior to those of state-of-the-art methods. }, author = {Torresani, Lorenzo and Vladimir Kolmogorov and Rother, Carsten}, pages = {596 -- 609}, publisher = {Springer}, title = {{Feature correspondence via graph matching: Models and global optimization}}, doi = {10.1007/978-3-540-88688-4_44}, volume = {5303}, year = {2008}, } @inproceedings{3224, abstract = {We propose a new mode of operation, enciphered CBC, for domain extension of length-preserving functions (like block ciphers), which is a variation on the popular CBC mode of operation. Our new mode is twice slower than CBC, but has many (property-preserving) properties not enjoyed by CBC and other known modes. Most notably, it yields the first constant-rate Variable Input Length (VIL) MAC from any length preserving Fixed Input Length (FIL) MAC. This answers the question of Dodis and Puniya from Eurocrypt 2007. Further, our mode is a secure domain extender for PRFs (with basically the same security as encrypted CBC). This provides a hedge against the security of the block cipher: if the block cipher is pseudorandom, one gets a VIL-PRF, while if it is "only" unpredictable, one "at least" gets a VIL-MAC. Additionally, our mode yields a VIL random oracle (and, hence, a collision-resistant hash function) when instantiated with length-preserving random functions, or even random permutations (which can be queried from both sides). This means that one does not have to re-key the block cipher during the computation, which was critically used in most previous constructions (analyzed in the ideal cipher model). }, author = {Dodis, Yevgeniy and Krzysztof Pietrzak and Puniya, Prashant}, pages = {198 -- 219}, publisher = {Springer}, title = {{A new mode of operation for block ciphers and length preserving MACs}}, doi = {10.1007/978-3-540-78967-3_12}, volume = {4965}, year = {2008}, } @inproceedings{3225, abstract = {A robust multi-property combiner for a set of security properties merges two hash functions such that the resulting function satisfies each of the properties which at least one of the two starting functions has. Fischlin and Lehmann (TCC 2008) recently constructed a combiner which simultaneously preserves collision-resistance, target collision-resistance, message authentication, pseudorandomness and indifferentiability from a random oracle (IRO). Their combiner produces outputs of 5n bits, where n denotes the output length of the underlying hash functions. In this paper we propose improved combiners with shorter outputs. By sacrificing the indifferentiability from random oracles we obtain a combiner which preserves all of the other aforementioned properties but with output length 2n only. This matches a lower bound for black-box combiners for collision-resistance as the only property, showing that the other properties can be achieved without penalizing the length of the hash values. We then propose a combiner which also preserves the IRO property, slightly increasing the output length to 2n + ω(logn). Finally, we show that a twist on our combiners also makes them robust for one-wayness (but at the price of a fixed input length). }, author = {Fischlin, Marc and Lehmann, Anja and Krzysztof Pietrzak}, number = {PART 2}, pages = {655 -- 666}, publisher = {Springer}, title = {{Robust multi property combiners for hash functions revisited}}, doi = {10.1007/978-3-540-70583-3_53}, volume = {5126}, year = {2008}, } @inproceedings{3226, abstract = {A family of functions is weakly pseudorandom if a random member of the family is indistinguishable from a uniform random function when queried on random inputs. We point out a subtle ambiguity in the definition of weak PRFs: there are natural weak PRFs whose security breaks down if the randomness used to sample the inputs is revealed. To capture this ambiguity we distinguish between public-coin and secret-coin weak PRFs. We show that the existence of a secret-coin weak PRF which is not also a public-coin weak PRF implies the existence of two pass key-agreement (i.e. public-key encryption). So in Minicrypt, i.e. under the assumption that one-way functions exist but public-key cryptography does not, the notion of public- and secret-coin weak PRFs coincide. Previous to this paper all positive cryptographic statements known to hold exclusively in Minicrypt concerned the adaptive security of constructions using non-adaptively secure components. Weak PRFs give rise to a new set of statements having this property. As another example we consider the problem of range extension for weak PRFs. We show that in Minicrypt one can beat the best possible range expansion factor (using a fixed number of distinct keys) for a very general class of constructions (in particular, this class contains all constructions that are known today). }, author = {Krzysztof Pietrzak and Sjödin, Johan}, number = {PART 2}, pages = {423 -- 436}, publisher = {Springer}, title = {{Weak pseudorandom functions in minicrypt}}, doi = {10.1007/978-3-540-70583-3_35}, volume = {5126}, year = {2008}, } @article{3227, abstract = {Large amount of data management can cause a lot of troubles which can be solved by dedicated computer system. To facilitate management of measurement data which are gathered in Institute of Power Engineering - Insulation Department a special system called Elektrowiz® was developed. It allows storing measurement results which concern partial discharges in insulation of turbo- and hydrogenerators in power stations. Multilayer architecture of the system allows reaching gathered data independently on user localization. There are possible different access methods to the system and dependency on current requirements data exploration can be realized with read-only or edit rights.}, author = {Zubielik, Piotr and Nadaczny, Jerzy and Krzysztof Pietrzak and Lawenda, Marcin}, journal = {Przeglad Elektrotechniczny}, number = {10}, pages = {239 -- 242}, publisher = {SIGMA-NOT}, title = {{Elektrowiz – system of measurement data management}}, volume = {84}, year = {2008}, } @inproceedings{3228, abstract = { A black-box combiner for collision resistant hash functions (CRHF) is a construction which given black-box access to two hash functions is collision resistant if at least one of the components is collision resistant. In this paper we prove a lower bound on the output length of black-box combiners for CRHFs. The bound we prove is basically tight as it is achieved by a recent construction of Canetti et al [Crypto'07]. The best previously known lower bounds only ruled out a very restricted class of combiners having a very strong security reduction: the reduction was required to output collisions for both underlying candidate hash-functions given a single collision for the combiner (Canetti et al [Crypto'07] building on Boneh and Boyen [Crypto'06] and Pietrzak [Eurocrypt'07]). Our proof uses a lemma similar to the elegant "reconstruction lemma" of Gennaro and Trevisan [FOCS'00], which states that any function which is not one-way is compressible (and thus uniformly random function must be one-way). In a similar vein we show that a function which is not collision resistant is compressible. We also borrow ideas from recent work by Haitner et al. [FOCS'07], who show that one can prove the reconstruction lemma even relative to some very powerful oracles (in our case this will be an exponential time collision-finding oracle). © 2008 Springer-Verlag Berlin Heidelberg.}, author = {Krzysztof Pietrzak}, pages = {413 -- 432}, publisher = {Springer}, title = {{Compression from collisions or why CRHF combiners have a long output}}, doi = {10.1007/978-3-540-85174-5_23}, volume = {5157}, year = {2008}, } @inproceedings{3229, abstract = {We construct a stream-cipher S whose implementation is secure even if a bounded amount of arbitrary (adversarially chosen) information on the internal state ofS is leaked during computation. This captures all possible side-channel attacks on S where the amount of information leaked in a given period is bounded, but overall can be arbitrary large. The only other assumption we make on the implementation of S is that only data that is accessed during computation leaks information. The stream-cipher S generates its output in chunks K1, K2, . . . and arbitrary but bounded information leakage is modeled by allowing the adversary to adaptively chose a function fl : {0,1}* rarr {0, 1}lambda before Kl is computed, she then gets fl(taul) where taul is the internal state ofS that is accessed during the computation of Kg. One notion of security we prove for S is that Kg is indistinguishable from random when given K1,..., K1-1,f1(tau1 ),..., fl-1(taul-1) and also the complete internal state of S after Kg has been computed (i.e. S is forward-secure). The construction is based on alternating extraction (used in the intrusion-resilient secret-sharing scheme from FOCS'07). We move this concept to the computational setting by proving a lemma that states that the output of any PRG has high HILLpseudoentropy (i.e. is indistinguishable from some distribution with high min-entropy) even if arbitrary information about the seed is leaked. The amount of leakage lambda that we can tolerate in each step depends on the strength of the underlying PRG, it is at least logarithmic, but can be as large as a constant fraction of the internal state of S if the PRG is exponentially hard.}, author = {Dziembowski, Stefan and Krzysztof Pietrzak}, pages = {293 -- 302}, publisher = {IEEE}, title = {{Leakage resilient cryptography}}, doi = {10.1109/FOCS.2008.56}, year = {2008}, } @article{3291, abstract = {The filamentous fungus Aspergillus fumigatus is responsible for a lethal disease called Invasive Aspergillosis that affects immunocompromised patients. This disease, like other human fungal diseases, is generally treated by compounds targeting the primary fungal cell membrane sterol. Recently, glucan synthesis inhibitors were added to the limited antifungal arsenal and encouraged the search for novel targets in cell wall biosynthesis. Although galactomannan is a major component of the A. fumigatus cell wall and extracellular matrix, the biosynthesis and role of galactomannan are currently unknown. By a targeted gene deletion approach, we demonstrate that UDP-galactopyranose mutase, a key enzyme of galactofuranose metabolism, controls the biosynthesis of galactomannan and galactofuranose containing glycoconjugates. The glfA deletion mutant generated in this study is devoid of galactofuranose and displays attenuated virulence in a low-dose mouse model of invasive aspergillosis that likely reflects the impaired growth of the mutant at mammalian body temperature. Furthermore, the absence of galactofuranose results in a thinner cell wall that correlates with an increased susceptibility to several antifungal agents. The UDP-galactopyranose mutase thus appears to be an appealing adjunct therapeutic target in combination with other drugs against A. fumigatus. Its absence from mammalian cells indeed offers a considerable advantage to achieve therapeutic selectivity. }, author = {Philipp Schmalhorst and Krappmann, Sven and Vervecken, Wouter and Rohde, Manfred and Müller, Meike and Braus, Gerhard H. and Contreras, Roland and Braun, Armin and Bakker, Hans and Routier, Françoise H}, journal = {Eukaryotic Cell}, number = {8}, pages = {1268 -- 1277}, publisher = {American Society for Microbiology}, title = {{Contribution of galactofuranose to the virulence of the opportunistic pathogen Aspergillus fumigatus}}, doi = {10.1128/EC.00065-08}, volume = {7}, year = {2008}, } @article{3307, abstract = {A complete mitochondrial (mt) genome sequence was reconstructed from a 38,000 year-old Neandertal individual with 8341 mtDNA sequences identified among 4.8 Gb of DNA generated from ∼0.3 g of bone. Analysis of the assembled sequence unequivocally establishes that the Neandertal mtDNA falls outside the variation of extant human mtDNAs, and allows an estimate of the divergence date between the two mtDNA lineages of 660,000 ± 140,000 years. Of the 13 proteins encoded in the mtDNA, subunit 2 of cytochrome c oxidase of the mitochondrial electron transport chain has experienced the largest number of amino acid substitutions in human ancestors since the separation from Neandertals. There is evidence that purifying selection in the Neandertal mtDNA was reduced compared with other primate lineages, suggesting that the effective population size of Neandertals was small.}, author = {Green, Richard E and Malaspinas, Anna-Sapfo and Krause, Johannes and Briggs, Adrian W and Johnson, Philip L and Caroline Uhler and Meyer, Matthias and Good, Jeffrey M and Maricic, Tomislav and Stenzel, Udo and Prüfer, Kay and Siebauer, Michael F and Burbano, Hernän A and Ronan, Michael T and Rothberg, Jonathan M and Egholm, Michael and Rudan, Pavao and Brajković, Dejana and Kućan, Željko and Gušić, Ivan and Wikström, Mårten K and Laakkonen, Liisa J and Kelso, Janet F and Slatkin, Montgomery and Pääbo, Svante H}, journal = {Cell}, pages = {416 -- 426}, publisher = {Cell Press}, title = {{A complete neandertal mitochondrial genome sequence determined by highhhroughput sequencing}}, doi = {10.1016/j.cell.2008.06.021}, volume = {134}, year = {2008}, } @article{3409, abstract = {With the introduction of single-molecule force spectroscopy (SMFS) it has become possible to directly access the interactions of various molecular systems. A bottleneck in conventional SMFS is collecting the large amount of data required for statistically meaningful analysis. Currently, atomic force microscopy (AFM)-based SMFS requires the user to tediously 'fish' for single molecules. In addition, most experimental and environmental conditions must be manually adjusted. Here, we developed a fully automated single-molecule force spectroscope. The instrument is able to perform SMFS while monitoring and regulating experimental conditions such as buffer composition and temperature. Cantilever alignment and calibration can also be automatically performed during experiments. This, combined with in-line data analysis, enables the instrument, once set up, to perform complete SMFS experiments autonomously.}, author = {Struckmeier, Jens and Wahl, Reiner and Leuschner, Mirko and Nunes, Joao and Harald Janovjak and Geisler, Ulrich and Hofmann, Gerd and Jähnke, Torsten and Mueller, Daniel J}, journal = {Nanotechnology}, number = {38}, publisher = {IOP Publishing Ltd.}, title = {{Fully automated single-molecule force spectroscopy for screening applications}}, doi = {10.1088/0957-4484/19/38/384020}, volume = {19}, year = {2008}, } @misc{3410, abstract = {Membrane proteins are involved in essential biological processes such as energy conversion, signal transduction, solute transport and secretion. All biological processes, also those involving membrane proteins, are steered by molecular interactions. Molecular interactions guide the folding and stability of membrane proteins, determine their assembly, switch their functional states or mediate signal transduction. The sequential steps of molecular interactions driving these processes can be described by dynamic energy landscapes. The conceptual energy landscape allows to follow the complex reaction pathways of membrane proteins while its modifications describe why and how pathways are changed. Single-molecule force spectroscopy (SMFS) detects, quantifies and locates interactions within and between membrane proteins. SMFS helps to determine how these interactions change with temperature, point mutations, oligomerization and the functional states of membrane proteins. Applied in different modes, SMFS explores the co-existence and population of reaction pathways in the energy landscape of the protein and thus reveals detailed insights into local mechanisms, determining its structural and functional relationships. Here we review how SMFS extracts the defining parameters of an energy landscape such as the barrier position, reaction kinetics and roughness with high precision.}, author = {Harald Janovjak and Sapra, Tanuj K and Kedrov, Alexej and Mueller, Daniel J}, booktitle = {ChemPhysChem}, number = {7}, pages = {954 -- 966}, publisher = {Wiley-Blackwell}, title = {{From valleys to ridges: Exploring the energy landscape of single membrane proteins}}, doi = {10.1002/cphc.200700662}, volume = {9}, year = {2008}, } @article{844, abstract = {Mutation rate varies greatly between nucleotide sites of the human genome and depends both on the global genomic location and the local sequence context of a site. In particular, CpG context elevates the mutation rate by an order of magnitude. Mutations also vary widely in their effect on the molecular function, phenotype, and fitness. Independence of the probability of occurrence of a new mutation's effect has been a fundamental premise in genetics. However, highly mutable contexts may be preserved by negative selection at important sites but destroyed by mutation at sites under no selection. Thus, there may be a positive correlation between the rate of mutations at a nucleotide site and the magnitude of their effect on fitness. We studied the impact of CpG context on the rate of human-chimpanzee divergence and on intrahuman nucleotide diversity at non-synonymous coding sites. We compared nucleotides that occupy identical positions within codons of identical amino acids and only differ by being within versus outside CpG context. Nucleotides within CpG context are under a stronger negative selection, as revealed by their lower, proportionally to the mutation rate, rate of evolution and nucleotide diversity. In particular, the probability of fixation of a non-synonymous transition at a CpG site is two times lower than at a CpG site. Thus, sites with different mutation rates are not necessarily selectively equivalent. This suggests that the mutation rate may complement sequence conservation as a characteristic predictive of functional importance of nucleotide sites.}, author = {Schmidt, Steffen and Gerasimova, Anna and Fyodor Kondrashov and Adzuhbei, Ivan A and Kondrashov, Alexey S and Sunyaev, Shamil R}, journal = {PLoS Genetics}, number = {11}, publisher = {Public Library of Science}, title = {{Hypermutable non-synonymous sites are under stronger negative selection}}, doi = {10.1371/journal.pgen.1000281}, volume = {4}, year = {2008}, } @article{8480, abstract = {The KIX domain of the transcription co-activator CBP is a three-helix bundle protein that folds via rapid accumulation of an intermediate state, followed by a slower folding phase. Recent NMR relaxation dispersion studies revealed the presence of a low-populated (excited) state of KIX that exists in equilibrium with the natively folded form under non-denaturing conditions, and likely represents the equilibrium analog of the folding intermediate. Here, we combine amide hydrogen/deuterium exchange measurements using rapid NMR data acquisition techniques with backbone 15N and 13C relaxation dispersion experiments to further investigate the equilibrium folding of the KIX domain. Residual structure within the folding intermediate is detected by both methods, and their combination enables reliable quantification of the amount of persistent residual structure. Three well-defined folding subunits are found, which display variable stability and correspond closely to the individual helices in the native state. While two of the three helices (α2 and α3) are partially formed in the folding intermediate (to ∼ 50% and ∼ 80%, respectively, at 20 °C), the third helix is disordered. The observed helical content within the excited state exceeds the helical propensities predicted for the corresponding peptide regions, suggesting that the two helices are weakly mutually stabilized, while methyl 13C relaxation dispersion data indicate that a defined packing arrangement is unlikely. Temperature-dependent experiments reveal that the largest enthalpy and entropy changes along the folding reaction occur during the final transition from the intermediate to the native state. Our experimental data are consistent with a folding mechanism where helices α2 and α3 form rapidly, although to different extents, while helix α1 consolidates only as folding proceeds to complete the native state-structure.}, author = {Schanda, Paul and Brutscher, Bernhard and Konrat, Robert and Tollinger, Martin}, issn = {0022-2836}, journal = {Journal of Molecular Biology}, keywords = {Molecular Biology}, number = {4}, pages = {726--741}, publisher = {Elsevier}, title = {{Folding of the KIX domain: Characterization of the equilibrium analog of a folding intermediate using 15N/13C relaxation dispersion and fast 1H/2H amide exchange NMR spectroscopy}}, doi = {10.1016/j.jmb.2008.05.040}, volume = {380}, year = {2008}, } @article{8481, abstract = {The copK gene is localized on the pMOL30 plasmid of Cupriavidus metallidurans CH34 within the complex cop cluster of genes, for which 21 genes have been identified. The expression of the corresponding periplasmic CopK protein is strongly upregulated in the presence of copper, leading to a high periplasmic accumulation. The structure and metal-binding properties of CopK were investigated by NMR and mass spectrometry. The protein is dimeric in the apo state with a dissociation constant in the range of 10- 5 M estimated from analytical ultracentrifugation. Mass spectrometry revealed that CopK has two high-affinity Cu(I)-binding sites per monomer with different Cu(I) affinities. Binding of Cu(II) was observed but appeared to be non-specific. The solution structure of apo-CopK revealed an all-β fold formed of two β-sheets in perpendicular orientation with an unstructured C-terminal tail. The dimer interface is formed by the surface of the C-terminal β-sheet. Binding of the first Cu(I)-ion induces a major structural modification involving dissociation of the dimeric apo-protein. Backbone chemical shifts determined for the 1Cu(I)-bound form confirm the conservation of the N-terminal β-sheet, while the last strand of the C-terminal sheet appears in slow conformational exchange. We hypothesize that the partial disruption of the C-terminal β-sheet is related to dimer dissociation. NH-exchange data acquired on the apo-protein are consistent with a lower thermodynamic stability of the C-terminal sheet. CopK contains seven methionine residues, five of which appear highly conserved. Chemical shift data suggest implication of two or three methionines (Met54, Met38, Met28) in the first Cu(I) site. Addition of a second Cu(I) ion further increases protein plasticity. Comparison of the structural and metal-binding properties of CopK with other periplasmic copper-binding proteins reveals two conserved features within these functionally related proteins: the all-β fold and the methionine-rich Cu(I)-binding site.}, author = {Bersch, Beate and Favier, Adrien and Schanda, Paul and van Aelst, Sébastien and Vallaeys, Tatiana and Covès, Jacques and Mergeay, Max and Wattiez, Ruddy}, issn = {0022-2836}, journal = {Journal of Molecular Biology}, keywords = {Molecular Biology}, number = {2}, pages = {386--403}, publisher = {Elsevier}, title = {{Molecular structure and metal-binding properties of the periplasmic CopK protein expressed in Cupriavidus metallidurans CH34 during copper challenge}}, doi = {10.1016/j.jmb.2008.05.017}, volume = {380}, year = {2008}, } @article{8482, abstract = {The SOFAST-HMQC experiment [P. Schanda, B. Brutscher, Very fast two-dimensional NMR spectroscopy for real-time investigation of dynamic events in proteins on the time scale of seconds, J. Am. Chem. Soc. 127 (2005) 8014–8015] allows recording two-dimensional correlation spectra of macromolecules such as proteins in only a few seconds acquisition time. To achieve the highest possible sensitivity, SOFAST-HMQC experiments are preferably performed on high-field NMR spectrometers equipped with cryogenically cooled probes. The duty cycle of over 80% in fast-pulsing SOFAST-HMQC experiments, however, may cause problems when using a cryogenic probe. Here we introduce SE-IPAP-SOFAST-HMQC, a new pulse sequence that provides comparable sensitivity to standard SOFAST-HMQC, while avoiding heteronuclear decoupling during 1H detection, and thus significantly reducing the radiofrequency load of the probe during the experiment. The experiment is also attractive for fast and sensitive measurement of heteronuclear one-bond spin coupling constants.}, author = {Kern, Thomas and Schanda, Paul and Brutscher, Bernhard}, issn = {1090-7807}, journal = {Journal of Magnetic Resonance}, keywords = {Nuclear and High Energy Physics, Biophysics, Biochemistry, Condensed Matter Physics}, number = {2}, pages = {333--338}, publisher = {Elsevier}, title = {{Sensitivity-enhanced IPAP-SOFAST-HMQC for fast-pulsing 2D NMR with reduced radiofrequency load}}, doi = {10.1016/j.jmr.2007.11.015}, volume = {190}, year = {2008}, } @article{8509, abstract = {The goal of this paper is to present to nonspecialists what is perhaps the simplest possible geometrical picture explaining the mechanism of Arnold diffusion. We choose to speak of a specific model—that of geometric rays in a periodic optical medium. This model is equivalent to that of a particle in a periodic potential in ${\mathbb R}^{n}$ with energy prescribed and to the geodesic flow in a Riemannian metric on ${\mathbb R}^{n} $.}, author = {Kaloshin, Vadim and Levi, Mark}, issn = {0036-1445}, journal = {SIAM Review}, keywords = {Theoretical Computer Science, Applied Mathematics, Computational Mathematics}, number = {4}, pages = {702--720}, publisher = {Society for Industrial & Applied Mathematics}, title = {{Geometry of Arnold diffusion}}, doi = {10.1137/070703235}, volume = {50}, year = {2008}, } @article{8510, abstract = {In this paper, using the ideas of Bessi and Mather, we present a simple mechanical system exhibiting Arnold diffusion. This system of a particle in a small periodic potential can be also interpreted as ray propagation in a periodic optical medium with a near-constant index of refraction. Arnold diffusion in this context manifests itself as an arbitrary finite change of direction for nearly constant index of refraction.}, author = {Kaloshin, Vadim and Levi, Mark}, issn = {0273-0979}, journal = {Bulletin of the American Mathematical Society}, keywords = {Applied Mathematics, General Mathematics}, number = {3}, pages = {409--427}, publisher = {American Mathematical Society}, title = {{An example of Arnold diffusion for near-integrable Hamiltonians}}, doi = {10.1090/s0273-0979-08-01211-1}, volume = {45}, year = {2008}, } @phdthesis{4415, abstract = {Many computing applications, especially those in safety critical embedded systems, require highly predictable timing properties. However, time is often not present in the prevailing computing and networking abstractions. In fact, most advances in computer architecture, software, and networking favor average-case performance over timing predictability. This thesis studies several methods for the design of concurrent and/or distributed embedded systems with precise timing guarantees. The focus is on flexible and compositional methods for programming and verification of the timing properties. The presented methods together with related formalisms cover two levels of design: (1) Programming language/model level. We propose the distributed variant of Giotto, a coordination programming language with an explicit temporal semantics—the logical execution time (LET) semantics. The LET of a task is an interval of time that specifies the time instants at which task inputs and outputs become available (task release and termination instants). The LET of a task is always non-zero. This allows us to communicate values across the network without changing the timing information of the task, and without introducing nondeterminism. We show how this methodology supports distributed code generation for distributed real-time systems. The method gives up some performance in favor of composability and predictability. We characterize the tradeoff by comparing the LET semantics with the semantics used in Simulink. (2) Abstract task graph level. We study interface-based design and verification of applications represented with task graphs. We consider task sequence graphs with general event models, and cyclic graphs with periodic event models with jitter and phase. Here an interface of a component exposes time and resource constraints of the component. Together with interfaces we formally define interface composition operations and the refinement relation. For efficient and flexible composability checking two properties are important: incremental design and independent refinement. According to the incremental design property the composition of interfaces can be performed in any order, even if interfaces for some components are not known. The refinement relation is defined such that in a design we can always substitute a refined interface for an abstract one. We show that the framework supports independent refinement, i.e., the refinement relation is preserved under composition operations.}, author = {Matic, Slobodan}, pages = {1 -- 148}, publisher = {University of California, Berkeley}, title = {{Compositionality in deterministic real-time embedded systems}}, year = {2008}, } @inproceedings{4452, abstract = {We describe Valigator, a software tool for imperative program verification that efficiently combines symbolic computation and automated reasoning in a uniform framework. The system offers support for automatically generating and proving verification conditions and, most importantly, for automatically inferring loop invariants and bound assertions by means of symbolic summation, Gröbner basis computation, and quantifier elimination. We present general principles of the implementation and illustrate them on examples.}, author = {Thomas Henzinger and Hottelier, Thibaud and Kovács, Laura}, pages = {333 -- 342}, publisher = {Springer}, title = {{Valigator: A verification tool with bound and invariant generation}}, doi = {10.1007/978-3-540-89439-1_24}, volume = {5330}, year = {2008}, } @article{4509, abstract = {I discuss two main challenges in embedded systems design: the challenge to build predictable systems, and that to build robust systems. I suggest how predictability can be formalized as a form of determinism, and robustness as a form of continuity.}, author = {Thomas Henzinger}, journal = {Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences}, number = {1881}, pages = {3727 -- 3736}, publisher = {Royal Society of London}, title = {{Two challenges in embedded systems design: Predictability and robustness}}, doi = {10.1098/rsta.2008.0141}, volume = {366}, year = {2008}, } @inproceedings{4521, abstract = {The search for proof and the search for counterexamples (bugs) are complementary activities that need to be pursued concurrently in order to maximize the practical success rate of verification tools.While this is well-understood in safety verification, the current focus of liveness verification has been almost exclusively on the search for termination proofs. A counterexample to termination is an infinite programexecution. In this paper, we propose a method to search for such counterexamples. The search proceeds in two phases. We first dynamically enumerate lasso-shaped candidate paths for counterexamples, and then statically prove their feasibility. We illustrate the utility of our nontermination prover, called TNT, on several nontrivial examples, some of which require bit-level reasoning about integer representations.}, author = {Ashutosh Gupta and Thomas Henzinger and Majumdar, Ritankar S and Rybalchenko, Andrey and Xu, Ru-Gang}, pages = {147 -- 158}, publisher = {ACM}, title = {{Proving non-termination}}, doi = {10.1145/1328438.1328459}, year = {2008}, } @phdthesis{4524, abstract = {Complex requirements, time-to-market pressure and regulatory constraints have made the designing of embedded systems extremely challenging. This is evident by the increase in effort and expenditure for design of safety-driven real-time control-dominated applications like automotive and avionic controllers. Design processes are often challenged by lack of proper programming tools for specifying and verifying critical requirements (e.g. timing and reliability) of such applications. Platform based design, an approach for designing embedded systems, addresses the above concerns by separating requirement from architecture. The requirement specifies the intended behavior of an application while the architecture specifies the guarantees (e.g. execution speed, failure rate etc). An implementation, a mapping of the requirement on the architecture, is then analyzed for correctness. The orthogonalization of concerns makes the specification and analyses simpler. An effective use of such design methodology has been proposed in Logical Execution Time (LET) model of real-time tasks. The model separates the timing requirements (specified by release and termination instances of a task) from the architecture guarantees (specified by worst-case execution time of the task). This dissertation proposes a coordination language, Hierarchical Timing Language (HTL), that captures the timing and reliability requirements of real-time applications. An implementation of the program on an architecture is then analyzed to check whether desired timing and reliability requirements are met or not. The core framework extends the LET model by accounting for reliability and refinement. The reliability model separates the reliability requirements of tasks from the reliability guarantees of the architecture. The requirement expresses the desired long-term reliability while the architecture provides a short-term reliability guarantee (e.g. failure rate for each iteration). The analysis checks if the short-term guarantee ensures the desired long-term reliability. The refinement model allows replacing a task by another task during program execution. Refinement preserves schedulability and reliability, i.e., if a refined task is schedulable and reliable for an implementation, then the refining task is also schedulable and reliable for the implementation. Refinement helps in concise specification without overloading analysis. The work presents the formal model, the analyses (both with and without refinement), and a compiler for HTL programs. The compiler checks composition and refinement constraints, performs schedulability and reliability analyses, and generates code for implementation of an HTL program on a virtual machine. Three real-time controllers, one each from automatic control, automotive control and avionic control, are used to illustrate the steps in modeling and analyzing HTL programs.}, author = {Ghosal, Arkadeb}, pages = {1 -- 210}, publisher = {University of California, Berkeley}, title = {{A hierarchical coordination language for reliable real-time tasks}}, year = {2008}, } @inproceedings{4527, abstract = {We introduce bounded asynchrony, a notion of concurrency tailored to the modeling of biological cell-cell interactions. Bounded asynchrony is the result of a scheduler that bounds the number of steps that one process gets ahead of other processes; this allows the components of a system to move independently while keeping them coupled. Bounded asynchrony accurately reproduces the experimental observations made about certain cell-cell interactions: its constrained nondeterminism captures the variability observed in cells that, although equally potent, assume distinct fates. Real-life cells are not “scheduled”, but we show that distributed real-time behavior can lead to component interactions that are observationally equivalent to bounded asynchrony; this provides a possible mechanistic explanation for the phenomena observed during cell fate specification. We use model checking to determine cell fates. The nondeterminism of bounded asynchrony causes state explosion during model checking, but partial-order methods are not directly applicable. We present a new algorithm that reduces the number of states that need to be explored: our optimization takes advantage of the bounded-asynchronous progress and the spatially local interactions of components that model cells. We compare our own communication-based reduction with partial-order reduction (on a restricted form of bounded asynchrony) and experiments illustrate that our algorithm leads to significant savings.}, author = {Fisher, Jasmin and Thomas Henzinger and Maria Mateescu and Piterman, Nir}, pages = {17 -- 32}, publisher = {Springer}, title = {{Bounded asynchrony: Concurrency for modeling cell-cell interactions}}, doi = {10.1007/978-3-540-68413-8_2}, volume = {5054}, year = {2008}, } @article{4532, abstract = {We consider the equivalence problem for labeled Markov chains (LMCs), where each state is labeled with an observation. Two LMCs are equivalent if every finite sequence of observations has the same probability of occurrence in the two LMCs. We show that equivalence can be decided in polynomial time, using a reduction to the equivalence problem for probabilistic automata, which is known to be solvable in polynomial time. We provide an alternative algorithm to solve the equivalence problem, which is based on a new definition of bisimulation for probabilistic automata. We also extend the technique to decide the equivalence of weighted probabilistic automata.}, author = {Doyen, Laurent and Thomas Henzinger and Raskin, Jean-François}, journal = {International Journal of Foundations of Computer Science}, number = {3}, pages = {549 -- 563}, publisher = {World Scientific Publishing}, title = {{Equivalence of labeled Markov chains}}, doi = {10.1142/S0129054108005814 }, volume = {19}, year = {2008}, } @inproceedings{4533, abstract = {Interface theories have been proposed to support incremental design and independent implementability. Incremental design means that the compatibility checking of interfaces can proceed for partial system descriptions, without knowing the interfaces of all components. Independent implementability means that compatible interfaces can be refined separately, maintaining compatibility. We show that these interface theories provide no formal support for component reuse, meaning that the same component cannot be used to implement several different interfaces in a design. We add a new operation to interface theories in order to support such reuse. For example, different interfaces for the same component may refer to different aspects such as functionality, timing, and power consumption. We give both stateless and stateful examples for interface theories with component reuse. To illustrate component reuse in interface-based design, we show how the stateful theory provides a natural framework for specifying and refining PCI bus clients.}, author = {Doyen, Laurent and Thomas Henzinger and Jobstmann, Barbara and Tatjana Petrov}, pages = {79 -- 88}, publisher = {ACM}, title = {{Interface theories with component reuse}}, doi = {10.1145/1450058.1450070}, year = {2008}, } @article{4534, abstract = {A stochastic graph game is played by two players on a game graph with probabilistic transitions. We consider stochastic graph games with ω-regular winning conditions specified as parity objectives, and mean-payoff (or limit-average) objectives. These games lie in NP ∩ coNP. We present a polynomial-time Turing reduction of stochastic parity games to stochastic mean-payoff games.}, author = {Krishnendu Chatterjee and Thomas Henzinger}, journal = {Information Processing Letters}, number = {1}, pages = {1 -- 7}, publisher = {Elsevier}, title = {{Reduction of stochastic parity to stochastic mean-payoff games}}, doi = {10.1016/j.ipl.2007.08.035}, volume = {106}, year = {2008}, } @inproceedings{4546, abstract = {We propose the notion of logical reliability for real-time program tasks that interact through periodically updated program variables. We describe a reliability analysis that checks if the given short-term (e.g., single-period) reliability of a program variable update in an implementation is sufficient to meet the logical reliability requirement (of the program variable) in the long run. We then present a notion of design by refinement where a task can be refined by another task that writes to program variables with less logical reliability. The resulting analysis can be combined with an incremental schedulability analysis for interacting real-time tasks proposed earlier for the Hierarchical Timing Language (HTL), a coordination language for distributed real-time systems. We implemented a logical-reliability-enhanced prototype of the compiler and runtime infrastructure for HTL.}, author = {Krishnendu Chatterjee and Ghosal, Arkadeb and Thomas Henzinger and Iercan, Daniel and Kirsch, Christoph M and Pinello, Claudio and Sangiovanni-Vincentelli, Alberto}, pages = {909 -- 914}, publisher = {IEEE}, title = {{Logical reliability of interacting real-time tasks}}, doi = {10.1145/1403375.1403595}, year = {2008}, } @article{4548, abstract = {The value of a finite-state two-player zero-sum stochastic game with limit-average payoff can be approximated to within ε in time exponential in a polynomial in the size of the game times polynomial in logarithmic in 1/ε, for all ε > 0.}, author = {Krishnendu Chatterjee and Majumdar, Ritankar S and Thomas Henzinger}, journal = {International Journal of Game Theory}, number = {2}, pages = {219 -- 234}, publisher = {Springer}, title = {{Stochastic limit-average games are in EXPTIME}}, doi = {10.1007/s00182-007-0110-5}, volume = {37}, year = {2008}, } @inproceedings{4568, abstract = {We present and evaluate a framework and tool for combining multiple program analyses which allows the dynamic (on-line) adjustment of the precision of each analysis depending on the accumulated results. For example, the explicit tracking of the values of a variable may be switched off in favor of a predicate abstraction when and where the number of different variable values that have been encountered has exceeded a specified threshold. The method is evaluated on verifying the SSH client/server software and shows significant gains compared with predicate abstraction-based model checking.}, author = {Beyer, Dirk and Thomas Henzinger and Théoduloz, Grégory}, pages = {29 -- 38}, publisher = {ACM}, title = {{Program analysis with dynamic change of precision}}, doi = {10.1109/ASE.2008.13}, year = {2008}, } @article{517, author = {Barton, Nicholas H}, journal = {Genetical Research}, number = {5-6}, pages = {475 -- 477}, publisher = {Cambridge University Press}, title = {{Identity and coalescence in structured populations: A commentary on 'Inbreeding coefficients and coalescence times' by Montgomery Slatkin}}, doi = {10.1017/S0016672308009683}, volume = {89}, year = {2008}, } @article{581, abstract = {We have detected a spin-dependent displacement perpendicular to the refractive index gradient for photons passing through an air-glass interface. The effect is the photonic version of the spin Hall effect in electronic systems, indicating the universality of the effect for particles of different nature. Treating the effect as a weak measurement of the spin projection of the photons, we used a preselection and postselection technique on the spin state to enhance the original displacement by nearly four orders of magnitude, attaining sensitivity to displacements of ∼1 angstrom. The spin Hall effect can be used for manipulating photonic angular momentum states, and the measurement technique holds promise for precision metrology.}, author = {Onur Hosten and Kwiat, Paul}, journal = {Science}, number = {5864}, pages = {787 -- 790}, publisher = {American Association for the Advancement of Science}, title = {{Observation of the spin hall effect of light via weak measurements}}, doi = {10.1126/science.1152697}, volume = {319}, year = {2008}, } @article{7320, abstract = {A comparative, experimental diffusivity study of gas diffusion layer (GDL) materials for polymer electrolyte fuel cells (PEFC) is presented for the first time. The GDL plays an important role for electrochemical losses due to gas transport limitations at high current densities. Characterization and optimization of these layers is therefore essential to improve power density. A recently developed method which allows for fast diffusimetry is applied and data compared to the literature values. Measurements are made as a function of direction and compression and the effect of different binder structures and hydrophobic treatments on effective diffusivities are discussed. A better understanding of the results is gained by including novel GDL cross-section images and a meaningful unit cell model for the interpretation of the data. The diffusivity data is valuable for GDL manufacturers and future PEFC models. The study reveals that a binder–fiber ratio larger than 50% has a negative impact on the effective diffusion properties. The hydrophobic treatment which is necessary to improve the water management can impede diffusion and thus reduce the power density. Furthermore binder has an isotropic effect while compression pronounces the in-plane orientation of the fibers.}, author = {Flückiger, Reto and Freunberger, Stefan Alexander and Kramer, Denis and Wokaun, Alexander and Scherer, Günther G. and Büchi, Felix N.}, issn = {0013-4686}, journal = {Electrochimica Acta}, number = {2}, pages = {551--559}, publisher = {Elsevier}, title = {{Anisotropic, effective diffusivity of porous gas diffusion layer materials for PEFC}}, doi = {10.1016/j.electacta.2008.07.034}, volume = {54}, year = {2008}, } @article{7321, abstract = {Cell interaction phenomena in polymer electrolyte fuel cell stacks that arise from imbalance between adjacent cells are investigated in detail experimentally and theoretically. A specialized two-cell stack with advanced localized diagnostics was developed and used to analyze the mechanism and effect of cell-to-cell coupling as a result of operationally relevant variations in reactant feed flow. Contributions to overall and local voltage changes with respect to uniformly operated cells are scrutinized. Unequal operation of the cells causes in-plane current in the bipolar plate to redistribute current and result in inhomogeneous polarization. Both increasing and decreasing polarization along the air-flow path reduces cell power as compared to isopotential operation. A two-dimensional model based on a commercial computational fluid dynamics code is used to back and extend the experimental results to more general cases. Furthermore, the experimental setup presented allowed for the first time to perform simultaneous localized electrochemical impedance spectroscopy beyond the single-cell level. The mechanism of mutual cell interaction on local and integral spectra is revealed. Results show that virtually identical operation of the cells is essential to obtain meaningful integral spectra.}, author = {Freunberger, Stefan Alexander and Schneider, Ingo A. and Sui, Pang-Chieh and Wokaun, Alexander and Djilali, Nedjib and Büchi, Felix N.}, issn = {0013-4651}, journal = {Journal of The Electrochemical Society}, number = {7}, publisher = {The Electrochemical Society}, title = {{Cell interaction phenomena in polymer electrolyte fuel cell stacks}}, doi = {10.1149/1.2913095}, volume = {155}, year = {2008}, } @article{7322, abstract = {The gas diffusion layers (GDLs) of a membrane electrode assembly (MEA) serve as link between flow field and porous electrode within a polymer electrolyte fuel cell. Beside ensuring sufficient electrical and thermal contact between the whole electrode area and the flow field, these typically 200–400 μm thick porous structures enable the access of educts to the electrode area which would be occluded by the flow field lands if the flow field is directly attached to the electrode. Hence, the characterisation of properties pertaining to mass transport of educts and products through these structures is indispensable whilst examining the contribution of the GDLs to the overall electrochemical characteristics of a MEA. A fast and cost effective method to measure the effective diffusivity of a GDL is presented. Electrochemical impedance spectroscopy is applied to measure the effective ionic conductivity of an electrolyte-soaked GDL. Taking advantage of the analogy between Ficks and Ohms law, this provides a measure for the effective diffusivity. The method is described in detail, including experimental as well as theoretical aspects, and selected results, highlighting the anisotropy and dependence on the degree of compression, are shown. Moreover, a two-dimensional model consisting of regularly spaced ellipses is developed to represent the porous structure of the GDL, and by using conformal maps, the agreement between this model and experiment with respect to the sensitivity of the effective diffusivity towards compression is shown.}, author = {Kramer, Denis and Freunberger, Stefan Alexander and Flückiger, Reto and Schneider, Ingo A. and Wokaun, Alexander and Büchi, Felix N. and Scherer, Günther G.}, issn = {1572-6657}, journal = {Journal of Electroanalytical Chemistry}, number = {1}, pages = {63--77}, publisher = {Elsevier}, title = {{Electrochemical diffusimetry of fuel cell gas diffusion layers}}, doi = {10.1016/j.jelechem.2007.09.014}, volume = {612}, year = {2008}, } @inproceedings{7425, abstract = {The propagation of single cell performance losses to adjacent cells in a polymer electrolyte fuel cell stack is studied by means of local current density measurements in a two cell stack. In this stack, the working conditions of adjacent cells can be controlled independently in order to deliberately change the performance of one cell (inducing cell) and study the coupling effects to the adjacent cell (response cell), while keeping the working conditions of the later one unchanged. The experiments have shown that changes in the current density distribution caused by lowering of the air stoichiometry in the inducing cell cause changes in the current density distribution of the response cell in the order of 60% of the change of the inducing cell, even when the air stoichiometry of the response cell is kept constant. The losses in cell voltage of the inducing cell cause losses in cell voltage of the response cell in a magnitude between 30 and 50%.}, author = {Santis, Marco and Freunberger, Stefan Alexander and Papra, Matthias and Büchi, Felix N.}, booktitle = {3rd International Conference on Fuel Cell Science, Engineering and Technology}, isbn = {0791837645}, location = {Ypsilanti, MI, United States}, pages = {763--765}, publisher = {ASMEDC}, title = {{Experimental investigation of the propagation of local current density variations to adjacent cells in PEFC stacks}}, doi = {10.1115/fuelcell2005-74116}, year = {2008}, } @inproceedings{753, abstract = {This paper addresses the following question: what is the minimum-sized synchronous window needed to solve consensus in an otherwise asynchronous system? In answer to this question, we present the first optimally-resilient algorithm ASAP that solves consensus as soon as possible in an eventually synchronous system, i.e., a system that from some time GST onwards, delivers messages in a timely fashion. ASAP guarantees that, in an execution with at most f failures, every process decides no later than round GST + f + 2, which is optimal.}, author = {Alistarh, Dan-Adrian and Gilbert, Seth and Guerraoui, Rachid and Travers, Corentin}, pages = {32 -- 46}, publisher = {Springer}, title = {{How to solve consensus in the smallest window of synchrony}}, doi = {10.1007/978-3-540-87779-0_3}, volume = {5218 LNCS}, year = {2008}, } @article{7752, author = {Robinson, Matthew Richard and Pilkington, Jill G. and Clutton-Brock, Tim H. and Pemberton, Josephine M. and Kruuk, Loeske. E.B.}, issn = {0960-9822}, journal = {Current Biology}, number = {10}, pages = {751--757}, publisher = {Elsevier}, title = {{Environmental heterogeneity generates fluctuating selection on a secondary sexual trait}}, doi = {10.1016/j.cub.2008.04.059}, volume = {18}, year = {2008}, } @article{1717, abstract = {Two key processes are in the basis of morphogenesis: the spatial allocation of cell types in fields of naïve cells and the regulation of growth. Both are controlled by morphogens, which activate target genes in the growing tissue in a concentration-dependent manner. Thus the morphogen model is an intrinsically quantitative concept. However, quantitative studies were performed only in recent years on two morphogens: Bicoid and Decapentaplegic. This review covers quantitative aspects of the formation and precision of the Decapentaplegic morphogen gradient. The morphogen gradient concept is transitioning from a soft definition to a precise idea of what the gradient could really do.}, author = {Anna Kicheva and González-Gaitán, Marcos A}, journal = {Current Opinion in Cell Biology}, number = {2}, pages = {137 -- 143}, publisher = {Elsevier}, title = {{The Decapentaplegic morphogen gradient a precise definition}}, doi = {10.1016/j.ceb.2008.01.008}, volume = {20}, year = {2008}, } @article{1719, abstract = {We study the mechanics of tissue growth via cell division and cell death (apoptosis). The rearrangements of cells can on large scales and times be captured by a continuum theory which describes the tissue as an effective viscous material with active stresses generated by cell division. We study the effects of anisotropies of cell division on cell rearrangements and show that average cellular trajectories exhibit anisotropic scaling behaviors. If cell division and apoptosis balance, there is no net growth, but for anisotropic cell division the tissue undergoes spontaneous shear deformations. Our description is relevant for the study of developing tissues such as the imaginal disks of the fruit fly Drosophila melanogaster, which grow anisotropically.}, author = {Bittig, Thomas and Wartlick, Ortrud and Anna Kicheva and González-Gaitárr, Marcos and Julicher, Frank}, journal = {New Journal of Physics}, publisher = {IOP Publishing Ltd.}, title = {{Dynamics of anisotropic tissue growth}}, doi = {10.1088/1367-2630/10/6/063001}, volume = {10}, year = {2008}, } @article{1749, abstract = {Scanning probe microscopy; Semiconductor quantum dots; Composition gradients; Composition profiles; Nanotomography; Single quantum dots; Strained sige/si; Three-dimensional (3D); Wet-chemical etchings; X-ray scattering measurements; quantum dot; methodology; nanotechnology; optical tomography; scanning probe microscopy; three dimensional imaging; Imaging, Three-Dimensional; Materials Testing; Microscopy, Scanning Probe; Nanotechnology; Quantum Dots; Tomography,}, author = {Rastelli, Armando and Stoffel, Mathieu and Malachias, Ângelo S and Merdzhanova, Tsvetelina and Georgios Katsaros and Kern, Klaus and Metzger, Till H and Schmidt, Oliver G}, journal = {Nano Letters}, number = {5}, pages = {1404 -- 1409}, publisher = {American Chemical Society}, title = {{Three-dimensional composition profiles of single quantum dots determined by scanning-probe-microscopy-based nanotomography}}, doi = {10.1021/nl080290y}, volume = {8}, year = {2008}, } @article{1751, abstract = {When strained Stranski-Krastanow islands are used as "self-assembled quantum dots," a key goal is to control the island position. Here we show that nanoscale grooves can control the nucleation of epitaxial Ge islands on Si(001), and can drive lateral motion of existing islands onto the grooves, even when the grooves are very narrow and shallow compared to the islands. A position centered on the groove minimizes energy. We use as prototype grooves the trenches which form naturally around islands. During coarsening, the shrinking islands move laterally to sit directly astride that trench. In subsequent growth, we demonstrate that islands nucleate on the "empty trenches" which remain on the surface after complete dissolution of the original islands.}, author = {Georgios Katsaros and Tersoff, Jerry and Stoffel, Mathieu and Rastelli, Armando and Acosta-Diaz, P and Kar, Gouranga S and Costantini, Giovanni and Schmidt, Oliver G and Kern, Klaus}, journal = {Physical Review Letters}, number = {9}, publisher = {American Physical Society}, title = {{Positioning of strained islands by interaction with surface nanogrooves}}, doi = {10.1103/PhysRevLett.101.096103}, volume = {101}, year = {2008}, } @article{1763, abstract = {The field of cavity quantum electrodynamics (QED), traditionally studied in atomic systems, has gained new momentum by recent reports of quantum optical experiments with solid-state semiconducting and superconducting systems. In cavity QED, the observation of the vacuum Rabi mode splitting is used to investigate the nature of matter-light interaction at a quantum-mechanical level. However, this effect can, at least in principle, be explained classically as the normal mode splitting of two coupled linear oscillators. It has been suggested that an observation of the scaling of the resonant atom-photon coupling strength in the Jaynes-Cummings energy ladder with the square root of photon number n is sufficient to prove that the system is quantum mechanical in nature. Here we report a direct spectroscopic observation of this characteristic quantum nonlinearity. Measuring the photonic degree of freedom of the coupled system, our measurements provide unambiguous spectroscopic evidence for the quantum nature of the resonant atom-field interaction in cavity QED. We explore atom-photon superposition states involving up to two photons, using a spectroscopic pump and probe technique. The experiments have been performed in a circuit QED set-up, in which very strong coupling is realized by the large dipole coupling strength and the long coherence time of a superconducting qubit embedded in a high-quality on-chip microwave cavity. Circuit QED systems also provide a natural quantum interface between flying qubits (photons) and stationary qubits for applications in quantum information processing and communication.}, author = {Johannes Fink and Göppl, M and Baur, Matthias P and Bianchetti, R and Leek, Peter J and Blais, Alexandre and Wallraff, Andreas}, journal = {Nature}, number = {7202}, pages = {315 -- 318}, publisher = {Nature Publishing Group}, title = {{Climbing the Jaynes-Cummings ladder and observing its √n nonlinearity in a cavity QED system}}, doi = {10.1038/nature07112}, volume = {454}, year = {2008}, } @article{1764, abstract = {Quantum theory predicts that empty space is not truly empty. Even in the absence of any particles or radiation, in pure vacuum, virtual particles are constantly created and annihilated. In an electromagnetic field, the presence of virtual photons manifests itself as a small renormalization of the energy of a quantum system, known as the Lamb shift. We present an experimental observation of the Lamb shift in a solid-state system. The strong dispersive coupling of a superconducting electronic circuit acting as a quantum bit (qubit) to the vacuum field in a transmission-line resonator leads to measurable Lamb shifts of up to 1.4% of the qubit transition frequency. The qubit is also observed to couple more strongly to the vacuum field than to a single photon inside the cavity, an effect that is explained by taking into account the limited anharmonicity of the higher excited qubit states.}, author = {Fragner, A and Göppl, M and Johannes Fink and Baur, Matthias P and Bianchetti, R and Leek, Peter J and Blais, Alexandre and Wallraff, Andreas}, journal = {Science}, number = {5906}, pages = {1357 -- 1360}, publisher = {American Association for the Advancement of Science}, title = {{Resolving vacuum fluctuations in an electrical circuit by measuring the lamb shift}}, doi = {10.1126/science.1164482}, volume = {322}, year = {2008}, } @article{1765, abstract = {High quality on-chip microwave resonators have recently found prominent new applications in quantum optics and quantum information processing experiments with superconducting electronic circuits, a field now known as circuit quantum electrodynamics (QED). They are also used as single photon detectors and parametric amplifiers. Here we analyze the physical properties of coplanar waveguide resonators and their relation to the materials properties for use in circuit QED. We have designed and fabricated resonators with fundamental frequencies from 2 to 9 GHz and quality factors ranging from a few hundreds to a several hundred thousands controlled by appropriately designed input and output coupling capacitors. The microwave transmission spectra measured at temperatures of 20 mK are shown to be in good agreement with theoretical lumped element and distributed element transmission matrix models. In particular, the experimentally determined resonance frequencies, quality factors, and insertion losses are fully and consistently explained by the two models for all measured devices. The high level of control and flexibility in design renders these resonators ideal for storing and manipulating quantum electromagnetic fields in integrated superconducting electronic circuits.}, author = {Göppl, M and Fragner, A and Baur, Matthias P and Bianchetti, R and Filipp, Stefan and Johannes Fink and Leek, Peter J and Puebla, G and Steffen, L. Kraig and Wallraff, Andreas}, journal = {Journal of Applied Physics}, number = {11}, publisher = {American Institute of Physics}, title = {{Coplanar waveguide resonators for circuit quantum electrodynamics}}, doi = {10.1063/1.3010859}, volume = {104}, year = {2008}, } @article{1826, abstract = {Proliferating cell populations at steady-state growth often exhibit broad protein distributions with exponential tails. The sources of this variation and its universality are of much theoretical interest. Here we address the problem by asymptotic analysis of the population balance equation. We show that the steady-state distribution tail is determined by a combination of protein production and cell division and is insensitive to other model details. Under general conditions this tail is exponential with a dependence on parameters consistent with experiment. We discuss the conditions for this effect to be dominant over other sources of variation and the relation to experiments.}, author = {Tamar Friedlander and Brenner, Naama}, journal = {Physical Review Letters}, number = {1}, publisher = {American Physical Society}, title = {{Cellular properties and population asymptotics in the population balance equation}}, doi = {10.1103/PhysRevLett.101.018104}, volume = {101}, year = {2008}, } @article{1967, abstract = {Complex I of respiratory chains transfers electrons from NADH to ubiquinone, coupled to the translocation of protons across the membrane. Two alternative coupling mechanisms are being discussed, redox-driven or conformation-driven. Using "zero-length" cross-linking reagent and isolated hydrophilic domains of complex I from Escherichia coli and Thermus thermophilus, we show that the pattern of cross-links between subunits changes significantly in the presence of NADH. Similar observations were made previously with intact purified E. coli and bovine complex I. This indicates that, upon reduction with NADH, similar conformational changes are likely to occur in the intact enzyme and in the isolated hydrophilic domain (which can be used for crystallographic studies). Within intact E. coli complex I, the cross-link between the hydrophobic subunits NuoA and NuoJ was abolished in the presence of NADH, indicating that conformational changes extend into the membrane domain, possibly as part of a coupling mechanism. Unexpectedly, in the absence of any chemical cross-linker, incubation of complex I with NADH resulted in covalent cross-links between subunits Nqo4 (NuoCD) and Nqo6 (NuoB), as well as between Nqo6 and Nqo9. Their formation depends on the presence of oxygen and so is likely a result of oxidative damage via reactive oxygen species (ROS) induced cross-linking. In addition, ROS- and metal ion-dependent proteolysis of these subunits (as well as Nqo3) is observed. Fe-S cluster N2 is coordinated between subunits Nqo4 and Nqo6 and could be involved in these processes. Our observations suggest that oxidative damage to complex I in vivo may include not only side-chain modifications but also protein cross-linking and degradation.}, author = {Berrisford, John M and Thompson, Christopher J and Leonid Sazanov}, journal = {Biochemistry}, number = {39}, pages = {10262 -- 10270}, publisher = {ACS}, title = {{Chemical and NADH-induced, ROS-dependent, cross-linking between sublimits of complex I from Escherichia coli and Thermus thermophilus}}, doi = {10.1021/bi801160u}, volume = {47}, year = {2008}, } @article{1968, abstract = { Complex I (NADH:ubiquinone oxidoreductase) is the largest protein complex of bacterial and mitochondrial respiratory chains. The first three-dimensional structure of bacterial complex I in vitrified ice was determined by electron cryo-microscopy and single particle analysis. The structure of the Escherichia coli enzyme incubated with either NAD+ (as a reference) or NADH was calculated to 35 and 39 Å resolution, respectively. The X-ray structure of the peripheral arm of Thermus thermophilus complex I was docked into the reference EM structure. The model obtained indicates that Fe-S cluster N2 is close to the membrane domain interface, allowing for effective electron transfer to membrane-embedded quinone. At the current resolution, the structures in the presence of NAD+ or NADH are similar. Additionally, side-view class averages were calculated for the negatively stained bovine enzyme. The structures of bovine complex I in the presence of either NAD+ or NADH also appeared to be similar. These observations indicate that conformational changes upon reduction with NADH, suggested to occur by a range of studies, are smaller than had been thought previously. The model of the entire bacterial complex I could be built from the crystal structures of subcomplexes using the EM envelope described here.}, author = {Morgan, David J and Leonid Sazanov}, journal = {Biochimica et Biophysica Acta - Bioenergetics}, number = {7-8}, pages = {711 -- 718}, publisher = {Elsevier}, title = {{Three-dimensional structure of respiratory complex I from Escherichia coli in ice in the presence of nucleotides}}, doi = {10.1016/j.bbabio.2008.03.023}, volume = {1777}, year = {2008}, } @article{1982, abstract = {In the bacterium Escherichia coli, the Min proteins oscillate between the cell poles to select the cell center as division site. This dynamic pattern has been proposed to arise by self-organization of these proteins, and several models have suggested a reaction-diffusion type mechanism. Here, we found that the Min proteins spontaneously formed planar surface waves on a flat membrane in vitro. The formation and maintenance of these patterns, which extended for hundreds of micrometers, required adenosine 5′-triphosphate (ATP), and they persisted for hours. We present a reaction-diffusion model of the MinD and MinE dynamics that accounts for our experimental observations and also captures the in vivo oscillations.}, author = {Martin Loose and Fischer-Friedrich, Elisabeth and Ries, Jonas and Kruse, Karsten and Schwille, Petra }, journal = {Science}, number = {5877}, pages = {789 -- 792}, publisher = {American Association for the Advancement of Science}, title = {{Spatial regulators for bacterial cell division self-organize into surface waves in vitro}}, doi = {10.1126/science.1154413}, volume = {320}, year = {2008}, } @article{6146, abstract = {Homeostasis of internal carbon dioxide (CO2) and oxygen (O2) levels is fundamental to all animals. Here we examine the CO2 response of the nematode Caenorhabditis elegans. This species inhabits rotting material, which typically has a broad CO2 concentration range. We show that well fed C. elegans avoid CO2 levels above 0.5%. Animals can respond to both absolute CO2 concentrations and changes in CO2 levels within seconds. Responses to CO2 do not reflect avoidance of acid pH but appear to define a new sensory response. Sensation of CO2 is promoted by the cGMP-gated ion channel subunits TAX-2 and TAX-4, but other pathways are also important. Robust CO2 avoidance in well fed animals requires inhibition of the DAF-16 forkhead transcription factor by the insulin-like receptor DAF-2. Starvation, which activates DAF-16, strongly suppresses CO2 avoidance. Exposure to hypoxia (<1% O2) also suppresses CO2 avoidance via activation of the hypoxia-inducible transcription factor HIF-1. The npr-1 215V allele of the naturally polymorphic neuropeptide receptor npr-1, besides inhibiting avoidance of high ambient O2 in feeding C. elegans, also promotes avoidance of high CO2. C. elegans integrates competing O2 and CO2 sensory inputs so that one response dominates. Food and allelic variation at NPR-1 regulate which response prevails. Our results suggest that multiple sensory inputs are coordinated by C. elegans to generate different coherent foraging strategies.}, author = {Bretscher, A. J. and Busch, K. E. and de Bono, Mario}, issn = {0027-8424}, journal = {Proceedings of the National Academy of Sciences}, number = {23}, pages = {8044--8049}, publisher = {Proceedings of the National Academy of Sciences}, title = {{A carbon dioxide avoidance behavior is integrated with responses to ambient oxygen and food in Caenorhabditis elegans}}, doi = {10.1073/pnas.0707607105}, volume = {105}, year = {2008}, } @article{6148, author = {Kammenga, Jan E. and Phillips, Patrick C. and de Bono, Mario and Doroszuk, Agnieszka}, issn = {0168-9525}, journal = {Trends in Genetics}, number = {4}, pages = {178--185}, publisher = {Elsevier}, title = {{Beyond induced mutants: using worms to study natural variation in genetic pathways}}, doi = {10.1016/j.tig.2008.01.001}, volume = {24}, year = {2008}, } @article{895, abstract = {Background. The arginine vasopressin V1a receptor (V1aR) modulates social cognition and behavior in a wide variety of species. Variation in a repetitive microsatellite element in the 5′ flanking region of the V1aR gene (AVPR1A) in rodents has been associated with variation in brain V1aR expression and in social behavior. In humans, the 5′ flanking region of AVPR1A contains a tandem duplication of two ∼350 bp, microsatellite-containing elements located approximately 3.5 kb upstream of the transcription start site. The first block, referred to as DupA, contains a polymorphic (GT) 25microsatellite; the second block, DupB, has a complex (CT) 4-(TT)-(CT)8-(GT)24polymorphic motif, known as RS3. Polymorphisms in RS3 have been associated with variation in sociobehavioral traits in humans, including autism spectrum disorders. Thus, evolution of these regions may have contributed to variation in social behavior in primates. We examined the structure of these regions in six ape, six monkey, and one prosimian species. Results. Both tandem repeat blocks are present upstream of the AVPR1A coding region in five of the ape species we investigated, while monkeys have only one copy of this region. As in humans, the microsatellites within DupA and DupB are polymorphic in many primate species. Furthermore, both single (lacking DupB) and duplicated alleles (containing both DupA and DupB) are present in chimpanzee (Pan troglodytes) populations with allele frequencies of 0.795 and 0.205 for the single and duplicated alleles, respectively, based on the analysis of 47 wild-caught individuals. Finally, a phylogenetic reconstruction suggests two alternate evolutionary histories for this locus. Conclusion. There is no obvious relationship between the presence of the RS3 duplication and social organization in primates. However, polymorphisms identified in some species may be useful in future genetic association studies. In particular, the presence of both single and duplicated alleles in chimpanzees provides a unique opportunity to assess the functional role of this duplication in contributing to variation in social behavior in primates. While our initial studies show no signs of directional selection on this locus in chimps, pharmacological and genetic association studies support a potential role for this region in influencing V1aR expression and social behavior.}, author = {Donaldson, Zoe R and Fyodor Kondrashov and Putnam, Andrea S and Bai, Yaohui and Stoinski, Tara S and Hammock, Elizabeth A and Young, Larry}, journal = {BMC Evolutionary Biology}, number = {1}, publisher = {BioMed Central}, title = {{Evolution of a behavior-linked microsatellite-containing element in the 5′ flanking region of the primate AVPR1A gene}}, doi = {10.1186/1471-2148-8-180}, volume = {8}, year = {2008}, } @article{907, abstract = {The most common form of protein-coding gene overlap in eukaryotes is a simple nested structure, whereby one gene is embedded in an intron of another. Analysis of nested protein-coding genes in vertebrates, fruit flies and nematodes revealed substantially higher rates of evolutionary gains than losses. The accumulation of nested gene structures could not be attributed to any obvious functional relationships between the genes involved and represents an increase of the organizational complexity of animal genomes via a neutral process.}, author = {Assis, Raquel and Kondrashov, Alexey S and Koonin, Eugene V and Fyodor Kondrashov}, journal = {Trends in Genetics}, number = {10}, pages = {475 -- 478}, publisher = {Elsevier}, title = {{Nested genes and increasing organizational complexity of metazoan genomes}}, doi = {10.1016/j.tig.2008.08.003}, volume = {24}, year = {2008}, } @article{1296, abstract = {The crystalline-like structure of the optic lobes of the fruit fly Drosophila melanogaster has made them a model system for the study of neuronal cell-fate determination, axonal path finding, and target selection. For functional studies, however, the small size of the constituting visual interneurons has so far presented a formidable barrier. We have overcome this problem by establishing in vivo whole-cell recordings [1] from genetically targeted visual interneurons of Drosophila. Here, we describe the response properties of six motion-sensitive large-field neurons in the lobula plate that form a network consisting of individually identifiable, directionally selective cells most sensitive to vertical image motion (VS cells [2, 3]). Individual VS cell responses to visual motion stimuli exhibit all the characteristics that are indicative of presynaptic input from elementary motion detectors of the correlation type [4, 5]. Different VS cells possess distinct receptive fields that are arranged sequentially along the eye's azimuth, corresponding to their characteristic cellular morphology and position within the retinotopically organized lobula plate. In addition, lateral connections between individual VS cells cause strongly overlapping receptive fields that are wider than expected from their dendritic input. Our results suggest that motion vision in different dipteran fly species is accomplished in similar circuitries and according to common algorithmic rules. The underlying neural mechanisms of population coding within the VS cell network and of elementary motion detection, respectively, can now be analyzed by the combination of electrophysiology and genetic intervention in Drosophila.}, author = {Maximilian Jösch and Plett, Johannes and Borst, Alexander and Reiff, Dierk F}, journal = {Current Biology}, number = {5}, pages = {368 -- 374}, publisher = {Cell Press}, title = {{Response properties of motion sensitive visual interneurons in the Lobula plate of Drosophila melanogaster}}, doi = {10.1016/j.cub.2008.02.022}, volume = {18}, year = {2008}, } @article{1460, abstract = {We calculate the E-polynomials of certain twisted GL(n,ℂ)-character varieties Mn of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type GL(n, q) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,ℂ)-character variety. The calculation also leads to several conjectures about the cohomology of Mn: an explicit conjecture for its mixed Hodge polynomial; a conjectured curious hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n=2.}, author = {Tamas Hausel and Rodríguez Villegas, Fernando}, journal = {Inventiones Mathematicae}, number = {3}, pages = {555 -- 624}, publisher = {Springer}, title = {{Mixed Hodge polynomials of character varieties: With an appendix by Nicholas M. Katz}}, doi = {10.1007/s00222-008-0142-x}, volume = {174}, year = {2008}, } @article{1036, abstract = {We report on the control of interaction-induced dephasing of Bloch oscillations for an atomic Bose-Einstein condensate in an optical lattice. We quantify the dephasing in terms of the width of the quasimomentum distribution and measure its dependence on time for different interaction strengths which we control by means of a Feshbach resonance. For minimal interaction, the dephasing time is increased from a few to more than 20 thousand Bloch oscillation periods, allowing us to realize a BEC-based atom interferometer in the noninteracting limit.}, author = {Gustavsson, Mattias and Haller, Elmar and Mark, Manfred and Danzl, Johann G and Rojas Kopeinig, Gabriel and Nägerl, Hanns}, journal = {Physical Review Letters}, number = {8}, publisher = {American Physical Society}, title = {{Control of interaction-induced dephasing of bloch oscillations}}, doi = {10.1103/PhysRevLett.100.080404}, volume = {100}, year = {2008}, } @article{1037, abstract = {We experimentally demonstrate Cs2 Feshbach molecules well above the dissociation threshold, which are stable against spontaneous decay on the time scale of 1s. An optically trapped sample of ultracold dimers is prepared in a high rotational state and magnetically tuned into a region with a negative binding energy. The metastable character of these molecules arises from the large centrifugal barrier in combination with negligible coupling to states with low rotational angular momentum. A sharp onset of dissociation with increasing magnetic field is mediated by a crossing with a lower rotational dimer state and facilitates dissociation on demand with a well-defined energy.}, author = {Knoop, Steven and Mark, Michael and Ferlaino, Francesca and Danzl, Johann G and Kraemer, Tobias and Nägerl, Hanns and Grimm, Rudolf}, journal = {Physical Review Letters}, number = {8}, publisher = {American Physical Society}, title = {{Metastable feshbach molecules in high rotational states}}, doi = {10.1103/PhysRevLett.100.083002}, volume = {100}, year = {2008}, } @article{1039, abstract = {Molecular cooling techniques face the hurdle of dissipating translational as well as internal energy in the presence of a rich electronic, vibrational, and rotational energy spectrum. In our experiment, we create a translationally ultracold, dense quantum gas of molecules bound by more than 1000 wave numbers in the electronic ground state. Specifically, we stimulate with 80% efficiency, a two-photon transfer of molecules associated on a Feshbach resonance from a Bose-Einstein condensate of cesium atoms. In the process, the initial loose, long-range electrostatic bond of the Feshbach molecule is coherently transformed into a tight chemical bond. We demonstrate coherence of the transfer in a Ramsey-type experiment and show that the molecular sample is not heated during the transfer. Our results show that the preparation of a quantum gas of molecules in specific rovibrational states is possible and that the creation of a Bose-Einstein condensate of molecules in their rovibronic ground state is within reach.}, author = {Danzl, Johann G and Haller, Elmar and Gustavsson, Mattias and Mark, Manfred and Hart, Russell and Bouloufa, Nadia and Dulieu, Olivier and Ritsch, Helmut and Nägerl, Hanns}, journal = {Science}, number = {5892}, pages = {1062 -- 1066}, publisher = {American Association for the Advancement of Science}, title = {{Quantum gas of deeply bound ground state molecules}}, doi = {10.1126/science.1159909}, volume = {321}, year = {2008}, } @article{10392, abstract = {Protonated formylmetallocenes [M(C5H5)(C5H4-CHOH)]+ (M = Fe, Ru) and their isomers have been studied at the BP86 and B3LYP levels of density functional theory. Oxygen-protonated isomers are the most stable forms in each case, with a plethora of ring- or metal-protonated species at least ca. 14 and 10 kcal/mol higher in energy for M = Fe and Ru, respectively. The computed rotational barriers around the C−C bond connecting the cyclopentadienyl and protonated formyl moieties, ca. 18 kcal/mol, are indicative of substantial conjugation between these moieties. Some of the ring- and iron-protonated species are models for possible intermediates in Friedel–Crafts acylation of ferrocene, and the computations provide further evidence that exo attack is clearly favored over endo attack of the electrophile in this reaction. The structures of the most stable mono- and diprotonated formylferrocenes are corroborated by the good agreement between GIAO-B3LYP-computed and experimental NMR chemical shifts.}, author = {Šarić, Anđela and Vrček, Valerije and Bühl, Michael}, issn = {1520-6041}, journal = {Organometallics}, keywords = {Inorganic Chemistry, Organic Chemistry, Physical and Theoretical Chemistry}, number = {3}, pages = {394--401}, publisher = {American Chemical Society}, title = {{Density functional study of protonated formylmetallocenes}}, doi = {10.1021/om700916f}, volume = {27}, year = {2008}, } @article{2065, abstract = {Population genetics models show that, under certain conditions, the X chromosome is expected to be under more efficient selection than the autosomes. This could lead to 'faster-X evolution', if a large proportion of mutations are fixed by positive selection, as suggested by recent studies in Drosophila. We used a multispecies approach to test this: Muller's element D, an autosomal arm, is fused to the ancestral X chromosome in Drosophila pseudoobscura and its sister species, Drosophila affinis. We tested whether the same set of genes had higher rates of non-synonymous evolution when they were X-linked (in the D. pseudoobscura/D. affinis comparison) than when they were autosomal (in Drosophila melanogaster/Drosophila yakuba). Although not significant, our results suggest this may be the case, but only for genes under particularly strong positive selection/weak purifying selection. They also suggest that genes that have become X-linked have higher levels of codon bias and slower synonymous site evolution, consistent with more effective selection on codon usage at X-linked sites.}, author = {Beatriz Vicoso and Haddrill, Penelope R and Charlesworth, Brian}, journal = {Genetical Research}, number = {5}, pages = {421 -- 431}, publisher = {Cambridge University Press}, title = {{A multispecies approach for comparing sequence evolution of X-linked and autosomal sites in Drosophila}}, doi = {10.1017/S0016672308009804}, volume = {90}, year = {2008}, } @inproceedings{2078, abstract = {This paper presents a novel method for real-time animation of highly-detailed facial expressions based on a multi-scale decomposition of facial geometry into large-scale motion and fine-scale details, such as expression wrinkles. Our hybrid animation is tailored to the specific characteristics of large- and fine-scale facial deformations: Large-scale deformations are computed with a fast linear shell model, which is intuitively and accurately controlled through a sparse set of motion-capture markers or user-defined handle points. Fine-scale facial details are incorporated using a novel pose-space deformation technique, which learns the correspondence of sparse measurements of skin strain to wrinkle formation from a small set of example poses. Our hybrid method features real-time animation of highly-detailed faces with realistic wrinkle formation, and allows both large-scale deformations and fine-scale wrinkles to be edited intuitively. Furthermore, our pose-space representation enables the transfer of facial details to novel expressions or other facial models.}, author = {Bickel, Bernd and Lang, Manuel and Botsch, Mario and Otaduy, Miguel and Gross, Markus}, pages = {57 -- 66}, publisher = {ACM}, title = {{Pose-space animation and transfer of facial details}}, doi = {10.2312/SCA/SCA08/057-066}, year = {2008}, } @article{2120, abstract = {We consider the linear stochastic Cauchy problem dX (t) =AX (t) dt +B dWH (t), t≥ 0, where A generates a C0-semigroup on a Banach space E, WH is a cylindrical Brownian motion over a Hilbert space H, and B: H → E is a bounded operator. Assuming the existence of a unique minimal invariant measure μ∞, let Lp denote the realization of the Ornstein-Uhlenbeck operator associated with this problem in Lp (E, μ∞). Under suitable assumptions concerning the invariance of the range of B under the semigroup generated by A, we prove the following domain inclusions, valid for 1 < p ≤ 2: Image omitted. Here WHk, p (E, μinfin; denotes the kth order Sobolev space of functions with Fréchet derivatives up to order k in the direction of H. No symmetry assumptions are made on L p.}, author = {Jan Maas and van Neerven, Jan M}, journal = {Infinite Dimensional Analysis, Quantum Probability and Related Topics}, number = {4}, pages = {603 -- 626}, publisher = {World Scientific Publishing}, title = {{On the domain of non-symmetric Ornstein-Uhlenbeck operators in banach spaces}}, doi = {10.1142/S0219025708003245}, volume = {11}, year = {2008}, } @article{2121, abstract = {Let H be a separable real Hubert space and let double struck F sign = (ℱt)t∈[0,T] be the augmented filtration generated by an H-cylindrical Brownian motion (WH(t))t∈[0,T] on a probability space (Ω, ℱ ℙ). We prove that if E is a UMD Banach space, 1 ≤ p < ∞, and F ∈ double struck D sign1,p(Ω E) is ℱT-measurable, then F = double struck E sign(F) + ∫0T Pdouble struck F sign(DF) dW H, where D is the Malliavin derivative of F and P double struck F sign is the projection onto the F-adapted elements in a suitable Banach space of Lp-stochastically integrable ℒ(H, E)-valued processes.}, author = {van Neerven, Jan M and Jan Maas}, journal = {Electronic Communications in Probability}, pages = {151 -- 164}, publisher = {Institute of Mathematical Statistics}, title = {{A Clark-Ocone formula in UMD Banach spaces}}, volume = {13}, year = {2008}, } @article{2146, abstract = {We present an analytic model of thermal state-to-state rotationally inelastic collisions of polar molecules in electric fields. The model is based on the Fraunhofer scattering of matter waves and requires Legendre moments characterizing the “shape” of the target in the body-fixed frame as its input. The electric field orients the target in the space-fixed frame and thereby effects a striking alteration of the dynamical observables: both the phase and amplitude of the oscillations in the partial differential cross sections undergo characteristic field-dependent changes that transgress into the partial integral cross sections. As the cross sections can be evaluated for a field applied parallel or perpendicular to the relative velocity, the model also offers predictions about steric asymmetry. We exemplify the field-dependent quantum collision dynamics with the behavior of the Ne–OCS(Σ1) and Ar–NO(Π2) systems. A comparison with the close-coupling calculations available for the latter system [Chem. Phys. Lett.313, 491 (1999)] demonstrates the model’s ability to qualitatively explain the field dependence of all the scattering features observed.}, author = {Mikhail Lemeshko and Friedrich, Břetislav}, journal = {Journal of Chemical Physics}, number = {2}, publisher = {American Institute of Physics}, title = {{An analytic model of rotationally inelastic collisions of polar molecules in electric fields}}, doi = {10.1063/1.2948392}, volume = {129}, year = {2008}, } @misc{2147, abstract = {We present the physics of the quantum Zeno effect, whose gist is often expressed by invoking the adage "a watched pot never boils". We review aspects of the theoretical and experimental work done on the effect since its inception in 1977, and mention some applications. We dedicate the article - with our very best wishes - to Rudolf Zahradnik at the occasion of his great jubilee. Perhaps Rudolf's lasting youthfulness and freshness are due to that he himself had been frequently observed throughout his life: until the political turn-around in 1989 by those who wished, by their surveillance, to prevent Rudolf from spoiling the youth by his personal culture and his passion for science and things beautiful and useful in general. This attempt had failed. Out of gratitude, the youth has infected Rudolf with its youthfulness. Chronically. Since 1989, Rudolf has been closely watched by the public at large. For the same traits of his as before, but with the opposite goal and for the benefit of all generations. We relish keeping him in sight...}, author = {Mikhail Lemeshko and Friedrich, Břetislav}, booktitle = {Chemicke Listy}, number = {10}, pages = {880 -- 883}, publisher = {Czech Society of Chemical Engineering}, title = {{Kvantový Zenonův jev aneb co nesejde z očí, nezestárne}}, volume = {102}, year = {2008}, } @article{2148, abstract = {Despite the growing geological evidence that fluid boiling and vapour-liquid separation affect the distribution of metals in magmatic-hydrothermal systems significantly, there are few experimental data on the chemical status and partitioning of metals in the vapour and liquid phases. Here we report on an in situ measurement, using X-ray absorption fine structure (XAFS) spectroscopy, of antimony speciation and partitioning in the system Sb2O3-H2O-NaCl-HCl at 400°C and pressures 270–300 bar corresponding to the vapour-liquid equilibrium. Experiments were performed using a spectroscopic cell which allows simultaneous determination of the total concentration and atomic environment of the absorbing element (Sb) in each phase. Results show that quantitative vapour-brine separation of a supercritical aqueous salt fluid can be achieved by a controlled decompression and monitoring the X-ray absorbance of the fluid phase. Antimony concentrations in equilibrium with Sb2O3 (cubic, senarmontite) in the coexisting vapour and liquid phases and corresponding SbIII vapour-liquid partitioning coefficients are in agreement with recent data obtained using batch-reactor solubility techniques. The XAFS spectra analysis shows that hydroxy-chloride complexes, probably Sb(OH)2Cl0, are dominant both in the vapour and liquid phase in a salt-water system at acidic conditions. This first in situ XAFS study of element fractionation between coexisting volatile and dense phases opens new possibilities for systematic investigations of vapour-brine and fluid-melt immiscibility phenomena, avoiding many experimental artifacts common in less direct techniques.}, author = {Pokrovski, Gleb S and Roux, Jacques L and Hazemann, Jean L and Borisova, Anastassia Y and Gonchar, Anastasia A and Mikhail Lemeshko}, journal = {Mineralogical Magazine}, number = {2}, pages = {667 -- 681}, publisher = {Mineralogical Society}, title = {{In situ X-ray absorption spectroscopy measurement of vapour-brine fractionation of antimony at hydrothermal conditions}}, doi = {10.1180/minmag.2008.072.2.667 }, volume = {72}, year = {2008}, } @article{224, abstract = {Let n ≥ 4 and let Q ∈ [X1, ..., Xn] be a non-singular quadratic form. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q = 0, and when Q is positive definite we provide improved upper bounds for the greatest positive integer k for which the equation Q = k is insoluble in integers, despite being soluble modulo every prime power.}, author = {Timothy Browning and Dietmann, Rainer}, journal = {Proceedings of the London Mathematical Society}, number = {2}, pages = {389 -- 416}, publisher = {John Wiley and Sons Ltd}, title = {{On the representation of integers by quadratic forms}}, doi = {10.1112/plms/pdm032}, volume = {96}, year = {2008}, } @article{225, abstract = {We revisit recent work of Heath-Brown on the average order of the quantity r(L1(x))⋯r(L4(x)), for suitable binary linear forms L1,...,L4, as x=(x1,x2) ranges over quite general regions in ℤ2. In addition to improving the error term in Heath-Browns estimate, we generalise his result to cover a wider class of linear forms.}, author = {de la Bretèche, Régis and Timothy Browning}, journal = {Compositio Mathematica}, number = {6}, pages = {1375 -- 1402}, publisher = {Cambridge University Press}, title = {{Binary linear forms as sums of two squares}}, doi = {10.1112/S0010437X08003692}, volume = {144}, year = {2008}, } @inproceedings{2331, abstract = {We present a review of recent work on the mathematical aspects of the BCS gap equation, covering our results of Ref. 9 as well our recent joint work with Hamza and Solovej and with Frank and Naboko, respectively. In addition, we mention some related new results.}, author = {Hainzl, Christian and Robert Seiringer}, pages = {117 -- 136}, publisher = {World Scientific Publishing}, title = {{ Spectral properties of the BCS gap equation of superfluidity}}, doi = {10.1142/9789812832382_0009}, year = {2008}, } @inproceedings{2332, abstract = {We present a rigorous proof of the appearance of quantized vortices in dilute trapped Bose gases with repulsive two-body interactions subject to rotation, which was obtained recently in joint work with Elliott Lieb.14 Starting from the many-body Schrödinger equation, we show that the ground state of such gases is, in a suitable limit, well described by the nonlinear Gross-Pitaevskii equation. In the case of axially symmetric traps, our results show that the appearance of quantized vortices causes spontaneous symmetry breaking in the ground state.}, author = {Robert Seiringer}, pages = {241 -- 254}, publisher = {World Scientific Publishing}, title = {{Vortices and Spontaneous Symmetry Breaking in Rotating Bose Gases}}, doi = {10.1142/9789812832382_0017}, year = {2008}, } @article{2374, abstract = {A lower bound is derived on the free energy (per unit volume) of a homogeneous Bose gas at density Q and temperature T. In the dilute regime, i.e., when a3 1, where a denotes the scattering length of the pair-interaction potential, our bound differs to leading order from the expression for non-interacting particles by the term 4πa(2 2}-[ - c]2+). Here, c(T) denotes the critical density for Bose-Einstein condensation (for the non-interacting gas), and [ · ]+ = max{ ·, 0} denotes the positive part. Our bound is uniform in the temperature up to temperatures of the order of the critical temperature, i.e., T ~ 2/3 or smaller. One of the key ingredients in the proof is the use of coherent states to extend the method introduced in [17] for estimating correlations to temperatures below the critical one.}, author = {Robert Seiringer}, journal = {Communications in Mathematical Physics}, number = {3}, pages = {595 -- 636}, publisher = {Springer}, title = {{Free energy of a dilute Bose gas: Lower bound}}, doi = {10.1007/s00220-008-0428-2}, volume = {279}, year = {2008}, } @article{2376, abstract = {We derive upper and lower bounds on the critical temperature Tc and the energy gap Ξ (at zero temperature) for the BCS gap equation, describing spin- 1 2 fermions interacting via a local two-body interaction potential λV(x). At weak coupling λ 1 and under appropriate assumptions on V(x), our bounds show that Tc ∼A exp(-B/λ) and Ξ∼C exp(-B/λ) for some explicit coefficients A, B, and C depending on the interaction V(x) and the chemical potential μ. The ratio A/C turns out to be a universal constant, independent of both V(x) and μ. Our analysis is valid for any μ; for small μ, or low density, our formulas reduce to well-known expressions involving the scattering length of V(x).}, author = {Hainzl, Christian and Robert Seiringer}, journal = {Physical Review B - Condensed Matter and Materials Physics}, number = {18}, publisher = {American Physical Society}, title = {{Critical temperature and energy gap for the BCS equation}}, doi = {10.1103/PhysRevB.77.184517}, volume = {77}, year = {2008}, } @article{2377, abstract = {We prove that the critical temperature for the BCS gap equation is given by T c = μ ( 8\π e γ-2+ o(1)) e π/(2μa) in the low density limit μ→ 0, with γ denoting Euler's constant. The formula holds for a suitable class of interaction potentials with negative scattering length a in the absence of bound states.}, author = {Hainzl, Christian and Robert Seiringer}, journal = {Letters in Mathematical Physics}, number = {2-3}, pages = {99 -- 107}, publisher = {Springer}, title = {{The BCS critical temperature for potentials with negative scattering length}}, doi = {10.1007/s11005-008-0242-y}, volume = {84}, year = {2008}, } @article{2378, abstract = {We derive a lower bound on the ground state energy of the Hubbard model for given value of the total spin. In combination with the upper bound derived previously by Giuliani (J. Math. Phys. 48:023302, [2007]), our result proves that in the low density limit the leading order correction compared to the ground state energy of a non-interacting lattice Fermi gas is given by 8πaσ uσ d , where σ u(d) denotes the density of the spin-up (down) particles, and a is the scattering length of the contact interaction potential. This result extends previous work on the corresponding continuum model to the lattice case.}, author = {Robert Seiringer and Yin, Jun}, journal = {Journal of Statistical Physics}, number = {6}, pages = {1139 -- 1154}, publisher = {Springer}, title = {{Ground state energy of the low density hubbard model}}, doi = {10.1007/s10955-008-9527-x}, volume = {131}, year = {2008}, } @article{2379, author = {Frank, Rupert L and Lieb, Élliott H and Robert Seiringer}, journal = {Journal of the American Mathematical Society}, number = {4}, pages = {925 -- 950}, publisher = {American Mathematical Society}, title = {{Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators}}, doi = {10.1090/S0894-0347-07-00582-6}, volume = {21}, year = {2008}, } @article{2380, abstract = {The Bardeen-Cooper-Schrieffer (BCS) functional has recently received renewed attention as a description of fermionic gases interacting with local pairwise interactions. We present here a rigorous analysis of the BCS functional for general pair interaction potentials. For both zero and positive temperature, we show that the existence of a non-trivial solution of the nonlinear BCS gap equation is equivalent to the existence of a negative eigenvalue of a certain linear operator. From this we conclude the existence of a critical temperature below which the BCS pairing wave function does not vanish identically. For attractive potentials, we prove that the critical temperature is non-zero and exponentially small in the strength of the potential.}, author = {Hainzl, Christian and Hamza, Eman and Robert Seiringer and Solovej, Jan P}, journal = {Communications in Mathematical Physics}, number = {2}, pages = {349 -- 367}, publisher = {Springer}, title = {{The BCS functional for general pair interactions}}, doi = {10.1007/s00220-008-0489-2}, volume = {281}, year = {2008}, } @article{2381, abstract = {We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in the rearrangement inequality in fractional Sobolev spaces.}, author = {Frank, Rupert L and Robert Seiringer}, journal = {Journal of Functional Analysis}, number = {12}, pages = {3407 -- 3430}, publisher = {Academic Press}, title = {{Non-linear ground state representations and sharp Hardy inequalities}}, doi = {10.1016/j.jfa.2008.05.015}, volume = {255}, year = {2008}, } |
35abe932b5c4f5b1 | Demystifying Quantum Physics: You Need it for Your Faith
· Religion & Science
He (Allah) is the First and the Last, and the Manifest and the Hidden, and He knows every little detail fully well. (Al Quran 57:4)
Quantum physics has come to symbolize complexity among other things and most of us try to shy away from it. But, the fundamental reality is that if we put the mathematics aside and find the right teachers, following arguments in Quantum physics is not any harder than any other scientific, religious, philosophical, logical and political argument. Often what it means is picking up, which expert is giving a fair and balanced understanding and which one is blinded by his or her ideological concerns.
Napoleon, in one of the most notable conversations in the history of science, asked the French scientist Pierre Simon Laplace about the role of God in his scientific world view. It is said that Laplace had presented Napoleon with a copy of his work, who had heard that the book contained no mention of God. Napoleon, who was fond of imposing embarrassment, received it with the remark, “Laplace, they tell me you have written this large book on the system of the universe, and have never even mentioned its Creator.” Laplace is said to have replied, “Sir, I have no need of that hypothesis.” And so it goes. The apparent so called self-sufficiency of our physical universe has caused many a scientist to move away from the idea of a Creator of the universe or the God Hypothesis. But is it really so?
Laplace is one of the seventy two people to have their names on the Eiffel Tower. So strong was his belief in determinism and the scientific process that he said that given the knowledge of every atomic motion, the entire future of the universe could be mapped out. This was precisely the reason why Einstein did not believe in free will or accountability except for the horrific crimes of the Nazis. Laplace wrote:
Atheist physicist and philosophers want to continue to read determinism in physics despite the discoveries of Quantum physics, in the twentieth century, to rule out human soul, human free will and Providence of God. Read Carl Sagan as he rightfully sings praises of science, but, implies to rule out prayer and Providence, by bracketing them with quackery and witchcraft:
You can go to the witch doctor to lift the spell that causes your pernicious anemia, or you can take vitamin B12. If you want to save your child from polio, you can pray or you can inoculate. If you’re interested in the sex of your unborn child, you can consult plumb-bob danglers all you want (leftright, a boy; forward-back, a girl-or maybe it’s the other way around), but they’ll be right, on average, only one time in two. If you want real accuracy (here, ninety-nine percent accuracy), try amniocentesis and sonograms. Try science.
If our world is deterministic then the claims of atheist scientists are true. There is no room for Islam or Christianity or any other religion. If hard determinism is true then God does not exist and our claims about human soul are no more than those in previous decades about Santa Clause and in previous centuries about witches. Encyclopedia Britannica tells us about determinism and its implications:
So, if determinism is true there is no need to invoke human soul, human free will and Providence of God. These three become agents that simply cannot influence our world. In Wikipedia we can read:
Determinism is a philosophy stating that for everything that happens there are conditions such that, given them, nothing else could happen. Different versions of this theory depend upon various alleged connections, and interdependencies of things and events, asserting that these hold without exception. Deterministic theories throughout the history of philosophy have sprung from diverse motives and considerations, some of which overlap. They can be understood in relation to their historical significance and alternative theories. Some forms of determinism can be tested empirically with ideas stemming from physics and the philosophy of physics. The opposite of determinism is some kind of indeterminism (otherwise called nondeterminism). Determinism is often contrasted with free will. Determinism is often taken to mean simply causal determinism: an idea known in physics as cause-and-effect. It is the concept that events within a given paradigm are bound by causality in such a way that any state (of an object or event) is completely determined by prior states. This can be distinguished from other varieties of determinism mentioned below. Other debates often concern the scope of determined systems, with some maintaining that the entire universe (or multiverse) is a single determinate system and others identifying other more limited determinate systems. Within numerous historical debates, many varieties and philosophical positions on the subject of determinism exist. This includes debates concerning human action and free will, where opinions might be sorted as compatibilistic and incompatibilistic.[2]
I as a Muslim believe in free will and deny Hard Determinism. The Philosophers have created four different combinations of belief or disbelief in free will and determinism. They have argued that either Determinism is true or Indeterminism is true, but also that Free Will either exists or it does not. This creates four possible positions. Compatibilism refers to the view that free will is, in some sense, compatible with Determinism. The three Incompatibilist positions, on the other hand, deny this possibility. They instead suggest there is a dichotomy between determinism and free will (only one can be true).
According to this classification, I am arguing for Metaphysical Libertarianism and I believe that for that we need proper understanding of Quantum physics, otherwise we cannot argue for it. The principle of free will has religious, ethical, and scientific implications. For example, in the religious realm, free will implies that individual will and choices can coexist with an omnipotent divinity. In ethics, it may hold implications for whether individuals can be held morally accountable for their actions.
Like dominoes fall in a deterministic fashion, if Laplace or causal determinism is true then our choices are predetermined and we are not free to make them and hence do not have free will. But, I believe that the twentieth century physics, as opposed to earlier physics has shown us that our world is indeterministic. Quantum physics developed in the first 3-4 decades of twentieth century provides explanation and avenue not only for free will but also for God’s Providence.
The Miracle of Light – An Every Day Metaphor to Appreciate Quantum Physics
God said let there be light and there was the Holy Quran! The Quran describes Allah as Manifest as well as Transcendent and Hidden at the same time, in the verse quoted in the beginning. It is in this duality that the relationship of religion and science is to be understood. If Laplace had been right in predicting the future accurately, not only there would have been no Personal God but also no ‘free will’ for mankind. But something beautiful yet common place, namely, each and every ray of light, defies the tall claims of Laplace.
The scientific conflict between particle and wave models of light has permeated the history of science for several centuries. The issue dates back to at least Newton. His careful investigations into the properties of light in the 1660s led to his discovery that white light consists of a mixture of colors. He struggled with a formulation of the nature of light, ultimately asserting in Opticks (1704) that light consists of a stream of ‘corpuscles,’ or particles. The wave model explains certain observed phenomena but the photoelectric phenomena are best explained by ‘corpuscle’ nature of light.
If you have ever held a metal wire over a gas flame, you have borne witness to one of the great secrets of the universe. As the wire gets hotter, it begins to glow, to give off light. And the color of that light changes with temperature. A cooler wire gives off a reddish glow, while the hottest wires shine with a blue-white brilliance. What you are watching, as any high school physics student can tell you, is the transformation of one kind of energy (heat) into another (light). As the wire gets hotter and hotter, it gets brighter. That’s because if there is more heat energy available, more light energy can be given off, which makes sense.
Why does the color of that light change with temperature? Throughout the nineteenth century, that deceptively simple question baffled the best minds of classical physics. As the wire gets hotter and hotter, the atoms within it move more rapidly. Maybe that causes the color (the wavelength) of the light to change? Well, that’s true, but there’s more to it. Every time classical physicists used their understanding of matter and energy to try to predict exactly which wavelengths of light should be given off by a hot wire, they got it wrong. At high temperatures, those classical predictions were dramatically wrong. Something didn’t make sense.
Max Planck, a German physicist, found a way to solve the problem. Physicists had always assumed that light, being a wave, could be emitted from an object at any wavelength and in any amount. Planck realized that for this phenomenon the particulate nature as suggested by Newton was the key. He proposed that light could only be released in little packets containing a precise amount of energy. He called these packets or ‘corpuscles’ of Newton as ‘quanta.’ All of a sudden, everything fell into place.
It was known that when some solids were struck by light, they emitted electrons. This phenomenon is called the photoelectric effect. Albert Einstein offered the best explanation of the photoelectric effect in a brilliant paper that eventually won him his Nobel Prize. He seized on the dual nature of light. Light was not only a waveform but is composed of individual quanta later called photons. This understanding of the dual nature of light was needed to explain some of the phenomena that had been observed in study of light. The wave theory of light did not explain the photoelectric effect but conceptualizing the light to be also particle, beautifully solved this riddle. Einstein proposed that the energy to eject a single electron from the plate came from a single quantum of light. That’s why a more intense light (more quanta) just ejects more electrons. But the energy in each of those packets, the quantum wallop if you will, is determined by the wavelength, the color, of the light. With one stroke, of genius, Einstein had shown that Planck’s quanta were not just theoretical constructs. Light really could behave as if it were made of a stream of particles, today known as photons. He was awarded the 1921 Nobel Prize for Physics for this work.
Prof. Kenneth R Miller wrote in his popular book, Finding Darwin’s God:
All of this might have been sensible and comforting were it not for the fact that light was already known to behave as if it were a wave! So many experiments already had shown that light could be diffracted, that light had a frequency and a wavelength, that light spread out like a wave on the surface of a pond. Could all those experiments be wrong? No, they were not. All of those experiments were right. Light was both a particle and a wave. It was both a continuous stream and a shower of discrete quantum packets. And that nonsensical result was just the beginning.
Classical physics had prepared everyone to think of physical events as governed by fixed laws, but the quantum revolution quickly destroyed this Newtonian certainty. An object as simple as a mirror can show us why. A household mirror reflects about ninety-five percent of light hitting it. The other five percent passes right through. As long as we think of light as a Wave, a continuous stream of energy, it’s easy to visualize ninety-five percent reflection. But photons are indivisible-each individual photon must either be reflected or pass through the surface of the mirror. That means that for one hundred photons fired at the surface, ninety-five will bounce off but five will pass right through.
If we fire a series of one hundred photons at the mirror, can we tell in advance which will be the five that are going to pass through? Absolutely not. All photons of a particular wavelength are identical; there is nothing to distinguish one from the other. If we rig up an experiment in which we fire a single photon at our mirror, we cannot predict in advance what will happen, no matter how precise our knowledge of the system might be. Most of the time, that photon is going to come bouncing off; but one time out of twenty, on average, it’s going to go right through the mirror. There is nothing we can do, not even in principle, to figure out when that one chance in twenty is going to come up. It means that the outcome of each individual experiment is unpredictable in principle.”[2]
Any hopes that the strange uncertainty of quantum behavior would be confined to light were quickly destroyed when it became clear that the quantum theory had to be applied to explain the behavior of electrons also. Their behavior in any individual encounter, just like the photon fired at the mirror, cannot be predicted, not even in principle. The photo electric effect was leading the physics community to quantum mechanics.
Just as the invention of the telescope dramatically broadened exploration of the Cosmos, so too the invention of the microscope opened the intricate world of the cell. The analysis of the frequencies of light emitted and absorbed by atoms was a principal impetus for the development of quantum mechanics. What had begun as a tiny loose end, a strange little problem in the relationship between heat and light, now is understood to mean that nothing is quite the way it had once seemed. The unfolding of quantum mechanics was and still is a drama of high suspense, as Heisenberg himself wrote:
I remember discussions with Bohr (in 1927) which went through many hours till very late at night and ended almost in despair, and when at the end of the discussion I went alone for a walk in the neighboring park, I repeated to myself again and again the question: ‘Can nature possibly be absurd as it seemed to us in these atomic experiments?’[3]
One hundred years after the discovery of the quantum, we can say that the answer is yes, that is exactly what nature is like. Just because science can explain so many unknowns doesn’t mean that it can explain everything, or that it can vanquish the unknowable. At its very core, in the midst of the ultimate constituents of matter and energy, the predictable causality that once formed the heart of classical physics breaks down. Deep down the nature is unknowable as the Transcendent God is Unknowable. It may be, this is where the finite meets the Infinite, and by the very nature of the meeting point, it is hidden in mystery and awe, an enigma or a riddle never to be solved!
Double slit experiment: An easy way to appreciate the mysteries of the Quantum world
The double-slit experiment, sometimes called Young’s experiment (after Young’s interference experiment), is a demonstration that matter and energy can display characteristics of both waves and particles, and demonstrates the fundamentally probabilistic nature of quantum mechanical phenomena.
In the basic version of the experiment, a coherent light source such as a laser beam illuminates a thin plate pierced by two parallel slits, and the light passing through the slits is observed on a screen behind the plate. The wave nature of light causes the light waves passing through the two slits to interfere, producing bright and dark bands on the screen — a result that would not be expected if light consisted strictly of particles. However, on the screen, the light is always found to be absorbed as though it were composed of discrete particles or photons.[1][2]
This result establishes the principle known as wave–particle duality. Additionally, the detection of individual photons is observed to be inherently probabilistic, which is inexplicable using classical mechanics.[3]
The following short video, in a very easy manner, not only explains the double split experiment, but, its implications on the indeterminacy of our quantum world. After all there are limits to what humans can know and those limits will not go away with technological advances:
If you do not believe our cartoon professor, then you can read the same details in the first chapter of a book, by Prof. James Al-Khalili, who is Professor of Theoretical Physics and Chair in the Public Engagement in Science at the University of Surrey, Quantum: A Guide for the Perplexed.
Quantum Physics and Uncertainty Principle
Lot of time the complexity that Quantum physicist have to deal with is calculations for each electron based on Schrödinger equation, which gives the first time derivative of the quantum state. That is, it explicitly and uniquely predicts the development of the wave function with time.
The complexity of the equation is obvious at the first glance, but if we can bypass this then life is not too tough to bear.
At one time, it was assumed in the physical sciences that if the behavior observed in a system cannot be predicted, the problem is due to lack of fine-grained information, so that a sufficiently detailed investigation would eventually result in a deterministic theory (“If you knew exactly all the forces acting on the dice, you would be able to predict which number comes up”).
In fact there are two sources of quantum indeterminism:
1. the Heisenberg uncertainty principle prevents the simultaneous accurate measurement of all a particle’s properties; and
2. the collapse of the wave function, in which the state of a system upon measurement cannot be predicted.
The latter kind of indeterminism is not only a feature of the Copenhagen interpretation, with its observer-dependence, but also of objective collapse theories.
Opponents of quantum indeterminism suggested that determinism could be restored by formulating a new theory in which additional information, so-called hidden variables ,[28] would allow definite outcomes to be determined. For instance, in 1935, Einstein, Podolsky and Rosen wrote a paper titled Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? arguing that such a theory was in fact necessary.
The double-slit experiment (and its variations), conducted with individual particles, has become a classic thought experiment for its clarity in expressing the central puzzles of quantum mechanics. Because it demonstrates the fundamental limitation of the observer to predict experimental results, Richard Feynman called it “a phenomenon which is impossible … to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery [of quantum mechanics].”[3], and was fond of saying that all of quantum mechanics can be gleaned from carefully thinking through the implications of this single experiment[4]. Časlav Brukner and Anton Zeilinger have succinctly expressed this limitation as follows:
[T]he observer can decide whether or not to put detectors into the interfering path. That way, by deciding whether or not to determine the path through the two-slit experiment, he/she can decide which property can become reality. If he/she chooses not to put the detectors there, then the interference pattern will become reality; if he/she does put the detectors there, then the beam path will become reality. Yet, most importantly, the observer has no influence on the specific element of the world that becomes reality. Specifically, if he/she chooses to determine the path, then he/she has no influence whatsoever over which of the two paths, the left one or the right one, nature will tell him/her is the one in which the particle is found. Likewise, if he/she chooses to observe the interference pattern, then he/she has no influence whatsoever over where in the observation plane he/she will observe a specific particle. Both outcomes are completely random.[5]
If there is a ‘Personal God’ that hears human prayers then there has to be a way for the Deity to influence the physical world without breaking the laws of nature and making the study of science futile. Quantum physics may be the magical wand, whereby ‘Personal God’ can influence our world, without breaking the laws of nature. In His infinite wisdom, the Omniscient God provided for infinite means, at the quantum level, to maintain His divinity! He says in the Holy Quran, in Sura Hadid:
Quantum physics is the magical wand, by which Allah has established His divinity on each and every quark, photon and boson. In so doing He has not only provided for His Providence but also for our free will, while ensuring predictability and reign of laws of nature at macroscopic level. If Laplace had been right, he would have not only ruled out God, but, also our free will and personal responsibility.
[2] Kenneth R Miller. Finding Darwin’s God. Cliff Street Books (Harper Collins), paper back edition 2000, p. 199-200.
[3] David Pepper, Frank Webster and George Revill. Environmentalism: Critical Concepts. Routledge, 2003. Page 148.
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458ff164b4456329 | Quark nuggets of wisdom
Article title: “Dark Quark Nuggets”
Reference: arXiv:1810.04360
Dark QCD?
Forming a nugget
How do we know they could be there?
References and further reading:
Beauty-full exotic bound states at the LHC
Article: Beauty-full Tetraquarks
Authors: Yang Bai, Sida Lu, and James Osborn
Reference: https://arxiv.org/abs/1612.00012
Good Day Nibblers,
As you probably already know, a single quark in isolation has never been observed in Nature. The Quantum Chromo Dynamics (QCD) strong force prevents this from happening by what is called ‘confinement. This refers to the fact that when quarks are produced in a collision for example, instead of flying off alone each to be detected separately, the strong force very quickly forces them to bind into composite states of two or more quarks called hadrons. These multi-quark bound states were first proposed in 1964 by Murray Gell-Mann as a way to explain observations at the time.
The quarks are bound together by QCD via the exchange of gluons (e.g. see Figure 1) and there is an energy associated with how strongly they are bound together. This binding energy between the quarks contributes to the ‘effective mass’ for the composite states and in fact it is what is largely responsible for the mass of ordinary matter (Footnote 1). Most of the theoretical and experimental progress has been in two or three quark bound states, referred to as mesons and baryons respectively. The most familiar examples of quark bound states are the neutron and proton, both of which are baryons composed of three quarks bound together and form the basis for atomic nuclei.
Figure 1: Bound state of four bottom quarks (blue) held together by the QCD strong force which is transmitted via the exchange of gluons (pink).
Of course four and even more quark bound states are possible and some have been observed, but things get much trickier theoretically in these cases. For four quark bound states (called tetra-quarks) the theoretical progress had been largely limited to the case where at least one of the quarks was a light quark, like an up or a down quark.
The paper highlighted here takes a step towards understanding four quark bound states in the case where all four quarks are heavy. These heavy four body systems are extra tricky because they cannot be decomposed into pairs of two body systems which we could solve much more easily. Instead, one must solve the Schrödinger equation for the full four body system for which approximation methods are needed. The example the current authors focus on is the four bottom quark bound state or 4b state for short (see Figure 1). In this paper they use sophisticated numerical methods to solve the non-relativistic Schrödinger equation for a four-body system bound together by QCD. Specifically they solve for the energy of the ground state, or lowest energy state, of the 4b system. This lowest energy state can effectively be interpreted as the mass of the 4b composite state.
In the ground state the four bottom quarks arrange themselves in such a way that the composite system appears as spin-0 particle. So in effect the authors have computed the mass of a composite spin-0 particle which, as opposed to being an elementary scalar like the Standard Model Higgs boson, is made up of four bottom quarks bound together. They find the ground state energy, and thus the mass of the 4b state, to be about 18.7 GeV. This is a bit below the sum of the masses of the four (elementary) bottom quarks which means the binding energy between the quarks actually lowers the effective mass of the composite system.
The interesting thing about this study is that so far no tetra-quark states composed only of heavy quarks (like the bottom and top quarks) have been discovered at colliders. The prediction of the mass of the 4b resonance is exciting because it means we know where we should look at the LHC and can optimize a search strategy accordingly. This of course increases the prospects of observing a new state of matter when the 4b state decays, which it can potentially do in a number of ways.
For instance it can decay as a spin-0 particle (depicted as \varphi in Figure 2) into two bound states of pairs of b quarks, which themselves are referred to as \Upsilon mesons. These in turn can be observed in their decays to light Standard Model particles giving many possible signatures at the LHC. As the authors point out, one such signature is the four lepton final state which, as I’ve discussed before, is a very precisely measured channel with small backgrounds. The light mass of the 4b state also allows for it to potentially be produced in large rates at the LHC via the strong force. This sets up the exciting possibility that a new composite state could be discovered at the LHC before long simply by looking at events with four leptons with total energy around 18 – 19 GeV.
Figure 2: Production of a four bottom quark bound state (\varphi) which then decays to two bound states of bottom quark pairs called \Upsilon mesons.
Of course, one could argue this is less exciting than discovering a new elementary particle since if the 4b state is observed it won’t be the discovery of a new particle but instead of yet another manifestation of the QCD strong force. At the end of the day though, it is still an exotic state of nature which has never been observed. Furthermore, these exotic states could be interesting testing grounds for beyond the Standard Model theories which include new forces that communicate with the bottom quark.
We’ll have to wait and see if the QCD strong force can indeed manifest itself as a four bottom quark bound state and if the prediction of its mass made by the authors indeed turns out to be correct. In the meantime, it gives plenty of motivation to experimentalists at the LHC to search for these and other exotic bound states and gives us perhaps some hope for finding physics beyond the Standard Model at the LHC.
Footnote 1: I know what you are thinking, but I thought the Higgs gave mass to matter!? Well yes, but…The Higgs gives mass to the elementary particles of the Standard Model. But most of the matter (that is not dark!) in the universe is not elementary, but instead made up of protons and neutrons which are composed of three quarks bound together. The mass of protons and neutrons is dominated by the binding and kinetic energy of the three quarks systems and therefore it is this that is largely responsible for the mass of normal matter we see in the universe and not the Higgs mechanism.
Other recent studies on heavy quark bound states:
1) https://arxiv.org/abs/1601.02092
2) https://arxiv.org/abs/1605.01647
3) https://arxiv.org/abs/1611.00348
Further reading and video:
1) TASI 2014 has some great introductory lectures and notes on QCD: https://physicslearning.colorado.edu/tasi/tasi_2014/tasi_2014.htm
A Quark Gluon Plasma Primer
Artist's rendition of a proton breaking down into free quarks after a critical temperature. Image credit Lawrence Berkeley National Laboratory.
Figure 1: Artist’s rendition of a proton breaking down into free quarks after a critical temperature. Image credit Lawrence Berkeley National Laboratory.
Quark gluon plasma, affectionately known as QGP or “quark soup”, is a big deal, attracting attention from particle, nuclear, and astrophysicists alike. In fact, scrolling through past ParticleBites, I was amazed to see that it hadn’t been covered yet! So consider this a QGP primer of sorts, including what exactly is predicted, why it matters, and what the landscape looks like in current experiments.
To understand why quark gluon plasma is important, we first have to talk about quarks themselves, and the laws that explain how they interact, otherwise known as quantum chromodynamics. In our observable universe, quarks are needy little socialites who can’t bear to exist by themselves. We know them as constituent particles in hadronic color-neutral matter, where the individual color charge of a single quark is either cancelled by its anticolor (as in mesons) or by two other differently colored quarks (as with baryons). But theory predicts that at a high enough temperature and density, the quarks can rip free of the strong force that binds them and become deconfined. This resulting matter is thus composed entirely of free quarks and gluons, and we expect it to behave as an almost perfect fluid. Physicists believe that in the first few fleeting moments after the Big Bang, all matter was in this state due to the extremely high temperatures. In this way, understanding QGP and how particles behave at the highest possible temperatures will give us a new insight into the creation and evolution of the universe.
The history of experiment with QGP begins in the 80s at CERN with the Super Proton Synchrotron (which is now used as the final injector into the LHC.) Two decades into the experiment, CERN announced in 2000 that it had evidence for a ‘new state of matter’; see Further Reading #3 for more information. Since then, the LHC and the Brookhaven Relativistic Heavy Ion Collider (RHIC) have taken up the search, colliding heavy lead or gold ions and producing temperatures on the order of trillions of Kelvin. Since then, both experiments have released results claiming to have produced QGP; see Figure 2 for a phase diagram that shows where QGP lives in experimental space.
Phases of QCD and the energy scales probed by experiment.
Phases of QCD and the energy scales probed by experiment.
All this being said, the QGP story is not over just yet. Physicists still want a better understanding of how this new matter state behaves; evidence seems to indicate that it acts almost like a perfect fluid (but when has “almost” ever satisfied a physicist?) Furthermore, experiments are searching to know more about how QGP transitions into a regular hadronic state of matter, as shown in the phase diagram. These questions draw in some other kinds of physics, including statistical mechanics, to examine how bubble formation or ‘cavitation’ occurs when chemical potential or pressure is altered during QGP evolution (see Further Reading 5). In this sense, observation of a QGP-like state is just the beginning, and heavy ion collision experiments will surely be releasing new results in the future.
Further Reading:
1. “The Quark Gluon Plasma: A Short Introduction”, arXiv hep-ph 1101.3937
2. “Evidence for a New State of Matter”, CERN press release
3. “Hot stuff: CERN physicists create record-breaking subatomic soup”, Nature blog
4. “The QGP Discovered at RHIC”, arXiv nucl-th 0403.032
5. “Cavitation in a quark gluon plasma with finite chemical potential and several transport coefficients”, arXiv hep-ph 1505.06335 |
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English Français
On the one hand, quantum systems have remarkable specificities such as the coherence manifesting itself in various wave phenomena or in the quantization of transverse (Hall) conductivities at low temperatures. On the other hand, quantum systems also sustain relaxation, decoherence, transport, and stochastic processes, as well as their classical analogues. We contribute to the development of theoretical methods of statistical physics and transport theory to understand these intertwined aspects.
Selected publications
Stochastic Schrödinger equation:
NonMarkovian stochastic Schrödinger equation,
P. Gaspard and M. Nagaoka, J. Chem. Phys. 111, 5676 (1999).
Fluctuation theorem and counting statistics in quantum systems:
Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems,
M. Esposito, U. Harbola, and S. Mukamel, Rev. Mod. Phys. 81, 1665 (2009).
Quantum theory of nonequilibrium steady states:
Schrödinger equation for current carrying states,
D. S. Kosov, J. Chem. Phys. 116, 6368 (2002).
Kohn-Sham equations for nanowires with direct current,
D. S. Kosov, J. Chem. Phys. 119, 1 (2003).
Quantum transport in nanoscale molecular systems:
Nature of well-defined conductance of amine anchored molecular junctions: Density functional calculations,
Z. Li and D. S. Kosov, Phys. Rev. B 76, 035415 (2007).
Ultracold atomic gases and the quantum Hall effects:
Non-Abelian optical lattices: Anomalous quantum Hall effect and Dirac Fermions,
N. Goldman, A. Kubasiak, A. Bermudez, P. Gaspard, M. Lewenstein and M.A. Martin-Delgado, Phys. Rev. Lett. 103, 035301 (2009).
Ultracold atomic gases in non-Abelian gauge potentials:
The case of constant Wilson loop
N. Goldman, A. Kubasiak, P. Gaspard, and M. Lewenstein, Phys. Rev. A 79, 023624 (2009).
Quantum Hall-like effect for cold atoms in non-Abelian gauge potentials,
N. Goldman and P. Gaspard, Europhys. Lett. 78, 60001 (2007). |
d86ed3316047d5f5 | • AP Magazine
An alternative way to explore and explain the mysteries of our world. "Published since 1985, online since 2001."
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Letters to the Editor—Alternate Perceptions Magazine, November 2018
Hi, Brent.
Thanks so much for your April AP issue. I especially enjoyed your Reality Checking review of the March Discover magazine article Down the Quantum Rabbit Hole. As you noted, the question of consciousness, of its ultimate nature and origin, is certainly one of the new frontiers in modern physics. The connection between the two has been hypothesized from the earliest years of quantum mechanics. Now, physicists are collaborating with biologists, neuroscientists, computer scientists, and other disciplines to try to decipher the mystery of consciousness.
Sir Roger Penrose has stated in some of his books that quantum physics can help neuroscience better understand consciousness. What I've learned from researching the ideas I'm developing in the overview of my book Subtle Realms: The Jung-Pauli Search for the Nature of Reality: Time, Causality, and the Linear Mind is that Penrose's claim is true but, conversely, and even more importantly, I believe, is that neuroscience can provide critical assistance in explaining the gaps in understanding between micro and macro-level physics, particularly in regard to the quantum measurement problem which I’ll address near the end of this email.
You did a beautiful job explaining the ideas espoused by Penrose and Hameroff and their theory of microtubules as a driving force of consciousness. A number of questions came to mind when reading your article. To find answers, I thought it might be helpful to read the original Discover piece. After doing so, I was left with even more questions. I think most of these questions stem from the various notions of consciousness which abound in scientific and lay communities. It's difficult talking about consciousness when it means different things to different people.
Although not explicitly stated, the researchers interviewed or mentioned in the Discover article seemed to view consciousness as a kind of cognitive or information-processing function or, at an even more elementary level, as a state of simply being awake as opposed to asleep or comatose. For most of us, though, consciousness is synonymous with awareness of self and one's relationship to the external environment; with a sense of one's own existence; and with reflective thought. To many, it is believed to transcend death as you mentioned in Reality Checking. To others, it is a force or energy which imbues the universe. It occurred to me halfway through writing this email that I should practice what I preach and define how I plan to use the term. For the sake of limiting the scope of my argument addressing what I believe are the limitations of the Penrose-Hameroff theory, consciousness will be used to mean self-awareness unless otherwise stated.
The author of Down the Quantum Rabbit Hole, Steve Volk, provides many helpful photographs and schematics to explain how microtubules and quantum physics may be the basis of consciousness. This leads to the initial question I had when reading your article: specifically, how microtubules facilitate conscious awareness. Volk states that the Penrose-Hameroff theory postulates that "consciousness originates from microtubules and actions inside neurons, rather than the connections between neurons". Stuart Hameroff has speculated, however, that the single-celled paramecium has rudimentary consciousness but no neurons which seems inconsistent with their theory. So, I'm wondering which cells and their associated microtubules are involved in the process? Are neurons the only ones or can any type of cell potentially facilitate consciousness? If neurons are the only cells involved, where do they reside? Do only cortical neurons facilitate consciousness?
In any case, I would argue that a paramecium is not capable of self-awareness or information-processing as we humans know it. What appears to be decision-making is actually a stimulus-response reaction to its environment. That said, the paramecium is undoubtedly a fascinating entity in its own right. I am wondering how a paramecium can respond to its environment if it contains no neurons unless it itself is a neuron, not like those in more sophisticated life forms yet still capable of performing basic survival activities. The cilia may function as pseudo axons and/or dendrites, conveying information from the environment to the cell body, while the cell body performs basic activities such as metabolism and reproduction just as cells in higher life forms do. Although the paramecium may be a bit of a mystery, we do know that as organisms become more highly evolved, differentiation of body parts increases to support diverse functions beyond those needed for survival. Very simple organisms require fewer specialized parts. A limited number of cell structures can serve multiple functions, allowing single-celled organisms to be fully self-contained and function on auto-pilot. This is merely an interesting aside, however. So, moving on …
From everything neuroscience has learned to date, self-awareness and sensory perception can only be experienced as such in the brain's cerebral cortex. An electrical impulse must travel from the sensory receptor where it originates to various cortical structures, sometimes quite distant, where it finally culminates in conscious awareness in the prefrontal cortex. Here, along with support from neural feedback loops with memory and emotional centers in the limbic system, the electrical signal is integrated with other signals from the same and different receptors to provide context, meaning, interpretation, association, and timing. These impulses must be transported from neuron to neuron across numerous synaptic junctions on their way to prefrontal and other cortical structures. So, from the standpoint of neuroscience, neuronal connections do matter. If electrical signals cannot be transported across synaptic clefts, they will not reach the cerebral cortex where they are experienced as meaningful sight, sound, taste, pain, and other sensations.
This leads to my next question regarding the Penrose-Hameroff theory of consciousness: What is it about neurons in the cerebral cortex that facilitates self-awareness? Does consciousness arise spontaneously in these neurons without interaction with other neurons as the Penrose-Hameroff theory would suggest? Doesn’t our sensory apparatus play a role in the development of consciousness? If we were deprived of our basic five senses and a sixth, proprioception, would we still be self-aware? Would we be capable of self-reflection? Would we know we exist? Would the concept of existence even be meaningful?
As you noted, Hameroff, an anesthesiologist, believes that the key to consciousness lies in the field of anesthesiology. He noticed that the brain functions of an anesthetized patient continue more or less as they normally would, that neurons continue to fire, and pain signals travel their normal paths. I question how he knows that. Has he observed or recorded these pain signals as they traverse the distance to the prefrontal cortex where they are experienced as pain?
Normally, only a tiny fraction of electrical signals are able to complete their journey from sensory receptors to the cerebral cortex. The transmission of an electrical impulse from neuron to neuron across synaptic junctions can fail anywhere along the sensory pathway, even in the prefrontal cortex and other cortical structures. Appropriate neurotransmitters must be present in the correct mix and balance to convey the tiny, delicate electron across the gap to the next neuron.
There are also brain structures which can suppress the transmission of electrical signals in a process called sensory gating. The gating mechanism serves an important function in mental health. It acts like a filter, ensuring that the conscious mind is not flooded with noise from extraneous or overwhelming sensory input. Two disorders which currently receive major research funding, autism and schizophrenia, are believed to be caused in part by the breakdown of sensory gating. Without sensory gating, our normal conscious lives would be overwhelmed with visual, auditory, olfactory, pain and other stimuli including motor impulses.
This normal modulation of sensory and motor signals may provide insight into how general anesthesia works and fails to work in the situation where a patient is believed to be anesthetized but is actually fully conscious and experiencing pain during surgery. In this latter situation, the patient is paralyzed (from a paralytic component of the anesthesia) and cannot cry out to alert the anesthesiologist.
This filtering process is important because it is the means by which sensory and motor input is single-threaded through sensory and motor pathways, preventing everything from happening at once. It separates a quantum flood of simultaneity into a linear, step-by-step progression of one signal followed by another. The end product of this process is what neuroanatomists call the linear mind. The fact that events are serially created in the conscious mind forms the basis for our notions of time and causality and the forward arrow of time.
Another neurological phenomenon which further focuses attention and limits our view of reality is called neuronal pruning. This process involves the mass die-off of fetal brain cells beginning one or two months before birth and continuing for a month or so after. Pruning also occurs during adolescence and early adulthood. The purpose of pruning is to eliminate neurons which fail to form appropriate connections or form no connections at all. In this process, targeted neurons are gobbled up by phagocytic immune cells. Sometimes pruning can go awry and certain neurons are eliminated which shouldn’t be or too few or too many neurons are eliminated. Post-mortem examination of brain tissue of schizophrenics has revealed a marked loss of gray matter and fewer neural connections (signs of inappropriate pruning) in the prefrontal cortex. Inappropriate neuronal pruning is also implicated as a possible cause of autism.
Sensory gating and neuronal pruning create the linear mind which determines the scope and nature of reality as we humans, and probably all mammalian, avian, and even lower forms of life, generally perceive it. This concept is important because it addresses what I believe is the weakest leg of the Penrose-Hameroff theory of consciousness: the role played by quantum activity inside a neuron’s microtubules which is thought to be the impetus for consciousness. According to the Discover article, Penrose and Hameroff have proposed that the collapse of all possible quantum states of an electron into a single state inside a neuron produces a conscious moment. To assess the plausibility of this theory, we need to understand what is meant by a quantum state and how the hypothetical collapse is thought to work. What I’m going to discuss next is probably stuff you already know but which I need to put out on the plate to support why I think the Penrose-Hameroff theory misses the mark.
A quantum state is a set of attributes describing an isolated quantum system such as an electron at any point in time. The set of attributes consists of the following quantum variables: position in 3 dimensional space, momentum, angular momentum, energy, and spin. These attributes can be expressed as a set of quantum numbers which define a particular state of a particle. A particle can have many possible quantum states and it exists in all of its states simultaneously. This property is known as superposition.
A measurement can be taken in a classical experimental setting to determine the specific quantum state of a particle at a given moment in time. However, because of difficulties inherent in measuring a quantum particle in a classical setting, mathematically described in the Heisenberg uncertainty principle, a precise determination of a quantum state can never be achieved. The conundrum associated with such a measurement is that as one attribute, say location, becomes well-defined, another attribute, say momentum, becomes less well-defined. At best, the measurement can only predict the probability of a particle being in any particular state at any given time. A series of measurements can be taken to produce a probability distribution of quantum states as the quantum system evolves over time. This probability distribution is described by Schrödinger’s wavefunction equation mentioned in the Discover article.
After Schrödinger developed his equation in 1926, the concept of wavefunction collapse was added by other physicists to account for what investigators find in the classical experimental setting. In this hypothesis, the constantly fluctuating probabilities associated with the quantum state of a particle are believed to collapse into a single probability corresponding to a specific set of attributes associated with the particle when it is measured or observed. Quantum position can be used to illustrate this principle. In its natural setting, a particle exists everywhere at once in a potentially infinite number of locations. When a particle is subjected to measurement or observation, the probabilities associated with all possible locations comprising its wavefunction appear to collapse into a single probability that the particle now exists in a specific location corresponding to what is seen or measured.
Although the concept of wavefunction collapse is routinely taught by educators and espoused in popular books on quantum mechanics, it is by no means universally accepted as representing what actually happens at the quantum level. This brings us to the quantum measurement problem I mentioned earlier in the second paragraph of my email. This problem arose from questions regarding how the hypothetical collapse actually works. Since the Schrödinger equation does not hypothesize wavefunction collapse and assumes that a particle will be in any number of mutually-exclusive states simultaneously, what occurs during measurement to induce the hypothetical collapse? What is special about the measurement procedure that causes the particle to cease existing in all its possible states? The ambient environment of a particle normally includes many things: people, furniture, equipment, movement of other things in its environment, particularly other particles with which it interacts including photons and dark matter and dark energy. Presumably, these interactions do not cause the quantum states of the particle to collapse so why should observation or measurement? More importantly, attempting to capture a specific quantum state from the rapidly fluctuating probabilities using relatively sluggish classical measurement procedures could never truly reflect the attributes at a given point in time. And finally, what happens to the other quantum states after the supposed collapse?
Many innovative theorists in quantum mechanics have rejected the theory of collapse. Besides Schrödinger, others included John Wheeler and two of his graduate students, Richard Feynman and Hugh Everett, at Princeton’s Institute for Advanced Study. A number of alternate theories have been proposed to better describe what actually happens at the quantum level.
I think the collapse hypothesis became popular because it is intuitive and an easy way around the measurement problem. I do not believe it accurately describes what is really happening at the quantum level, however. Rather, it reflects what appears to be happening in the human mind. I believe that quantum states do not collapse into a single state when measured or observed and that the particle still exists in all possible states just as it did before measurement. Just because the other states are not measured or simultaneously perceived by a human being, does not mean they don’t exist. Here’s an example, again using the quantum attribute of location, to illustrate what I mean:
From the perspective of the human observer doing the measurement, the particle ends up in location x, y, z at the moment the measurement occurs. From the perspective of the particle, anthropomorphically-speaking, it still exists simultaneously in all locations in which it had ever existed, currently exists, and ever will exist – in other words, without regard to time. That is why mutually-exclusive states can exist simultaneously in the quantum realm: because at this level, time does not exist, it is not a constraint, and mutual exclusivity is an illusion imposed by the human mind.
It seems to me that physicists who espouse the collapse hypothesis are looking at particle behavior from the perspective of classical physics, not quantum mechanics, and that the collapse of all possible quantum states into one occurs only at the macro level in the mind and instrumentation of the observer. In the collapse hypothesis, the behavior of a particle is being forced into a classical mold. I would agree that the acts of observation or measurement, the presence of measuring instrumentation, or even the human mind do change the particle’s behavior as we know it at the macro level and as it is measured, but not intrinsically. By definition, measurement or observation must occur at the macro level since the investigator and equipment exist at that level. Wavefunction equations were designed to formalize what is observed in the real world of macro-level objects and behavior.
I think some of the issues expressed in the quantum measurement problem arise from the inherent difficulty of conducting quantum experiments in a classical setting, of attempting to bring quantum behavior up to a level where it can be analyzed by human perception. Experimenters must contend with the fact that the quantum realm is fundamentally random and acausal and the universe is nonlocal, quite different from the way the human brain perceives matters. Probability is not meaningful at the quantum level. It is a classical methodology used to predict the evolution of a particle’s state as perceived by a human observer in a time-based framework.
This brings us back full-circle to the linear mind and the huge limitations it imposes on our study of quantum behavior and the quest to understand consciousness. Even if the collapse hypothesis were true, the Penrose-Hameroff theory, as outlined in the Discover article, does not explain specifically how the collapse creates consciousness. That question has not been answered.
Antonio Damasio, Professor of Neurobiology at the University of Southern California and author of several popular books including Descartes’ Error, has specifically addressed Roger Penrose’s theory of consciousness. In the December, 1999 issue of Scientific American, he has this to say in an article titled How the Brain Creates the Mind:
The quantum level of operations might help explain how we have a mind, but I regard it as unnecessary to explain how we know that we own that mind – the issue I regard as most critical for a comprehensive account of consciousness.
I apologize for my lengthy reply to your excellent article. I’ve consumed a lot of time explaining why I don’t agree with the Penrose-Hameroff theory of consciousness but haven’t offered much in the way of my own insight to the problem. I suppose I have a fairly generalized view of consciousness as it relates to quantum theory. It is my understanding that quantum activity underlies everything in the universe – animate and inanimate; animal, vegetable, and mineral – not just the few quantum-level biological, chemical, and physical processes we’ve discovered at the classical level. Quantum life is universal; it is everything, everywhere simultaneously, beyond time. It is the foundation of the universe as we currently understand it.
On that note, I am reminded of the response the current historical Buddha, Siddhartha Gautama, gave to his cousin Ananda when asked about the possibility of a supreme being. Gautama Buddha’s reply has been paraphrased thus:
There is, Ananda, the unoriginated, undifferentiated, unconstituted which is ultimately unknowable.
Consider the meaning of those words – unoriginated: comes from nowhere, has no beginning, has existed from all eternity; undifferentiated: cannot be broken into parts; unconstituted: unformed, composed of nothing, not created or equivalent to something else. In my view, this is a perfect description of consciousness, in the broader sense, and possibly even quantum reality.
Perhaps the quantum realm itself is consciousness and the narrower our focus in trying to understand it, the further removed we become from grasping its true meaning. That said, self-awareness is definitely part of the equation and easier to get a handle on. I love the definition of consciousness I found in the book The Serpent Power by Arthur Avalon, the pen name of Sir John Woodroffe, an early twentieth-century scholar of shaktic and tantric thought. Woodroffe states that consciousness is “the power of matter to know itself”. The thought of matter in this context is striking. It is so amazing because that’s what we are – vibrating, spinning particles imbued with a mind that is self-reflective and self-aware, bounded by its invention of time!
I am haunted by one of Jill Bolte Taylor’s quotes in her book My Stroke of Insight. Jill is a neuroanatomist who experienced a massive hemorrhage-induced stroke in her left cerebral hemisphere while preparing to leave for work at the Harvard Brain Tissue Resource Center. She describes how this stroke caused her brain to function non-linearly and what it was like living in this state. In her words:
And here, deep within the absence of earthly temporality, the boundaries of my earthly body dissolved and I melted into the universe.
How close does that come to experiencing Universal Consciousness?!!
Before closing, I’d like to credit a few of the sources I’ve relied on to formulate my ideas regarding the interconnection of neuroscience and quantum mechanics. I see these two disciplines fitting together like two pieces of a jigsaw puzzle. They interlock perfectly. A complete understanding of one cannot be achieved without a full understanding of the other, just like Niels Bohr’s concept of complementarity.
Here are some of my sources:
Brian Greene, The Fabric of the Cosmos
John Ratey, A User’s Guide to the Brain
Jill Bolte Taylor, My Stroke of Insight
Stanford Encyclopedia of Philosophy – Philosophical Issues in Quantum Theory
Journals such as Scientific American, Nature, Science, Current Biology
Online university courses in quantum mechanics
Online university courses in neurology and neural pathways
Abstracts for research grants on neurological problems associated with sensory gating
Roger Penrose, Shadows of the Mind
Roger Penrose, The Road to Reality
I must say how impressed I am with the writings of Roger Penrose. He writes with humility and an easy style, not aggressively pushing his ideas. He has stated on numerous occasions that Einstein’s classical theories of special and general relativity are on solid ground. They have stood the test of time and continue to be verified experimentally. Conversely, he believes that quantum mechanics has a long way to go before it can be fully understood. He is disturbed with the gaping holes in quantum theory and feels that many of the theoretical underpinnings may end up being reworked. I very much agree with his philosophy and believe that the inherent difficulties of trying to study quantum behavior at the macro, classical level are the root of some of these problems. The quantum measurement problem is a case in point.
Thanks for your patience and all the great articles you’ve sent our way. They’ve certainly been an inspiration for thought on my part. Take care.
Mary Kerfoot
Path of Souls
Edgar Cayces Atlantis
On the Edge of Reality
The Search of Edgar Crace's Atlantis DvD
The Yucatan Hall of Records
Ancient Mound Builders
Alien Energy: UFOs, Ritual Landscapes and the Human Mind
Path of Souls
New Book
The Illustrated Encyclopedia of Native American Indian Mounds & Earthworks
Path of Souls
Visitors from Hidden Realms
Ancient South America
The ARE's Search for Atlantis
Freedom To Change: Why You Are The Way You Are and What You Can Do About It
Native American Mounds in Alabama: An Illustrated Guide to Public Sites
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I. What is Analogy? 1. The Common Meaning of the Word Analogy 2. Analogy and Logic 3. Analogy and Metaphysics - II. Analogy in Aristotelian-Thomistic Logic and Metaphysics 1. Analogy of Attribution or Simple Proportion 2. Analogy of Proper or Intrinsic Proportionality 3. Analogy of Improper, Extrinsic, or Metaphorical Proportionality 4. Analogia Entis 5. The Crisis of Analogy - III. Analogy and Theology 1. The Knowledge of God and the Divine Names 2. Examples of Analogy in the Scriptures. 3. Uses of Analogy in Theology 4. Analogia fidei - IV. Analogy and Science 1. Analogy and Scientific Theory: The Experimental Sciences 2. Analogy and Scientific Theory: The Mathematical Sciences 3. Analogy within Scientific Theories 4. The First Steps towards a Theory of Analogy - V. The “Profundity” of Analogy.
I. What is Analogy?
1. The Common Meaning of the Word “Analogy.” The word “analogy” in its usual sense in modern English means “a form of reasoning in which one thing is inferred to be similar to another thing in a certain respect, on the basis of the known similarity between the things in other respects” (Random House Unabridged Dictionary [Random House, Inc., 2006]). Recently, the adjectival form of the world “analogy,” “analog,” has come to be frequently used in a technical sense, denoting electronic devices that work in a way different from “digital” or “numerical” electronic devices. The origin of the word “analogy,” as the Greek root (analoghía) suggests, is ancient and is based on the mathematical concept of “proportion” (a:b = c:d), which establishes a similarity based on the equivalence of ratios. One could think, for instance, of the similarity of two triangles whose sides stand in a fixed ratio. The transfer of the word “analogy” from mathematics to logic and philosophy dates back to Plato (427-347 B.C.) who, however, never devised a theory of analogy. Aristotle (384-322 B.C.) was the first to give a systematic formulation of it in the field of logic. In the Middle Ages, Thomas Aquinas brought Aristotle’s work to perfection with both a philosophical and theological aim. Later, beginning with the Nominalists, analogy became less and less understood. It was gradually abandoned in the fields of logic and philosophy and restricted in its scope to the point of becoming a simple literary “metaphor.” It is in this sense that the term is used today in the context of hermeneutics.
2. Analogy and Logic. The need for introducing analogy into Greek thought seems to have arisen from two kinds of problems: the first was strictly “logical-linguistic,” while the second one was more properly “metaphysical.” From the logical-linguistic point of view, Aristotle, and later Thomas Aquinas, began with the observation that in common language—which expresses, and therefore is a sign of, the structure of how thought proceeds—the same term (or “predicate”) can be attributed to different subjects in a “univocal,” “equivocal,” or “analogous” way. In the first case of univocity, the predicate has exactly the same meaning for the entire class of subjects to which it is attributed: For example, when we say, “Tom is a man” and “Dick is a man,” the term “man” corresponds to the same definition “rational animal” in both instances. In the second case of equivocity, on the other hand, the same term is used with completely different and uncorrelated meanings, as when one says, “this animal is a bull” and “this document is a papal bull.” In this case, the word “bull” corresponds to different definitions in each of the two examples. In the first example, it involves an “adult male bovine,” in the second, “a text written by the Pope.” Consequently, the use of the same word to signify different things is adopted purely by convention, so much so that equivocity is related to the language one uses and is lost in translation to another language. Finally, in the third case of analogy, the same term is used with different meanings but in such a way that they have a real correlation, and therefore the use of the same term indicates a real similarity and not a mere choice of convention. An example of this would be when one says “Einstein was clever” and “the theory of general relativity is clever.” Properly speaking, only a man can be clever, but a theory can be said to be clever in so far as it is an expression and a “real effect” of the cleverness of its author (rather than a theory being considered clever merely by convention).
3. Analogy and Metaphysics. The second class of problems which have led to the idea of analogy is not purely logical and linguistic but more properly metaphysical, in that analogy is inherent in things and is successively transferred to the thought and language with which one attempts to understand reality. Greek thinkers confronted the problem of reconciling two seemingly contradictory facts of experience, namely, the being of things versus their “becoming” (or in physical terms, their “motion”). The “monistic” solution—that is, a solution based on the assumption that reality is founded on only one constitutive principle (be it material or immaterial)—requires that one take one of the two facts of experience as apparent: If one admits only the reality of being, as a single undifferentiated state, “being” can never be other than itself, as it cannot change into something different from itself. On the other hand, in adopting this approach, one cannot explain the phenomenon of motion that we observe in everyday experience, as the passage from one state to another. Therefore, one would have to say that this passage is not real but purely apparent (this was the solution proposed by Parmenides, 6th-5th century, B.C.). We are then left with the problem of understanding what produces this illusion in us. If, on the other hand, one only admits the reality of becoming, it is then necessary to admit the contradiction that becoming, by the very fact that it is, coincides with being, that multiplicity coincides with oneness, that nothingness (that is, non-being) is a state of being, and that becoming is a continuous oscillation between these two contradictory states. But, admitting this contradiction implies, in the end, that knowledge is impossible (this was the extreme consequence of Cratilus, following in the footsteps of Heraclitus, 6th-5th century B.C.). In order to explain human experience completely, it is necessary to hypothesize that being may exist according to “differentiated states” that constitute a spectrum of modes of being lying somewhere between being in its absolute fullness (God, Pure Act) and in its complete absence (nothingness). To correctly understand the analogy of being, we need the help of the accurate Latin terminology: Ens means “being” as a subject capable of being, while esse is the property of “being.” Being (esse) is the principle by which a being (ens) is: “Being” (ens) is a term which is predicated in a differentiated but not equivocal way of different subjects.
The notion of analogy of being corresponds, from the logical point of view, to the metaphysical fact that assumes that being (esse) is actuated in differentiated modes and degrees in existing things (or, to say it another way, that things participate in being to varying degrees). Thus, the logical theory of analogy corresponds to the metaphysical theory of participation.
II. Analogy in Aristotelian-Thomistic Logic and Metaphysics
In Aristotelian-Thomistic logic, three types of analogy are possible (although further distinctions have been introduced by later schools): analogy of “attribution” or “simple proportion,” analogy of “proper” or “intrinsic proportionality,” and analogy of “improper,” or “extrinsic,” or “metaphorical proportionality.”
1. Analogy of Attribution or Simple Proportion. Analogy of attribution is usually presented with a classic example: “Tom is ‘healthy,’ his complexion is ‘healthy,’ this food is ‘healthy,’ the air is ‘healthy.’” By observing this example, we note that the characteristic of being “healthy” is proper only to Tom, who is the only subject that can be said to enjoy good health, as he is the only living being of the things considered in this example. One cannot properly speak of the other things as being “healthy” because they are not living beings. One can say that in a certain sense these non-living beings are “healthy” only in reference to the good health of Tom, who alone is the subject of the predicate “health” in the proper sense. For this reason, Tom is called the summum analogatum or primum analogatum.
As for the other subjects, one can single out the relationship they have with the healthy state of being of Tom: His healthy complexion is a sign of his good health, in so far as it is an “effect” of his good health. Healthy food is that which favors Tom’s good health as one of its “causes.” It must be understood that the reference to the summum analogatum is neither conventional nor accidental, but is instead founded on reality and confirmed by experience (from the fact that healthy food really contributes to the good health of someone who eats such food, and that a healthy complexion is really a sign of good health, and so on and so forth). For this reason, food, complexion, and climate are referred to as the analogata inferiora. It is this reference, which is founded on reality, that makes the concept of attribution more than just “equivocal.” These things and realities are and remain different, but the common name of the predicate expresses qualities which, even if they are in themselves different, have, under a certain aspect, a direct relationship with the quality of the primum analogatum (cf. Thomas Aquinas, Summa Theologiae, I, q. 13, a. 5).
2. Analogy of Proper or Intrinsic Proportionality. Even this second kind of analogy is usually illustrated with a classic example that consists in comparing sight with intelligence. We often use the idea of “vision” either in reference to “eyesight” or in reference to the “mind’s understanding.” Thus, we use the expressions, for example, “the light of truth illuminates the mind,” “to understand at first glance,” and “a philosophical vision of reality.” In these examples, we have a term which expresses an action (seeing) which we attribute to two different subjects (the eye and the mind). In this type of analogy, the similarity is established between the “relations” between predicate and subjects rather than between different senses of the same predicate attributed to different subjects. This similarity between the relations can be summarized by a formula which recalls that of a mathematical proportion: “Seeing” is to the “eye” as “understanding” is to the “mind.” Nevertheless, when we write a mathematical proportion, we establish two “equal” relations (2:3 = 4:6), whereas in the case of the analogy of proportionality, we state that two subject-predicate relations are not the same, but “similar” (cf. Thomas Aquinas, De Veritate, q. 2, a. 11). It must be emphasized that the action attributed to the subjects is really connected with each of them. The faculty of seeing is intrinsic to the eye, and the faculty of understanding is intrinsic to the mind: In both cases, we are dealing with a natural capacity, a proper and therefore really possessed faculty. For this reason, one speaks of analogy of “proper” or “intrinsic” proportionality. We note that in this type of analogy there exists neither a primum analogatum nor analogata inferiora: We have instead a subject-quality relationship which can be applied, in the proper sense, to a subject (the eye in the case of vision) and in a “similar” sense to the other subject (the mind). Seeing is proper to the eye, not the mind. One can therefore say that, in a certain sense, what takes the place of the primum analogatum is not the subject to which the predicate is properly attributed, but a relation between the subject (the eye) and the predicate (able to see).
3. Analogy of Improper, Extrinsic, or Metaphoric Proportionality. The third type of analogy is that of the “metaphor.” It involves a kind of analogy in which, unlike the two preceding cases, there is no real basis for similarity. It is a kind of analogy which is founded instead on a similarity discovered by the knowing subject who does not see any cause-effect relation in the nature of the subjects and the predicate, nor any real similarity in their relations. Properly speaking, it is not a real analogy, but we can consider it as such in a loose or improper sense. A typical example used to illustrate the concept of this kind of analogy is the following: “Tom has the courage of a lion.” Even in this case there is implicitly a kind of proportion: We can, in fact, reformulate this example in the following terms: “Tom is as courageous as a lion is courageous.” We see immediately that the quality “courageous” through which Tom can be likened to a lion is a quality that can be found in its highest degree in a lion: In a certain sense, this recalls the analogy of attribution. Nevertheless, there is a fundamental difference: There is no cause-effect relation between the courage of the lion and that of Tom, in that Tom is not courageous in virtue of a supposed participation in the courage of the lion. We cannot therefore speak of an analogy of proportion. It is instead a similarity that the knowing subject recognizes, as an external observer, between the courage of Tom and the courage of the lion. In this case, we have a similarity of relations between the subject and its quality, as in the case of the analogy of proportionality. Nevertheless, one cannot even speak of a true analogy of proper proportionality. In fact, in order to have an analogy of “proper” proportionality, the proportion that one wishes to establish would have to be: Tom is to the courage (of Tom) as the lion is to the courage (of the lion), whereas in the analogy of improper proportionality the same quality of courage proper to the lion (lion-like courage) is attributed to both Tom and the lion. Properly speaking, Tom has a human courage, while “lion-like courage” is attributed to him. We are dealing with a kind of “extrinsic” attribution, in that one attributes a character which is natural and proper to a lion to a natural endowment of Tom (cf. Thomas Aquinas, Summa Theologiae, I, q. 13, a. 3, 1um).
4. Analogia Entis. The fundamental discovery of the metaphysics of antiquity has probably been that of the analogy of being (analogia entis). Unlike the different genera which, from the logical point of view, are formalized in “universal” concepts predicated in a univocal way of various subjects, as “man” is said with the same meaning of Tom, Dick, and Harry—“being” is predicated in an analogous way of several subjects and rises above the genera and universal concepts which describe them (cf. Aristotle, Metaphysics, 998b, 22-27).
We note here two relevant aspects of the issue: First, in particular, “being” is said according to an analogy of proper proportionality of an object (substance) and its properties (accidents). This is a result of the fact that a property is always a property of something and can exist only in something else and not alone. A color, a length, a temperature, etc., exist always and only in an object, while an object possesses an autonomous existence. Thus, one must say that a property is to its mode of being as an object is to its mode of being, but the two modes are not identical, though they may have in common the fact of being. Second, in addition, “being” is said of a finite object, which has being by participation, and is said to be so according to an analogy of proportion with respect to Pure Act which is being in itself and is the cause of the being of a finite object. A similar property to that of “being” is also characteristic of the super-universal notions of “true,” “one,” and “good,” which, together with “being,” are called the “transcendentals.”
5. The Crisis of Analogy. The concept of analogy, which finds its most complete development and use in the philosophy of Thomas Aquinas, contains, beginning with Thomas Aquinas’ contemporaries, the seeds of its future downfall. In fact, from as early as the 13th century, the two great schools of philosophical-theological thought in Paris, where Albert the Great (1200-c.1280) and later his disciple Thomas Aquinas flowered, and in Oxford, with Roger Bacon (1214-1252), Robert Grosseteste (1174-1253), and later John Duns Scotus (1275-1308) and William of Ockham (1280-1349), were in opposition and would follow two different paths without ever coming to a mutual understanding. The Aristotelian path of Albertus Magnus and Thomas Aquinas would become of great importance especially for Catholic theology and, three centuries later, would be officially recognized in large part by the Council of Trent (1545-1563). The Platonic path, prevalent in Oxford, would concentrate on the problem of the mathematical formulation of the sciences, beginning with Roger Bacon, creating the methodological premises for the development of modern science.
In this way, there arose an ever more univocal and mathematical scientific way of thinking that took root and departed from a metaphysical and theological analogy-based thought. Duns Scotus would resolve the analogy of being in a multiplicity of univocals, just as William of Ockham would dissolve the reality of universals into pure names (Nominalism) by denying them a real existence outside of the mind. This development would then have an influence on the philosophical thought of Descartes (1596-1650), and later on Kant (1724-1805) and the success of Galilean and Newtonian science, and would eventually lead to the end of the very possibility of metaphysics as a science and consequently of theology as a systematic science. Nevertheless, in the last few decades, we have witnessed a new trend in the sciences which seem to be seeking, and in a certain sense discovering anew, the concept of analogy, with the aim of confronting new problems related to theories regarding the logical and mathematical foundations of the sciences and the complexity of self-organizing structures. Even if it is too early to judge, one could say that the concept of analogy, which was initially excluded from scientific thought for fear of equivocity, has now claimed its place. New disciplines like “formal ontology” seem to open up a new perspective, a sort of scientific approach to metaphysics. It is an approach that is claimed by modern mathematical logic and even by the technologies related to electronic computing and “artificial intelligence.”
III. Analogy and Theology
Recourse to the concept of analogy in theology is necessary for many reasons. It cannot be otherwise since human reason, which is by its very nature creaturely, is able to approach the mystery of God only if it maintains a distance between creature and Creator by acknowledging that one can speak of God only by analogy and not in a univocal or equivocal way. In the context of the metaphysics of being, the analogia entis allows one to approach the problem of God’s existence as the foundation of the being of all things and to predicate God’s attributes and perfections that are present, in a participatory way, in God’s works. But it is the very language of Revelation as presented in Sacred Scripture which uses analogy in its various forms, be they proper or improper, as for example in metaphor and even in “parable,” expressing, through human concepts, that which would otherwise remain transcendent and ineffable in itself. The language of analogy is then used by theologians in their attempts to approach, through recourse to images and comparisons, the mysteries of the faith, and it is also used in order to discern relations between such things, thereby grasping a deeper, inner coherence of God’s plan of salvation.
1. The Knowledge of God and Divine Names. The various applications of the concept of analogy to theology lie on different levels. The first question one asks concerns the knowledge of God, either through human reason alone or by faith in what God has revealed about Himself. Theologians have traditionally taken two paths to this goal. The first is the “apophatic” or “negative” way, typical of Eastern Christianity, which emphasizes the fact that we can only know with certainty what God is not, rather than what He is. Following this approach, such characteristics as composition, corporeality, finitude, and so on, are excluded from the notion of God. In addition to negative theology, and inspired by a scriptural passage from the Book of Wisdom (cf. Wis 13:5) in which explicit reference is made to the concept of analogy, Western Christianity developed a positive theology. On the basis of the analogy of simple proportion, it allows one to recognize in God a similarity with the perfections found in creation, as effects whose summum analogatum is God Himself (cf. Thomas Aquinas, Summa Theologiae, I, q. 12). This involves a cognitive approach which certainly does not dissolve mystery in that, as the Fourth Lateran Council (1215) recalled, “between Creator and creature, there is always a greater difference than likeness” (DH 806; Fides et Ratio, n. 19).
Another classical theological problem that is closely tied to the problem of the knowledge of God is that of the titles one can correctly attribute to God (the “divine names”). This theme, treated by pseudo-Dionysius in De Divinis Nominibus, was taken up and given a complete treatment by Thomas Aquinas for whom analogy would play a decisive role. First of all, he maintained that the names that denote what God most certainly is not (imperfections or ontological or moral limits) cannot be attributed to God. He then states that we can attribute to God the words we use to describe the perfections of creatures, but only by analogy, as our language refers mainly to what we know of creatures. These are in fact an effect of which God is the cause, a cause that cannot be known directly by us. We cannot speak of Him univocally because God is a cause that is infinitely higher than His effects and transcends their natures as He does not belong to any genus. We cannot speak of Him equivocally, since there is a cause-effect relation, which is a real relation from the creatures towards God. Thus, the names signifying God’s perfections are used by analogy of proportion, God being here the summum analogatum. When one says that something is good, one says this most properly of God, who is good in and of Himself, rather than of creatures, who are good only by participation. Other names can be attributed to God only metaphorically. This happens either when one signifies a perfection by means of a name describing a creature who possesses it or when, instead of the name of a certain perfection, the creature’s name is attributed to God, with the intention of attributing that perfection to Him. This happens, for example, when in Holy Scripture God is called a “rock” or “lion,” with the intention of attributing the perfections of a rock and a lion to him (cf. Summa Theologiae, I, q. 13).
2. Examples of Analogy in the Scriptures. It is proper to the language of Holy Scripture to offer, through different literary genres, a treasure trove of analogies and metaphors. This is due, as already mentioned above, to the need for expressing with human words, which are used primarily to describe creatures, contents regarding the transcendent reality of God, which reason alone cannot reach and which is not an object of common experience. It is God who communicates His will and His plan through images based on analogy. Abraham is asked to try to conceive of the immense number of descendents of whom he is called to be the father by an analogy with the great number of stars in the sky and grains of sand in the sea (cf. Gen 15:5 and 22:17). Another example is the prophet Jeremiah, who is invited by God to look at the renewal that God will bring about in the house of Israel (Jer 18:1-4) by considering the analogy of the potter who forms and then destroys the work of his hands in order to make it anew. The prophets themselves were the ones who spoke to the people through numerous images and analogies, drawing from what happens in nature, in their own history, and in the story of different peoples (Ez 31:1-14; Hos 1:2-9; Dan 2:31-45).
Jesus spoke in “parables” rather frequently to describe the reality of the Kingdom with effective and coherent images, in order to make it more understandable to his audience. The expression “The Kingdom of Heaven is similar to” frequently recurs in the Gospels (cf. Mt 13:1-41; Mk 4:1-34; Lk 8:4-18). This comparison is based on the “analogy of proportionality.” The use of images and metaphors establishes a simile between a known reality and an unknown or difficult to understand one, allowing the transposition of properties and relations from the better known to the lesser known image. The parable is often told in the form of a story whose argumentative force consists in the narration of a fact (a fictitious but true-to-life fact) that the audience can understand well, and through which the audience can draw logical conclusions. Such conclusions, by dint of analogy, can be then applied to the initially unknown reality so as to understand some of its most important characteristics. The language of metaphor and parable, or if you prefer, of “narration,” is particularly fitting to the human mind. By the use of it, we find ourselves in a situation in which it is possible to identify a series of unchanging relations between human beings and things, or between human beings themselves, that goes beyond the changing objects of experience. These relations can be used as logical, cosmological, and anthropological coordinates in order to communicate a certain message. It is not surprising that the Word of God, which has also taken on the history and logic of such communicative and cognitive structures (which were taken on together with the true humanity of Christ) makes recourse to it as a kind of “fundamental human language.”
From a hermeneutic point of view, the language of analogy in Scripture has a special role, which must be distinguished from the symbolic one, which is also present. In the case of analogy an analogate is always referred to, whereas symbolic language refers to a reality beyond the limits of human discourse and language that requires completely new, non-analogous categories. But symbol remains incomplete without the help of analogy, since it recalls a reality independent of symbol itself, which carries the risk of mentally conceiving an infinite chain of symbols that never attains its real object.
3. Uses of Analogy in Theology. Analogies are widely used in Ecclesiology when speaking of the Church by resorting to “figures,” as used for example by the Magisterium during the Second Vatican Council (cf. Lumen Gentium, 6). The mystery of the Church, in fact, participates in the richness and transcendence of God, since she has her origin in the mystery of God the Father’s plan of salvation, and is revealed and accomplished through the missions of the Son and the Holy Spirit. In order to be expressed by words, the reality of the Church needs the analogy of intrinsic and extrinsic proportionality. Based on Sacred Scripture and the teachings of the Fathers of the Church, theology employs different images for the Church: a flock led by a shepherd, the Lord’s vine, a house built on a keystone which is Christ, the Kingdom, the family and abode of God, and, above all, God’s people and the Body of Christ. It should also be observed that one must use this last analogy not in a metaphorical, but in a proper, sense (cf. Lumen Gentium, 7; Pius XII, Mystici Corporis, June 29, 1943). The relationship between Christ and His Church is likened, in addition, to the relationship between bride and bridegroom, and also to the relationship of the head to its body. The peculiarity of such analogy-based images lies in the fact that none of them alone is adequate enough to express the mystery of the Church (she is visible and invisible, temporal and eternal, one, yet present in many places, distinct from her Bridegroom, and yet one with her Head, etc.), whereas all of them together play their parts in clarifying her character and properties.
Classical examples of the applications of analogy can be found in the teaching concerning the sacraments. As stages of the “Christian life,” they can be compared to the various phases of “natural life,” whether individual or social, according to an analogy of proper proportionality. In this way, Baptism is like the “birth” of Christian life, Confirmation is like “becoming an adult,” the Eucharist is like nourishment for one’s spiritual journey, and so on (cf. Thomas Aquinas, Summa Theologiae, III, q. 65). In the life of grace, then, sin is compared to death, so that one can understand its effects on the spiritual soul, in an analogy with what death brings about in the body. Even though such uses come with the limitations inherent in any type of comparison, they have undoubtedly aided our understanding of the mysteries of the faith and facilitated its diffusion.
Concerning the relationship between scientific thought and religious faith, the theological analogies used throughout history to clarify the relationship between faith and reason (or between philosophy and theology) are worthy of note. In medieval thought, philosophy is spoken of as the handmaiden of theology. Such a comparison, which has not infrequently been presented in a reductive and instrumental way, elicited an ironic response from Kant. Kant remarked that the handmaiden should have preceded her mistress, like a torch, in order to light the way. But the relationship between faith and reason has also been viewed as a marriage relationship (a typical image also used to describe the relationship between nature and grace, but one which stresses the greater dignity of the faith-husband pole). Contemporary theology in particular uses Marian and Christological analogies. For example, there is an analogy of the faith-word-Spirit that is accepted and embraced by an analogy of the reason-listening-Mary, thus “generating” the fruit of Theology (theology is used here in the strong sense of a wisdom which participates, by dint of Revelation, in the uncreated Wisdom of Christ). In a Christological analogy, reason and faith are seen in relation to each other as the human nature is seen in relation to the divine nature within the Person of the Divine Word made man (see Jesus Christ, Incarnation and doctrine of the Logos). As Christ’s humanity gives visible and historical expression to the divine nature and person, so philosophy and reason give theology and faith an indispensable language to express, in a clearly limited and incomplete, but authentic, way that which one knows by faith as belonging to the transcendence of God.
Concerning the history of theology and its relationship with scientific thought, Joseph Butler’s essay (1692-1752) titled The Analogy of Natural and Revealed Religion in the Constitution and Course of Nature (1736) must be mentioned. In it, the author presents the course of nature and of human history as a great analogy for the purpose of understanding the language and meaning of Christian revelation. This work became famous for its great influence on the thought of John Henry Newman (1801-1890) who often cited it in his books.
4. Analogia Fidei. A different meaning for the word analogy, at least when compared with its counterpart in Aristotelian-Thomistic philosophy, is that present in the expression “analogy of faith” (analogia fidei). It is first found in the letter of St. Paul to the Romans (“Let he who has the gift of prophecy make use of it according to the measure of faith,” Rom 12:6), where the Greek term analoghía is used in the sense of “measure” or “proportion.” In the Catholic tradition, this expression has taken on a technical character and signifies the inner coherence and harmony between the truths of faith that cannot contradict each other. The Catechism of the Catholic Church defines it today in the following way: “By ‘analogy of faith’ we mean the coherence of the truths of faith among themselves and within the whole plan of Revelation” (CCC 114). The analogy of faith guides us in our interpretation of the Old Testament in light of the New Testament. It is essential, indeed for a correct understanding of what the “development of dogma” means. Under the guidance of analogy, such development must not be viewed as a change in the content of truth but as the consistent deepening of understanding of the same revealed truth. Classic sources for this understanding can be found in St. Vincent of Lerins (cf. Commonitorium, 53: PL 50, 668) and in John Henry Newman (cf. An Essay on the Development of Christian Doctrine, 1845).
Reformed theologians, especially Karl Barth (1886-1968), made use of the expression analogia fidei to indicate the one and only source of knowledge about God, that of Divine Revelation, as opposed to analogia entis understood as the foundation of the path that allows natural reason to reach a non-revealed knowledge of God, a path that the Lutheran view rejects. Refusing the possibility that there could be an analogy-based knowledge of God arising from the experience of creatures, such theologians attempt to base the possibility and intelligibility of Revelation solely on the gift of grace. According to Karl Barth, “our human concepts and our human terms, in so far as they are ours and human, are totally incapable of expressing God and His mysteries; their aptitude for adequate and correct expression comes only from revelation.” One may say of God only what God says of Himself, that is, his Word, Christ. It should be observed, however, that such a perspective does not seem to solve in a convincing way the problem of how to ground the intelligibility and understanding of the revealed word, in that, even though we are helped by grace, our understanding of God is always expressed through our own words, which are the only words we have at our disposal. “It remains true that the notions chosen by Christ to introduce us to the divine mystery are still human notions. Christ borrowed them from human language, from the whole range of created realities. And it is on the basis of these realities, objects of human experience, that is effected a purification and development of meaning which are dictated by the necessities of revelation [...]. If Christ can utilize all the resources of the created universe to make us know God and the ways to God, it is because the word of creation has preceded and left a foundation for the word of revelation; it is because both one and the other have their principle in the same interior Word of God. The revelation of Christ presupposes the truth of analogy” (R. Latourelle, Theology of Revelation [New York: Alba House, 1966], pp. 366-367).
IV. Analogy and Science
Up until now, the concept of analogy has never been a part of any scientific theory, even though it has always in fact accompanied the progress of science from the outside, suggesting new avenues of research and new interpretations of results. This can be understood by considering the fact that modern science, which employs the Galilean method, is as mathematical as possible. In mathematics, as it has been developed up to now, every symbol used in the same proof must unambiguously correspond to a single definition. In the second place, even when direct use is not made of mathematics, univocity is systematically adopted so as to avoid the possibility of ambiguity or of error. It is, however, interesting to observe that in the last decades, research concerning the science of complexity and self-reference in different fields seems to demonstrate the theoretical limits of univocity and to suggest an analogy-based approach.
1. Analogy and Scientific Theory: The Experimental Sciences. The word “analogy” is often used by scientists in their qualitative descriptions of their results, even though it has never been a part of any scientific theory. In particular, analogies have proven to be useful throughout the history of science and have been used for a two-fold purpose: (a) to suggest a way to build a theory (a heuristic purpose), and; (b) to aid in interpreting an already developed theory which is similar to another theory because it has a similar mathematical structure (a hermeneutic or interpretative purpose). In both cases, analogy, however, does not play a direct part in the mathematical formulation of the theory, in that the symbols used continue to have an unambiguous definition. And, it must be emphasized that from the Aristotelian-Thomistic point of view, we are dealing with “analogies of proper proportionality,” that is, with similarities between relations. These similarities lie at the root of any possible model describing certain facts of experience. In particular, analogies, thus understood, can be said to be “material,” i.e., concerned with the “physical structure” of the systems to be described, or “formal,” i.e., concerned with the “mathematical laws” that describe and explain the determined behavior of physical systems.
“Material analogies” are useful in describing the properties of a system of which the internal structure is still unknown: One assumes that the unknown structure of the system might be similar to that of another well-known system and governed by a known law. In such cases, a “model” is proposed for the system to be to described. A familiar example, in physics, is provided by the model of “elastic rigid balls,” which is adopted as an approximate description of the behavior of gas molecules. In instances such as these, the similarity between the model and the physical phenomenon is supposed on the level of the structure of the material components; consequently, it is also expected that there will be a behavior that is similar to both systems, and similar laws supposed to govern both. This involves analogy of proper proportionality, which can be expressed by the following statement: “The rigid balls are to their dynamics as the molecules are to their own dynamics.” A similarity between the relationships (balls-dynamics and molecules-dynamics) is supposed, which is so tight as to legitimate the use of the same law to describe both systems within an acceptable margin of error.
On the other hand, “formal analogies” are not based on a model of the physical constituents of a certain system but on mathematical equations capable of describing its behavior without any hypothesis regarding the material structure governed by such laws (cf. Nagel, 1961). This way of proceeding is less natural to those who are not used to representing things in mathematical terms, whereas it is completely obvious to the mathematical physicist, accustomed to substituting the physical object in his or her mind with the mathematical equations that govern its behavior. In such cases, the similarity lies at the level of the “physical laws” governing the systems, which are supposed to be represented by the same equations within an acceptable range of error. In some cases, the formal equivalence of certain equations (which, however, have different physical interpretations of the same mathematical symbols) lead to new theories that are difficult to formulate without the aid of such a formal analogy. The most significant example of this is found in wave mechanics: The Schrödinger equation, which is the fundamental equation of Quantum mechanics, is obtained through an analogy between geometrical optics and classical analytical mechanics.
Aside from the heuristic aspect of analogy in the sciences, there is also a hermeneutic aspect. Analogy, in fact, can aid in the interpretation or explanation of the behavior of a system for which a certain model is adopted because it serves the purpose of reducing a lesser known phenomenon to a better known one. Suffice it to think of all of the microscopic models developed to explain the behavior of a macroscopic system: Kinetic theory, for example, gives, as a mechanical-statistical model of a thermodynamic macroscopic system, a detailed understanding of the macroscopic processes involving the state variables that characterize the system. In this case, the analogy which one forms is the following: “The kinetic model is to the laws of statistical mechanics as the thermodynamic system is to the laws of thermodynamics.” If we accept this analogy and assume that it is possible to identify the laws of kinetic theory with those of thermodynamics within an acceptable margin of error, we can obtain a relationship between the kinetic theory quantities and those of thermodynamics and thereby obtain a kinetic interpretation of the latter. One might think, for example, of the conceptual identification of the absolute thermodynamic temperature with the average translational kinetic energy of the molecules in a gas. In this case, analogy proves to be advantageous since it leads to a new understanding.
2. Analogy and Scientific Theory: The Mathematical Sciences. If in physics analogy does not play a direct role, except as a methodology that suggests from the outside how to build and interpret theories, formal analogy has a similar role in the development of new mathematical structures. The latter are intended to be based on simpler models for which one looks for a generalization that keeps some of their formal properties. It is important to keep in mind that in both physics and mathematics, analogy does not directly come into play as an “internal” element of the theoretical system but rather plays a role in the building and interpretation of science. It is true that in the internal structure of mathematics there are biunivocal relations between elements of distinct sets (isomorphisms, homeomorphisms, diffeomorphisms, etc.), but we are not dealing, in this case, with real analogies of proper proportionality in the sense above, but instead with structural identities. In these cases, there is a complete identification, and not only a similarity, between the relations. For this reason, such sets are indistinguishable as far as the properties of the structure are concerned, and it can be said that each of these sets is a “model” for the structure under consideration. In Aristotelian-Thomistic language, one could say that these models are like the “species” of the same “genus.” A well-known example is found in the so-called “Euclidean models” of non-Euclidean geometries and, more generally, in any mathematical model with an abstract structure. A non-Euclidean geometry, for example, can be thought of as abstractly defined by its axioms, regardless of the fact that there are different realizations of any one of its models. Nevertheless, as soon as we realize these models, they are not simply analogous but completely isomorphic to each other. This is because every relation between the elements of the model corresponds to an identical, not just a similar, relation between the elements of the other model. In the example of non-Euclidean geometries, we might think of the hyperbolic geometry of Bolyai that can have as a Euclidean model the Klein model in the plane (cf. Courant and Robbins, 1996).
Another well-known example of two mathematical models with the same structure is found in quantum mechanics, which admits a two-fold representation in two isomorphic Hilbert spaces; that is, the Schrödinger picture, formulated in terms of wave-functions in an L2 Hilbert space (square integrable functions), and that of Heisenberg, expressed in terms of l2 vectors expanded on an orthonormal basis of eigenfunctions (cf. Fano, 1971).
3. Analogy within Scientific Theory. Interest in analogy and research devoted to the development of a “scientific theory of analogy” and a “method of demonstration” based on the latter, seem to emerge inevitably from the study of systems (whether they are biological, chemical, physical, mathematical, logical, etc.) that are organized according to “hierarchical levels.” Some of these levels cannot be reduced to more elementary ones (cf. Cini, 1994) because they differ not only “quantitatively” but “qualitatively.” They have different natures but, at the same time, something real in common. In this case, it seems possible and useful to invoke the analogy of simple proportion or that of proper proportionality.
Up until now, the sciences have involved the search for components that act as fundamental “parts” or “building blocks” to explain the structure of the universe as a “whole,” assuming that the parts have the same nature as the whole (matter-radiation). In this scheme, the “building blocks” of the whole, according to the Standard Model, are “quarks” and the “gluons” that bind them, which form particles once believed to be elementary and which in turn form nuclei and atoms, which then form molecules, and finally, living cells and more complex living organisms. Every level of this scale is considered perfectly homogeneous with the other levels, made of the same matter, and considered of the same nature. In a sense that seems to contradict this way of framing the problem, qualitatively diversified (and, hence, irreducible to each other) levels have a tendency of emerging in the same system. If in fact one of these levels of organization (the “higher level”) were in some way deconstructable to other, more elementary ones (the “lower levels”), and if it could be reconstructed through an appropriate reconstruction of the latter, the higher level would not be “qualitatively” different but a simple “superimposition” of the lower levels. These different levels do not represent absolutely disparate properties that cannot be compared to each other, but constitute, instead, different ways of manifesting and realizing the same property, which can therefore be actuated in varying ways (that is, not univocally), but according to differentiated ways which are really related to each other (that is, analogically). In particular, we are faced with a two-fold modality in the relationship between the whole and its parts. On the one hand, we have a whole that is not reducible to the sum of its parts but possesses a new informative and unifying element that characterizes it as a whole. On the other hand, we have parts in which there exists something similar to the whole. Scientists commonly describe such a structure as “complex” (cf. Nicolis and Prigogine, 1989).
This situation is encountered today in every scientific discipline: The irreducibility of the levels is none other than a sign of the insufficiency of reductionism in formulating scientific theories that deal with complex systems (cf. Dalla Porta Xydias, 1997). The biological sciences, for example, have always dealt with properties of living beings that are not shared by non-living beings, even from the chemical and physical point of view. The behavior of a living being, even the simplest, cannot be described entirely by its constituent parts. On this level, the analysis of the constituent parts is no longer enough, and a study of the new level of the whole is necessary. A thorough study of a somewhat complex molecule, such as those found in a crystal lattice, or a study of the impurities in a crystal that determine the electrical properties of an entire semiconductor, to cite a few examples, have shown that even in the chemistry of non-living objects, the properties of the whole of a complex, composite structure cannot be deduced from the properties of the atoms that comprise it. The existence of molecular orbitals of fully shared electrons no longer allows us to think of those electrons as belonging to a single atom. In an electric conductor, the conduction electrons are in fact shared among all the atoms of the lattice. In the fields of physics and mathematics, the problem of the whole and of the parts is clearly of relevance in the two senses alluded to above: In particular, the “non-reducibility of the whole to the sum of the parts” is a consequence of the “non-linearity” of the differential equations that govern complex physical systems, whereas the self-replication of the whole in each of its parts is none other than a sign of “self-reference,” which is of great relevance to the logician and to the computer scientist. In fact, it seems that computer scientists were the ones to revive the by now classical problems of mathematical logic. Take, for instance, the problems related to Godel’s theorem concerning the consistency and completeness of axiomatic systems, or the problem of displaying fractal sets, in all their beauty, on the computer screen, which up to then had seemed to be “mathematical monsters” due to their infinitely winding boundary (as the Julia sets). Benôit Mandelbrot’s work served to rekindle interest in these problems. The field of fractal geometry began to develop when computers were utilized as laboratories in which mathematical experiments could be performed, in a way similar to the manner in which Archimedes, more than two thousand years ago, performed mechanical experiments so as to catch a glimpse of geometrical properties; only later would he seek a logical demonstration of such properties beginning with a set of axioms. Research in the field of artificial intelligence, in addition, has afforded an understanding of the fact that information can be found on various levels and that there can be different hierarchies of information. The lower level lies in the hardware of the machine, and the higher levels in the software. The programming language, in turn, contains the higher-level information that is meaningful for the programmer, which implies, in turn, lower level instructions mechanically executable by the machine, which cannot perceive their higher-level significance.
The program itself, as a whole, involves higher-level information related to the goal for which it was written (which lies in the mind of the programmer and in that of the user, and so on and so forth). In every scientific discipline, there seems to be a hierarchical structure of information related to the degree of complexity, and therefore of the unity of the structure studied. It therefore seems necessary to widen the scope of current scientific methodology and rationality so that the sciences can overcome the barriers erected by impossibility theorems such as that of Gödel (cf. De Giorgi et al., 1995).
The need for such a widening of scope is felt, first of all, in the study of “non-linearity.” From the mathematical point of view, and therefore from the point of view of all mathematical sciences, the impossibility of conceiving the whole as the sum of parts that are homogeneous with the whole (reductionism) is encountered in the field of non-linear differential equations for which, as it is well known, the sum of two or more solutions is not a solution, and conversely, for which every solution cannot be written as a linear combination of simpler solutions (which is the case with linear differential equations). Therefore, it is not possible, in general, to reduce the study of any given solution to simpler and already determined solutions in a non-linear system. Moreover, nature herself is described in great part by systems of non-linear equations, and linear solutions are only a first approximation. Non-linearity, therefore, introduces the concept of the “irreducibility” of certain solutions to simpler ones. The different solutions, however, have something in common: They are all solutions of the same equation.
In the second place, the problem of self-reference must be considered. By “self-referring,” a term originating in the field of logic but which is now universally used, one means an operation or system whose “whole” is completely replicated, i.e., is completely identical to itself, in its parts. Self-reference was discovered by the logicians of Ancient Greece who viewed it as a possible source of contradictions: One thinks of the famous “liar’s paradox” in its varying versions. For the same reason, modern logicians and mathematicians have carefully kept self-reference out of their axiomatic systems. Betrand Russell (1903) excluded it from his set theory, where it had emerged, for example, in the idea of the “self-inclusion” of certain sets of elements, which contain themselves. Kurt Gödel (1931) had succeeded, on the contrary, in exploiting precisely the possibility of creating paradoxes through self-reference for the purpose of proving the non-decidability of certain propositions of formal systems, such as the Principia Mathematica. He deduced the incompleteness of such a system and the impossibility of demonstrating its consistency from within the system. The use of the computer, which makes wide use of recursive algorithms, once again brought up the problem of self-reference in the fields of logic and mathematics. If it is clear that self-reference can lead to contradictions, it is likewise just as clear that this does not always, and does not necessarily, happen. We have a contradictory self-referring proposition when the predicate negates the truth of the proposition itself. For example: “This proposition is not true.” In like manner, we have a contradiction in set theory when we restrict the set of all sets not to include itself: “The set of all sets that do not contain themselves” is contradictory because the definition implies the set contains itself and does not contain itself at the same time. Nevertheless, certain contradictions can be avoided if one has a clear idea as to how self-reference can be applied to “differentiated levels” of the same object, and if one understands that it must be interpreted in an analogous, and not a univocal, sense. In this case, the “whole” cannot replicate into copies that are “identical to itself” but only “similar to itself.”
4. The First Steps towards a Theory of Analogy. In this subsection, I will set forth a few examples. The first example involves acknowledging a hierarchy of levels. Where does the contradiction lie in the self-referential proposition, “This proposition is not true,” or in the definition of the “set of all sets which do not contain themselves”? The contradiction arises because the “proposition” (“this proposition is not true”) and the subject “this proposition” are identified with one another, whereas, in reality, they are not the same proposition. They share the fact of being propositions in common, but they differ in the “manner” in which they are propositions. Likewise, the “set of all sets which do not contain themselves” is not a set in the same manner as the “sets which do not contain themselves.” The fact of identifying them (univocity) does not take into account the difference in the mode of being of the sets and therefore gives rise to the contradiction. In order to eliminate this contradiction at its root, Russell proposed clasifying the sets into “sets of differentiated types.” Sets of simple elements (that is, elements which cannot themselves be sets) belong to the first level (or type). Sets whose elements can only be sets of the first type belong to the second level (or type). Sets of the third type are those whose elements are sets of the second type, and so on and so forth. In this manner, one obtains a hierarchy of sets belonging to different well-defined levels. Thus, the term “set” can be said in different senses depending on whether or not one is speaking of sets of the first, of the second, or of another level. In a similar manner Gödel proposed a solution to the paradox of the universal class (the class of all sets) by distinguishing two types of classes: the “proper” ones that, by definition, are not allowed to be contained in wider classes, and the “ improper “classes (or sets) that may belong to a wider class. According to this, two different ways of being a class, both the universal class and the Russell class, result in being a proper class and they are no longer paradoxical (Gödel [1938], 1990, p. 38).
A similar classification is made for propositions. To summarize, we can say that one has made the first small step towards the concept of analogy due to a need arising from within the system. And, this first step consists of introducing levels, or differentiated ways, according to which the same term can be predicated, and the same object can exist, as, in our case, a set or a proposition. It must be observed in this kind of analogy that it is possible to establish similarities between relations of different types of sets, in a way similar to what happens in the analogy of proper proportionality.
Connected to the topic of self-reference, another important direction can be found in the field of fractal geometry. Fractals are geometrical structures that often have the noteworthy property of being “self-similar,” that is, they replicate themselves infinitely in each of their parts. In certain cases, as the curve of von Koch, such self-similarity is so perfect that it is impossible to determine the scale of magnification of a given level, since the replicated form is always the same in every part (cf. Peitgen and Richter, 1986). In other cases, such as the Mandelbrot set, there is not a complete self-similarity, but an infinite replication of itself into “similar” copies that are not exactly the same as the whole. Unlike what happens with sets or propositions, each of the parts of a fractal that replicate the whole are not, however, identical to the whole. But, though being distinct from the whole, it is nevertheless similar in form to it. In this case, it is preferable to speak of “self-reference” instead of “self-referentiality.” The latter geometrical example, even if it only gives a geometrical representation and is only an informal model, allows us to make a few considerations: (a) The geometrical structure is “similar” in its whole and in its parts, even if such a structure is actualized in slightly different ways in each part. Therefore, one cannot speak of complete identity, but only of similarity, as it so happens in the analogy of terms; (b) Every replicate is not properly speaking separable from the whole, but always subsists as a part of the primary whole. For this reason, the whole can be compared to a sort of “analogatum primum” (as in the “analogy of proportion”) on which every part physically depends, and; (c) One can establish relational correspondences between the parts and the whole, and among the parts with each other, as in the “analogy of proportionality.”
A further step can be made if we acknowledge the difference between “essence” and “existence.” The decisive leap, which is needed for analogy in the strict sense, is to begin thinking of “objects” (as the scientist would say) or of “entities” (as the philosopher would say) that are “similar” but irreducible to the same “mode of existence.” In order to characterize different “modes of existence,” one needs to avoid reducing existence to a simple logical “non-contradiction,” as is the tendency in formal logic. This kind of reduction makes the very notion of existence univocal, as it postulates that which is not contradictory, that is, that which is thinkable, exists, and exists only because it is not contradictory and only according to a single mode determined by its non-contradictory nature. In philosophical language, this position is equivalent to that of “the identity of essence and existence.” Gödel’s theorem has shown this kind of mathematical approach to be insufficient. The first attempt to refute mathematical formalism through the distinction between existence and essence can be found in the intuitionistic approach (cf. Basti and Perrone, 1996). The intuitionistic approach goes to an extreme position that denies the universal role of essence and overemphasizes that of existence. In fact, intuitionism posits the distinction between essence and existence by denying the “principle of the excluded middle”: In this way, proofs by contradiction are insufficient to prove the existence of a mathematical entity and are only capable of showing its logical impossibility. Existence must be proved with a constructive, finite method. Only what can be constructed with a finite number of operations exists. In other words, only this or that particular model can be constructed, and therefore the universal cannot be reached and remains a pure name (Nominalism). It is interesting to observe how both formalism and intuitionism assume a univocist mind-set, whereas the analogy-based solution, which acknowledges differentiated modes of existence of the universal and the particular, seems to be more appropriate (cf. Ibid., pp. 220-223). Research in this direction is still in the developmental phase.
Another scientific field in which the concept of analogy is being used is that of artificial intelligence, or better yet (and more generally), that of cognitive science, a wider field of study which involves not only problems dealing with machine learning but more generally problems in psychology, such as the mind-body relationship and the relationship between the mind and the brain (see Mind-body relationship). It is important to stress the effort made to overcome Cartesian dualism, a philosophical position according to which the mind and body are two separate “objects” joined together in a completely extrinsic way (cf. Basti, 1991, p. 105). On the one hand, computer science has in practice forced the revision of such a dualistic-mechanistic view. In fact, information inserted in a machine by means of software and input peripherals, which allows the machine to interact with the external world, is not a “thing” to be placed on the same footing as the hardware, but lies on a higher plane. The stratification of different levels of information allows one to establish relationships between entities of different levels (which recalls the analogy of proportion) and relations between these relationships (which recalls the analogy of proportionality). In this way, a structure of information emerges which is in a certain sense analogous. On the other hand, the experimental study of the mind-body relationship of human cognitive processes has convinced several scientists that the human mind works by analogy and not simply through an accumulation or extraction of information from a kind of data base (cf. Hofstadter et al., 1998). Consequently, with the aim of imitating human intelligence with a computer, a way is sought to reproduce this kind of analogy-based operation rather than simply a way to store a lot of specific information concerning the problem that the machine is to solve according to a reductionist mind-set that isolates single parts of an object from all the rest. Certainly, it is not enough to found a theory on a merely intuitive notion of analogy taken from its everyday meaning in common language. A rigorous theory of analogy is therefore needed.
V. The “Profundity” of Analogy
In conclusion, the genius of analogy, about which scientific interest is gradually increasing, lies in two fundamental aspects: (a) the fact that it distinguishes between qualitatively different, but really related, levels of the same entity; (b) the fact that it is inseparable from a true extra-mental reality that participates in the being. The Aristotelian-Thomistic concept of analogy, as we have striven to point out, acknowledges different hierarchical levels of being that differ by their very nature. For this reason, there are “things” and “principles” that allow these things “to be” and “to be what they are.” The “principles” and “things” are irreducible to each other for the very reason that they have different natures. At the same time, they are not completely heterogeneous with one another since they constitute different modes of the same being they possess in a differentiated way. In Latin terminology, quod indicates the “thing” and quo the principles by which the thing “is” and “is what it is,” that is, they possess their own characterizing properties. In the language of modern physics, we would say that that which is “observable” is a quod, whereas the quo is not only unobservable in practice, since it is in a certain sense confined in virtue of a certain infinite potential barrier (as a quark in an infinitely deep potential well), but it is also not observable due to a theoretical reason, since it is of a completely different nature from the observable. For example, if the “thing” is a particle, its constitutive “principle” is not a particle, or at least not in the same way, but in an analogous way. For this reason, the “principle” is not observable. The unobservable quo is introduced, not as a superfluous element of the theory (as if it were a hidden variable that could be eliminated), but as a simple principle which is in a certain sense necessary and inevitable in order to account for the observable phenomenon. It is clear that the mathematical sciences, in their current version, are not yet in a position to introduce into their language a quo that is irreducible by nature to a quantitative and relational quod. Nevertheless, in a “broad enough theory,” such an introduction seems possible and plausible. In this way, one can broaden a reductionist theory to a non-reductionist one that is able to accommodate principles that are irreducible and analogous to each other, without falling short of the demands for the rigor of a formal theory.
The second characteristic that we cannot afford to ignore in the theory of analogy is the close tie between logic and truth, or in other words, the relationship between, on the one side what is thought and, on the other side, what is extra-mental reality. Analogy can be fully understood only if it is considered a logical description of what does happen in the extra-mental reality of things, since it is capable of describing on the logical level what is a reality on the ontological level. Consequently, a broad theory with which one can formalize analogy in the sense understood here must be able to accommodate the distinction between both a purely logical-formal mode of existence (non-contradiction) and different real modes of existence (extra-mental) through the distinction between essence and existence.
Analogy is one of the tools that allow us to understand why essence and existence are not reducible to each other. In a certain way, it constitutes a response to the incompleteness of existential philosophy (the truth of the thing leads only to its emergence in the stream of existence and not to other questions) and of the essentialist philosophy (the truth of the thing consists only of the explanation of what it is, that is, its essence). Analogy also serves as a guide aiding us in the correct use of language and symbols as it prevents language from ending up in a continuous regress with no epistemological basis.
Documents of the Catholic Church related to the subject:
Scientific Aspects: T.F. ARECCHI, Lexicon of complexity (Firenze: Studio Editoriale Fiorentino, 1996); G. BASTI, Il rapporto mente-corpo nella filosofia e nella scienza (Bologna: Edizioni Studio Domenicano, 1991); J.M. BOCHENSKI, Ancient Formal Logic (Amsterdam: North-Holland, 1968); J.M. BOCHENSKI, A History of Formal Logic (New York: Chelsea Publishing Co., 1970); N.B. COCCHIARELLA, “Conceptual Realism as a Formal Ontology,” in Formal Ontology, R. POLI, P. SIMONS (eds.) (Dordrecht: Kluwer Academic Press, 1996), pp. 27-60; R. COURANT, H. ROBBINS, What is Mathematics? An Elementary Approach to Ideas and Methods (New York: Oxford University Press, 1996); E. DE GIORGI, M. FORTI, G. LENZI, and V.M. TORTORELLI, “Calcolo dei predicati e concetti metateorici in una teoria base dei Fondamenti della Matematica,” in Rend. Mat. Acc. Lincei vol. 6, n. 9 (1995); G. FANO, Mathematical Methods of Quantum Mechanics (New York: McGraw-Hill, 1971); K. GÖDEL, The Collected Works of Kurt Gödel. Volume II: Publications 1938-1974, S. FEFERMAN (ed.) (Oxford: Oxford University Press, 1990); D.R. HOFSTADTER, Gödel, Escher, Bach: An Eternal Golden Braid (London: Penguin, 2000); D.R. HOFSTADTER and the Fluid Analogies Research Group, Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought (London: Penguin, 1998); B.B. MANDELBROT, Fractals: Form, Chance and Dimension (San Francisco: W.H. FREEMAN & Co., 1977); E. Mendelson, Introduction to Mathematical Logic (London: Chapman & Hall, 1997); E. NAGEL, J. R. NEWMAN, Gödel’s Proof (London: Routledge, 1989); G. NICOLIS, I. PRIGOGINE, Exploring Complexity: An Introduction (New York: W.H. Freeman, 1989); H.O. PEITGEN, P.H. RICHTER, The Beauty of Fractals: Images of Complex Dynamical Systems (Berlin: Springer, 1986); R. PENROSE, Shadows of the Mind: A Search for the Missing Science of Consciousness (Reading: Vintage, 1995); M. RIGHETTI A. STRUMIA, L’arte del pensare. Appunti di logica (Bologna: Edizioni Studio Domenicano, 1998); B. RUSSEL, The Principles of Mathematics (New York: W.W. NORTON 1980).
Philosophy and History of Science: G. BASTI, A.L. PERRONE, Le radici forti del pensiero debole (Padova: Il Poligrafo, 1996); F. BERTELE', A. Olmi, A. Salucci, and A. STRUMIA, Scienza, analogia, astrazione. Tommaso d’Aquino e le scienze della complessità (Padova: Il Poligrafo, 1999); J. BOCHENSKI, “On Analogy,” in The Thomist, vol. XI (1948), pp. 424-447; M. CINI, Un paradiso perduto. Dall’universo delle leggi naturali al mondo dei processi evolutivi (Milano: Feltrinelli, 1994); J. MARITAIN, Distinguish to Unite: Or, The Degrees of Knowledge (Notre Dame, IN: University of Notre Dame Press, 1995); E. NAGEL, The Structure of Science: Problems in the Logic of Scientific Explanation (London: Routledge & Kegan Paul, 1961); A. STRUMIA, Introduzione alle filosofia delle scienze (Bologna: Edizioni Studio Domenicano, 1992).
Philosophy and Theology: V. FUSCO, “Parabola / Parabole,” in NDTB (1988), pp. 1081-1097; E. GILSON, The Spirit of Mediaeval Philosophy (Notre Dame, IN: University of Notre Dame Press, 1991); C. GRECO, S. MURATORE (eds.), La conoscenza simbolica (Cinisello Balsamo: San Paolo, 1998); G.P. KLUBERTANZ, St. Thomas Aquinas on Analogy: A Textual Analysis and Systematic Synthesis (Chicago: Loyola University Press, 1960); R. MCINERNY, Aquinas and Analogy (Washington, D.C.: The Catholic University of America Press, 1996); R. MCINERNY, The Logic of Analogy: An Interpretation of St. Thomas (The Hague: M. Nijhoff, 1961); B. MONDIN, The Principle of Analogy in Protestant and Catholic Theology (The Hauge: M. Nijhoff, 1963); J.H. NEWMAN, An Essay on the Development of Christian Doctrine (Notre Dame, IN: University of Notre Dame Press, 1989); P.A. SEQUERI, “Analogia,” in Dizionario Teologico Interdisciplinare (Torino: Marietti, 1977-1978), vol. I, pp. 341-351; T. TYN, Metafisica della sostanza. Partecipazione e analogia entis (Bologna: Edizioni Studio Domenicano, 1991). |
e0c3e24c6500902f | Vincent B. McKoy
Professor of Theoretical Chemistry, Emeritus
Prof B Vincent McKoy
B.S., Nova Scotia Technical University, 1960; Ph.D., Yale University, 1964. Noyes Research Instructor in Chemistry, Caltech, 1964-66; Assistant Professor of Theoretical Chemistry, 1967-69; Associate Professor, 1969-75; Professor, 1975-2016.
Research Areas: Chemistry
Research Interests
Rotationally resolved photoelectron spectra of molecules
Assistant: Elisha Okawa
The McKoy group's research focuses on the interactions of slow electrons with large molecules, and especially biomolecules such as the bases and other constituents of DNA. Low-energy electrons are known to produce single and double strand-break lesions in DNA, but the mechanisms behind this damage are not yet clear. Experimental evidence indicates that the electrons are trapped to form temporary negative ions, but major questions remain about the sites of trapping, the nature of the anion states formed, and the bond-breaking process. Are the electrons initially captured on the bases and subsequently transferred to the phosphate-sugar backbone, or do they attach directly to the backbone? Does the trapping happen in the ground electronic state, or must the electron first excite the molecule to a higher state? And which bonds actually break? Our goal is to use high-level computational methods to help answer these and related questions in electron-molecule dynamics.
To address such questions, one must solve the many-electron Schrödinger equation for the electron-molecule collision system. In this sense, what we do is very similar to the electronic structure studies of bound states that are carried out by many chemists using standard software packages like Gaussian. However, because we have a free electron, the boundary conditions on the solutions are different. To deal with the many complications that arise in dealing with an unbound electron, special-purpose programs are required. Moreover, because we are interested in large molecules such as nucleotides, we must design programs that run efficiently on parallel computer systems consisting of hundreds of processors.
For the past few years, we have doing initial studies of electron collisions with the DNA and RNA bases and with the related nucleosides and nucleotides, as well as with components of the DNA backbone. Although much remains to be done, these studies have provided useful insight into electron-capture mechanisms important at low collision energies. One surprising result was that some of the temporary ions formed by attaching electrons to the ground electronic state appear to be able to decay readily into triplet excited states, which may be one way that electrons promote disruption of DNA. Through continued program development and application, we intend to carry forward these studies to make closer connection to conditions in the living cell by considering larger moieties such as nucleotide pairs and by incorporating waters of hydration. |
917385674fe3fcda | On the cover: Harriss spiral
The golden ratio (1.6180339…) has a rather overblown reputation as a mathematical path to aesthetic beauty. It is often claimed that this number is a magic constant hidden in everything from flowers to human faces. In truth, this is an exaggeration, but the number does however have some beautiful properties.
The golden ratio, often written $\phi$, is equal to $(1+\sqrt5)/2$, and is one of the solutions of the equation $x^2=x+1$. The other solution of the equation is $(1-\sqrt5)/2$, or $-1/\phi$. One of the nicest properties of the golden ratio is self-similarity: if a square is removed from a golden rectangle (a rectangle with side lengths in the golden ratio), then the remaining rectangle will also be golden. By repeatedly drawing these squares on the remaining rectangle, we can draw a golden spiral.
Left: The large rectangle is golden. If a square (blue) is removed, then the remaining rectangle (green) is also golden. Right: A golden spiral. Image: Chalkdust.
Numbers that are a solution of a polynomial equation with integer coefficients are called algebraic numbers: the golden ratio is algebraic as it is a solution of $x^2=x+1$. At this point, it’s natural to wonder whether you can create interesting spirals like this with other algebraic numbers. Unsurprisingly (as otherwise, we wouldn’t be writing this article), there are other numbers that lead to pretty pictures.
The plastic ratio, $\rho=1.3247179…$, is the real solution of the equation $x\hspace{.5pt}^3=x+1$. Its exact value is
A plastic rectangle—a rectangle with side lengths in the plastic ratio—can be split into a square and two plastic rectangles. If this splitting is repeated on the smaller plastic rectangles and two arcs are drawn in each square, a spiral is formed. These particular arcs are chosen so that they line up with the corresponding arcs drawn in the smaller rectangles.
Left: The large rectangle is plastic and can be split into a square (blue) and two plastic rectangles (red) and (green). Centre: The two arcs drawn in each square. Right: A Harriss spiral.
This spiral is called the Harriss spiral, and is named after its creator Edmund Harriss. It is the shape that appears on the cover of this issue of Chalkdust, and we think its resemblance to a tree in bloom makes it perfect for spring-time. We also believe that its beauty shows that the golden ratio is a gateway into a world of mathematical creativity, not an end point. There must be other nice algebraic spirals out there, buried in the roots of polynomials. If you unearth a prize-winning specimen, let us know. You may even see it on the cover of a future issue!
On the cover: Hydrogen orbitals
Quantum mechanics has a reputation.
It’s notorious for being obtuse, difficult, confusing, and unintuitive. That reputation is… entirely deserved. I work on quantum systems full time for my job and I feel like I’ve barely scratched the surface of the mysteries it contains. But one other feature of quantum mechanics that’s often overlooked is how beautiful it can be.
So, for the cover of this issue, I wanted to share one aspect of quantum mechanics that I think is stunning. It’s a certain set of solutions to a differential equation: the orbitals of an electron in a hydrogen atom.
In school, you’re taught that electrons orbit the nucleus of an atom like a planet orbiting a star. This is mostly wrong. The main problem is that electrons, protons and neutrons aren’t little billiard balls, they exist as ‘clouds’ of probability.
To understand what a hydrogen atom really looks like, imagine a cloud of something whizzing around a single proton. The proton’s positive charge attracts and traps the negatively-charged something in what we call the proton’s potential well. Imagine that cloud is denser in some places and sparser in others. That cloud of something can be just one electron whose position has been smeared out. The density of the cloud at a point represents the probability of finding the electron at that point in space. The electron’s position may be smeared out over all space, but it has different odds of being found at different points in space. In fact, it’s usually exponentially less likely to be found outside the small, confined volume of the potential well.
The mathematical explanation for this is that our system is obeying the Schrödinger equation. For our case, it looks like this:
\Big(\dfrac{-\hbar^{\hspace{0.3mm}2}}{2m}\nabla^2 + V(\mathbf{r})\Big)\psi(\mathbf{r}) = E\psi(\mathbf{r}).
The Schrödinger equation is the foundational equation of quantum mechanics. It’s used to determine the wavefunction, $\psi(\mathbf{r})$, and the energy, $E$, of the components of the system. In this case the wavefunction represents the electron (with mass $m$) trapped in the electric potential well of the proton, which is represented by $V(\mathbf{r})$. The reduced Planck constant, $\hbar$, (often called “h-bar”) is a fundamental physical constant, and $\nabla^2$ is the Laplacian operator, which sums second derivatives over all the coordinates. The modulus squared of the wavefunction, $\vert\psi(\mathbf{r})\vert^2$, tells you what the density of that probability cloud is like: where are you more likely to find the electron?
Most people who do quantum mechanics for a living spend their time solving this equation and its variants, myself included. The problem is that this is really, really hard. The Schrödinger equation for a hydrogen atom has analytic solutions you can write down, but with almost all other physical systems, you aren’t so lucky. Once you have more than one electron, the complexity skyrockets. Understanding the analytic solutions form an important part of a physics undergraduate’s introduction to quantum mechanics, especially in my field of research. I work on finding approximate solutions to the Schrödinger equation for more complex systems.
To solve the Schrödinger equation, you can separate the wavefunction to get a radial part which is a function of the distance from the nucleus, $r$, and an angular part which is a function of the angles $(\theta, \phi)$. Both parts have multiple solutions, and it turns out that you need three labels to identify these solutions. We call these labels quantum numbers. Here, the three are called $n$, $l$, and $m$. Putting these two concepts together, we can say:
E\psi_{nlm}(\mathbf{r}) = R_n(r)Y_{lm}(\theta, \phi).
Plots of the solution with $n=0$, $l=5$ and $m=0$ to $m=5$
There’s lots of constraints on the allowed values of $n$, $l$, and $m$, but the most important one is that each number take whole number values only. This is where the ‘quantum’ in quantum mechanics comes from!
The quantum numbers each have physical interpretations: they loosely correspond to the three spatial coordinates. Here, $n$ corresponds to energy. Higher values of $n$ mean the electron has a larger amount of energy, which, due to how electric fields work, also exactly corresponds to a larger distance from the nucleus. That means $n$ is associated with the radial coordinate: the higher $n$ is, the further from the nucleus the electron can be.
Meanwhile, $l$ and $m$ correspond to angular momentum, and so they are associated with the angular coordinates. Roughly speaking, higher values of $l$ correspond to the electron ‘orbiting’ around the nucleus with greater energy (in a weird, quantum mechanical way that doesn’t really look like a planet orbiting a star). Changing $m$ means changing exactly how it orbits for a given value of $l$.
What this all means in practice is that by varying the three quantum numbers you get a huge variety of electron distributions. For instance, $n=1$, $l=0$, $m=0$ means that the electron isn’t orbiting the nucleus at all, so it’s most likely to be found right on top of the nucleus – opposite charges attract! When $n$, $l$, and $m$ are all large you get things like concentric sets of lobes of varying shapes and sizes.
Bringing it back to the cover, the pictures were all generated by making a 2D slice through the full 3D distribution at $y=0$. The brighter a given point is shaded, the higher the value of $\vert\psi(\mathbf{r})\vert^2$ is there—the higher the odds of finding the electron there are. The full 3D versions look like spheres, balloons, lobes, and other wild shapes. The 2D slices have a different sort of haunting beauty to them. The distributions can be concentric rings, orange slices, weird lobes, insect-like segments, and more.
The front cover is the 2D slice of the solution for $n=9$, $l=4$, $m=1$. The back cover contains all the allowed solutions from $n=1$, $l=0$, $m=0$ up to $n=9$, $l=7$, $m=7$.
These orbitals are beautiful by themselves as pieces of abstract maths, but they also provide profound insights into the strange quantum nature of our reality. They’re a testament to the amazing power physics and mathematics can have when they work together to help us understand our universe.
On the cover: Euclidean Egg III
Throughout my life I have made an informal study of natural phenomena, through drawing or just looking, in a spirit of curiosity. This long but unsystematic practice has given me an impression of the world around us as a dynamic and fertile system, driven by a ubiquitous tendency for spontaneous pattern formation (best understood in terms of the laws of physics) mitigated by an equally strong tendency for seemingly random variation.
It could be argued that the evolutionary process itself is driven by this tension between pattern and randomness, structure and chaos, order and disorder, theme and variation; without random mutation there would be stasis.
A bilaterally symmetric scorpion. Image: Rosa Pineda, CC BY-SA 3.0
In nature, we often see this ordering principle manifest itself as various kinds of symmetry or repetition. Most animate creatures exhibit external bilateral symmetry; insects, crustaceans, fish, birds and animals including ourselves all tend to be bilaterally symmetric.
In common with other sentient creatures, we humans navigate and comprehend the world both spatially and temporally through pattern recognition, and being highly social creatures we are particularly attuned to reading expression and meaning in faces and bodies. It is therefore no surprise that bilaterally symmetric shapes seem to have a unique sense of potential meaning and emotional impact for us.
Whilst mirror image symmetry gives structure, the actual pattern being reflected is often far more chaotic. Like a kaleidoscope, the coloured shards are arranged at random; order is created by repetition of these random arrangements. Think of the patterns on moths, butterflies, shield bugs, ladybirds and beetles, there is often very little order in the arrangement of marks on one half, the exquisitely satisfying order of the whole is created by reflection.
Euclidean Egg III, our featured cover art this issue. Image: Peter Randall-Page
In the ‘Euclidean Egg’ series of drawings, as with much of my other work, I have chosen to use a working process which has an inherent element of chance and randomness.
There are two ordering principles in these drawings: one is bilateral symmetry, the other is Euclidean geometry. I constructed a series of geometric egg shapes in such a way as to create a seamless curve where two arcs meet. The result is a faint line drawing of an egg shape together with the construction lines needed in order to create such a taut and smooth curve. These geometric eggs by their very nature have mirror image symmetry around a vertical axis.
Folding the paper along this vertical axis and using paint introduces an element of chance. Using a pipette dropper, I spread ochre paint onto one of the areas between the construction lines on one half of the drawing. Folding the paper in half along the axis of symmetry creates two identical blobs of paint which, whilst roughly contained within the construction lines, inevitably have a somewhat random outline, reminiscent of the inlets and peninsulas of a Scandinavian island. I then add another blob of paint and continue the process, gradually building the drawing; blot by blot, fold by fold.
This process is akin to the psychoanalytic evaluation technique developed by the Swiss psychoanalyst Hermann Rorschach in 1921. Rorschach’s theory was predicted on our psychological sensitivity to bilateral symmetric shapes. He developed a series of 10 mirror image ink blots which are shown to the subject, who is then asked to say what they see in them. Their observations are then used as a way of analysing the subjects subjective response to what are effectively totally random, but highly symmetric, shapes.
Rorschach’s ink blot test has gone in and out of favour as a psychoanalytic tool during the last century but for me, our reaction to his ambiguous symmetric forms reveals something about the way in which our perception of the world is driven by subjective projection of feeling as well as objective analysis and observation. We read meaning into the world as well as taking meaning from what we perceive.
A construction of the simplest Euclidean egg
My fundamental concern in making art is an exploration of what makes us tick, the emotional subtext to our everyday experience. The world enters our consciousness as emotion and expression as well as information and knowledge. We respond to shapes and colours, forms and spaces, poetry and music in ways which can be difficult to analyse or quantify.
Whilst we have so many ways of communicating with one another (not least language itself), the medium of visual art is uniquely capable of exploring these often intangible emotional responses.
In this particular drawing I am attempting to reconcile order and randomness, Euclid and Rorschach. My attention is concentrated on making a satisfactory balance between the ‘theory’ of pure abstract geometry with the ‘practice’ of what happens in the real world (in this case, the viscosity of the paint as well as the texture and absorbency of the paper are all determining factors).
Being preoccupied with my attempt to reconcile these polarities is strangely liberating. The task involves innumerable decisions and appraisals which is conducive to a spontaneous and playful approach. In fact, play is an important concept for me. Play can be unselfconscious and create fresh associations and ideas. In order to play well, however, one needs a playground. Football without rules and a finite pitch would neither be fun to play nor interesting to watch.
Although rooted in a study of natural phenomena, my work is less concerned with reproducing existing forms than with trying to grasp the underlying dynamics which determine the shapes and forms we see around us and to use these dynamic processes to create new objects which are both novel and familiar.
In the words of the philosopher and art historian Ananda K Coomaraswamy in his 1956 essay The Transformation of Nature in Art, “art is ideal in the mathematical sense like nature, not in appearance but in operation.”
On the cover: dragon curves
Take a long strip of paper. Fold it in half in the same direction a few times. Unfold it and look at the shape the edge of the paper makes. If you folded the paper $n$ times, then the edge will make an order $n$ dragon curve, so called because it faintly resembles a dragon. Each of the curves shown on the cover of issue 05 of Chalkdust, and in the header box above, is an order 10 dragon curve.
Left: Folding a strip of paper in half four times leads to an order four dragon curve (after rounding the corners). Right: A level 10 dragon curve resembling a dragon
Continue reading
Mathematics: queen of the arts?
In the brief tradition of Chalkdust cover articles there is a developing discussion of how mathematics and art are related.
Art is simply the making of representations. Art happens when a person has an idea or a vision that exists in their imagination (the mind’s eye) and is impelled to communicate said idea by making a visible manifestation (representation) of it in the material world. The idea or vision on its own is not art. Art occurs amid the struggle to make a representation of the idea that the artist can show to other people. Art may be relatively `fine’ or popular, conceptual or objective, highbrow or applied, yet still fall within this definition. Judgements about the quality of art are made largely by consensus among the cognoscenti in a given art milieu. These judgements are subject to change over time as the perception of works of art are always modified by the current `cultural environment’ and fashion.
The image Central Quadratic explains itself, I hope, as a celebration of analytic geometry.
So artists and mathematicians share the `having of ideas’. But what then? Mathematicians communicate their ideas—yes. But ideas in maths take the form of theorems or conjectures about numbers, space or other abstract entities. The quality of these ideas is first assessed by proof. Can the idea be shown to be true? And second, if the idea is true, is it interesting? That is, does it usefully contribute to the mass of existing mathematics? Communication of mathematical ideas may require the invention of new symbols or diagrammatic forms, etc, but these are in the nature of being a new language, not art.
In my view then, art and mathematics share the magical process of `idea getting’ but essentially differ in where they go with those ideas. If maths is to be considered an art, it would have to be a sort of `super-art’ or art `to a higher power’. Easier, I think, to class mathematics as the science of number, space, shape and structure, etc—the abstract entities that exist in our minds.
Imagine an intelligent alien’s perception of our arts and our mathematics. Our art would be more or less incomprehensible, depending on how alien the being was; but our maths would be as true for the alien as it is for us. Furthermore, good mathematics will not diminish with time or go out of fashion.
There is an affinity between some mathematicians and some artists. Certainly, it is a most pernicious error that scientific and artistic talent exclude each other—an idea unfortunately common among school counsellors. The common ground between art and science/maths that leads us to the `getting of ideas’ is the activity we call play. The thing of it—the thrilling thing, the magical thing—is the moment when one discovers a new idea, or pattern, or conceptual framework, or whatever: the eureka moment! And are these moments not usually approached through playing in the mind with new combinations and orderings of existing mental constructs?
In ray tracing, each ray is used to decide the colour of a pixel on the image plane.
Spheres was created using ray tracing.
Spheres was created using ray tracing.
I had one of my most memorable eureka moments sometime in 1971 while sitting on a dead tree in Epping forest. At that time, I had been collaborating on an automatic projective line drawing program with `hidden line removal’, going where Autocad later arrived. I was considering algorithmic approaches to colouring surfaces in projective drawings. I realised that if I thought of objects in the scene as being represented mathematically as arrays of vertices and planes in some coordinate space, then I could solve for the equation of the line going from an eyepoint through a particular pixel in the image plane and into the scene (as in the diagram). From the equation of the line, I could find the closest surface along the path and then compute the colour and illumination value for that pixel based on the defined colour on the surface, along with its relationship to any light source or other light-emitting surfaces. And so I had invented ray tracing—the foundation of all computer generated synthetic imaging for special effects in cinema, television and gaming. Of course, I neither invented it first nor alone—and I certainly had neither the persistence nor vision to pursue ray tracing to practical or rewarding development. But its discovery was a thrill, as were the few simple pictures I made using the technique in a primitive manner on the pen plotter available.
As spheres have become my most persistent motif, I will end with two more related works that play on the division and articulation of spherical surfaces: Sphere Architecture and Star Sphere.
[ Pictures: Central Quadratic: Used with permission from UCL Art Museum, University College London; Spheres, Sphere Architecture and Star Sphere: Used with permission from John Crabtree ]
Spherical Dendrite by Mark J Stock
We are surrounded by complex structures and systems that appear to be lawless and disorderly. Mathematicians try to look for patterns in the seemingly chaotic behaviour and build models that are simple, and yet have the capacity to accurately predict the reality around us. But can a scientific or mathematical model have any artistic value? It seems that the answer is yes. There is a group of digital and algorithmic artists that use science and computational mathematics to create visual art. However, there is an even smaller group of people whose art and science coincide. Meet Mark J Stock. Continue reading
Fermat Point by Suman Vaze
Fermat Point by Suman Vaze
Fermat Point by Suman Vaze
Suman Vaze sits on her small balcony in crowded, bustling Hong Kong, with a view, just about, of a beautiful Chinese Banyan tree tenaciously growing on a steep stony slope, and paints mathematics. Inspired by the abstract expressionism of Rothko, the radical and influential work of Picasso, and the experimental models of Calder, she fully embodies Hardy’s belief that mathematicians are ‘maker[s] of patterns’. Our front cover is one of her pieces: the bold colours proclaim the eponymous Fermat Point – the point that minimises the total distance to each vertex of a triangle – along with its geometrical construction. Add an equilateral triangle to each side of the original triangle then draw a line connecting the new vertex of the equilateral triangle to the opposite vertex of the original: the intersection of these lines gives the Fermat point. Not only do these lines all have the same length, but the circumscribed circles of the three equilateral triangles will also intersect at the Fermat point.
Continue reading
The Symposium of the Muses
This issue’s cover picture is a creation of Anthony Lee, a young British artist, who has always been fascinated by exploring the possibilities of creating images through light. In Anthony’s eyes, this experimental process is the result of “the idea of an ephemeral substance or state, the idea that the captured moment was never intended to last or be repeated. In my light images neither the light nor the shape can last and yet they stay captured in the image I present.”
It is interesting to notice where both the artistic and scientific processes intersect and interact with each other – and where they do not. The artist, Anthony, is looking for a way to use scientific knowledge to express his personal emotions and inner thrills; and the resultant art is the outcome and purpose that elevates and distinguishes the science. And yet Anthony is bending and filling reality with his own meanings – his “ephemeral” ideas of light and shape – that are changeable and unique to him. Contrast this with the aims of scientists, who look for permanent truths that affect every observer, irrespective of their uniqueness in this space-time continuum.
Continue reading |
ad9c58cf6af7e36a | Hydrogen atom
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Hydrogen atom, 1H
Hydrogen 1.svg
Name, symbolprotium,1H
Nuclide data
Natural abundance99.985%
Isotope mass1.007825 u
Excess energy7288.969± 0.001 keV
Binding energy0.000± 0.0000 keV
Isotopes of hydrogen
Complete table of nuclides
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 baryonic mass of the universe.[1]
In everyday life on Earth, isolated hydrogen atoms (called "atomic hydrogen") are extremely rare. Instead, a hydrogen atom tends to combine with other atoms in compounds, or with another hydrogen atom 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).
Atomic spectroscopy shows that there is a discrete infinite set of states in which a hydrogen (or any) atom can exist, contrary to the predictions of classical physics. Attempts to develop a theoretical understanding of the states of the hydrogen atom have been important to the history of quantum mechanics, since all other atoms can be roughly understood by knowing in detail about this simplest atomic structure.
The most abundant isotope, hydrogen-1, protium, or light hydrogen, contains no neutrons and is simply a proton and an electron. Protium is stable and makes up 99.985% of naturally occurring hydrogen atoms.[2]
Deuterium contains one neutron and one proton. Deuterium is stable and makes up 0.0156% of naturally occurring hydrogen[2] and is used in industrial processes like nuclear reactors and Nuclear Magnetic Resonance.
Tritium contains two neutrons and one proton and is not stable, decaying with a half-life of 12.32 years. Because of the short half life, tritium does not exist in nature except in trace amounts.
Higher isotopes of hydrogen are only created in artificial accelerators and reactors and have half lives around the order of 10−22 (0.0000000000000000000001) second.
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.
Hydrogen ion[edit]
Lone neutral hydrogen atoms are rare under normal conditions. However, neutral hydrogen is common when it is covalently bound to another atom, and hydrogen atoms can also exist in several cationic and anionic forms.
If a neutral hydrogen atom loses its electron, it becomes a cation. The resulting atom, which consists solely of a proton for the usual isotope, is written as "H+" and sometimes called hydron. Free protons are common in the interstellar medium, and solar wind. In the context of aqueous solutions of classical Brønsted–Lowry acids, such as hydrochloric acid, it is actually hydronium, H3O+, that is meant. Instead of a literal ionized single hydrogen atom being formed, the acid transfers the hydrogen to H2O, forming H3O+.
If instead a hydrogen atom gains a second electron, it becomes an anion. The hydrogen anion is written as "H" and called hydride.
Theoretical analysis[edit]
Failed classical description[edit]
Experiments by Ernest Rutherford in 1909 showed the structure of the atom to be a dense, positive nucleus with a tenuous negative charge cloud around it. This immediately raised questions about how such a system could be stable. Classical electromagnetism had shown that any accelerating charge radiates energy, as shown by the Larmor formula. If the electron is assumed to orbit in a perfect circle and radiates energy continuously, the electron would rapidly spiral into the nucleus with a fall time of:[3]
Where is the Bohr radius and is the classical electron radius. If this were true, all atoms would instantly collapse, however atoms seem to be stable. Furthermore, the spiral inward would release a smear of electromagnetic frequencies as the orbit got smaller. Instead, atoms were observed to only emit discrete frequencies of radiation. The resolution would lie in the development of quantum mechanics.
Bohr–Sommerfeld Model[edit]
In 1913, Niels Bohr obtained the energy levels and spectral frequencies of the hydrogen atom after making a number of simple assumptions in order to correct the failed classical model. The assumptions included:
1. Electrons can only be in certain, discrete circular orbits or stationary states, thereby having a discrete set of possible radii and energies.
2. Electrons do not emit radiation while in one of these stationary states.
3. An electron can gain or lose energy by jumping from one discrete orbital to another.
Bohr supposed that the electron's angular momentum is quantized with possible values:
and is Planck constant over . He also supposed that the centripetal force which keeps the electron in its orbit is provided by the Coulomb force, and that energy is conserved. Bohr derived the energy of each orbit of the hydrogen atom to be:[4]
where is the electron mass, is the electron charge, is the vacuum permittivity, and is the quantum number (now known as the principal quantum number). Bohr's predictions matched experiments measuring the hydrogen spectral series to the first order, giving more confidence to a theory that used quantized values.
For , the value
is called the Rydberg unit of energy. It is related to the Rydberg constant of atomic physics by
The exact value of the Rydberg constant assumes that the nucleus is infinitely massive with respect to the electron. For hydrogen-1, hydrogen-2 (deuterium), and hydrogen-3 (tritium) which have finite mass, the constant must be slightly modified to use the reduced mass of the system, rather than simply the mass of the electron. This includes the kinetic energy of the nucleus in the problem, because the total (electron plus nuclear) kinetic energy is equivalent to the kinetic energy of the reduced mass moving with a velocity equal to the electron velocity relative to the nucleus. However, since the nucleus is much heavier than the electron, the electron mass and reduced mass are nearly the same. The Rydberg constant RM for a hydrogen atom (one electron), R is given by
where is the mass of the atomic nucleus. For hydrogen-1, the quantity 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.
There were still problems with Bohr's model:
1. it failed to predict other spectral details such as fine structure and hyperfine structure
2. it could only predict energy levels with any accuracy for single–electron atoms (hydrogen–like atoms)
3. the predicted values were only correct to , where is the fine-structure constant.
Most of these shortcomings were resolved by Arnold Sommerfeld's modification of the Bohr model. Sommerfeld introduced two additional degrees of freedom, allowing an electron to move on an elliptical orbit characterized by its eccentricity and declination with respect to a chosen axis. This introduced two additional quantum numbers, which correspond to the orbital angular momentum and its projection on the chosen axis. Thus the correct multiplicity of states (except for the factor 2 accounting for the yet unknown electron spin) was found. Further, by applying special relativity to the elliptic orbits, Sommerfeld succeeded in deriving the correct expression for the fine structure of hydrogen spectra (which happens to be exactly the same as in the most elaborate Dirac theory). However, some observed phenomena, such as the anomalous Zeeman effect, remained unexplained. These issues were resolved with the full development of quantum mechanics and the Dirac equation. It is often alleged that the Schrödinger equation is superior to the Bohr–Sommerfeld theory in describing hydrogen atom. This is not the case, as most of the results of both approaches coincide or are very close (a remarkable exception is the problem of hydrogen atom in crossed electric and magnetic fields, which cannot be self-consistently solved in the framework of the Bohr–Sommerfeld theory), and in both theories the main shortcomings result from the absence of the electron spin. It was the complete failure of the Bohr–Sommerfeld theory to explain many-electron systems (such as helium atom or hydrogen molecule) which demonstrated its inadequacy in describing quantum phenomena.
Schrödinger equation[edit]
The Schrödinger equation allows one to calculate the stationary states and also the time evolution of quantum systems. Exact analytical answers are available for the nonrelativistic hydrogen atom. Before we go to present a formal account, here we give an elementary overview.
Given that the hydrogen atom contains a nucleus and an electron, quantum mechanics allows one to predict the probability of finding the electron at any given radial distance . It is given by the square of a mathematical function known as the "wavefunction," which is a solution of the Schrödinger equation. The lowest energy equilibrium state of the hydrogen atom is known as the ground state. The ground state wave function is known as the wavefunction. It is written as:
Here, is the numerical value of the Bohr radius. The probability of finding the electron at a distance in any radial direction is the squared value of the wavefunction:
The wavefunction is spherically symmetric, and the surface area of a shell at distance is , so the total probability of the electron being in a shell at a distance and thickness is
It turns out that this is a maximum at . That is, the Bohr picture of an electron orbiting the nucleus at radius is recovered as a statistically valid result. However, although the electron is most likely to be on a Bohr orbit, there is a finite probability that the electron may be at any other place , with the probability indicated by the square of the wavefunction. Since the probability of finding the electron somewhere in the whole volume is unity, the integral of is unity. Then we say that the wavefunction is properly normalized.
As discussed below, the ground state is also indicated by the quantum numbers . The second lowest energy states, just above the ground state, are given by the quantum numbers ; ; and . These states all have the same energy and are known as the and states. There is one state:
and there are three states:
An electron in the or state is most likely to be found in the second Bohr orbit with energy given by the Bohr formula.
The Hamiltonian of the hydrogen atom is the radial kinetic energy operator and Coulomb attraction force between the positive proton and negative electron. Using the time-independent Schrödinger equation, ignoring all spin-coupling interactions and using the reduced mass , the equation is written as:
Expanding the Laplacian in spherical coordinates:
3D illustration of the eigenstate . Electrons in this state are 45% likely to be found within the solid body shown.
is the reduced Bohr radius, ,
is a generalized Laguerre polynomial of degree n − 1, and
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,[6] and Mathematica.[7] In other places, the Laguerre polynomial includes a factor of ,[8] or the generalized Laguerre polynomial appearing in the hydrogen wave function is instead.[9]
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:
where is the state represented by the wavefunction in Dirac notation, and is the Kronecker delta function.[10]
The wavefunctions in momentum space are related to the wavefunctions in position space through a Fourier transform
which, for the bound states, results in [11]
where denotes a Gegenbauer polynomial and is in units of .
The solutions to the Schrödinger equation for hydrogen are analytical, giving a simple expression for the hydrogen energy levels and thus the frequencies of the hydrogen spectral lines and fully reproduced the Bohr model and went beyond it. It also yields two other quantum numbers and 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.
Since the Schrödinger equation is only valid for non-relativistic quantum mechanics, the solutions it yields for the hydrogen atom are not entirely correct. The Dirac equation of relativistic quantum theory improves these solutions (see below).
Results of Schrödinger equation[edit]
The solution of the Schrödinger equation (wave equation) 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 an (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.
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 fine structure expression:[12]
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. This formula represents a small correction to the energy obtained by Bohr and Schrödinger as given above. 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). It is worth noting that this expression was first obtained by A. Sommerfeld in 1916 based on the relativistic version of the old Bohr theory. Sommerfeld has however used different notation for the quantum numbers.
Coherent states[edit]
The coherent states have been proposed as[13]
which satisfies and takes the form
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).
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 equation in spherical coordinates.)
The quantum numbers determine the layout of these nodes.[14] There are:
• total nodes,
• of which are angular nodes:
• angular nodes go around the axis (in the xy plane). (The figure above does not show these nodes since it plots cross-sections through the xz-plane.)
• (the remaining angular nodes) occur on the (vertical) axis.
• (the remaining non-angular nodes) are radial nodes.
Features going beyond the Schrödinger solution[edit]
There are several important effects that are neglected by the Schrödinger equation and which are responsible for certain small but measurable deviations of the real spectral lines from the predicted ones:
• 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.
• Even when there is no external magnetic field, in the inertial frame of the moving electron, the electromagnetic field of the nucleus has a magnetic component. The spin of the electron has an associated magnetic moment which interacts with this magnetic field. This effect is also explained by special relativity, and it leads to the so-called spin-orbit coupling, i.e., an interaction between the electron's orbital motion around the nucleus, and its spin.
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).
For these developments, it was essential that the solution of the Dirac equation for the hydrogen atom could be worked out exactly, such that any experimentally observed deviation had to be taken seriously as a signal of failure of the theory.
Alternatives to the Schrödinger theory[edit]
In the language of Heisenberg's matrix mechanics, the hydrogen atom was first solved by Wolfgang Pauli[15] using a rotational symmetry in four dimensions [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.[16]
In 1979 the (non relativistic) hydrogen atom was solved for the first time within Feynman's path integral formulation of quantum mechanics.[17][18] This work greatly extended the range of applicability of Feynman's method.
See also[edit]
1. ^ Palmer, D. (13 September 1997). "Hydrogen in the Universe". NASA. Archived from the original on 29 October 2014. Retrieved 23 February 2017.
2. ^ a b Housecroft, Catherine E.; Sharpe, Alan G. (2005). Inorganic Chemistry (2nd ed.). Pearson Prentice-Hall. p. 237. ISBN 0130-39913-2.
3. ^ Olsen, James; McDonald, Kirk (7 March 2005). "Classical Lifetime of a Bohr Atom" (PDF). Joseph Henry Laboratories, Princeton University.
4. ^ "Derivation of Bohr's Equations for the One-electron Atom" (PDF). University of Massachusetts Boston.
5. ^ Eite Tiesinga, Peter J. Mohr, David B. Newell, and Barry N. Taylor (2019), "The 2018 CODATA Recommended Values of the Fundamental Physical Constants" (Web Version 8.0). Database developed by J. Baker, M. Douma, and S. Kotochigova. Available at http://physics.nist.gov/constants, National Institute of Standards and Technology, Gaithersburg, MD 20899. Link to R, Link to hcR
6. ^ Messiah, Albert (1999). Quantum Mechanics. New York: Dover. p. 1136. ISBN 0-486-40924-4.
7. ^ LaguerreL. Wolfram Mathematica page
8. ^ Griffiths, p. 152
9. ^ Condon and Shortley (1963). The Theory of Atomic Spectra. London: Cambridge. p. 441.
10. ^ Griffiths, Ch. 4 p. 89
11. ^ Bransden, B. H.; Joachain, C. J. (1983). Physics of Atoms and Molecules. Longman. p. Appendix 5. ISBN 0-582-44401-2.
12. ^ Sommerfeld, Arnold (1919). Atombau und Spektrallinien [Atomic Structure and Spectral Lines]. Braunschweig: Friedrich Vieweg und Sohn. ISBN 3-87144-484-7. German English
13. ^ Klauder, John R (21 June 1996). "Coherent states for the hydrogen atom". Journal of Physics A: Mathematical and General. 29 (12): L293–L298. arXiv:quant-ph/9511033. doi:10.1088/0305-4470/29/12/002. Retrieved 18 June 2019.
14. ^ Summary of atomic quantum numbers. Lecture notes. 28 July 2006
15. ^ Pauli, W (1926). "Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik". Zeitschrift für Physik. 36 (5): 336–363. Bibcode:1926ZPhy...36..336P. doi:10.1007/BF01450175.
16. ^ Kleinert H. (1968). "Group Dynamics of the Hydrogen Atom" (PDF). Lectures in Theoretical Physics, edited by W.E. Brittin and A.O. Barut, Gordon and Breach, N.Y. 1968: 427–482.
17. ^ Duru I.H., Kleinert H. (1979). "Solution of the path integral for the H-atom" (PDF). Physics Letters B. 84 (2): 185–188. Bibcode:1979PhLB...84..185D. doi:10.1016/0370-2693(79)90280-6.
18. ^ Duru I.H., Kleinert H. (1982). "Quantum Mechanics of H-Atom from Path Integrals" (PDF). Fortschr. Phys. 30 (2): 401–435. Bibcode:1982ForPh..30..401D. doi:10.1002/prop.19820300802.
External links[edit]
(none, lightest possible)
Hydrogen atom is an
isotope of hydrogen
Decay product of:
free neutron
Decay chain
of hydrogen atom
Decays to: |
3273ac6d7c1c774d | A potted history of quantum mechanics
There’s some ambiguity when it comes to quantum mechanics. Some people apply the term widely, others apply it to the theory that was developed in the 1920s to replace the old quantum theory. There’s some ambiguity with that too, in that the old quantum theory was primarily an atomic model proposed by Niels Bohr in 1913 and extended by Arnold Sommerfeld in 1916. It didn’t include the quantum nature of light, which arguably began with Max Planck’s black-body paper in 1900. Or with Albert Einstein’s photoelectric paper in 1905. Bohr didn’t agree with the quantum nature of light, and was still arguing about it in the early 1920s. He was proven wrong, and the old quantum theory was replaced by a new quantum theory which included aspects of an even older quantum theory. As for what it’s called, Max Born called it quantum mechanics, so that will do for me.
Compton scattering
As for when this quantum mechanics began, it’s hard to say. Wolfgang Pauli was talking about the Bohr magneton in 1920. Arthur Compton was talking about electron spin in his paper the magnetic electron in 1921. He referred to the Parson electron or magneton which featured a rotation with a “peripheral velocity of the order of that of light”. Compton said “we may suppose with Nicholson that instead of being a ring of electricity, the electron has a more nearly isotropic form”. The Stern-Gerlach experiment was performed in 1922. It demonstrates that particles possess an intrinsic angular momentum that is closely analogous to the angular momentum of a classically spinning object, but that takes only certain quantized values”. 1922 was also when Arthur Compton discovered what’s now known as Compton scattering. Peter Debye was doing similar work at much the same time, so it’s sometimes referred to as the Compton-Debye effect. But see Compton’s 1923 paper A Quantum Theory of the Scattering of X-rays by Light Elements. The scattered X-rays had an increased wavelength, and the increase in the wavelength depended on the scattering angle. The results matched a model wherein one light quantum interacted with one electron:
Image from Rod Nave’s hyperphysics
Compton had demonstrated the quantum nature of light. See his Nobel lecture for a nice slice of history. He described how X-rays were once thought to be comprised of “streams of little bullets called ‘neutrons’, and weren’t thought to be a form of light. But then he referred to Barkla and Stenström and others, and said “it would take a bold man indeed to suggest, in light of these experiments, that they differ in nature from ordinary light”. X-rays were light. But to account for the change in wavelength of the scattered rays, however, we have had to adopt a wholly different picture of the scattering process, as shown in Fig. 9. Here we do not think of the X-rays as waves but as light corpuscles, quanta, or, as we may call them, photons”. Light has a wavelength, but emission is directed. The energy of light is discontinuously distributed in space, as energy quanta which move without dividing. Einstein was right. Light consists of photons.
The wave nature of matter
Another important step took place in 1923 when Louis de Broglie sent a letter to Nature on waves and quanta. He said he’d ”been able to show that the stability conditions of the trajectories in Bohr’s atom express that the wave is tuned with the length of the closed path”. His thesis followed in 1924. See the English translation by Al Kracklauer. It’s on the theory of quanta, and it includes a useful historical survey. De Broglie referred to Christiaan Huygens and his undulatory theory of light, and to Isaac Newton and his corpuscular theory of light. He also said when Augustin-Jean Fresnel developed his “beautiful” elastic theory of light propagation, Newton’s ideas lost credibility. De Broglie then talked about Max Planck and the energy exchange between resonator and radiation taking place in integer multiples of hν. He talked about Einstein and the photoelectric effect, saying Einstein instinctively understood that one must consider the corpuscular nature of light. He referred to his brother Maurice De Broglie along with Rutherford and Ellis and said photo-electric studies “have further substantiated the corpuscular nature of radiation”. But he also referred to von Laue, Debye, and W L Bragg and said “the wave picture can also point to successes”. He referred to Compton too, and said “the time appears to have arrived, to attempt to unify the corpuscular and undulatory approaches”.
Bohr-de Broglie atom by Kenneth Snelson
De Broglie said the fundamental idea pertaining to quanta is the impossibility to consider an isolated quantity of energy without associating a particular frequency to it”. And that the phase-wave concept permits explanation of Einstein’s condition. He also said propagation is analogous to a liquid wave in a closed channel, wherein the length of the channel was resonant with the wave. And that this can be applied to the closed circular Bohr orbits in an atom. De Broglie was talking about corpuscles and phase waves, but saying “a corpuscle and its phase wave are not separate physical realities”. That doesn’t seem to square with what people say about pilot waves. However that’s perhaps because he “left the definitions of phase waves and the periodic phenomena for which such waves are a realization, as well as the notion of a photon, deliberately vague”. That’s a shame, but nevertheless he’d planted a seed.
The Pauli exclusion principle
Meanwhile in 1924 Wolfgang Pauli proposed a fourth quantum number to explain the anomalous Zeeman effect. See Wolfgang Pauli announces the exclusion principle written by Ernie Tretkoff in 2007. Pauli had spent a year in Copenhagen with Bohr, and later said “the question, as to why all electrons for an atom in its ground state were not bound in the innermost shell, had already been emphasized by Bohr as a fundamental problem in his earlier works”. Also see the Wikipedia Pauli exclusion principle article, which says he found an essential clue in a 1924 paper by Edmund Stoner. This led Pauli to realize that the numbers of electrons in closed shells can be simplified to one electron per state, provided electron states were defined using the three existing quantum numbers plus a new two-valued fourth state. In early 1925 Pauli wrote a paper on the connexion between the completion of electron groups in an atom with the complex structure of spectra. That’s where he said cases are excluded “where both electrons have m1 = ½ or both have m1 = -½; rather, we can only have m1 = ½ for the first electron and m1 = -½ for the second electron”. The Pauli exclusion principle was born. Pauli later said “physicists found it difficult to understand the exclusion principle, since no meaning in terms of a model was given”. But he also said the gap was filled by Uhlenbeck and Goudsmit’s idea of electron spin.
It means that the electron has a spin, that it rotates
There’s a reader-friendly article on the discovery of the electron spin where Samuel Goudsmit gives the history: When the day came I had to tell Uhlenbeck about the Pauli principle – of course using my own quantum numbers – then he said to me: But don’t you see what this implies? It means that there is a fourth degree of freedom for the electron. It means that the electron has a spin, that it rotates”. Goudsmit knew about the spectra, and said “if one now allows the electron to be magnetic with the appropriate magnetic moment, then one can understand all those complicated Zeeman-effects. They come out naturally, as well as the Landé formulae and everything, it works beautifully”.
Image from Princeton modern understanding originally from General Chemistry 3rd edition, by Hill and Petrucci
Goudsmit also said the man who never cared to believe in spin was Pauli. And that Llewellyn Thomas sent him a letter saying Ralph Kronig had had the idea a year previously. See the Wikipedia Ralph Kronig article for more. It says Kronig proposed electron spin in January 1925 after hearing Pauli in Tübingen, but Heisenberg and Pauli hated the idea, so Kronig didn’t publish. However Goudsmit and George Uhlenbeck didn’t stop to ask. Their paper was published in November 1925, and the spin ½ electron was born. Their subsequent paper on spinning electrons and the structure of spectra was printed in Nature. It was followed by a note from Neils Bohr who said this: “In my article expression was given to the view that these difficulties were inherently connected with the limited possibility of representing the stationary states of the atom by a mechanical model. The situation seems, however, to be somewhat altered by the introduction of the hypothesis of the spinning electron which, in spite of the incompleteness of the conclusions that can be derived from models, promises to be a very welcome supplement to our ideas of atomic structure”. Llewellyn Thomas was of course responsible for Thomas precession, see his April 1926 paper on the motion of the spinning electron.
Matrix mechanics
Meanwhile Werner Heisenberg was joining the fray. Like Pauli he was a Sommerfeld student, but unlike summa cum laude Pauli, he had angered experimentalist Wilhelm Wien and almost flunked his doctorate. See The Sad Story of Heisenberg’s Doctoral Oral Exam by David Cassidy. Also see the Wikipedia article on Heisenberg’s entryway to matrix mechanics along with the matrix mechanics article. After a seven-month stint with Bohr in Copenhagen, Heisenberg took note of Hendrik Kramer’s 1924 paper The Law of Dispersion and Bohr’s Theory of Spectra. Then he famously travelled from Göttingen to Helgoland in June 1925 to combat his hay fever, and came back in July with a paper called Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations. In it Heisenberg said “it seems more reasonable to try to establish a theoretical quantum mechanics, analogous to classical mechanics, but in which only the relations between observable quantities occur”. His paper was described as magic, as if it was some mathematical retrofit. It offered no description of what was going on inside the hydrogen atom, and it didn’t refer to de Broglie at all. In addition, whilst it’s usually described as Heisenberg’s matrix mechanics paper, it neither uses nor even mentions matrices”. That was down to Max Born, who recognised the underlying formalism, and who wrote a paper on quantum mechanics with his assistant Pascual Jordan.
Five of the seven ideas were Jordan’s
There’s an English translation courtesy of David Delphenich, and an abridged version in Bartel van der Waerden’s 1967 book Sources of Quantum Mechanics. The latter is online courtesy of the internet archive, and contains a significant number of references to Fourier, plus some useful history – Pauli was cold and sarcastic, and five out of seven ideas were Jordan’s. Two months later in November 1925, Born, Jordan, and Heisenberg finished their follow-up Dreimännerarbeit “three-man work” paper on quantum mechanics II. They expressed their conviction that difficulties could only be surmounted via a mathematical system which “would entirely consist of relations between quantities that are in principle observable”. And that such a system “would labour under the disadvantage of not being directly amenable to a geometrically visualizable interpretation”. They also said the motions of electrons could not be described in terms of the familiar concepts of space and time. Van der Waerden’s book also gives Pauli’s paper on the hydrogen spectrum from the standpoint of the new quantum mechanics. It says this showed that the hydrogen spectrum can be derived from the new theory, and that Pauli said “Heisenberg’s form of quantum theory completely avoids a mechanical-kinematic visualization of the motion of electrons in the stationary states of the atom”. As if it was a virtue.
Debye was right
It wasn’t just Heisenberg and co joining the fray, it was Dirac too, and Schrödinger. See Heisenberg and the early days of quantum mechanics by Felix Bloch. It’s a charming recount: “at the end of a colloquium I heard Debye saying something like: “Schrödinger, you are not working right now on very important problems anyway. Why don’t you tell us some time about that thesis of de Broglie, which seems to have attracted some attention”. So, in one of the next colloquia, Schrödinger gave a beautifully clear account of how de Broglie associated a wave with a particle and how he could obtain the quantization rules of Niels Bohr and Sommerfeld by demanding that an integer number of waves should be fitted along a stationary orbit. When he had finished, Debye casually remarked that he thought this way of talking was rather childish. As a student of Sommerfeld he had learned that, to deal properly with waves, one had to have a wave equation. It sounded quite trivial and did not seem to make a great impression, but Schrödinger evidently thought a bit more about the idea afterwards. Just a few weeks later he gave another talk in the colloquium which he started by saying: “My colleague Debye suggested that one should have a wave equation; well, I have found one!” And then he told us essentially what he was about to publish under the title “Quantization as Eigenvalue Problem” as a first paper of a series in the Annalen der Physik. I was still too green to really appreciate the significance of this talk, but from the general reaction of the audience I realized that something rather important had happened, and I need not tell you what the name of Schrödinger has meant from then on. Many years later, I reminded Debye of his remark about the wave equation; interestingly enough he claimed that he had forgotten about it and I am not quite sure whether this was not the subconscious suppression of his regret that he had not done it himself. In any event, he turned to me with a broad smile and said: “Well, wasn’t I right?”
Wave mechanics
He was. Erwin Schrödinger had a head start because he’d written a paper in 1922 on a remarkable property of the quantum orbits of a single electron. He had a heads-up too. See Foundations of Quantum Mechanics in the Light of New Technology where you can read Chen-Ning Yang’s 1997 paper Complex Phases in Quantum Mechanics. Yang said Einstein alerted Schrödinger to de Broglie’s thesis, and Schrödinger wrote back to Einstein in November 1925. Schrödinger said “the de Broglie interpretation of the quantum rules seems to me to be related in some ways to my note in the Zs. F. Phys. 12 13 1922 where a remarkable property of the Weyl ‘gauge factor’ exp[-ϕdx] along each quasiperiod is shown”. Schrödinger submitted the first part of his four-part paper in January 1926. It was called quantization as a problem of proper values, part I. He talked of the hydrogen atom and said integralness arises in the same natural way as the node numbers of a vibrating string. He also talked of the azimuthal quantum number and said “the splitting up of this number through a closer definition of the surface harmonic can be compared with the resolution of the azimuthal quantum number into an ‘equatorial’ and a ‘polar’ quantum’”. He said “It is, of course, strongly suggested that we should try to connect the function ψ with some vibration process within the atom”. And that he was led to these deliberations by the suggestive papers of M Louis de Broglie. But that the “main difference is that de Broglie thinks of progressive waves, while we are led to stationary proper vibrations”. Schrödinger was talking about spherical harmonics, standing waves, and atomic orbitals:
Image from The Star Garden article Sommerfeld’s atom by Dr Helen Klus
He also said “It is hardly necessary to emphasize how much more congenial it would be to imagine that at a quantum transition the energy changes from one form of vibration to another, than to think of a jumping electron”. His second paper quantization as a problem of proper values, part II was more of the same. He talked about wavefunction and phase and geometrical optics, and on page 18 said classical mechanics fails for very small dimensions of the path and for very great curvature. He talked of a wave system consisting of sine waves where the frequency works out to be ν=E/h, and said we can attempt of build up a wave group which will have relatively small dimensions in every dimension. He said “let us think of a wave group of the nature described above, which in some way gets into a small closed ‘path’, whose dimensions are of the order of the wave length”. And that the wave group not only fills the whole path domain all at once but also stretches far beyond it in all directions. He said this: “All these assertions systematically contribute to the relinquishing of the ideas of “place of the electron” and “path of the electron”. If these ideas are not given up, contradictions remain. This contradiction has been so strongly felt that it has even been doubted that what goes on in the atom could ever be described within the scheme of space and time. From the philosophical standpoint, I would consider a conclusive decision in this sense as equivalent to complete surrender”.
Light rays show the most remarkable curvatures
He also said light rays “show, even in homogeneous media, the most remarkable curvatures, and obviously mutually influence one another”. That’s on page 27. I think it’s visionary stuff myself, though Schrödinger’s quantization as a problem of proper values, part 3 is arguably more mundane. It’s about perturbation theory and the Stark effect. But it does say this: “since then I have learned what is lacking from the from the most important publications of G E Uhlenbeck and S Goudsmit”. Schrödinger refers to the angular moment of the electron which gives it a magnetic moment, and says “the introduction of the paradoxical yet happy conception of the spinning electron will be able to master the disquieting difficulties which have latterly begun to accumulate”. But he also says that in the present paper the taking over of the idea is not yet attempted. He didn’t attempt it in his part 4 either. Or in the condensed English version in Physical Review. That was called An Undulatory Theory of the Mechanics of Atoms and Molecules. He ended up saying “The deficiency must be intimately connected with Uhlenbeck-Goudsmit’s theory of the spinning electron. But in what way the electron spin has to be taken into account in the present theory is yet unknown”. That’s why there is no spin in the Schrödinger equation. But no matter, there’s some great stuff in there. Like material points consist of, or are nothing but, wave-systems. What’s not to like? For Neils Bohr, plenty.
Schrödinger goes to Copenhagen
Take a look at page 192 of Walter Moore’s 1989 book Schrödinger, life and thought: “Schrödinger wanted to find the structure of such waves when they are refracted sufficiently to travel in one of the Bohr orbits”. Also see page 209. Schrödinger had sent Max Planck a preprint of his first paper, which Planck read “like an eager child”. Planck also showed it to Einstein, who wrote to Schrödinger in April 1926 saying “the idea of your work springs from pure genius”. Ten days later Einstein wrote again. He said “I am convinced that you have made a decisive advance with your formulation of the quantum condition, just as I am convinced that the Heisenberg-Born method is misleading”. By then Schrödinger had written a paper on the relation of the Heisenberg-Born-Jordan quantum mechanics to mine. He said “it is very strange that these two new theories agree with one another”. Along with “I refer in particular to the peculiar ‘half-integralness’ which arises in connection with the oscillator and rotator”. It comes across as civil and diplomatic, but there’s perhaps an undercurrent that says something like this is proper physics, not mysticism.
Their true feelings were not concealed
On page 221 Moore talks about the relationship between Schrödinger and Heisenberg. He says they were diplomatic in printed papers, but “in personal letters their true feelings were not concealed”. He quotes words like monstrous and abominable, and bullshit. He tells how Schrödinger was invited to lecture in Germany, and travelled to Stuttgart then Berlin where he stayed with the Plancks. And how the older generation of Berlin physicists such as Einstein, Laue, Nernst and Planck were impressed, so much so that “Planck began to consider seriously his plans to bring Schrödinger to Berlin as his successor”. Schrödinger then went to Jena thence Munich, where he repeated his Berlin lecture. Heisenberg was in the audience, and in the question-and-answer session he asked how Schrödinger ever hoped to explain the photoelectric effect and black-body radiation. But before Schrödinger could reply, “Willy Wien angrily broke in and, as Heisenberg reported to Pauli, ‘almost threw me out of the room’”. Moore says Heisenberg was upset and immediately wrote to Bohr. Whereupon Bohr wrote to Schrödinger to invite him to Copenhagen for some “serious discussions”. They did not go well. See page 222 of Manjit Kumar’s 2008 book Quantum: Einstein, Bohr and the Great Debate About the Nature of Reality. He’s referring to Heisenberg’s book The Part and the Whole. Bohr met Schrödinger at the station in late September 1926: “after the exchange of pleasantries, battle began almost at once, and according to Heisenberg, ‘continued daily from early morning until late at night’”. Bohr appeared even to Heisenberg to be a “remorseless fanatic, one who was not prepared to make the least concession or grant that he could ever be mistaken”. When Schrödinger took ill and took to bed, Bohr sat on the edge of the bed and continued the argument. The words Bohr and bully seem to go together.
The Copenhagen Interpretation
By then Max Born had written his paper on the quantum mechanics of collisions. He’d come up with what’s now known as the Born rule. On page 3 he said this: “If one translates this result into terms of particles, only one interpretation is possible. Φn,m(α, β, γ) gives the probability for the electron, arriving from the z-direction, to be thrown out into the direction designated by the angles α, β, γ, with the phase change δ”. He was saying the only possible interpretation is that the Schrödinger wave equation described probabilities rather than something that was actually there. This was the beginning of the Copenhagen interpretation. Andrew Zimmerman Jones gives an overview in his 2017 essay on The Copenhagen Interpretation of Quantum Mechanics. He says it’s a combination of Born’s probabilistic statistical interpretation, Heisenberg’s uncertainty principle, and Bohr’s concept of complementarity:
Copenhagen Interpretation image from Andrew Friedman’s website, see http://afriedman.org/
Heisenberg came up with his uncertainty principle in his March 1927 paper on the actual content of quantum theoretical kinematics and mechanics. He said “canonically conjugated variables can be determined simultaneously only with a characteristic uncertainty”. He also said the interpretation of quantum mechanics is still full of internal contradictions, “which become apparent in the battle of opinions on the theory of continuums and discontinuums, corpuscles and waves. This alone tempts us to believe that an interpretation of quantum mechanics is not going to be possible in the customary terms of kinematic and mechanical concepts”. Heisenberg was promoting the viewpoint he shared with Born, Jordan, and Bohr, even though his paper used the word “wave” 36 times. Hence the Wikipedia uncertainty principle article strikes a chord when it says this: It has since become clearer, however, that the uncertainty principle is inherent in the properties of all wave-like systems, and that it arises in quantum mechanics simply due to the matter wave nature of all quantum objects”. What’s not clear however is why Heisenberg didn’t associate what came to be known as wavefunction collapse with the optical Fourier transform:
Image from Steven Lehar’s intuitive explanation of Fourier theory
I should mention that he didn’t actually use the word collapse, he said at the instant of the determination of its position, the electron discontinuously changes its impulse. But the meaning is the same, and it’s closely related to Bohr’s complementarity. Heisenberg referred to that in an addendum to his paper, and Bohr talked about it in September 1927 in Como. You can find the details in his Nature paper on The Quantum Postulate and the Recent Development of Atomic Theory. I would say the message is “you can never hope to understand it”. Interestingly while all this was going on Pauli wrote a paper on the quantum mechanics of magnetic electrons. That’s where Pauli referred to Yakov Frenkel’s 1926 paper on the electrodynamics of rotating electrons, the one where Frenkel said the electron will thus be treated simply as a point. Pauli then wondered “whether such a formulation of the theory is even possible at all as long as one retains the idealization of the electron by an infinitely small magnetic dipole”. And “whether a more precise model of the electron is required for such a theory”. But he didn’t pursue it, more’s the pity. Because a few months later he could have talked about it at the Solvay conference.
The 1927 Solvay conference
The 1927 Solvay conference was given the title Electrons and Photons. But it wasn’t about electrons and photons at all. It was all about the struggle between Einstein and the scientific realists” against “Bohr and the instrumentalists”. It is said that the latter won the argument. As to the truth of it, take a look at Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference written by Guido Bacciagaluppi and Antony Valentini in 2006. They give a good description of proceedings, including the reports by Bragg, Compton, de Broglie, Heisenberg and Born, and Schrödinger.
Fifth Solvay conferance image, see Wikipedia
They say “according to widespread historical folklore, the deep differences of opinion among the leading physicists of the day led to intense debates, which were satisfactorily resolved by Bohr and Heisenberg around the time of the 1927 Solvay meeting. But in fact, at the end of 1927, a significant number of the main participants (in particular de Broglie, Einstein, and Schrödinger) remained unconvinced, and the deep differences of opinion were never resolved”. They also say that “there has also been criticism – on the part of historians as well as physicists – of the tactics used by Bohr and others to propagate their views in the late 1920s”. That “a sense of unease lingers of the whole subject”. And that “’shut up and calculate’ emerged as the working rule among the vast majority”. See the Wikipedia article on the Einstein-Bohr debates for more. Bohr advocated a probabilistic quantum mechanics where there was no point trying to understand what was really going on inside the atom. Einstein said this meant quantum mechanics was incomplete, and that uncertainty was no substitute for understanding. Bohr defended his position by claiming that an independent reality in the ordinary physical sense can neither be ascribed to the phenomena nor to the agencies of observation. And that It is wrong to think the task of physics is to find out how nature is. Einstein said What we call science, has the sole purpose of determining what is. I agree one hundred percent with that.
The Copenhagen interpretation became mainstream orthodoxy
But amazingly, incredibly, the Copenhagen interpretation somehow prevailed. The Copenhagen interpretation became the mainstream orthodoxy. Despite the hard scientific facts. In 1917 the Einstein-de Haas effect demonstrated that spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies as conceived in classical mechanics. In 1922 the Stern-Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. In 1927 the Davisson-Germer experiment along with diffraction experiments by George Paget Thomson and Andrew Reid proved the wave nature of matter. In 1931 the Experimental proof of the spin of the photon was provided by Chandrasekhara Raman and Suri Bhagavantam. In 1932 Carl Anderson discovered the positron. In 1933 Patrick Blackett and Giuseppe Occhialini were the first to observe pair production. We could make electrons and positrons out of light, the electron had a different spin to the photon, and yet quantum mechanics surpasseth all human understanding? What happened? See the particle physics quantum theory timeline. It says in 1930 Max Born, after learning of the Dirac equation, said “physics as we know it will be over in six months”. In a way, a bad way, he was right.
What happened?
Paul Dirac made a significant contribution to quantum mechanics. See Kurt Gottfried’s 2010 essay P.A.M. Dirac and the Discovery of Quantum Mechanics for an overview. But note this on page 5: “Heisenberg, Dirac et al were hostile to wave mechanics because they thought it gave the misleading impression that the classical concepts of continuity and visualizability had survived the revolution, whereas they believed that it was a central virtue of their abstract theory that it did not evoke such delusions”. Perhaps that’s why Dirac’s 1927 paper on the physical interpretation of the quantum dynamics didn’t do what it said on the can. And why his 1928 paper the quantum theory of the electron doesn’t deliver a picture of the electron. The Dirac equation is said to describe the electron, but try explaining it to your grandmother, and you realise it doesn’t. Dirac wrote a paper in 1930 on the annihilation of electrons and protons, only they don’t. In a theory of electrons and protons he said negative kinetic energy appeared to have no physical meaning. That suggests he didn’t understand binding energy. He said the electron energy changes from positive to negative and energy of 2mc² is emitted. That suggests he didn’t understand E=mc². He said the Pauli exclusion principle prevents electrons decaying, and the “holes” in the distribution of negative-energy electrons are protons. That suggests he was talking out of his hat. As did his 1962 paper an extensible model of the electron, which depicted the electron as a charged conducting sphere with a surface tension. Yet in the Wikipedia timeline of quantum mechanics you can read that Dirac’s 1930 textbook Principles of Quantum Mechanics became a standard reference book that is still used today. Then Schrödinger’s cat, which illustrated the absurdity of the Copenhagen interpretation, was hijacked by the peddlers of mysticism to demonstrate just how “spooky” quantum physics is. Such people even advocate the many-worlds multiverse. What happened? More to the point, what didn’t? What didn’t happen at the 1927 Solvay conference was a discussion of what the photon was, or what the electron was. Then Bohr sold his pup to the world, and it was all downhill from there.
This Post Has 5 Comments
1. thx for dropping by to comment on my latest major effort. https://vzn1.wordpress.com/2018/05/25/fluid-paradigm-shift-2018/
am impressed youve already got some substantial dialogue going in your comments. would be very interested to hear from your readers also.
the quote by assistant/ acolyte heisenberg on his mentor bohr is really startling. “Bohr appeared even to Heisenberg to be a “remorseless fanatic, one who was not prepared to make the least concession or grant that he could ever be mistaken”. ” modern adherents/ acolytes of “shut up and calculate” are not much different than their cult leader. (using strong language here because hey, its the Duffield blog, wink)
its striking how some of physics history has been swept under the rug in the standard accounts bordering on dogma. this reminds me eg some of newton vs alchemy…
1. My pleasure vzn. Yes, there’s been some airbrushing of history going on. When it comes to gravity I’ve long known that people appeal to Einstein’s authority whilst flatly contradicting the guy. Then if you quote Einstein and refer to the digital papers, they’ll dismiss it as out of context or cherry picking. Now I also know that the same sort of thing has been going on in other sectors of physics. For example people peddle the mystery of wave/particle duality, but Pascual Jordan solved that mystery in 1925. The strong language is only there to show the enmity that existed back in the twenties, and which IMHO ended up causing some big problems.
PS: I started getting “adult” spam yesterday, so I turned comment moderation on. I really don’t like doing that, because I believe in free speech in science.
2. I think Schrödinger more or less nailed it with this:
“classical mechanics fails for very small dimensions of the path and for very great curvature”.
“let us think of a wave group… which in some way gets into a small closed ‘path’, whose dimensions are of the order of the wave length”.
“Light rays show, even in homogeneous media, the most remarkable curvatures, and obviously mutually influence one another”.
3. Bolches yarboclos, Batman!!!
What a useful post.
1. Thanks Poak. I hope everybody finds this and the other posts useful, When you dig into the history, your find some absolute treasure. Then you come to appreciate that some of the contemporary stuff is absolute junk.
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df36332e70d23aa1 | Dirac interaction picture
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The “interaction picture” in quantum physics is a way to decompose solutions to the Schrödinger equation and more generally the construction of quantum field theories into a free field theory-part and the interaction part that acts as a perturbation of the free theory. Therefore the interaction picture lends itself to the construction of perturbative quantum field theory, and in fact the only mathematically rigorous such construction scheme that is known, namely causal perturbation theory, proceeds this way.
Dynamics in physics affects both observables and, dually, states; this is most well known in quantum physics but applies equally well to classical physics. The different “pictures” of physics differ in how the dynamics is explicitly formalized:
The pictures are named after those physicists who first used or popularised these approaches to quantum physics.
In quantum mechanics
In quantum mechanics, let \mathcal{H} be some Hilbert space and let
H=H free+V H = H_{free} + V
be Hermitian operator, thought of as a Hamiltonian, decomposed as the sum of a free part (kinetic energy) and an interaction part (potential energy).
For example for a non-relativistic particle of mass mm propagating on the line subject to a potential energy V pot:V_{pot} \colon \mathbb{R} \to \mathbb{R}, then =L 2(X)\mathcal{H} = L^2(X) is the Hilbert space space of square integrable functions and
H= 22m 2 2xH free+V, H = \underset{H_{free}}{\underbrace{\tfrac{- \hbar^2}{2m} \frac{\partial^2}{\partial^2 x}}} + V \,,
where V=V pot(x)V = V_{pot}(x) is the operator of multiplying square integrable functions with the given potential energy function.
Now for
|ψ() t |ψ(t) \array{ \mathbb{R} &\overset{\vert \psi (-)\rangle }{\longrightarrow}& \mathcal{H} \\ t &\mapsto& \vert \psi(t) \rangle }
a one-parameter family of quantum states, the Schrödinger equation for this state reads
ddt|ψ(t)=1iH|ψ. \frac{d}{d t} \vert \psi(t) \rangle \;=\; \tfrac{1}{i \hbar} H \vert \psi\rangle \,.
It is easy to solve this differential equation formally via its Green function: for |ψ\vert \psi \rangle \in \mathcal{H} any state, then the unique solution |ψ()\vert \psi(-) \rangle to the Schrödinger equation subject to |ψ(0)=|ψ\vert \psi(0) \rangle = \vert \psi \rangle is
|ψ(t) Sexp(tiH)|ψ. \vert \psi(t)\rangle_S \coloneqq \exp( \tfrac{t}{i \hbar} H ) \vert \psi \rangle \,.
(One says that this is the solution “in the Schrödinger picture”, whence the subscript.)
However, if HH is sufficiently complicated, it may still be very hard to extract from this expression a more explicit formula for |ψ(t)\vert \psi(t) \rangle, such as, in the example of the free particle on the line, its expression as a function (“wave function”) of xx and tt.
But assume that the analogous expression for H freeH_{free} alone is well understood, hence that the operator
U S,free(t 1,t 2)exp(t 2t 1iH free) U_{S,free}(t_1, t_2) \coloneqq \exp\left({\tfrac{t_2 - t_1}{i \hbar} H_{free}}\right)
is sufficiently well understood. The “interaction picture” is a way to decompose the Schrödinger equation such that its dependence on VV gets separated from its dependence on H freeH_{free} in a way that admits to treat H intH_{int} in perturbation theory.
Namely define analogously
(1)|ψ(t) I exp(tiH free)|ψ(t) S =exp(tiH free)exp(+tiH)|ψ =exp(tiH free)exp(tiH free+tiV)|ψ. \begin{aligned} \vert \psi(t)\rangle_I &\coloneqq \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) \vert \psi(t)\rangle_S \\ & = \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) \exp\left({ \tfrac{+ t}{i \hbar} H} \right)\vert \psi \rangle \\ & = \exp\left({\tfrac{- t}{i \hbar} H_{free}}\right) \exp\left({\tfrac{t}{i \hbar} H_{free} + \tfrac{t}{i \hbar} V} \right)\vert \psi \rangle \end{aligned} \,.
This is called the solution of the Schrödinger equation “in the interaction picture”, whence the subscript. Its definition may be read as the result of propagating the actual solution |ψ() S\vert \psi(-)\rangle_S at time tt back to time t=0t = 0, but using just the free Hamiltonian, hence with “the interaction switched off”.
Notice that if the operator VV were to commute with H freeH_{free} (which it does not in all relevant examples) then we would simply have |ψ(t) I=exp(tiV)|ψ\vert \psi(t)\rangle_I = \exp( \tfrac{t}{i \hbar } V ) \vert \psi\rangle, hence then the solution (1) in the interaction picture would be the result of “propagating” the initial conditions using only the interaction. Now since VV may not be assumed to commute with H freeH_{free}, the actual form of |ψ() I\vert \psi(-) \rangle_{I} is more complicated. But infinitesimally it remains true that |ψ() I\vert \psi(-)\rangle_I is propagated this way, not by the plain operator VV, though, but by VV viewed in the Heisenberg picture of the free theory. This is the content of the differential equation (2) below.
But first notice that this will indeed be useful: If an explicit expression for the “state in the interaction picture(1) is known, then the assumption that also the operator exp(tiH free)\exp\left({\tfrac{t}{i \hbar} H_{free}}\right) is sufficiently well understood implies that the actual solution
|ψ(t) S=exp(tiH free)|ψ(t) I \vert \psi(t) \rangle_S = \exp\left({\tfrac{t}{i \hbar} H_{free}}\right) \vert \psi(t) \rangle_I
is under control. Hence the question now is how to find |ψ() I\vert \psi(-)\rangle_I given its value at some time tt. (It is conventional to consider this for t±t \to \pm \infty, see (3) below.)
Now it is clear from the construction and using the product law for differentiation, that |ψ() S\vert \psi(-)\rangle_S satisfies the following differential equation:
(2)ddt|ψ(t)=V I(t)|ψ(t) I, \frac{d}{d t} \vert \psi(t) \rangle \;=\; V_I(t) \vert \psi(t)\rangle_I \,,
V I(t)exp(tiH free)Vexp(+tiH free) V_I(t) \coloneqq \exp\left( -\tfrac{t}{i \hbar} H_{free} \right) V \exp\left( +\tfrac{t}{i \hbar} H_{free} \right)
is known as the interaction term VV “viewed in the interaction picture”. But in fact this is just VV “viewed in the Heisenberg picture”, but for the free theory. By our running assumption that the free theory is well understood, also V I(t)V_I(t) is well understood, and hence all that remains now is to find a sufficiently concrete solution to equation (2). This is the heart of working in the interaction picture.
Solutions to equations of the “parallel transport”-type such as (2) are given by time-ordering of Heisenberg picture operators, denoted TT, applied to the naive exponential solution as above. This is known as the Dyson formula:
|ψ(t) I=T(exp( t 0 tV I(t)dti))|ψ(t 0). \vert \psi(t)\rangle_I \;=\; T\left( \exp\left( \int_{t_0}^t V_I(t) \tfrac{d t}{i \hbar} \right) \right) \vert \psi(t_0)\rangle \,.
Here time-ordering means
T(V I(t 1)V I(t 2)){V I(t 1)V I(t 2) | t 1t 2 V I(t 2)V I(t 1) | t 2t 2. T( V_I(t_1) V_I(t_2) ) \;\coloneqq\; \left\{ \array{ V_I(t_1) V_I(t_2) &\vert& t_1 \geq t_2 \\ V_I(t_2) V_I(t_1) &\vert& t_2 \geq t_2 } \right. \,.
(This is abuse of notation: Strictly speaking time ordering acts on the tensor algebra spanned by the {V I(t)} t\{V_I(t)\}_{t \in \mathbb{R}} and has to be folllowed by taking tensor products to actual products. )
In applications to scattering processes one is interest in prescribing the quantum state/wave function far in the past, hence for tt \to - \infty, and computing its form far in the future, hence for tt \to \infty.
The operator that sends such “asymptotic ingoing-states” |ψ() I\vert \psi(-\infty) \rangle_I to “asymptic outgoing states” |ψ(+) I\vert \psi(+ \infty) \rangle_I is hence the limit
(3)SlimtT(exp( t tV I(t)dti)). S \;\coloneqq\; \underset{t \to \infty}{\lim} T\left( \exp\left( \int_{-t}^t V_I(t) \tfrac{d t}{i \hbar} \right) \right) \,.
This limit (if it exists) is called the scattering matrix or S-matrix, for short.
In quantum field theory
In perturbative quantum field theory the broad structure of the interaction picture in quantum mechanics (above) remains a very good guide, but various technical details have to be generalized with due care:
1. The algebra of operators in the Heisenberg picture of the free theory becomes the Wick algebra of the free field theory (taking into account “normal ordering” of field operators) defined on microcausal functionals built from operator-valued distributions with constraints on their wave front set.
2. The time-ordered products in the Dyson formula have to be refined to causally ordered products and the resulting product at coincident points has to be defined by point-extension of distributions – the freedom in making this choice is the renormalization freedom (“conter-terms”).
3. The sharp interaction cutoff in the Dyson formula that is hidden in the integration over [t 0,t][t_0,t] has to be smoothed out by adiabatic switching of the interaction (making the whole S-matrix an operator-valued distribution).
Together these three point are taken care of by the axiomatization of the “adiabatically switched S-matrix” according to causal perturbation theory.
The analogue of the limit tt \to \infty in the construction of the S-matrix (now: adiabatic limit) in general does not exist in field theory (“infrared divergencies”). But in fact it need not be taken: The field algebra in a bounded region of spacetime may be computed with any adiabatic switching that is constant on this region. Moreover, the algebras assigned to regions of spacetime this way satisfy causal locality by the causal ordering in the construction of the S-matrix. Therefore, even without taking the adiabtic limit in causal perturbation theory one obtains a field theory in the form of a local net of observables. This is the topic of locally covariant perturbative quantum field theory.
For instance
• Eberhard Zeidler, section 7.19.3 of Quantum field theory. A bridge between mathematicians and physicists – volume I Springer (2009) (web)
Last revised on January 9, 2018 at 09:35:16. See the history of this page for a list of all contributions to it. |
7ebad8b367619797 | The Megaphragma mymaripenne is the smallest animal with eyes, brain, wings, muscles, guts and genitals.
If by some miracle it could be still shrinked, how much smaller could it get before it starts to become largely aware of quantum mechanical effects such as tunneling?
By shrinking I mean wasp being made of less atoms but with similar "organs".
By affected I mean, if that wasp was by some miracle smart as humans, it would have understanding of quantum mechanical effects same as humans have understanding of naive physics.
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$\begingroup$ According to a guy named "Schrödinger", a cat should be small enough to be affected by quantum mechanics. $\endgroup$ – Nolonar Nov 3 '16 at 14:39
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$\begingroup$ One of the theories of olfaction (smell) includes pretty significant quantum effects. If that is true, humans (and most other animals) would fit as well. $\endgroup$ – Alice Nov 3 '16 at 15:37
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$\begingroup$ @Nolonar I don't know who you're talking about, but according to the physicist named Schrödinger, a cat definitely shouldn't be small enough. $\endgroup$ – JiK Nov 3 '16 at 15:58
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$\begingroup$ As it turns out, photosynthesis uses quantum tunneling to move electrons from the surface, deeper into the leaf without generating heat on the way. So some of the biggest living things on the planet are using "quantum mechanics" outside of chemistry. $\endgroup$ – Chris Becke Nov 4 '16 at 6:05
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$\begingroup$ Huh... as for Schrödinger... a lion is a cat... a 250kg cat. And now suddenly, it becomes obvious why sabertooths are extinct and we find their bones in limestone. They got stuck trying to tunnel through walls and starved. $\endgroup$ – Damon Nov 4 '16 at 11:17
12 Answers 12
It is hard to put an exact number on this, but it seems like the answer would be maybe 1000 atoms at most. From Wikipedia,
And that is just for superposition in location, not even getting to quantum tunneling like you mention in your question. Observing QM effects in anything larger than that has been notoriously difficult. However, some scientists have been trying to observe a small microbe in a superposition. I can't find anything indicating that the experiment was actually done, just lots of stuff about people trying to do it and thinking it will be done in the next few years. So maybe we will get small bacterium and viruses to experience QM effects relatively soon. That would probably set the upper limit on the size you are asking for. This source claims that even a 100nm microbe would be seriously difficult to observe in a superposition:
A recent proposal suggested “piggybacking” a tiny microbe (100 nanometres) on to a slightly less tiny (15 micrometres) aluminium drum, whose motion has been brought to the quantum level. While this experiment is feasible, the separation between the “two places at once” that the bacteria would find itself in is 100m times smaller than the bacterium itself.
Edit: Just to clarify my wording, everything always experiences quantum effects, they just become unobservably small as the object gets larger and larger (with rare exceptions, like the black body spectrum of the sun, but that is another matter entirely).
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$\begingroup$ I estimate about 2nm for diffraction effects, as a very rough analysis. Superposition is a slippery concept, but it's nice to see we're on about the same scale. $\endgroup$ – spraff Nov 4 '16 at 18:33
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$\begingroup$ The problem for superposition is to keep the system coherent. That typically mean a very cold system near the vacuum, definitely not physiological conditions. $\endgroup$ – Davidmh Nov 5 '16 at 15:13
You appear to have a misunderstanding of how physics works. Classical physics (i.e., the thing we generally refer to when discussing how things interact) is merely an approximation of quantum mechanics. There is no boundary that says "Only past this point are you affected by quantum mechanics."
But, if you are concerned with how this creature would behave, you would need to make it smaller than an atom, as only then does quantum mechanics predict different behavior than Newtonian physics.
Of course, one could simply look up quantum tunneling to see that it applies to particles, and not organisms, which are comprised of lots of particles.
There appears to be some concern about my third citation, and I completely agree. The user on Physics has no linked research, low reputation, and low votes. However, I don't pretend to be an expert in the field of quantum mechanics; I rely entirely on some basic ideas of what it is and the expertise of others. To sum up the above (and comments below): quantum mechanics dominates in the smallest scales, while Newtonian physics dominates in the largest scales, and no one knows why or what the tipping point is.
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$\begingroup$ Yay! No boundary to QM. My first thought seeing this question. Glad to see your answer. Macroscopic systems don't show quantum behaviour -- usually. Bose-Einstein condensates are one of the exceptions. Although modern electronics runs on QM properties the spooky effects don't flow into our everyday world. $\endgroup$ – a4android Nov 3 '16 at 13:00
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$\begingroup$ You need a tighter definition of "affected". Chemistry happens because of quantum mechanics. Deuterium is an imperfect substitute for hydrogen in biochemistry because of QM. We don't spontaneously ignite because of QM (triplet vs. singlet Oxygen energy levels). Differently, when our eyes are startlight-adapted we see a "grainy" low-resolution image. Your eyes are detecting individual light quanta. $\endgroup$ – nigel222 Nov 3 '16 at 15:25
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$\begingroup$ QM predicts different behavior even at macroscopic scales. See the Ultraviolet Catastrophe. True, statistical approximations could be used instead of raw QM, but the statistical approximation relies on quantization of photons... $\endgroup$ – Yakk Nov 3 '16 at 18:32
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$\begingroup$ Interference has been shown for molecules made of up to 430 atoms, so quite obviously quantum effects don't stop at the atomic scale. $\endgroup$ – celtschk Nov 3 '16 at 18:57
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$\begingroup$ This answer is incorrect. Just because using quantum mechanics on atoms is more correct than Newtonian doesn't make it the limit. Newtonian mechanics start not working on a larger scale, as @celtschk's example shows. Your "source" Isn't a source at all $\endgroup$ – Zach Saucier Nov 3 '16 at 19:36
In the world of processors 5nm was assumed as smallest size before quantum tunneling starts to be a problem. If you shrink your wasp 1000 times it will become 200nm long, since its legs are much smaller they will probably be affected by tunneling.
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$\begingroup$ You are misunderstanding this effect. The quantum tunneling becomes more prominent at 5nm because the gate oxide must shrink (but not because the linewidth is 5nm). As this reduces quantum effects will have significantly more effects (on the gate oxide). But this is not what the 5nm refers to (minimum printable linewidth, or minimum drain->source difference). $\endgroup$ – jbord39 Nov 3 '16 at 16:03
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$\begingroup$ Not to mention that quantum tunnelling is required for semiconductors to work. The charges simply don't have enough energy to cross the potential boundary - they need to tunnel through. The problem you're talking about is related to unwanted quantum tunnelling - the charges tunnelling through places we don't want them to tunnel through. $\endgroup$ – Luaan Nov 4 '16 at 12:58
My day job is (currently) designing the software/firmware/electronics for nanopositioning systems. With our current best kit, we can reliably and repeatably move something to 70pm accuracy over a 15um range.
This is a classical-mechanics chunk of metalwork moving. At that range we have significant challenges with material stiffness and other interesting mechanical effects, but the physics is still very much in the classical domain. So the basic chemistry of the wasp's body isn't something it needs to worry about just yet.
Of course quantum tunnelling could be an issue for the wasp's nervous system. Since that relies on electrical signals, it'll have the same issues as shrinking a processor die.
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$\begingroup$ My day job is (currently) designing the software/firmware/electronics for nanopositioning systems. Wow... I mean wow. :jaw drops: (Sorry for the comment spam.) $\endgroup$ – mg30rg Nov 4 '16 at 9:33
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$\begingroup$ @mg30rg Someone has to do it :) $\endgroup$ – Roman Nov 4 '16 at 11:50
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$\begingroup$ Of course, the semiconductors used in those electronics only work thanks to quantum electrodynamics, but that's kind of begging the question - classical physics is just an approximation, a model of the underlying reality. Quantum physics is a more accurate model of the underlying reality (and possibly actual reality - but how could we tell? :)). Things don't "start" or "stop" behaving classically - it's just that under different conditions, classical physics can be a better or worse approximation of reality as far as we care. Quantumness doesn't disappear when things get big. $\endgroup$ – Luaan Nov 4 '16 at 12:56
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$\begingroup$ Just to be clear it's not a typo, you can accurately position things to roughly the covalent bonding diameter of a hydrogen atom? $\endgroup$ – BenRW Nov 5 '16 at 21:20
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$\begingroup$ @BenRW With a best specimen of our top-line kit we can measure and position to around 70 picometres resolution. We aim for around 100pm, and we'd reject it if it's worse than about 150pm. Yes, this is insane stuff! There are caveats to this, of course. Temperature and pressure will affect the system if they're not insanely tightly controlled. Also hysteresis is a problem for run-to-run variation - you can't do that kind of fine movement in both directions. $\endgroup$ – Graham Nov 7 '16 at 11:30
Quite large animals are "affected" by quantum mechanics, because even large animals consist of small parts and many mechanisms at the smallest scales of animal bodies rely on quantum mechanics.
For example: the reason that geckos' feet stick to glass is because of quantum mechanics (Van der Waals forces to be precise: see here). For other examples see this Wikipedia article about quantum biology.
Your question is rather vague, in that you don't specify what you mean by "affected". Quantum mechanics can affect everything at the molecular level. By that logic, even blue whales are affected by quantum mechanics.
For example:
Vision relies on quantized energy in order to convert light signals to an action potential in a process called phototransduction. In phototransduction, a photon interacts with a chromophore in a light receptor. The chromophore absorbs the photon and undergoes photoisomerization. This change in structure induces a change in the structure of the photo receptor and resulting signal transduction pathways lead to a visual signal. However, the photoisomerization reaction occurs at a rapid rate, <200 fs, with high yield. Models suggest the use of quantum effects in shaping the ground state and excited state potentials in order to achieve this efficiency.
Other examples on that Wikipedia page include:
• Studies show that long distance electron transfers between redox centers through quantum tunneling plays important roles in enzymatic activity of photosynthesis and cellular respiration.
• Magnetoreception refers to the ability of animals to navigate using the magnetic field of the earth. A possible explanation for magnetoreception is the radical pair mechanism.
• Other examples of quantum phenomena in biological systems include olfaction, the conversion of chemical energy into motion, DNA mutation and brownian motors in many cellular processes.
Regarding DNA mutation:
DNA’s twisted ladder structure requires rungs of hydrogen bonds to hold it together; each bond is essentially made up of a single hydrogen atom that unites two molecules. This means sometimes a single atom can determine whether a gene mutates. And single atoms are vulnerable to quantum weirdness. Usually the single atom sits closer to a molecule on one side of the DNA ladder than the other. Al-Khalili and McFadden dug out a long-forgotten proposal made back in 1963 that suggested DNA mutates when this hydrogen atom tunnels, quantum-mechanically, to the “wrong” half of its rung. The pair built on this by arguing that, thanks to the property of superposition, before it is observed, the atom will simultaneously exist in both a mutated and non-mutated state — that is, it would sit on both sides of the rung at the same time.
• $\begingroup$ Well that's a 200: success answer! +1! $\endgroup$ – RudolfJelin Nov 4 '16 at 19:05
The world we know, macroscopically, would not be without quantum mechanics. Even solid matter wouldn't stay in cohesion without it. The sun wouldn't shine, chemical reactions wouldn't exist etc.
You might say: "yeah, but these are things we are used to. They make sense." Exactly. That's the point. We see these things all the time, so they don't sound "quantum", but they are.
Quantum mechanics are everywhere, and if some people say they appear only at some microscopic size, that's only because some "unusual" stuff happens then. Of course it is unusual! We are not that small to see them with our own eyes.
So the answer to the question: "how small should an animal be to show unusual quantum behaviour" would be:. Smaller than you can see (even with a microscope), because that's the definition of "unusual". It turns out to be of the order of hundreds of atoms.
Note that some systems, prepared in "coherent states" can exhibit similar properties because all atoms "beat" at the same rate. Their contributions add up to macroscopic scale.
Now, interesting studies suggest the quantum randomness of the world, one of the most amazing things in quantum mechanics, may be the cause of usual randomness (like flipping a coin). This is a big deal in my opinion:
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$\begingroup$ Best answer so far IMO. Of that randomness article I think not much though – you don't need to resort to quantum effects to explain e.g. fluctuations in gases, such fluctuations can even be observed in purely classical CFD simulations. Basically, any sufficiently chaotic system looks random if you don't have access to the full parameter space, even if the dynamics are actually completely deterministic. In fact this is even the case for quantum mechanics – the Schrödinger equation is perfectly deterministic and only if you introduce decoherence/measurements does it “cause randomness”. $\endgroup$ – leftaroundabout Nov 3 '16 at 20:53
• $\begingroup$ I don't have a definite opinion about that article. There are effects which are not fully explained through classical thermodynamics, like irreversibility, which might rely fundamentally on quantum randomness. But that's not very clear to me how much we don't know here, and I find the article interesting at least. Anyway, this is not really the point of the answer, but just a note. $\endgroup$ – fffred Nov 4 '16 at 14:16
Although it's correct to answer "QM happens at macroscopic scales and it affects humans", I'll try to answer in the spirit of the question.
What is a "quantum mechanical effect"? I'll pick one: matter diffraction. How big an animal can be and still diffract through a grating?
Larger particles (including composite particles) have smaller de Broglie wavelengths, and diffraction is most evident when the gap is about the same size as the wavelength. So to get the largest admissible animal, use the smallest admissible diffraction gate.
The de Broglie wavelength depends on momentum $mv=\frac{h}{\lambda}$ and as a coarse simplification, since we're dealing with small animals, pick $v=1~\mathrm{ms^{-1}}$ so $m=\frac{h}{\lambda}$.
Model the "particle" animal as a uniform sphere of "typical" density of $\rho\approx 10~\mathrm{ kg\cdot m^{-3}}$ so $m=\frac{4}{3}\rho\pi r^3\approx 4\rho r^3$ and as we said above, we are looking for $r=\lambda$ so $\frac{h}{r}\approx 4 \rho r^3$ and so...
$r \approx 2\times 10^{-9}~\mathrm m$
Animals significantly bigger than this can't produce diffraction patterns at normal animal speeds. This would be a difficult experiment to perform, since animals are not uniform spheres. You would get chaotic effects when legs broke off and such like, adding somewhat a lot of noise to the results.
You might be able to get larger animals to diffract successfully if they were moving on a tightly curved section of spacetime (they take up less space if they're stretched into the time direction somewhat) e.g. if their trajectory was the orbit of a small black hole, although I don't know enough GR to analyse this and relativistic velocities would shrink the limiting wavelength/radius further.
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$\begingroup$ I think animals the size of buckyball would move at speeds similar to velocities of particles in a gas (whether they want to or not), not “typical animal speeds”. $\endgroup$ – JDługosz Nov 5 '16 at 6:09
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$\begingroup$ Fair enough, but that factor doesn't change it much as an estimate when you take fourth-roots: $v=500ms^{-1}$ gives $4\rho r^3=\frac{h}{500\lambda}$, or $r^4\approx\frac{h}{100}$, $r\approx10^{-9}m$ $\endgroup$ – spraff Nov 6 '16 at 13:14
When I read "If by some miracle it could be still shrinked [sic]..." in the question, I wonder whether you really want to try to conform to "known" physics, especially if you're telling a story.
But that said, I haven't noticed the phrase "thermodynamic limit" being used in any answers yet. The reason human-sized object don't suddenly teleport is because along these lines:
(1) There's a probability of any given particle "suddenly showing up" anywhere in the known universe, as far as Shrodinger's equation can tell you.
(2) When you put multiple particles together, they behave as a "conjunctive event," in probability-speak. The short version is this: imagine you flip a coin. There's a 50% of either side landing, so neither outcome is a surprise. Now suppose you flip 6*10^23 coins and try to predict the outcome. (ex. "All heads!") Your probability of being right is the product of the probabilities of all the events that would make it up. That probability is minuscule enough that the entire lifespan of the universe (by current estimations) could easily elapse before you successfully guessed the outcome of such an event.
To get "teleportation," you'd need to probabilistic analogue of guessing such an outcome correctly. In other words, we don't see such things happen because the chemistry of the objects that we encounter in daily life (which is a consequence of quantum mechanics) makes is really unlikely for such things to happen during a time-span short enough for a human to observe it. (You'll note that this doesn't rule out such things...it's just says "don't spend your life waiting for it...you'll be bored.")
As an example of a a "thermodynamic limit as a conjunctive even of probabilistic events occurring as determined by quantum mechanics," imagine you have 6*10^23 particles, each with a 1% chance of showing up 1 meter away from where you last observed them, then as a "clump" they'll have a 0.01^(6*10^23) probability of appearing there. I don't think your calculator will be able to tell you what that number is....it's way, way too small of a probability.
This is the "first semester of quantum mechanics" answer, by the way. The afterword of your quantum mechanics textbook may then say, "So...entanglement plays a role in how this actually works, but that's beyond the scope of this book, and not entirely understood yet anyhow." (I guess my point is, don't expect to get the complete answer to this question without devoting your life to physics.)
By the way, if the number 6*10^23 doesn't ring any bells, check out Avogadro's number. (You'll also then have to consider how many multiples of Avogadro's number of molecules make up your lifeform in question.)
Let's point one more thing: A standard example in an introductory class on quantum mechanics (called "modern physics" when I took it) is that of radioactivity (in particular, that of alpha particles, I believe it was), and how quantum mechanics gives an explanation for why it can happen at all. (The answer is tunneling, although let's give it the definition of "a particle having a non-zero probability of suddenly existing away from the chemistry of its usual material, so it then continues its existence without being 'held in place' by all the other particles around it.") But radioactivity doesn't happen because your sample of uranium (for example) is small; it's just the chemistry of the material is such that the probability of a tunneling even is high enough that you can observe it over a time-frame that that people would consider pretty short.
Switching gears, let's get back to your story (or whatever prompted you to ask about this). Miniaturization, as it sounds like you're describing it, isn't really a real-world thing. The objects we encounter in a day-to-day lives are defined by their chemistry, and chemistry can't simply be 'shrunk.' (As an analogy: Build your dream house with Legos, then say "now I want to shrink this down to doll-house sized." To make that happen, you'd need the individual Legos to shrink. But the protons, neutrons and electrons that make up chemistry don't shrink. In fact, they don't vary in any way. Every electron is flawlessly identical to every other electron in the universe. (A physicists, I think John Wheeler, once made a probably-tongue-in-cheek quip about there only being one electron in the universe, doing the job of every electron we ever think exists. If you've every done object-oriented programming, you may find this reminiscent of defining an "electron" class, then instantiation it once every time for each electron that appears to exist in the universe. From the perspective, you might see why some content that the universe's construction seems oddly akin to a computer program.)
So, to actually miniaturize something, you construct something that behaves identically to the original object, but with fewer particles. Whether you can actually do this with a biological entity is probably not a question for the physicists anymore, unless they're physicists who do biological modeling. (As an aside, universities that have a medical school may have some biology-oriented classes in the physics department, probably oriented toward pre-med students that do their undergrad degree in physics. You may also find mathematicians doing things like neurological modeling at such universities.)
If it's sci-fi you're thinking about, you may want to look towards a couple possibilities:
(1) The 'miniaturization' process that you're describing could be more like "nanomachine recreations of biological organisms," which again would means that someone builds a device to try to duplicate the behavior of a given organism. Then you just have to find out a bit more about nanomachines, if you want to try to be accurate within its constraints.
(2) Look to the poorly-understand parts of physics for places where you can get creative. Regarding this...keep in mind that someone with a background in a a little chemistry and no physics may only think of three fundamental particles: protons, neutrons and electrons. (I suppose lots of people know about photons, but they overlook the fact that electrons are the "force mediators" for electrons.) That leads us to the place to dig deeper: If you crack open a particle physics textbook (or flip to the 'particle physics' chapter of a modern physics textbook), you'll see that there's a bunch more of these fundamental particles, some of which have been observed, some of which haven't. The "as of yet not understood" is a fertile place to find things you can make some 'informed speculation' for use in science fiction. (And if you're wondering about why the rest of the particles even exist....my not-particularly-informed response is "stars, stuff that comes from stars, 'mediation of physical effects' and then whatever machinery of the universe that we understood well enough to even suppose that it exists, but not well enough to explain it with any clarity.") Granted, I'm not suggesting that you try to make heads or tails of a particle physics textbook without having studied all the pre-requisites (eg. the usual year of calculus-based physics, intro to modern physics, intro to thermodynamics, undergrad Electricity and Magnetism, undergrad Quantum Mechanics; the in the preface to Griffith's Intro the Elementary Particles he suggests that 'most students in such a class' will have taken everything in that last, but he suggests that the last two don't need to be considered a strict prerequisite.) But unless you do, you'll probably have to fall back on 'informed speculation' ....but, of course, the less you know, the less informed your speculation will inevitably be.
Final note: If story-telling is your aim, don't forget that the primary device for not getting bogged down in "accuracy" is to simply not bring it up. (How much you can get away with that will depend on the story you're trying to tell, of course.)
• $\begingroup$ Sorry in advance for what I'm sure are copious typos. That answer wound up pretty long. (^^; $\endgroup$ – steve_0804 Nov 4 '16 at 18:00
• $\begingroup$ As it stands, it seems to me this is the best answer. $\endgroup$ – RudolfJelin Nov 4 '16 at 19:06
Humans are affected by quantum mechanics: some human eyes are able to detect a single quantum of light (a photon).
• $\begingroup$ Some human eyes? All human eyes can detect singular photons, since the wavelength of a photon is exactly what we have evolved to process/interpret. $\endgroup$ – Harry David Nov 4 '16 at 1:27
• $\begingroup$ @HarryDavid not native speaker here, lets say it other way - rods are capable to detect single photon at frequencies of visible light. from wiki: A photon is an elementary particle, the quantum of all forms of electromagnetic radiation including light. $\endgroup$ – MolbOrg Nov 4 '16 at 4:21
• $\begingroup$ @MolbOrg That would be "quantum" as "smallest unit", not necessarily as in "quantum physics". $\endgroup$ – a CVn Nov 5 '16 at 13:40
• $\begingroup$ @MichaelKjörling can't claim I understand you sentence in full, is that about quantum mechanics in answer - we(a human body) working because that quantum mechanics exists, and one of the reasons why it (CM) is interesting. $\endgroup$ – MolbOrg Nov 5 '16 at 14:19
• $\begingroup$ @MolbOrg A quantum is a smallest unit of something. Quantum physics is physics as it applies to those smallest units. When Wikipedia states that "a photon is ... the quantum of EM radiation", the claim made is that a photon is the smallest, non-divisible portion of EM radiation. See for example merriam-webster.com/dictionary/quantum. $\endgroup$ – a CVn Nov 5 '16 at 14:31
Proteins are the smallest machines of the cell that can do anything interesting (for some definition of interesting, but I work with proteins and I am biased). They are long chains of hundreds of aminoacids (thousands of atoms) do things like pumping water, nutrients, and waste in and out of the cells, guide chemical reactions, send signals, etc.
One of the tools to study them are molecular dynamics simulations. They pretty much use classical mechanics (replacing the atoms with a fancy version of soft balls) with minor numerical tweaks to reproduce quantum behaviour to a very accurate degree. The tweaks are mostly to avoid having to solve the full electrostatic problem of where are the electrons at each time step; but nothing of that would seem strange to a microscopical individual.
So, to get generally quantum-weird behaviour you have to go smaller than the basic functional unit of the life as we know it.
The actual question is : how big can a system be and still be quantic ?
Some theories say that if enough particles are entangled, then the wave function may spontaneously collapse, which means that for example it is not possible to entangle Shrödinger's cat to a decaying atom.
The limit for this would be also the size of this animal.
Your Answer
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1fe7992422f55b71 | Saturday, March 21, 2015
The Joy of Learning -OK Google: What is an Arrhenius acid?
This past Friday (3/20/15) I started my G-Chem class by getting out my cellphone and asking it: What is an Arrhenius acid? ....The phone replied: "According to an Arrhenius acid is a substance that when added to water ..." and continued with the whole definition including the definition of that of a base. So I asked my students: Am I here to tell you what an Arrhenius acid is? They moved their heads in the negative! Then I replied: "you are right I am here to tell you why you want and need to know about Arrhenius acids and bases and to help you make a connection between acid base chemistry with your whole life. This is one underlying principle of 'liberal arts' education. To see the context and to understand the relationships and connections of particular concepts within and without the topic on study.
Today's technology allows us to have instantaneous access to information, so information should not be the outcome of a lecture. It has been said that information is not knowledge, so class time should not be use to transmit information, it should be used to develop knowledge and to develop the skills necessary for oneself to create relevant knowledge. The teaching professor is there to guide inquiry and to set limits of time during the exercise of exploration. Learning science is complicated, I guess as learning anything that has many facets, but one can always try to stop the fragmentation of ideas through a holistic approach. Meaning that on can not separate individual steps of the solution of a problem with the overall context of the question being addressed. One can look at the solution of the problem as a simplified model or metaphor but one has to be conscientious of the fact that things are more complicated than that. Any particular and individualized solution of a problem has to be framed within a context and other consequences like secondary effects have to be at least noted, if not explored. This makes teaching science a difficult but enjoyable task, as challenges like puzzles are inherently attractive to the inquisitive mind. This is one important role of the science teacher: make challenging concepts appear like games in the journey that life is.
In my previous post, I mentioned the importance of 'joy' in learning, even to the point of saying: "If you are not having fun,... you are not learning!"
It seems simplistic in the light of many that believe that things that matter have to be hard to learn, difficult to understand, and that should take a long time to comprehend. I agree but have some reservations about the attitude that one must have while going through the process of learning. And I am including the activities of teaching as part of the learning process. The teacher must be having fun as s/he teaches or s/he will not be able to have and create the energy to deliver a well intended lesson. It might be said that this happens all the time with everything we do in our lives, that no one person that is successful has been doing the things that leaded to the success with an attitude contrary to his/her joy and satisfaction. A recent blog at "Class Teaching" use a perfect metaphor with playing a computer game called Manic Miner. In this post Shaun Allison @shaun_allison takes a step by step approach to make a parallel between playing a game with several levels of difficulty and learning. It sure is a great pedagogical insight.
Saturday, January 24, 2015
If You Are Not Having Fun You Are Not Learning
Once in a while I remind my students about the joy of learning. Remembering this is very important when you are having a hard time learning new ideas. Ideas that are complex and difficult by their own nature and by the fact that it's not easy to contextualize them with our daily lives.
I have used the poem by Wang Ken "Song of Joy" as an inspiration to encourage my students to enjoy learning. I stress and emphasize this so much in my classes that in fact I call homework "Homejoy!"
• Pleasure is the state of being Brought about by what you Learn.
• Learning is the process of Entering into the experience of this Kind of pleasure.
• No pleasure, no learning.
• No learning, no pleasure.
(Wang Ken, Song of Joy.)
Many books and articles have been written around this idea, one in particular is "The Power of Mindful Learning" by Ellen J. Langer. (For more link here.)
And recently a new edition of "Experiential Learning" by David A. Kolb. (link here to read more.)
Of course we must not forget the seriousness of learning and the fact that it can be hard to do, but keeping in mind that successful endeavors require more than just the material means to accomplish, we have to remind ourselves that attitude is critical for success.
Did you see the Seattle Seahawks game against the Green Bay's Packers?
A good example of how attitude -having fun- produces good results!
Sunday, November 23, 2014
Skepticism and Science
Framing a context for the value of content.
Being a skeptic is for scientist a core state, the value of skepticism is rooted in the need of science to ask questions and on having in mind that whatever model we have now to explain a phenomenon is only temporary an it can, and most likely, change in the future. The interconnectedness between the phenomenon and the surroundings does not allow the invention of models to be separated from the anthropomorphic view of the person creating the model. Therefore it is necessary to see what is the context of the people developing these ideas. Culture in general and language in particular restrict and guide the construction of hypothesis and theories.
Science education is more than teaching a set of rules given by theories or the transmission of content boxed in a set of models. Science education has to develop the connection with previous experiences in our society. These connections allow the student see how these ideas, hypothesis, and theories were developed and how they apply to our lives. As an example I can mention when teaching and explaining how the periodic table of the elements work I made the connection with my previous research on rare earths (aka Lanthanides) and the noble gases (aka inert gases). Not only teaching the names of these elements but having a story behind their nomenclature and behavior allowed the student get a feeling of discovery and a sense of awe of God’s creation. Knowing becomes an individual's integral status of relationship with his/her own history and environment.
What is necessary to know about the students when teaching science?
These students have gone to the traumatic experience of ‘directed’ education where ‘educators’ have induced in these students indoctrinated thinking void of ‘critical thinking’ which for the context of this writing is scientific skepticism. This scientific skepticism is so much needed in today’s society.
In his book "Think: Why You Should Question Everything" Guy P. Harrison (for a link to his website click here ) warns about the lack of critical thinking in our society and teaches us that thinking like a scientist is the only way to avoid being swindled by crooks, kooks, and demagogues selling all sort of silly, and wrong ideas. Including commercial products that are harmful to us and to our environment. Being critical thinkers is a matter of personal security and wellbeing.
The need to develop critical thinking, i.e. skepticism in my students is what drives me to be critical and skeptical, and to teach with a sense of awe and feelings of discovery at every step even when the topic at hand seems to be old and fully developed like the idea of the periodic table. We know that the periodic table as it is normally presented is not at all perfect and even though is highly useful it need some explanation and adaptation. At the same time students need to know that new ways of presenting the idea of 'periodicity' of the elements (in some cases by the use of a 'table') are currently being developed as this link shows. Click here for the link.
The question now becomes, how the context of an idea can be used to reflect on the value and accuracy of the model proposed by it?
Sunday, November 9, 2014
Difficult Concepts in Science
Learning scientific concepts has an inherent difficulty that arises from the fact that they are expressed in common language terminology but with a specific meaning. For example the word 'difference' that the dictionary definition would state as: "not equal", in mathematics is specific to the idea of a quantitative value 'A - B' "the result of arithmetic subtraction" (Mac's dictionary). In particular chemistry uses symbolism to express these differences, a capital Greek letter Δ (delta) for major differences like the difference in temperature, between two physical states; and lower case δ (delta) for minor/slight differences like the one encountered in electromagnetic polarities within the atom. These major differences are of extreme importance when looking at energy changes during physical and chemical reactions, and they can be expressed as difference in enthalpy, entropy, volume, or any other variable of state that only depends on the values at the end and beginning of the process not on the path that the change followed from initial to final state. Of course we can also apply the idea of big difference when dealing with non conservative phenomena that is dependent on the path followed, such as when dealing with friction generated loss of energy during a process.
It sure become critical in the discussion of these phenomena to keep in mind the definition of all variables and parameters in the process, and these is what makes these concepts difficult to understand.
So, I think, I have to start with the definition of definition!
From my Mac's Dictionary:
"definition |ˌdefəˈni sh ən|nouna statement of the exact meaning of a wordesp. in a dictionary.• an exact statement or description of the nature, scope, or meaningof something our definition of what constitutes poetry.• the action or process of defining something.the degree of distinctness in outline of an object, image, or sound, esp. of an image in a photograph or on a screen.• the capacity of an instrument or device for making images distinct in outline [in combination high-definition television.PHRASESby definition by its very nature; intrinsically underachievement, by definition, is not due to lack of talent.
A definition is astatement of the meaning of a term (awordphrase, or other set of symbols).[a] The term to be defined is the definiendum. The term may have many different senses and multiple meanings. For each meaning, a definiens is a cluster of words that defines that term (and clarifies the speaker's intention).
As a noun definition is a statement of the exact meaning of the word. Exact in the sense of providing meaning that not only is accurate but precise so one can use the meaning repetitively within different contexts. But as 2 above: provides a degree of distinctness characterized by its relationship to the topic. Within a metaphor the words "atomic view" and "microscopic view" can be interchanged without changing the intent of those words, while in the description of an item, atom and microscope are completely different.
With this in mind let's retake the idea of 'atom' for an initial analysis of what constitute a difficult concept in science. The last sentence in our definition of definition it is stated that additional details about etymology should be given, so atom mean without a parts from the Greek, so we infer it is the smallest part of the world, but we now know that the atom has parts, protons, neutrons, electrons, that themselves are made of smaller parts (subatomic) components such as muons, mesons, quarks, bosons, and others with a variable set of colors and flavors as you find out in Wikipedia.
So the question about understanding what an atom is becomes inherently complicated and a simple explanation of what an atom is becomes elusive. One can of course simplify with models or analogies but it must be understood that the simplification will undoubtedly produce inaccuracies and misinterpretations that can, if magnified lead to critical errors of understanding. One example of this could be the lack of understanding many people have regarding the significance of 'orbital' as a 'mathematical' description of the probable localization of the electron around the nucleus within the atom. An electron that is modeled as a small particle (dot in the drawing) but mathematically is represented by a wave or probability function as stated by the Schrödinger equationödinger_equation.
As an educator I have to make sure that the student understand the complexities of nature as well as the difficulties of concepts describing the behavior and properties of phenomena within nature while at the same time providing students with mechanisms, formulas, and procedures that will permit them apply their skill to the solution of basic problems, even without a full understanding of the deep meaning of the phenomena.
This is the art of making difficult concepts easy to understand.
Sunday, October 12, 2014
Online Content Education
As I think about the title of this post, "Online Content Education", I become aware of the apparent contradiction or stress between the words content and education. Transmitting information -bits of facts and data could be considered "Content Education" but is it education in the sense of a formative process? What about the need to think critically, or the ability to communicate complex ideas?
These require added context and have to be developed during the learning process.
Science teaching appears to be one of the topics where content is well defined, and measurable outcomes could be designed for specific subjects. For instance in chemistry one can teach the periodic table and assess learning outcomes by developing questions that directly reflect if the student understands the periodic table.
It seems like a simple task; understanding the periodic table seems like a topic that can be boxed into a simple set of questions. Questions that would have a 'right' answer, which can be stated within a multiple choice set of questions where all but one are wrong. We can do that today easily within an 'online' format expanding access, allowing students who otherwise wouldn't be able to learn.
On the other hand if content is not the only thing, how will online instruction be detrimental to learning? In today's The Oregonian I read a guest column by Ramin Farahmandpur (Professor in the Department of Educational Leadership and Policy in Portland State University's Graduate School of Education) that clearly articulates how students in online classes lose the opportunity given by classroom discussion and interaction. Prof. Farahmandpur uses the word 'shortchange' to describe the loss of learning opportunities during online instruction and mentions how Western Governors University (A well known online private non-for-profit organization) had in 2012 the lowest graduation rates according to the CBS Money Watch Report. To read more click the following link
Friday, October 10, 2014
Content and Context in Higher Ed
Science is supposed to be about content. Concepts, hypothesis, and theories are used to understand how the world works and to develop technology that is fundamental for the betterment of our society. Many would say that this last is why science is so important, and why we should as a society support its progress. Who could be against the advances of modern medicine, and engineering?
This view of science lead to the assumption that teaching science should be simply the transmission of ideas, the teaching of content. So we can always test that it is happening by a simple question: can the student solve such and such problem? Questions like "what is the temperature if .....?" are the standard questions in any assessment of student knowledge.
In a way this is OK, this will allow the student to be a "problem solver" but, will s/he be a "critical thinker"? I think that this is not enough. If we are not critical thinkers our ability to solve problems will be also impaired.
This week I'm teaching gas behavior in my general chemistry class. The mathematical expression that relates volume, pressure, amount, and temperature is known as the 'ideal gas law" PV = nRT. Working with this formula amounts to simple algebra, should not give much trouble. It looks like there is no context. So why should I talk about Robert Boyle a fellow of the Royal Society who in the XVII century developed what is now known as Boyle's law relating the volume and the pressure of a gas, or Jacques Charles a French aristocrat, member of the Paris Science Academy, who lived through the French Revolution and was probably the first to fly an unmanned balloon full of hydrogen in 1783. Charles Law relates temperature with volume of a gas and even though it was Gay-Lussac who published in 1802, Charles was given credit for his unpublished work.
It seems to me that this honesty in the scientific world has become less of a norm, I'm sad to say.
Then we have Avogadro (always concerned with the amounts of substances) lived the last part of the XVIII and first half of the XIX centuries. He of course saw the relationship between the amount of gas and the volume. Now we know this relationship as Avogadro's Law.
When in the late 1800's these laws where condensed into one: The Ideal Gas Law PV =nRT
Water vapor engineering was born. And "steam' energy became the driver of the second industrial revolution 1840-1870 by introducing "steam" engines to trains and boats transforming transportation.
Now the question I have is: why should students learn about all the history when learning how to solve problems with PV =nRT? Is the ideal gas law going to change if circumstances change? What can I learn from the fact that many minds where involved in the development of the "law"?
Are the answers to these questions self evident?
Wednesday, October 8, 2014
Opening Opportunities - Freedom to Flourish - A Counter System
It is a fact!
More and more students are coming out of high-school ill prepared. In my previous posting here I talked about an article in the Oregonian (10/7/14) where the average low SAT scores of Oregon high school graduates is mentioned. This is -as with any problem- an opportunity. And Warner Pacific College is stepping up to the challenge!
This is what WPC's president Dr. Andrea Cook has to say about it: "At Warner Pacific, we develop significant relationships with our students, and believe it’s an essential means of educating, challenging and serving students who might otherwise not finish their education. The reality is our educational system has been designed for advantaged people. In order to make education more fully accessible, we need to create a “counter system” that grants access to a wider population—that’s what we’re about." (Quote from the 'president's message in WPC website )
We are proud of the approach we are taking helping students that come from underserved cultures and backgrounds and helping them succeed and flourish. WPC is opening opportunities by recognizing the need for change in higher education. By embracing these challenges and turning them around making them opportunities.
The world is in dire need of STEM graduates in particular and in need of higher education in general, so this is how we can be part of the solution. Bringing the opportunity to study science to a population that is not normally served to do so is of great importance. It will of course create problems as these students are not well prepared for the rigor of the sciences curricula. But there are many things in favor of the success of these students, one is their eagerness to succeed, their gumption for life, their capacity for adventure, and their freedom to flourish! |
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Structure of the atom
Automatic translationAutomatic translation Category: matter and particles
Updated November 06, 2013
Everything we see is made up of atoms, many atoms. It was watching the smallest constituents of the matter that scientists have been able to explain, in the twentieth century, the operation of the entire universe. An atom is constituted by a core around which moves one or more electrons. What characterizes the core is its number of protons ( Z) ranging from 1 to 110, it is it who determines the element, such as iron (FE26) has 26 protons, 26 is the atomic number. The number of neutrons (N) ranging from 0 to 160, characteristic of the isotopes of the element, for example, hydrogen (H1) has a proton and no neutron, deuterium (H2) has a proton and a neutron, tritium (H3) has a proton and two neutrons. These three forms of hydrogen have only one electron, since there is only one electrical load, the single proton. Attention that it is only in the case of hydrogen that is given a different name to the isotopes of the element, in all other cases we indicate only the number of nucleons thereby find the number of neutrons. For example iron (FE26) has several isotopes Fe56, we understand that Fe56 has 30 neutrons, Fe57 has 31 neutrons, Fe58 32 neutrons, the number of neutrons differs well isotopes.
In the atom are electrons that give consistency to the material, yet it is very light, its mass is about 10-27 grams, the proton is 2000 times heavier and concentrates majority of the mass of the atom (99.99 % ). For stable atoms, the mass is between 1.674 × 10-24 g for Hydrogen and 3.953 × 10-22 g for uranium. Since 1811, we also know the approximate size of an atom, Amedeo Avogadro (1776-1856) estimated the size of atoms at 10-10 meter, i.e. a little more than 10 millionths of a millimeter.
In 1911, Ernest Rutherford (1871-1937) discovered the atomic nucleus and specifies the structure of the atom by bombarding gold foil with particles from the radioactive decay of uranium. He gives a nucleus size in the order of 10-14 meters. There is a little more than one hundred different atoms, these are the elements such as hydrogen, carbon, oxygen or iron. The New Zealand physicist had the idea of a representation of the atomic nucleus. Rutherford represents each atom as a mini solar system, the center and the nucleus like planets orbit the electrons. The nucleus itself is represented as a mature grains (picture to the right). This pictorial representation is false but has two advantages, it clearly differentiates the two particles, the proton and neutron and we understand that the core, very compact, is circumscribed within a defined volume. But since the advent of quantum mechanics in the 1920s, the nucleus image is disturbing, the nucleus is no longer a system of balls associated together. The nucleus is governed by quantum mechanics, in other words it exist than if it is observable but observe the protons and neutrons inside the nucleus as they are in the picture, is not possible because it would illuminate the particles with a light so intense that the nucleus would instantly disintegrate. This representation in form of grains blackberry hides the quantum concept of matter. It is the same for the electron, no longer represents the electron as a particle which rotates on a very regular orbit around the nucleus. The electron is both a wave and a particle, the wave-particle duality is the foundation of quantum mechanics. In quantum mechanics the electron does not follow a single path, it is located in a region around the nucleus, called the electronic cloud or atomic orbital.
classic representation of the atom
Image: Representation of the atomic nucleus, in the form of grains blackberry, has two advantages, we differentiate the two particles, the proton and neutron, and it is clear that the nucleus, very compact, is circumscribed within a defined volume. All nucleus and all isotopes have between 1 and 110 protons and between 0 and 160 neutrons. However, this representation is false because it hides the quantum concept. Since the 1920s, the nucleus is no longer a system of balls associated together, it is a quantum system much more troubling.
Electron cloud or atomic orbital
Since 1924, all matter is associated with a wave, is the assumption of Louis de Broglie (1892-1987). With this hypothese, he generalizes to all the particles of matter, wave-particle duality brought to light by Max Planck (1858-1947) in the early 20th century. All subatomic particles therefore have a wavelength.
The wavelength λ of a subatomic particle and momentum p are related by the equation: λ = h / p, where h is Planck's constant, the momentum p i.e. product of the mass by the speed vector (p = mv). Since we known, thanks to Einstein's famous formula that all matter has an associated energy (E = mc2). In other words, more the wavelength is small and more energy is high (E = h / λ). This energy will modify the shape of atoms. The foundations of quantum mechanics are then posed.
In short, matter is really composed of very small particles, fermions (electrons, neutrinos, quarks) that have mass, charge, energy, dimension, a wave, a spin. But what these particles resemble, in the world of the infinitely small ?
In 2013 we still can not see the particles of the atomic nucleus but only the outer layer of the atom i.e. the electron cloud. The electron cloud occupies all the spatial extent of the atom as it is about 10 000 times greater than its nucleus. In wave quantum mechanics, a particle is represented by a wave function, but it is very difficult to represent the fundamental concept of quantum mechanics or quantum state of a system. In 1927, Max Born (1882-1970) gave an interpretation of the wave function where the square of the wave function represents the probability, when you make a measurement, finding the particle at a specific location function.
A wave function is an amplitude or a probability density function of the presence system in a given position at a given instant. This function has a complex value. If a real number or an actual value is, for example, the length of a segment on a straight line, a complex value is represented by a vector in a plane, this vector has not only a length in space but also a phase which corresponds to the direction of the vector.
If you no longer represents the electron as a point particle on a regular orbit around a nucleus, how can you make a picture?
Well here, the electron does not follow a single path around the nucleus, it is somewhere in a vast region that we call the electron cloud or atomic orbital. The state of an electron is represented by the volume of the space around the nucleus in which it is localized. The fundamental state of hydrogen is about one angstrom is 10-10 meters. To represent the electron in this region just imagine a grain of rice of about 5 mm to move in a sphere about 50 meters in diameter. In addition, the shape of this region of atomic space depends on the energy of the electron and its momentum is what we see on the image cons. Thus the orbital of the electron characteristics may take various forms depending on the nature of the atom, for example, the orbital of the hydrogen atom on the first row at the top has a spherical shape, the orbital second row has the shape of two drops of water, the third row on the orbital in the form of four drops of water. In summary, in the orbiting region of the space where the electron is delocalized, the electron state is a superposition of all possible positions within the atomic orbital whose shape varies. The shape of the orbital changes when excited atom, as in the first row. If excites even more atom, the shape of the orbital change again as the second row or electronic layer. In a very excited state called "Rydberg state", electrons are delocalized in a torus "large radius" that can measure up to 1000 angstroms, the principal quantum number n (number of layer) is very high between 50 and 100.
nota: an electron attracted by the positive charge of the nucleus, can not "stick to the nucleus" because it would mean that the spatial extension of the wave function is reduced to a point. The Schrödinger equation says that an electron in the vicinity of the nucleus is in a orbital to geometry determined by the quantum numbers which satisfy this equation. In short, an electron is confined in the vicinity of the nucleus by the electrostatic potential well. When the potential energy is increasing, we say that the particle moves in a potential well.
electronic orbitals
Image: Representation of the first electronic orbitals of hydrogen based on the energy of the electron and its angular momentum, the energy level increases from top to bottom (n = 1 , 2, 3) and the momentum increases from left to right , (I = s , p, d , f , g) . This image shows the probability density of finding the electron, the black color represents the density 0, i.e. the area where the electron will never adventure. The white color represents the maximum density, i.e. the area where the electron passes most often. Between black and white in the orange-red zone, the probability density increases. Quantum numbers are represented by letters n, is the principal quantum number, it defines the energy level of the electron. I is the orbital quantum number, or secondary, as it defines the electronic sublayers, s (sharp) for l = 0 , p (principal) for l = 1 , d (diffuse) for l = 2 , f (Fundamental) for I = 3 , then (to the excited states ) g , h , i , ... the quantum number m is the magnetic quantum number or tertiary .
Credit image: GNU Free Documentation License.
See also
Quantum field theory
Fields of the real...
Reproduction prohibited without permission of the author.
nanoparticles, carbon nanotube
Scale of the nanoparticles...
Radioactivity of the Earth
Radioactivity of the Earth...
Neutrino, constituent of matter
and beta emission...
Magnetism and magnetization
Magnetic order
and magnetization...
Leap second
Leap second...
Electromagnetic spectrum
of the spectrum... |
06c2b93c3e0d4f4d | Give to MAT
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Multimodal Representation of Quantum Mechanics:
"The Hydrogen Atom"
Professor JoAnn Kuchera-Morin, Media Arts and Technology, professor Luca Peliti, Physics Department, and PhD student Lance Putnam, Media Arts and Technology
As the sciences increasingly rely on mathematical constructs to describe the invisible processes of nature, it is important to remain cognizant of the effectiveness of empirical observation towards gaining new insights. Digital systems provide not only a means of simulating models, but also a medium for communicating through image and sound.
This work interactively visualizes and sonifies the wavefunction of an electron of a single hydrogen atom. The atomic orbitals are modeled as solutions to the time-dependent Schrödinger equation with a spherically symmetric potential given by Coulomb's law of electrostatic force. Different orbitals of the electron can be combined in superposition to observe dynamic behaviors such as photon emission and absorption.
Hydrogen Atom
Video arrow
The interactive component of the simulation allows one to fly through the atom with a probe that emits "stream particles" that follow along the largest changes in the probability current and gradient of the electron. The electron probability amplitude is sonified by scanning through groups of stream particles in the space. The pitch can be adjusted by the rate at which a particular set of stream particles is scanned across. This allows us to give the sonification procedure a certain type of musicality, by assigning specific pitches to different features in the wavefunction.
This investigation is just the beginning of an effort to multimodally represent mathematical models used in physical and theoretical sciences. By finding a common meeting ground, artists and scientists can share insights and pursue similar fundamental questions about symmetry, pattern formation, and emergence.
Hydrogen Atom
Images by Lance Putnam
More Research Spotlights: << | >> |
3a64ee6c17453b07 | Physics & Astronomy Applets
I've written a small number of Java applications (sometimes applets that can be buried in a browsers, sometimes standalone applications) that I've used in classes over the years. A few of them are below.
These things require Java. If you try to run them as an applet in the browser, they require the Java plug-in; Google for it if you don't already have it in your browser. If you download them to run then, you'll need to have Java installed on your system; you may well have it already. If not, it's available for zero cost, and can be found at
Make sure you have at least Java version 1.6 (also known as Java 6) for everything below to work.
Spiral Galaxy Rotator
A very astute student who was in my first introductor astronomy class asked, when I told them about the rotation curve of Spiral Galaxies, "why don't the spiral arms wrap up"? I wrote this applet in an attempt to demonstrate to those who couldn't see it why this would seem to be an issue.
Wave Play
A toy for demonstrating wave interference. Set up sources of sin waves that propagate left and right, and watch them interfere!
1d-Schrödinger Solver
An application that uses the JSci libraries to solve the one-dimensional time-independent Schrödinger equation for a handful of built-in potentials. |
94f02788cb643ab7 | Partial differential equation
From Wikipedia, the free encyclopedia
Jump to: navigation, search
A visualisation of a solution to the heat equation on a two dimensional plane
In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives. (This is in contrast to ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.) PDEs are used to formulate problems involving functions of several variables, and are either solved by hand, or used to create a relevant computer model.
PDEs can be used to describe a wide variety of phenomena such as sound, heat, electrostatics, electrodynamics, fluid flow, elasticity, or quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations.
Partial differential equations (PDEs) are equations that involve rates of change with respect to continuous variables. The position of a rigid body is specified by six numbers, but the configuration of a fluid is given by the continuous distribution of several parameters, such as the temperature, pressure, and so forth. The dynamics for the rigid body take place in a finite-dimensional configuration space; the dynamics for the fluid occur in an infinite-dimensional configuration space. This distinction usually makes PDEs much harder to solve than ordinary differential equations (ODEs), but here again there will be simple solutions for linear problems. Classic domains where PDEs are used include acoustics, fluid flow, electrodynamics, and heat transfer.
A partial differential equation (PDE) for the function u(x_1, \cdots, x_n) is an equation of the form
f \left (x_1, \ldots, x_n, u, \frac{\partial u}{\partial x_1}, \ldots, \frac{\partial u}{\partial x_n}, \frac{\partial^2 u}{\partial x_1 \partial x_1}, \ldots, \frac{\partial^2 u}{\partial x_1 \partial x_n}, \ldots \right) = 0.
If f is a linear function of u and its derivatives, then the PDE is called linear. Common examples of linear PDEs include the heat equation, the wave equation, Laplace's equation, Helmholtz equation, Klein–Gordon equation, and Poisson's equation.
A relatively simple PDE is
\frac{\partial u}{\partial x}(x,y) = 0.~
This relation implies that the function u(x,y) is independent of x. However, the equation gives no information on the function's dependence on the variable y. Hence the general solution of this equation is
u(x,y) = f(y),
where f is an arbitrary function of y. The analogous ordinary differential equation is
\frac{\mathrm{d} u}{\mathrm{d} x}(x) = 0,
which has the solution
u(x) = c,
where c is any constant value. These two examples illustrate that general solutions of ordinary differential equations (ODEs) involve arbitrary constants, but solutions of PDEs involve arbitrary functions. A solution of a PDE is generally not unique; additional conditions must generally be specified on the boundary of the region where the solution is defined. For instance, in the simple example above, the function f(y) can be determined if u is specified on the line x = 0.
Existence and uniqueness[edit]
Although the issue of existence and uniqueness of solutions of ordinary differential equations has a very satisfactory answer with the Picard–Lindelöf theorem, that is far from the case for partial differential equations. The Cauchy–Kowalevski theorem states that the Cauchy problem for any partial differential equation whose coefficients are analytic in the unknown function and its derivatives, has a locally unique analytic solution. Although this result might appear to settle the existence and uniqueness of solutions, there are examples of linear partial differential equations whose coefficients have derivatives of all orders (which are nevertheless not analytic) but which have no solutions at all: see Lewy (1957). Even if the solution of a partial differential equation exists and is unique, it may nevertheless have undesirable properties. The mathematical study of these questions is usually in the more powerful context of weak solutions.
An example of pathological behavior is the sequence (depending upon n) of Cauchy problems for the Laplace equation
\frac{\part^2 u}{\partial x^2} + \frac{\part^2 u}{\partial y^2}=0,~
with boundary conditions
u(x,0) = 0,
\frac{\partial u}{\partial y}(x,0) = \frac{\sin (nx)}{n},
where n is an integer. The derivative of u with respect to y approaches 0 uniformly in x as n increases, but the solution is
u(x,y) = \frac{\sinh (ny) \sin (nx)}{n^2}.
This solution approaches infinity if nx is not an integer multiple of π for any non-zero value of y. The Cauchy problem for the Laplace equation is called ill-posed or not well posed, since the solution does not depend continuously upon the data of the problem. Such ill-posed problems are not usually satisfactory for physical applications.
In PDEs, it is common to denote partial derivatives using subscripts. That is:
u_x = {\partial u \over \partial x}
u_{xx} = {\part^2 u \over \partial x^2}
u_{xy} = {\part^2 u \over \partial y\, \partial x} = {\partial \over \partial y } \left({\partial u \over \partial x}\right).
Especially in physics, del or Nabla (∇) is often used to denote spatial derivatives, and \dot u\, \ddot u\, for time derivatives. For example, the wave equation (described below) can be written as
\ddot u=c^2\nabla^2u
\ddot u=c^2\Delta u
where Δ is the Laplace operator.
Heat equation in one space dimension[edit]
See also: Heat equation
The equation for conduction of heat in one dimension for a homogeneous body has
u_t = \alpha u_{xx}
where u(t,x) is temperature, and α is a positive constant that describes the rate of diffusion. The Cauchy problem for this equation consists in specifying u(0, x)= f(x), where f(x) is an arbitrary function.
General solutions of the heat equation can be found by the method of separation of variables. Some examples appear in the heat equation article. They are examples of Fourier series for periodic f and Fourier transforms for non-periodic f. Using the Fourier transform, a general solution of the heat equation has the form
u(t,x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty F(\xi) e^{-\alpha \xi^2 t} e^{i \xi x} d\xi, \,
where F is an arbitrary function. To satisfy the initial condition, F is given by the Fourier transform of f, that is
F(\xi) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(x) e^{-i \xi x}\, dx. \,
If f represents a very small but intense source of heat, then the preceding integral can be approximated by the delta distribution, multiplied by the strength of the source. For a source whose strength is normalized to 1, the result is
and the resulting solution of the heat equation is
u(t,x) = \frac{1}{2\pi} \int_{-\infty}^\infty e^{-\alpha \xi^2 t} e^{i \xi x} d\xi. \,
This is a Gaussian integral. It may be evaluated to obtain
u(t,x) = \frac{1}{2\sqrt{\pi \alpha t}} \exp\left(-\frac{x^2}{4 \alpha t} \right). \,
This result corresponds to the normal probability density for x with mean 0 and variance 2αt. The heat equation and similar diffusion equations are useful tools to study random phenomena.
Wave equation in one spatial dimension[edit]
The wave equation is an equation for an unknown function u(t, x) of the form
u_{tt} = c^2 u_{xx}.
Here u might describe the displacement of a stretched string from equilibrium, or the difference in air pressure in a tube, or the magnitude of an electromagnetic field in a tube, and c is a number that corresponds to the velocity of the wave. The Cauchy problem for this equation consists in prescribing the initial displacement and velocity of a string or other medium:
u(0,x) = f(x),
u_t(0,x) = g(x),
where f and g are arbitrary given functions. The solution of this problem is given by d'Alembert's formula:
u(t,x) = \tfrac{1}{2} \left[f(x-ct) + f(x+ct)\right] + \frac{1}{2c}\int_{x-ct}^{x+ct} g(y)\, dy.
This formula implies that the solution at (t,x) depends only upon the data on the segment of the initial line that is cut out by the characteristic curves
x - ct = \text{constant,} \quad x + ct = \text{constant},
that are drawn backwards from that point. These curves correspond to signals that propagate with velocity c forward and backward. Conversely, the influence of the data at any given point on the initial line propagates with the finite velocity c: there is no effect outside a triangle through that point whose sides are characteristic curves. This behavior is very different from the solution for the heat equation, where the effect of a point source appears (with small amplitude) instantaneously at every point in space. The solution given above is also valid if t < 0, and the explicit formula shows that the solution depends smoothly upon the data: both the forward and backward Cauchy problems for the wave equation are well-posed.
Generalised heat-like equation in one space dimension[edit]
Where heat-like equation means equations of the form:
\frac{\partial u}{\partial t} = \hat{H} u +f(x,t) u+g(x,t)
where \hat{H} is a Sturm–Liouville operator subject to the boundary conditions:
\hat{H} X_n = \lambda_n X_n
X_n (a) = X_n (b) =0
\dot{a}_n (t) - \lambda_n a_n (t) -\sum_m (X_n f(x,t),X_m) a_m (t) = (g(x,t),X_n)
a_n(0) = \frac{(h(x),X_n)}{(X_n,X_n)}
u(x,t) = \sum_{n} a_n (t) X_n(x)
(f,g)=\int_a^b f(x) g(x) w(x) \, dx.
Spherical waves[edit]
Spherical waves are waves whose amplitude depends only upon the radial distance r from a central point source. For such waves, the three-dimensional wave equation takes the form
u_{tt} = c^2 \left[u_{rr} + \frac{2}{r} u_r \right].
This is equivalent to
(ru)_{tt} = c^2 \left[(ru)_{rr} \right],
and hence the quantity ru satisfies the one-dimensional wave equation. Therefore a general solution for spherical waves has the form
u(t,r) = \frac{1}{r} \left[F(r-ct) + G(r+ct) \right],
where F and G are completely arbitrary functions. Radiation from an antenna corresponds to the case where G is identically zero. Thus the wave form transmitted from an antenna has no distortion in time: the only distorting factor is 1/r. This feature of undistorted propagation of waves is not present if there are two spatial dimensions.
Laplace equation in two dimensions[edit]
The Laplace equation for an unknown function of two variables φ has the form
\varphi_{xx} + \varphi_{yy} = 0.
Solutions of Laplace's equation are called harmonic functions.
Connection with holomorphic functions[edit]
Solutions of the Laplace equation in two dimensions are intimately connected with analytic functions of a complex variable (a.k.a. holomorphic functions): the real and imaginary parts of any analytic function are conjugate harmonic functions: they both satisfy the Laplace equation, and their gradients are orthogonal. If f=u+iv, then the Cauchy–Riemann equations state that
u_x = v_y, \quad v_x = -u_y,\,
and it follows that
u_{xx} + u_{yy} = 0, \quad v_{xx} + v_{yy}=0. \,
Conversely, given any harmonic function in two dimensions, it is the real part of an analytic function, at least locally. Details are given in Laplace equation.
A typical boundary value problem[edit]
A typical problem for Laplace's equation is to find a solution that satisfies arbitrary values on the boundary of a domain. For example, we may seek a harmonic function that takes on the values u(θ) on a circle of radius one. The solution was given by Poisson:
\varphi(r,\theta) = \frac{1}{2\pi} \int_0^{2\pi} \frac{1-r^2}{1 +r^2 -2r\cos (\theta -\theta')} u(\theta')d\theta'.\,
Petrovsky (1967, p. 248) shows how this formula can be obtained by summing a Fourier series for φ. If r < 1, the derivatives of φ may be computed by differentiating under the integral sign, and one can verify that φ is analytic, even if u is continuous but not necessarily differentiable. This behavior is typical for solutions of elliptic partial differential equations: the solutions may be much more smooth than the boundary data. This is in contrast to solutions of the wave equation, and more general hyperbolic partial differential equations, which typically have no more derivatives than the data.
Euler–Tricomi equation[edit]
The Euler–Tricomi equation is used in the investigation of transonic flow.
u_{xx} =xu_{yy}.
Advection equation[edit]
The advection equation describes the transport of a conserved scalar ψ in a velocity field u = (u, v, w). It is:
If the velocity field is solenoidal (that is, ∇⋅u = 0), then the equation may be simplified to
In the one-dimensional case where u is not constant and is equal to ψ, the equation is referred to as Burgers' equation.
Ginzburg–Landau equation[edit]
The Ginzburg–Landau equation is used in modelling superconductivity. It is
iu_t+pu_{xx} +q|u|^2u=i\gamma u
where p,qC and γ ∈ R are constants and i is the imaginary unit.
The Dym equation[edit]
The Dym equation is named for Harry Dym and occurs in the study of solitons. It is
u_t \, = u^3u_{xxx}.
Initial-boundary value problems[edit]
Many problems of mathematical physics are formulated as initial-boundary value problems.
Vibrating string[edit]
If the string is stretched between two points where x=0 and x=L and u denotes the amplitude of the displacement of the string, then u satisfies the one-dimensional wave equation in the region where 0 < x < L and t is unlimited. Since the string is tied down at the ends, u must also satisfy the boundary conditions
u(t,0)=0, \quad u(t,L)=0,
as well as the initial conditions
u(0,x)=f(x), \quad u_t(0,x)=g(x).
The method of separation of variables for the wave equation
u_{tt} = c^2 u_{xx}, \,
leads to solutions of the form
u(t,x) = T(t) X(x),\,
T'' + k^2 c^2 T=0, \quad X'' + k^2 X=0,\,
where the constant k must be determined. The boundary conditions then imply that X is a multiple of sin kx, and k must have the form
k= \frac{n\pi}{L},
where n is an integer. Each term in the sum corresponds to a mode of vibration of the string. The mode with n = 1 is called the fundamental mode, and the frequencies of the other modes are all multiples of this frequency. They form the overtone series of the string, and they are the basis for musical acoustics. The initial conditions may then be satisfied by representing f and g as infinite sums of these modes. Wind instruments typically correspond to vibrations of an air column with one end open and one end closed. The corresponding boundary conditions are
X(0) =0, \quad X'(L) = 0.
The method of separation of variables can also be applied in this case, and it leads to a series of odd overtones.
The general problem of this type is solved in Sturm–Liouville theory.
Vibrating membrane[edit]
If a membrane is stretched over a curve C that forms the boundary of a domain D in the plane, its vibrations are governed by the wave equation
\frac{1}{c^2} u_{tt} = u_{xx} + u_{yy},
if t>0 and (x,y) is in D. The boundary condition is u(t,x,y) = 0 if (x,y) is on C. The method of separation of variables leads to the form
u(t,x,y) = T(t) v(x,y),
which in turn must satisfy
\frac{1}{c^2}T'' +k^2 T=0,
v_{xx} + v_{yy} + k^2 v =0.
The latter equation is called the Helmholtz Equation. The constant k must be determined to allow a non-trivial v to satisfy the boundary condition on C. Such values of k2 are called the eigenvalues of the Laplacian in D, and the associated solutions are the eigenfunctions of the Laplacian in D. The Sturm–Liouville theory may be extended to this elliptic eigenvalue problem (Jost, 2002).
Other examples[edit]
The Schrödinger equation is a PDE at the heart of non-relativistic quantum mechanics. In the WKB approximation it is the Hamilton–Jacobi equation.
Except for the Dym equation and the Ginzburg–Landau equation, the above equations are linear in the sense that they can be written in the form Au = f for a given linear operator A and a given function f. Other important non-linear equations include the Navier–Stokes equations describing the flow of fluids, and Einstein's field equations of general relativity.
Also see the list of non-linear partial differential equations.
Some linear, second-order partial differential equations can be classified as parabolic, hyperbolic and elliptic. Others such as the Euler–Tricomi equation have different types in different regions. The classification provides a guide to appropriate initial and boundary conditions, and to smoothness of the solutions.
Equations of first order[edit]
Equations of second order[edit]
Assuming u_{xy}=u_{yx}, the general second-order PDE in two independent variables has the form
Au_{xx} + 2Bu_{xy} + Cu_{yy} + \cdots \mbox{(lower order terms)} = 0,
where the coefficients A, B, C etc. may depend upon x and y. If A^2 +B^2 + C^2 > 0 over a region of the xy plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section:
Ax^2 + 2Bxy + Cy^2 + \cdots = 0.
More precisely, replacing ∂x by X, and likewise for other variables (formally this is done by a Fourier transform), converts a constant-coefficient PDE into a polynomial of the same degree, with the top degree (a homogeneous polynomial, here a quadratic form) being most significant for the classification.
Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant B^2 - 4AC, the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by B^2 - AC, due to the convention of the xy term being 2B rather than B; formally, the discriminant (of the associated quadratic form) is (2B)^2 - 4AC = 4(B^2-AC), with the factor of 4 dropped for simplicity.
1. B^2 - AC < 0: solutions of elliptic PDEs are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of Laplace's equation are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where x < 0.
2. B^2 - AC = 0: equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where x = 0.
3. B^2 - AC > 0 : hyperbolic equations retain any discontinuities of functions or derivatives in the initial data. An example is the wave equation. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where x > 0.
If there are n independent variables x1, x2 , ..., xn, a general linear partial differential equation of second order has the form
L u =\sum_{i=1}^n\sum_{j=1}^n a_{i,j} \frac{\part^2 u}{\partial x_i \partial x_j} \quad \text{ plus lower-order terms} =0.
The classification depends upon the signature of the eigenvalues of the coefficient matrix ai,j..
1. Elliptic: The eigenvalues are all positive or all negative.
2. Parabolic : The eigenvalues are all positive or all negative, save one that is zero.
3. Hyperbolic: There is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.
4. Ultrahyperbolic: There is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues. There is only limited theory for ultrahyperbolic equations (Courant and Hilbert, 1962).
Systems of first-order equations and characteristic surfaces[edit]
The classification of partial differential equations can be extended to systems of first-order equations, where the unknown u is now a vector with m components, and the coefficient matrices Aν are m by m matrices for ν = 1, ..., n. The partial differential equation takes the form
Lu = \sum_{\nu=1}^{n} A_\nu \frac{\partial u}{\partial x_\nu} + B=0,
where the coefficient matrices Aν and the vector B may depend upon x and u. If a hypersurface S is given in the implicit form
\varphi(x_1, x_2, \ldots, x_n)=0, \,
where φ has a non-zero gradient, then S is a characteristic surface for the operator L at a given point if the characteristic form vanishes:
Q\left(\frac{\part\varphi}{\partial x_1}, \ldots,\frac{\part\varphi}{\partial x_n}\right) =\det\left[\sum_{\nu=1}^nA_\nu \frac{\partial \varphi}{\partial x_\nu}\right]=0.\,
The geometric interpretation of this condition is as follows: if data for u are prescribed on the surface S, then it may be possible to determine the normal derivative of u on S from the differential equation. If the data on S and the differential equation determine the normal derivative of u on S, then S is non-characteristic. If the data on S and the differential equation do not determine the normal derivative of u on S, then the surface is characteristic, and the differential equation restricts the data on S: the differential equation is internal to S.
1. A first-order system Lu=0 is elliptic if no surface is characteristic for L: the values of u on S and the differential equation always determine the normal derivative of u on S.
2. A first-order system is hyperbolic at a point if there is a space-like surface S with normal ξ at that point. This means that, given any non-trivial vector η orthogonal to ξ, and a scalar multiplier λ, the equation
Q(\lambda \xi + \eta) =0,
has m real roots λ1, λ2, ..., λm. The system is strictly hyperbolic if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form Q(ζ) = 0 defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has m sheets, and the axis ζ = λ ξ runs inside these sheets: it does not intersect any of them. But when displaced from the origin by η, this axis intersects every sheet. In the elliptic case, the normal cone has no real sheets.
Equations of mixed type[edit]
If a PDE has coefficients that are not constant, it is possible that it will not belong to any of these categories but rather be of mixed type. A simple but important example is the Euler–Tricomi equation
u_{xx} \, = xu_{yy},
which is called elliptic-hyperbolic because it is elliptic in the region x < 0, hyperbolic in the region x > 0, and degenerate parabolic on the line x = 0.
Infinite-order PDEs in quantum mechanics[edit]
In the phase space formulation of quantum mechanics, one may consider the quantum Hamilton's equations for trajectories of quantum particles. These equations are infinite-order PDEs. However, in the semiclassical expansion, one has a finite system of ODEs at any fixed order of ħ. The evolution equation of the Wigner function is also an infinite-order PDE. The quantum trajectories are quantum characteristics, with the use of which one could calculate the evolution of the Wigner function.
Analytical methods to solve PDEs[edit]
Separation of variables[edit]
Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. This technique rests on a characteristic of solutions to differential equations: if one can find any solution that solves the equation and satisfies the boundary conditions, then it is the solution (this also applies to ODEs). We assume as an ansatz that the dependence of a solution on the parameters space and time can be written as a product of terms that each depend on a single parameter, and then see if this can be made to solve the problem.[1]
In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ordinary differential equation if in one variable – these are in turn easier to solve.
This is possible for simple PDEs, which are called separable partial differential equations, and the domain is generally a rectangle (a product of intervals). Separable PDEs correspond to diagonal matrices – thinking of "the value for fixed x" as a coordinate, each coordinate can be understood separately.
This generalizes to the method of characteristics, and is also used in integral transforms.
Method of characteristics[edit]
In special cases, one can find characteristic curves on which the equation reduces to an ODE – changing coordinates in the domain to straighten these curves allows separation of variables, and is called the method of characteristics.
More generally, one may find characteristic surfaces.
Integral transform[edit]
An integral transform may transform the PDE to a simpler one, in particular a separable PDE. This corresponds to diagonalizing an operator.
An important example of this is Fourier analysis, which diagonalizes the heat equation using the eigenbasis of sinusoidal waves.
If the domain is finite or periodic, an infinite sum of solutions such as a Fourier series is appropriate, but an integral of solutions such as a Fourier integral is generally required for infinite domains. The solution for a point source for the heat equation given above is an example for use of a Fourier integral.
Change of variables[edit]
Often a PDE can be reduced to a simpler form with a known solution by a suitable change of variables. For example the Black–Scholes PDE
is reducible to the heat equation
\frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x^2}
by the change of variables (for complete details see Solution of the Black Scholes Equation at the Wayback Machine (archived April 11, 2008))
V(S,t) = K v(x,\tau)
x = \ln\left(\tfrac{S}{K} \right)
\tau = \tfrac{1}{2} \sigma^2 (T - t)
v(x,\tau)=\exp(-\alpha x-\beta\tau) u(x,\tau).
Fundamental solution[edit]
Main article: Fundamental solution
Inhomogeneous equations can often be solved (for constant coefficient PDEs, always be solved) by finding the fundamental solution (the solution for a point source), then taking the convolution with the boundary conditions to get the solution.
This is analogous in signal processing to understanding a filter by its impulse response.
Superposition principle[edit]
Because any superposition of solutions of a linear, homogeneous PDE is again a solution, the particular solutions may then be combined to obtain more general solutions. if u1 and u2 are solutions of a homogeneous linear pde in same region R, then u= c1u1+c2u2 with any constants c1 and c2 is also a solution of that pde in that same region....
Methods for non-linear equations[edit]
See also the list of nonlinear partial differential equations.
There are no generally applicable methods to solve non-linear PDEs. Still, existence and uniqueness results (such as the Cauchy–Kowalevski theorem) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of analysis). Computational solution to the nonlinear PDEs, the split-step method, exist for specific equations like nonlinear Schrödinger equation.
Nevertheless, some techniques can be used for several types of equations. The h-principle is the most powerful method to solve underdetermined equations. The Riquier–Janet theory is an effective method for obtaining information about many analytic overdetermined systems.
The method of characteristics (similarity transformation method) can be used in some very special cases to solve partial differential equations.
In some cases, a PDE can be solved via perturbation analysis in which the solution is considered to be a correction to an equation with a known solution. Alternatives are numerical analysis techniques from simple finite difference schemes to the more mature multigrid and finite element methods. Many interesting problems in science and engineering are solved in this way using computers, sometimes high performance supercomputers.
Lie group method[edit]
From 1870 Sophus Lie's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called Lie groups, be referred to a common source; and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of integration. He also emphasized the subject of transformations of contact.
A general approach to solve PDE's uses the symmetry property of differential equations, the continuous infinitesimal transformations of solutions to solutions (Lie theory). Continuous group theory, Lie algebras and differential geometry are used to understand the structure of linear and nonlinear partial differential equations for generating integrable equations, to find its Lax pairs, recursion operators, Bäcklund transform and finally finding exact analytic solutions to the PDE.
Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.
Semianalytical methods[edit]
The adomian decomposition method, the Lyapunov artificial small parameter method, and He's homotopy perturbation method are all special cases of the more general homotopy analysis method. These are series expansion methods, and except for the Lyapunov method, are independent of small physical parameters as compared to the well known perturbation theory, thus giving these methods greater flexibility and solution generality.
Numerical methods to solve PDEs[edit]
The three most widely used numerical methods to solve PDEs are the finite element method (FEM), finite volume methods (FVM) and finite difference methods (FDM). The FEM has a prominent position among these methods and especially its exceptionally efficient higher-order version hp-FEM. Other versions of FEM include the generalized finite element method (GFEM), extended finite element method (XFEM), spectral finite element method (SFEM), meshfree finite element method, discontinuous Galerkin finite element method (DGFEM), Element-Free Galerkin Method (EFGM), Interpolating Element-Free Galerkin Method (IEFGM), etc.
Finite element method[edit]
Main article: Finite element method
The finite element method (FEM) (its practical application often known as finite element analysis (FEA)) is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge–Kutta, etc.
Finite difference method[edit]
Finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.
Finite volume method[edit]
Main article: Finite volume method
Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, surface integrals in a partial differential equation that contain a divergence term are converted to volume integrals, using the Divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods are conservative.
See also[edit]
1. ^ Gershenfeld, Neil (2000). The nature of mathematical modeling (Reprinted (with corr.). ed.). Cambridge: Cambridge Univ. Press. p. 27. ISBN 0521570956.
• Adomian, G. (1994). Solving Frontier problems of Physics: The decomposition method. Kluwer Academic Publishers.
• Courant, R. & Hilbert, D. (1962), Methods of Mathematical Physics II, New York: Wiley-Interscience .
• Evans, L. C. (1998), Partial Differential Equations, Providence: American Mathematical Society, ISBN 0-8218-0772-2 .
• Holubová, Pavel Drábek ; Gabriela (2007). Elements of partial differential equations ([Online-Ausg.]. ed.). Berlin: de Gruyter. ISBN 9783110191240.
• Ibragimov, Nail H (1993), CRC Handbook of Lie Group Analysis of Differential Equations Vol. 1-3, Providence: CRC-Press, ISBN 0-8493-4488-3 .
• John, F. (1982), Partial Differential Equations (4th ed.), New York: Springer-Verlag, ISBN 0-387-90609-6 .
• Jost, J. (2002), Partial Differential Equations, New York: Springer-Verlag, ISBN 0-387-95428-7 .
• Lewy, Hans (1957), "An example of a smooth linear partial differential equation without solution", Annals of Mathematics. Second Series 66 (1): 155–158, doi:10.2307/1970121 .
• Liao, S.J. (2003), Beyond Perturbation: Introduction to the Homotopy Analysis Method, Boca Raton: Chapman & Hall/ CRC Press, ISBN 1-58488-407-X
• Olver, P.J. (1995), Equivalence, Invariants and Symmetry, Cambridge Press .
• Petrovskii, I. G. (1967), Partial Differential Equations, Philadelphia: W. B. Saunders Co. .
• Pinchover, Y. & Rubinstein, J. (2005), An Introduction to Partial Differential Equations, New York: Cambridge University Press, ISBN 0-521-84886-5 .
• Polyanin, A. D. (2002), Handbook of Linear Partial Differential Equations for Engineers and Scientists, Boca Raton: Chapman & Hall/CRC Press, ISBN 1-58488-299-9 .
• Polyanin, A. D. & Zaitsev, V. F. (2004), Handbook of Nonlinear Partial Differential Equations, Boca Raton: Chapman & Hall/CRC Press, ISBN 1-58488-355-3 .
• Polyanin, A. D.; Zaitsev, V. F. & Moussiaux, A. (2002), Handbook of First Order Partial Differential Equations, London: Taylor & Francis, ISBN 0-415-27267-X .
• Roubíček, T. (2013), Nonlinear Partial Differential Equations with Applications (2nd ed.), Basel, Boston, Berlin: Birkhäuser, ISBN 978-3-0348-0512-4, MR MR3014456
• Solin, P. (2005), Partial Differential Equations and the Finite Element Method, Hoboken, NJ: J. Wiley & Sons, ISBN 0-471-72070-4 .
• Solin, P.; Segeth, K. & Dolezel, I. (2003), Higher-Order Finite Element Methods, Boca Raton: Chapman & Hall/CRC Press, ISBN 1-58488-438-X .
• Stephani, H. (1989), Differential Equations: Their Solution Using Symmetries. Edited by M. MacCallum, Cambridge University Press .
• Wazwaz, Abdul-Majid (2009). Partial Differential Equations and Solitary Waves Theory. Higher Education Press. ISBN 90-5809-369-7.
• Zwillinger, D. (1997), Handbook of Differential Equations (3rd ed.), Boston: Academic Press, ISBN 0-12-784395-7 .
• Gershenfeld, N. (1999), The Nature of Mathematical Modeling (1st ed.), New York: Cambridge University Press, New York, NY, USA, ISBN 0-521-57095-6 .
• Krasil'shchik, I.S. & Vinogradov, A.M., Eds. (1999), Symmetries and Conserwation Laws for Differential Equations of Mathematical Physics, American Mathematical Society, Providence, Rhode Island,USA, ISBN 0-8218-0958-X .
• Krasil'shchik, I.S.; Lychagin, V.V. & Vinogradov, A.M. (1986), Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Gordon and Breach Science Publishers, New York, London, Paris, Montreux, Tokyo, ISBN 2-88124-051-8 .
• Vinogradov, A.M. (2001), Cohomological Analysis of Partial Differential Equations and Secondary Calculus, American Mathematical Society, Providence, Rhode Island,USA, ISBN 0-8218-2922-X .
External links[edit] |
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I am stuck on a QM homework problem. The setup is this:
enter image description here
(To be clear, the potential in the left and rightmost regions is $0$ while the potential in the center region is $V_0$, and the wavefunction vanishes when $|x|>b+a/2$.) I'm asked to write the Schrödinger equation for each region, find its solution, set up the BCs, and obtain the transcendental equations for the eigenvalues.
Where I'm at: I understand the infinite potential well easily and I have done a free particle going over a finite barrier before (which I understood less well, but I can deal with it it).
• The problem asks me to make use of "a symmetry" in the problem, which is a vague hint. Are they trying to get me to make $\psi$ an even function?
• I am supposed the condition for there to be one and only one bound state for $E<V_0$. How do I go about that?
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You seem to have trouble to understand the basic approach. Actually there is a systematic way to solve the Schrödinger equation for picewise constant potentials. Maybe this will give you some basic idea how to solve your problem:
Let be the potential given by $$V(z) = \begin{cases} \infty & z < z_1 \\ V_1 & z_1 <= z < z_2 \\ V_2 & z_2 <= z < z_3 \\ ... \end{cases}$$
• For the above potential the wavefunction for energy eigenvalue $E_n$ is given by $$\Psi_n(z) = \begin{cases} 0 & z < z_1 \\ A_1\exp(-i k_1 z) + B_1\exp(+i k_1 z) & z_1 <= z < z_2 \\ A_2\exp(-i k_2 z) + B_2\exp(+i k_2 z) & z_2 <= z < z_3 \\ ... \end{cases}$$ with $k_i = 2\pi/h \sqrt{2 m e (E_n-V_i)}$ and some (yet to be determined) constants $A_i$ and $B_i$. This is easily verified by plugging in. (In fact each "segment" is the solution to the Schrödinger equation with constant potential). Note that the $k_i$ can be real or imaginary, in which case the wavefunction in the respective segment is either sinusoidal or exponential.
• As required by physics the wavefunction must be continuous and continuously differentiable everywhere. Hence the constants $A_i$ and $B_i$ must be chosen so that this is fulfilled at each point where this possibly is violated (i.e. the points $z_i$).
• The above results in a linear equation system for the $A_i$ and $B_i$. This equation system now only contains the energy $E_n$ as remaining unknown. If you do it correctly the equation system contains as many unknowns as equations.
• Now you compute the determinant of the equation system and set it to zero to find the $E_n$ values for which it is solvable. This is the transcendetal equation for the eigenvalues. This equation has in your case infinitely many discrete solutions $E_n$ (each solution denoted by the running index $n$). For each $E_n$ there are sets of $A_i$ and $B_i$ (which solve the equation system) which give you the wavefunction. In case there is more than one set of linearly independent $A_i$ and $B_i$, you have more than one wavefunction to the same eigenvalue $E_n$. In that case the state is degenerate. (You have degenerate states in your problem!).
Regarding symmetry: The wavefunctions do not need to have the same symmetry as the potential. Of course if you have a solution wavefunction, then the mirrored wavefunction must be a solution as well (if the potential is symmetric as in your case). It needs to belong to the same energy eigenvalue.
Regarding the single bound state: Once you have calculated the $E_n$ you will see that there are conditions where $E_1 < V_0$ and $E_2 > V_0$ ($E_2$ the second largest eigenvalue). This depends on the geometry, i.e. width of your barrier and well. Generally speaking the energy states have higher spacing, if the well is smaller. So probably the single bound state condition will display itself as range specification for $a$ and $b$.
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Very good, thanks. Quite helpful. – Alexander Nikolas Gruber Oct 15 '12 at 23:37
The parity operator commutes with the Hamiltonian because of the symmetry in your potential. This says that all eigenstates of the Hamiltonian are eigenstates of the parity operator. Therefore, the only possible eigenstate solutions to the system are ones with even or odd parity. This fact will allow you to simplify the process of applying the boundary conditions mentioned by Andreas, as you can immediately conclude several things regarding the unknown coefficients.
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• richardmitnick 3:30 pm on March 21, 2022 Permalink | Reply
Tags: "The Evolving Quest for a Grand Unified Theory of Mathematics", , , , Quantum field theory,
From Scientific American: “The Evolving Quest for a Grand Unified Theory of Mathematics”
From Scientific American
March 21, 2022
Rachel Crowell
Credit: Boris SV/Getty Images.
More than 50 years after the seeds of a vast collection of mathematical ideas called the Langlands program began to sprout, surprising new findings are emerging.
Within mathematics, there is a vast and ever expanding web of conjectures, theorems and ideas called the Langlands program. That program links seemingly disconnected subfields. It is such a force that some mathematicians say it—or some aspect of it—belongs in the esteemed ranks of the Millennium Prize Problems, a list of the top open questions in math. Edward Frenkel, a mathematician at the University of California-Berkeley, has even dubbed the Langlands program “a Grand Unified Theory of Mathematics.”
The program is named after Robert Langlands, a mathematician at the Institute for Advanced Study in Princeton, N.J. Four years ago, he was awarded the Abel Prize, one of the most prestigious awards in mathematics, for his program, which was described as “visionary.”
Langlands is retired, but in recent years the project has sprouted into “almost its own mathematical field, with many disparate parts,” which are united by “a common wellspring of inspiration,” says Steven Rayan, a mathematician and mathematical physicist at the University of Saskatchewan. It has “many avatars, some of which are still open, some of which have been resolved in beautiful ways.”
Increasingly mathematicians are finding links between the original program—and its offshoot, geometric Langlands—and other fields of science. Researchers have already discovered strong links to physics, and Rayan and other scientists continue to explore new ones. He has a hunch that, with time, links will be found between these programs and other areas as well. “I think we’re only at the tip of the iceberg there,” he says. “I think that some of the most fascinating work that will come out of the next few decades is seeing consequences and manifestations of Langlands within parts of science where the interaction with this kind of pure mathematics may have been marginal up until now.” Overall Langlands remains mysterious, Rayan adds, and to know where it is headed, he wants to “see an understanding emerge of where these programs really come from.”
A Puzzling Web
The Langlands program has always been a tantalizing dance with the unexpected, according to James Arthur, a mathematician at the University of Toronto (CA). Langlands was Arthur’s adviser at Yale University, where Arthur earned his Ph.D. in 1970. (Langlands declined to be interviewed for this story.)
“I was essentially his first student, and I was very fortunate to have encountered him at that time,” Arthur says. “He was unlike any mathematician I had ever met. Any question I had, especially about the broader side of mathematics, he would answer clearly, often in a way that was more inspiring than anything I could have imagined.”
During that time, Langlands laid the foundation for what eventually became his namesake program. In 1969 Langlands famously handwrote a 17-page letter to French mathematician André Weil. In that letter, Langlands shared new ideas that later became known as the “Langlands conjectures.”
In 1969 Langlands delivered conference lectures in which he shared the seven conjectures that ultimately grew into the Langlands program, Arthur notes. One day Arthur asked his adviser for a copy of a preprint paper based on those lectures.
“He willingly gave me one, no doubt knowing that it was beyond me,” Arthur says. “But it was also beyond everybody else for many years. I could, however, tell that it was based on some truly extraordinary ideas, even if just about everything in it was unfamiliar to me.”
The Conjectures at the Heart of It All
Two conjectures are central to the Langlands program. “Just about everything in the Langlands program comes in one way or another from those,” Arthur says.
The reciprocity conjecture connects to the work of Alexander Grothendieck, famous for his research in algebraic geometry, including his prediction of “motives.” “I think Grothendieck chose the word [motive] because he saw it as a mathematical analogue of motifs that you have in art, music or literature: hidden ideas that are not explicitly made clear in the art, but things that are behind it that somehow govern how it all fits together,” Arthur says.
The reciprocity conjecture supposes these motives come from a different type of analytical mathematical object discovered by Langlands called automorphic representations, Arthur notes. “‘Automorphic representation’ is just a buzzword for the objects that satisfy analogues of the Schrödinger equation” from quantum physics, he adds. The Schrödinger equation predicts the likelihood of finding a particle in a certain state.
The second important conjecture is the functoriality conjecture, also simply called functoriality. It involves classifying number fields. Imagine starting with an equation of one variable whose coefficients are integers—such as x2 + 2x + 3 = 0—and looking for the roots of that equation. The conjecture predicts that the corresponding field will be “the smallest field that you get by taking sums, products and rational number multiples of these roots,” Arthur says.
Exploring Different Mathematical “Worlds”
With the original program, Langlands “discovered a whole new world,” Arthur says.
The offshoot, geometric Langlands, expanded the territory this mathematics covers. Rayan explains the different perspectives the original and geometric programs provide. “Ordinary Langlands is a package of ideas, correspondences, dualities and observations about the world at a point,” he says. “Your world is going to be described by some sequence of relevant numbers. You can measure the temperature where you are; you could measure the strength of gravity at that point,” he adds.
With the geometric program, however, your environment becomes more complex, with its own geometry. You are free to move about, collecting data at each point you visit. “You might not be so concerned with the individual numbers but more how they are varying as you move around in your world,” Rayan says. The data you gather are “going to be influenced by the geometry,” he says. Therefore, the geometric program “is essentially replacing numbers with functions.”
Number theory and representation theory are connected by the geometric Langlands program. “Broadly speaking, representation theory is the study of symmetries in mathematics,” says Chris Elliott, a mathematician at the University of Massachusetts Amherst.
Using geometric tools and ideas, geometric representation theory expands mathematicians’ understanding of abstract notions connected to symmetry, Elliot notes. That area of representation theory is where the geometric Langlands program “lives,” he says.
Intersections with Physics
The geometric program has already been linked to physics, foreshadowing possible connections to other scientific fields.
In 2018 Kazuki Ikeda, a postdoctoral researcher in Rayan’s group, published a Journal of Mathematical Physics study that he says is connected to an electromagnetic duality that is “a long-known concept in physics” and that is seen in error-correcting codes in quantum computers, for instance. Ikeda says his results “were the first in the world to suggest that the Langlands program is an extremely important and powerful concept that can be applied not only to mathematics but also to condensed-matter physics”—the study of substances in their solid state—“and quantum computation.”
Connections between condensed-matter physics and the geometric program have recently strengthened, according to Rayan. “In the last year the stage has been set with various kinds of investigations,” he says, including his own work involving the use of algebraic geometry and number theory in the context of quantum matter.
Other work established links between the geometric program and high-energy physics. In 2007 Anton Kapustin, a theoretical physicist at the California Institute of Technology, and Edward Witten, a mathematical and theoretical physicist at the Institute for Advanced Study, published what Rayan calls “a beautiful landmark paper” that “paved the way for an active life for geometric Langlands in theoretical high-energy physics.” In the paper, Kapustin and Witten wrote that they aimed to “show how this program can be understood as a chapter in quantum field theory.”
Elliott notes that viewing quantum field theory from a mathematical perspective can help glean new information about the structures that are foundational to it. For instance, Langlands may help physicists devise theories for worlds with different numbers of dimensions than our own.
Besides the geometric program, the original Langlands program is also thought to be fundamental to physics, Arthur says. But exploring that connection “may require first finding an overarching theory that links the original and geometric programs,” he says.
The reaches of these programs may not stop at math and physics. “I believe, without a doubt, that [they] have interpretations across science,” Rayan says. “The condensed-matter part of the story will lead naturally to forays into chemistry.” Furthermore, he adds, “pure mathematics always makes its way into every other area of science. It’s only a matter of time.”
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• richardmitnick 9:45 am on January 24, 2022 Permalink | Reply
Tags: "At the interface of physics and mathematics", , , Integrable model: equation that can be solved exactly., , , , Quantum field theory, String Theory-which scientists hope will eventually provide a unified description of particle physics and gravity., ,
From The Swiss Federal Institute of Technology in Zürich [ETH Zürich] [Eidgenössische Technische Hochschule Zürich] (CH): “At the interface of physics and mathematics”
Barbara Vonarburg
Sylvain Lacroix is a theoretical physicist who conducts research into fundamental concepts of physics – an exciting but intellectually challenging field of science. As an Advanced Fellow at ETH Zürich’s Institute for Theoretical Studies (ITS), he works on complex equations that can be solved exactly only thanks to their large number of symmetries.
“It was fascinating to learn abstract mathematical concepts and see them neatly applied in the realm of physics,” says Sylvain Lacroix, Advanced Fellow at the Institute for Theoretical Studies. Photo: Nicola Pitaro/ETH Zürich.
“I got hooked on the interplay of physics and mathematics while I was still at secondary school,” says 30-year-old Sylvain Lacroix, who was born and grew up near Paris. “It was fascinating to learn abstract mathematical concepts and see them neatly applied in the realm of physics.” During his studies at The University of Lyon [Université Claude Bernard Lyon 1] (FR), he devoted much of his energy and enthusiasm to physics problems that had highly complex underlying mathematical structures. So when it came to selecting a topic for his doctoral thesis, this area of research seemed like the obvious choice. He decided to explore the theory of what are known as integrable models – a subject he has continued to pursue up to the present day.
Lacroix readily acknowledges that most people outside his line of work find the term “integrable models” completely incomprehensible: “I have to admit that it’s probably not the simplest or most accessible field of physics,” he says, almost apologetically. That’s why he takes pains to explain it in layman’s terms: “We define a model as a body of laws, a set of equations that describe the behaviour of certain quantities, for example how the position of an object changes over time.” An integrable model is characterised by equations that can be solved exactly, which is by no means a given.
Symmetry is the key
Many of the equations used in modern physics – such as that practised at The European Organization for Nuclear Research [Organización Europea para la Investigación Nuclear][Organisation européenne pour la recherche nucléaire] [Europäische Organisation für Kernforschung](CH) [CERN] – are so complex that they can be solved only approximately. These approximation methods often serve their purpose well, for instance if there is only a weak interaction between two particles. However, other cases require exact calculations – and that’s where integrable models come in. But what makes them so exact? “That’s another aspect that is tricky to explain,” Lacroix says, “but it ultimately comes down to symmetry.” Take, for example, the symmetry of time or space: a physics experiment will produce the same results whether you perform it today or – under identical conditions – ten days from now, and whether it takes place in Zürich or New York. Consequently, the equation that describes the experiment must remain invariant even if the time or location changes. This is reflected in the mathematical structure of the equation, which contains the corresponding constraints. “If we have enough symmetries, this results in so many constraints that we can simplify the equation to the point where we get exact results,” says the physicist.
Integrable models and their exact solutions are actually very rare in mathematics. “If I chose a random equation, it would be extremely unlikely to have this property of exact solvability,” Lacroix says. “But equations of this kind really do exist in nature.” Some describe the movement of waves propagating in a channel, for example, while others describe the behaviour of a hydrogen atom. “But it’s important to note that my work doesn’t have any practical applications of that kind,” Lacroix says. “I don’t examine concrete physical models; instead, I study mathematical structures and attempt to find general approaches that will allow us to construct new exactly solvable equations.” Although some of these formulas may eventually find a real-world application, others probably won’t.
After completing his doctoral thesis, Lacroix spent three years working as a postdoc at The University of Hamburg [Universität Hamburg](DE), before finally moving to Zürich in September 2021. “I don’t have a family, so I had no problem making the switch,” he says. He is relieved that he can now spend five years at the ITS as an Advanced Fellow and focus entirely on his research without having to worry about the future. He admits it was a pleasure getting to know different countries as a postdoc and that he enjoyed moving from place to place. “But it makes it very hard to have any kind of stability in your life.”
A beautiful setting
Lacroix spends most of his time working in his office at the ITS, which is located in a stately building dating from 1882 not far from the ETH Main Building. “It’s a lovely place,” he says, glancing out the window at the green surroundings and the city beyond. “I feel very much at home here. Living in Zürich is wonderful, it’s such a great feeling being here.” In his spare time, he likes watching movies, reading books and socialising. “I love meeting up with friends in restaurants or cafés,” he says. He also feels fortunate that he didn’t start working in Zürich until after the Covid measures had been relaxed.
“I’m vaccinated and everyone’s very careful at ETH. We still have restrictions in place, but life is slowly getting back to normal – and that made it much easier to get to know my colleagues from day one,” he says. One of the greatest privileges of working at the ITS, Lacroix says, is that it offers an international environment that brings together researchers from all over the world. As well as offering a space for experts to exchange ideas and holding seminars where Fellows can present their work, the Institute also has a tradition of organising joint excursions. In the autumn of 2021, Lacroix joined his colleagues on a hike in the Flumserberg mountain resort for the first time: “I love hiking and it’s incredible to have the mountains so close.”
Normally, however, he can be found sitting at his desk jotting down a series of mostly abstract equations on a sheet of paper. Occasionally his computer comes in handy, he says, because it has become so much more than just a calculating device; today’s computers can also handle abstract mathematical concepts, which can be very useful. Most people don’t really understand much of what Lacroix puts down on paper, but that doesn’t bother him: “I’ve learned to live with that,” he says; “I don’t feel isolated in my research at all – at least not in the academic sphere.”
A better understanding of quantum field theory
Integrable models are extremely symmetrical models, Lacroix explains. The basic principle of symmetry plays an important role in modern physics, for example in quantum field theory – the theoretical basis of particle physics – as well as in string theory, which scientists hope will eventually provide a unified description of particle physics and gravity. So could such an all-encompassing unified field theory turn out to be an integrable model? “That would obviously be great, especially for me!” Lacroix says with a wry smile. “But it’s a bit optimistic to believe that whatever unified theory of physics finally emerges will have enough symmetries to make it completely exact.”
Even if the equations he studies don’t explain the world directly, he still believes they can help us achieve a better understanding of theoretical physics. For example, we can take advantage of so-called “toy models”, which have a particularly large number of symmetries, to simplify extremely complex equations in quantum field theory. “This gives us a better understanding of how the theory works, even if these models are too simplistic for the real world,” Lacroix says. Yet his primary interest lies in the purely mathematical questions that integrable models pose, and he admits that the equations they involve sometimes even appear in his dreams: “It’s hard to shake off what I’ve been thinking about the entire day. But I’ve never managed to solve a mathematical problem in my dreams – at least not so far!”
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ETH Zurich campus
The Swiss Federal Institute of Technology in Zürich [ETH Zürich] [Eidgenössische Technische Hochschule Zürich] (CH) is a public research university in the city of Zürich, Switzerland. Founded by the Swiss Federal Government in 1854 with the stated mission to educate engineers and scientists, the school focuses exclusively on science, technology, engineering and mathematics. Like its sister institution The Swiss Federal Institute of Technology in Lausanne [EPFL-École Polytechnique Fédérale de Lausanne](CH) , it is part of The Swiss Federal Institutes of Technology Domain (ETH Domain)) , part of the The Swiss Federal Department of Economic Affairs, Education and Research [EAER][Eidgenössisches Departement für Wirtschaft, Bildung und Forschung] [Département fédéral de l’économie, de la formation et de la recherche] (CH).
The university is an attractive destination for international students thanks to low tuition fees of 809 CHF per semester, PhD and graduate salaries that are amongst the world’s highest, and a world-class reputation in academia and industry. There are currently 22,200 students from over 120 countries, of which 4,180 are pursuing doctoral degrees. In the 2021 edition of the QS World University Rankings ETH Zürich is ranked 6th in the world and 8th by the Times Higher Education World Rankings 2020. In the 2020 QS World University Rankings by subject it is ranked 4th in the world for engineering and technology (2nd in Europe) and 1st for earth & marine science.
As of November 2019, 21 Nobel laureates, 2 Fields Medalists, 2 Pritzker Prize winners, and 1 Turing Award winner have been affiliated with the Institute, including Albert Einstein. Other notable alumni include John von Neumann and Santiago Calatrava. It is a founding member of the IDEA League and the International Alliance of Research Universities (IARU) and a member of the CESAER network.
ETH Zürich was founded on 7 February 1854 by the Swiss Confederation and began giving its first lectures on 16 October 1855 as a polytechnic institute (eidgenössische polytechnische Schule) at various sites throughout the city of Zurich. It was initially composed of six faculties: architecture, civil engineering, mechanical engineering, chemistry, forestry, and an integrated department for the fields of mathematics, natural sciences, literature, and social and political sciences.
It is locally still known as Polytechnikum, or simply as Poly, derived from the original name eidgenössische polytechnische Schule, which translates to “federal polytechnic school”.
ETH Zürich is a federal institute (i.e., under direct administration by the Swiss government), whereas The University of Zürich [Universität Zürich ] (CH) is a cantonal institution. The decision for a new federal university was heavily disputed at the time; the liberals pressed for a “federal university”, while the conservative forces wanted all universities to remain under cantonal control, worried that the liberals would gain more political power than they already had. In the beginning, both universities were co-located in the buildings of the University of Zürich.
From 1905 to 1908, under the presidency of Jérôme Franel, the course program of ETH Zürich was restructured to that of a real university and ETH Zürich was granted the right to award doctorates. In 1909 the first doctorates were awarded. In 1911, it was given its current name, Eidgenössische Technische Hochschule. In 1924, another reorganization structured the university in 12 departments. However, it now has 16 departments.
ETH Zürich, EPFL (Swiss Federal Institute of Technology in Lausanne) [École polytechnique fédérale de Lausanne](CH), and four associated research institutes form The Domain of the Swiss Federal Institutes of Technology (ETH Domain) [ETH-Bereich; Domaine des Écoles polytechniques fédérales] (CH) with the aim of collaborating on scientific projects.
Reputation and ranking
ETH Zürich is ranked among the top universities in the world. Typically, popular rankings place the institution as the best university in continental Europe and ETH Zürich is consistently ranked among the top 1-5 universities in Europe, and among the top 3-10 best universities of the world.
Historically, ETH Zürich has achieved its reputation particularly in the fields of chemistry, mathematics and physics. There are 32 Nobel laureates who are associated with ETH Zürich, the most recent of whom is Richard F. Heck, awarded the Nobel Prize in chemistry in 2010. Albert Einstein is perhaps its most famous alumnus.
In 2018, the QS World University Rankings placed ETH Zürich at 7th overall in the world. In 2015, ETH Zürich was ranked 5th in the world in Engineering, Science and Technology, just behind the Massachusetts Institute of Technology(US), Stanford University(US) and University of Cambridge(UK). In 2015, ETH Zürich also ranked 6th in the world in Natural Sciences, and in 2016 ranked 1st in the world for Earth & Marine Sciences for the second consecutive year.
In 2016, Times Higher Education World University Rankings ranked ETH Zürich 9th overall in the world and 8th in the world in the field of Engineering & Technology, just behind the Massachusetts Institute of Technology(US), Stanford University(US), California Institute of Technology(US), Princeton University(US), University of Cambridge(UK), Imperial College London(UK) and University of Oxford(UK) .
In the survey CHE ExcellenceRanking on the quality of Western European graduate school programs in the fields of biology, chemistry, physics and mathematics, ETH Zürich was assessed as one of the three institutions to have excellent programs in all the considered fields, the other two being Imperial College London(UK) and The University of Cambridge(UK), respectively.
• richardmitnick 12:13 pm on July 20, 2021 Permalink | Reply
Tags: "A Video Tour of the Standard Model", , , , , Quantum field theory,
From Quanta Magazine (US) via Symmetry: “A Video Tour of the Standard Model”
From Quanta Magazine
Symmetry Mag
July 16, 2021
Kevin Hartnett
Standard Model of Particle Physics. Credit: Quanta Magazine.
The Standard Model: The Most Successful Scientific Theory Ever.
Video: The Standard Model of particle physics is the most successful scientific theory of all time. In this explainer, Cambridge University physicist David Tong recreates the model, piece by piece, to provide some intuition for how the fundamental building blocks of our universe fit together.
Emily Buder/Quanta Magazine.
Kristina Armitage and Rui Braz for Quanta Magazine.
Recently, Quanta has explored the collaboration between physics and mathematics on one of the most important ideas in science: quantum field theory. The basic objects of a quantum field theory are quantum fields, which spread across the universe and, through their fluctuations, give rise to the most fundamental phenomena in the physical world. We’ve emphasized the unfinished business in both physics and mathematics — the ways in which physicists still don’t fully understand a theory they wield so effectively, and the grand rewards that await mathematicians if they can provide a full description of what quantum field theory actually is.
This incompleteness, however, does not mean the work has been unsatisfying so far.
For our final entry in this “Math Meets QFT” series, we’re exploring the most prominent quantum field theory of them all: the Standard Model. As the University of Cambridge (UK) physicist David Tong puts it in the accompanying video, it’s “the most successful scientific theory of all time” despite being saddled with a “rubbish name.”
The Standard Model describes physics in the three spatial dimensions and one time dimension of our universe. It captures the interplay between a dozen quantum fields representing fundamental particles and a handful of additional fields representing forces. The Standard Model ties them all together into a single equation that scientists have confirmed countless times, often with astonishing accuracy. In the video, Professor Tong walks us through that equation term by term, introducing us to all the pieces of the theory and how they fit together. The Standard Model is complicated, but it is easier to work with than many other quantum field theories. That’s because sometimes the fields of the Standard Model interact with each other quite feebly, as writer Charlie Wood described in the second piece in our series.
From Quanta Magazine : “Mathematicians Prove 2D Version of Quantum Gravity Really Works”
The Standard Model has been a boon for physics, but it’s also had a bit of a hangover effect. It’s been extraordinarily effective at explaining experiments we can do here on Earth, but it can’t account for several major features of the wider universe, including the action of gravity at short distances and the presence of dark matter and dark energy. Physicists would like to move beyond the Standard Model to an even more encompassing physical theory. But, as the physicist Davide Gaiotto put it in the first piece in our series, the glow of the Standard Model is so strong that it’s hard to see beyond it.
From Quanta Magazine : “The Mystery at the Heart of Physics That Only Math Can Solve”
And that, maybe, is where math comes in. Mathematicians will have to develop a fresh perspective on quantum field theory if they want to understand it in a self-consistent and rigorous way. There’s reason to hope that this new vantage will resolve many of the biggest open questions in physics.
The process of bringing QFT into math may take some time — maybe even centuries, as the physicist Nathan Seiberg speculated in the third piece in our series — but it’s also already well underway. By now, math and quantum field theory have indisputably met. It remains to be seen what happens as they really get to know each other.
From Quanta Magazine : “Nathan Seiberg on How Math Might Complete the Ultimate Physics Theory”
See the full article here .
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• richardmitnick 9:50 am on June 18, 2021 Permalink | Reply
Tags: "Brookhaven Lab Intern Returns to Continue Theoretical Physics Pursuit", Co-design Center for Quantum Advantage (C2QA), DOE Science Undergraduate Laboratory Internships, National Quantum Information Science Research Centers, , , , Quantum field theory, , Wenjie Gong recently received a Barry Goldwater Scholarship., Women in STEM-Wenjie Gong
From DOE’s Brookhaven National Laboratory (US) : Women in STEM-Wenjie Gong “Brookhaven Lab Intern Returns to Continue Theoretical Physics Pursuit”
From DOE’s Brookhaven National Laboratory (US)
June 14, 2021
Kelly Zegers
Wenjie Gong virtually visits Brookhaven for an internship to perform theory research on quantum information science in nuclear physics.
Wenjie Gong, who recently received a Barry Goldwater Scholarship. (Courtesy photo.)
Internships often help students nail down the direction they’d like to take their scientific pursuits. For Wenjie Gong, who just completed her junior year at Harvard University (US), a first look into theoretical physics last summer as an intern with the U.S. Department of Energy’s (DOE) Brookhaven National Laboratory made her want to dive further into the field.
Gong returns to Brookhaven Lab this summer for her second experience as a virtual DOE Science Undergraduate Laboratory Internships (SULI) participant to continue collaborating with Raju Venugopalan, a senior physicist and Nuclear Theory Group leader. Together, they will explore the connections between nuclear physics theory—which explores the interactions of fundamental particles—and quantum computing.
“I find theoretical physics fascinating as there are so many different avenues to explore and so many different angles from which to approach a problem,” Gong said. “Even though it can be difficult to parse through the technical underpinnings of different physical situations, any progress made is all the more exciting and rewarding.”
Last year, Gong collaborated with Venugopalan on a project exploring possible ways to measure a quantum phenomenon known as “entanglement” in the matter produced at high-energy collisions.
The physical properties of entangled particles are inextricably linked, even when the particles are separated by a great distance. Albert Einstein referred to entanglement as “spooky action at distance.”
Studying this phenomenon is an important part of setting up long-distance quantum computing networks—the topic of many of the experiments at Co-design Center for Quantum Advantage (C2QA). The center led by Brookhaven Lab is one of five National Quantum Information Science Research Centers and applies quantum principles to materials, devices and software co-design efforts to lay the foundation for a new generation of quantum computers.
“Usually, entanglement requires very precise measurements that are found in optics laboratories, but we wanted to look at how we could understand entanglement in high-energy particle collisions, which have much less of a controlled environment,” Gong said.
Venugopalan said the motivation behind thinking of ways to detect entanglement in high-energy collisions is two-fold, first asking the question: “Can we think of experimental measures in collider experiments that have comparable ability to extract quantum action-at-a distance just as the carefully designed tabletop experiments?”
“That would be interesting in itself because one might be inclined to think it unlikely,” he said.
Venugopalan said scientists have identified sub-atomic particle correlations of so-called Lambda hyperons, which have particular properties that may allow such an experiment. Those experiments would open up the question of whether entanglement persists if scientists change the conditions of the collisions, he said.
“If we made the collisions more violent, say, by increasing the number of particles produced, would the quantum action-at-a-distance correlation go away, just as you, and I, as macroscopic quantum states, don’t exhibit any spooky action-at-a-distance nonsense,” Venugopalan said. “When does such a quantum-to-classical transition take place?”
In addition, can such measurements teach us about the nature of the interactions of the building blocks of matter–quarks and gluons?
There may be more questions than answers at this stage, “but these questions force us to refine our experimental and computational tools,” Venugopalan said.
Gong will continue collaborating with Venugopalan to develop the project on entanglement this summer. She may also start a new project exploring quirky features of soft particles in the quantum theory of electromagnetism that also apply to the strong force of nuclear physics, Venugopalan said. While her internship is virtual again this year, she said she learned last summer that collaborating remotely can be productive and rewarding.
“Wenjie is the real deal,” Venugopalan said. “Even as a rising junior, she was functioning at the level of a postdoc. It’s a great joy to exchange ‘crazy’ ideas with her and work out the consequences. She shows great promise for an outstanding career in theoretical physics.”
Others have noticed Gong’s scientific talent. She was recently honored with a Barry M. Goldwater Scholarship. The prestigious award supports impressive undergraduates who plan to pursue a PhD in the natural sciences, mathematics, and engineering.
“I feel really honored and also very grateful to Raju, the Department of Energy (US) , and Brookhaven for providing me the opportunity to do this research—which I wrote about in my Goldwater essay,” Gong said.
Gong said she’s looking forward to applying concepts from courses she took at Harvard over the past year, including quantum field theory, which she found challenging but also rewarding.
Gong’s interest in physics started when she took Advanced Placement (AP) Physics in high school. The topic drew her in because it requires a way of thinking that’s different compared to other sciences because it explores the laws governing the motion of matter and existence, she said.
In addition to further exploring high energy theoretical physics research, Gong said she hopes to one day teach as a university professor. She’s currently a peer tutor at Harvard.
“I love teaching physics,” she said. “It’s really cool to see the ‘Ah-ha!’ moment when students go from not really understanding something to grasping a concept.”
The SULI program at Brookhaven is managed by the Lab’s Office of Educational Programs and sponsored by DOE’s Office of Workforce Development for Teachers and Scientists (WDTS) within the Department’s Office of Science.
See the full article here .
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One of ten national laboratories overseen and primarily funded by the DOE(US) Office of Science, DOE’s Brookhaven National Laboratory (US) conducts research in the physical, biomedical, and environmental sciences, as well as in energy technologies and national security. Brookhaven Lab also builds and operates major scientific facilities available to university, industry and government researchers. The Laboratory’s almost 3,000 scientists, engineers, and support staff are joined each year by more than 5,000 visiting researchers from around the world. Brookhaven is operated and managed for DOE’s Office of Science by Brookhaven Science Associates, a limited-liability company founded by Stony Brook University(US), the largest academic user of Laboratory facilities, and Battelle(US), a nonprofit, applied science and technology organization.
Research at BNL specializes in nuclear and high energy physics, energy science and technology, environmental and bioscience, nanoscience and national security. The 5,300 acre campus contains several large research facilities, including the Relativistic Heavy Ion Collider [below] and National Synchrotron Light Source II [below]. Seven Nobel prizes have been awarded for work conducted at Brookhaven lab.
BNL is staffed by approximately 2,750 scientists, engineers, technicians, and support personnel, and hosts 4,000 guest investigators every year. The laboratory has its own police station, fire department, and ZIP code (11973). In total, the lab spans a 5,265-acre (21 km^2) area that is mostly coterminous with the hamlet of Upton, New York. BNL is served by a rail spur operated as-needed by the New York and Atlantic Railway. Co-located with the laboratory is the Upton, New York, forecast office of the National Weather Service.
Major programs
Although originally conceived as a nuclear research facility, Brookhaven Lab’s mission has greatly expanded. Its foci are now:
Nuclear and high-energy physics
Physics and chemistry of materials
Environmental and climate research
Energy research
Structural biology
Accelerator physics
Brookhaven National Lab was originally owned by the Atomic Energy Commission(US) and is now owned by that agency’s successor, the United States Department of Energy (DOE). DOE subcontracts the research and operation to universities and research organizations. It is currently operated by Brookhaven Science Associates LLC, which is an equal partnership of Stony Brook University(US) and Battelle Memorial Institute(US). From 1947 to 1998, it was operated by Associated Universities, Inc. (AUI) (US), but AUI lost its contract in the wake of two incidents: a 1994 fire at the facility’s high-beam flux reactor that exposed several workers to radiation and reports in 1997 of a tritium leak into the groundwater of the Long Island Central Pine Barrens on which the facility sits.
Following World War II, the US Atomic Energy Commission was created to support government-sponsored peacetime research on atomic energy. The effort to build a nuclear reactor in the American northeast was fostered largely by physicists Isidor Isaac Rabi and Norman Foster Ramsey Jr., who during the war witnessed many of their colleagues at Columbia University leave for new remote research sites following the departure of the Manhattan Project from its campus. Their effort to house this reactor near New York City was rivalled by a similar effort at the Massachusetts Institute of Technology (US) to have a facility near Boston, Massachusettes(US). Involvement was quickly solicited from representatives of northeastern universities to the south and west of New York City such that this city would be at their geographic center. In March 1946 a nonprofit corporation was established that consisted of representatives from nine major research universities — Columbia University(US), Cornell University(US), Harvard University(US), Johns Hopkins University(US), Massachusetts Institute of Technology(US), Princeton University(US), University of Pennsylvania(US), University of Rochester(US), and Yale University(US).
Out of 17 considered sites in the Boston-Washington corridor, Camp Upton on Long Island was eventually chosen as the most suitable in consideration of space, transportation, and availability. The camp had been a training center from the US Army during both World War I and World War II. After the latter war, Camp Upton was deemed no longer necessary and became available for reuse. A plan was conceived to convert the military camp into a research facility.
On March 21, 1947, the Camp Upton site was officially transferred from the U.S. War Department to the new U.S. Atomic Energy Commission (AEC), predecessor to the U.S. Department of Energy (DOE).
Research and facilities
Reactor history
In 1947 construction began on the first nuclear reactor at Brookhaven, the Brookhaven Graphite Research Reactor. This reactor, which opened in 1950, was the first reactor to be constructed in the United States after World War II. The High Flux Beam Reactor operated from 1965 to 1999. In 1959 Brookhaven built the first US reactor specifically tailored to medical research, the Brookhaven Medical Research Reactor, which operated until 2000.
Accelerator history
In 1952 Brookhaven began using its first particle accelerator, the Cosmotron. At the time the Cosmotron was the world’s highest energy accelerator, being the first to impart more than 1 GeV of energy to a particle.
The Cosmotron was retired in 1966, after it was superseded in 1960 by the new Alternating Gradient Synchrotron (AGS).
The AGS was used in research that resulted in 3 Nobel prizes, including the discovery of the muon neutrino, the charm quark, and CP violation.
In 1970 in BNL started the ISABELLE project to develop and build two proton intersecting storage rings.
The groundbreaking for the project was in October 1978. In 1981, with the tunnel for the accelerator already excavated, problems with the superconducting magnets needed for the ISABELLE accelerator brought the project to a halt, and the project was eventually cancelled in 1983.
The National Synchrotron Light Source (US) operated from 1982 to 2014 and was involved with two Nobel Prize-winning discoveries. It has since been replaced by the National Synchrotron Light Source II (US) [below].
After ISABELLE’S cancellation, physicist at BNL proposed that the excavated tunnel and parts of the magnet assembly be used in another accelerator. In 1984 the first proposal for the accelerator now known as the Relativistic Heavy Ion Collider (RHIC)[below] was put forward. The construction got funded in 1991 and RHIC has been operational since 2000. One of the world’s only two operating heavy-ion colliders, RHIC is as of 2010 the second-highest-energy collider after the Large Hadron Collider(CH). RHIC is housed in a tunnel 2.4 miles (3.9 km) long and is visible from space.
On January 9, 2020, It was announced by Paul Dabbar, undersecretary of the US Department of Energy Office of Science, that the BNL eRHIC design has been selected over the conceptual design put forward by DOE’s Thomas Jefferson National Accelerator Facility [Jlab] (US) as the future Electron–ion collider (EIC) in the United States.
In addition to the site selection, it was announced that the BNL EIC had acquired CD-0 (mission need) from the Department of Energy. BNL’s eRHIC design proposes upgrading the existing Relativistic Heavy Ion Collider, which collides beams light to heavy ions including polarized protons, with a polarized electron facility, to be housed in the same tunnel.
Other discoveries
In 1958, Brookhaven scientists created one of the world’s first video games, Tennis for Two. In 1968 Brookhaven scientists patented Maglev, a transportation technology that utilizes magnetic levitation.
Major facilities
Relativistic Heavy Ion Collider (RHIC), which was designed to research quark–gluon plasma and the sources of proton spin. Until 2009 it was the world’s most powerful heavy ion collider. It is the only collider of spin-polarized protons.
Center for Functional Nanomaterials (CFN), used for the study of nanoscale materials.
BNL National Synchrotron Light Source II(US), Brookhaven’s newest user facility, opened in 2015 to replace the National Synchrotron Light Source (NSLS), which had operated for 30 years.[19] NSLS was involved in the work that won the 2003 and 2009 Nobel Prize in Chemistry.
Accelerator Test Facility, generates, accelerates and monitors particle beams.
Tandem Van de Graaff, once the world’s largest electrostatic accelerator.
Computational Science resources, including access to a massively parallel Blue Gene series supercomputer that is among the fastest in the world for scientific research, run jointly by Brookhaven National Laboratory and Stony Brook University.
Interdisciplinary Science Building, with unique laboratories for studying high-temperature superconductors and other materials important for addressing energy challenges.
NASA Space Radiation Laboratory, where scientists use beams of ions to simulate cosmic rays and assess the risks of space radiation to human space travelers and equipment.
Off-site contributions
It is a contributing partner to ATLAS experiment, one of the four detectors located at the Large Hadron Collider (LHC).
It is currently operating at CERN near Geneva, Switzerland.
Brookhaven was also responsible for the design of the SNS accumulator ring in partnership with Spallation Neutron Source at DOE’s Oak Ridge National Laboratory (US), Tennessee.
Brookhaven plays a role in a range of neutrino research projects around the world, including the Daya Bay Neutrino Experiment (CN) nuclear power plant, approximately 52 kilometers northeast of Hong Kong and 45 kilometers east of Shenzhen, China.
• richardmitnick 12:38 pm on June 11, 2021 Permalink | Reply
Tags: "The Mystery at the Heart of Physics That Only Math Can Solve", Even in this incomplete state QFT has prompted a number of important mathematical discoveries., Every idea that’s been used in physics over the past centuries had its natural place in mathematics., For millennia the physical world has been mathematics’ greatest muse., Mathematics does not admit new subjects lightly., Physicists realized in the 1930s that physics based on fields rather than particles resolved some of their most pressing inconsistencies., , Quantum field theory, Quantum field theory emerged as an almost universal language of physical phenomena but it’s in bad math shape., The accelerating effort to understand the mathematics of quantum field theory will have profound consequences for both math and physics., The distant relationship with math is a sign that there’s a lot more they need to understand about the theory they birthed., While QFT has been successful at generating leads for mathematics to follow its core ideas still exist almost entirely outside of mathematics.
From Quanta Magazine
June 10, 2021
Kevin Hartnett
Olena Shmahalo/Quanta Magazine.
Over the past century, quantum field theory has proved to be the single most sweeping and successful physical theory ever invented. It is an umbrella term that encompasses many specific quantum field theories — the way “shape” covers specific examples like the square and the circle. The most prominent of these theories is known as the Standard Model, and it is this framework of physics that has been so successful.
“It can explain at a fundamental level literally every single experiment that we’ve ever done,” said David Tong, a physicist at the University of Cambridge (UK).
But quantum field theory, or QFT, is indisputably incomplete. Neither physicists nor mathematicians know exactly what makes a quantum field theory a quantum field theory. They have glimpses of the full picture, but they can’t yet make it out.
“There are various indications that there could be a better way of thinking about QFT,” said Nathan Seiberg, a physicist at the Institute for Advanced Study (US). “It feels like it’s an animal you can touch from many places, but you don’t quite see the whole animal.”
Mathematics, which requires internal consistency and attention to every last detail, is the language that might make QFT whole. If mathematics can learn how to describe QFT with the same rigor with which it characterizes well-established mathematical objects, a more complete picture of the physical world will likely come along for the ride.
“If you really understood quantum field theory in a proper mathematical way, this would give us answers to many open physics problems, perhaps even including the quantization of gravity,” said Robbert Dijkgraaf, director of the Institute for Advanced Study (and a regular columnist for Quanta).
Nor is this a one-way street. For millennia the physical world has been mathematics’ greatest muse. The ancient Greeks invented trigonometry to study the motion of the stars. Mathematics turned it into a discipline with definitions and rules that students now learn without any reference to the topic’s celestial origins. Almost 2,000 years later, Isaac Newton wanted to understand Kepler’s laws of planetary motion and attempted to find a rigorous way of thinking about infinitesimal change. This impulse (along with revelations from Gottfried Leibniz) birthed the field of calculus, which mathematics appropriated and improved — and today could hardly exist without.
Now mathematicians want to do the same for QFT, taking the ideas, objects and techniques that physicists have developed to study fundamental particles and incorporating them into the main body of mathematics. This means defining the basic traits of QFT so that future mathematicians won’t have to think about the physical context in which the theory first arose.
The rewards are likely to be great: Mathematics grows when it finds new objects to explore and new structures that capture some of the most important relationships — between numbers, equations and shapes. QFT offers both.
“Physics itself, as a structure, is extremely deep and often a better way to think about mathematical things we’re already interested in. It’s just a better way to organize them,” said David Ben-Zvi, a mathematician at the University of Texas-Austin (US).
For 40 years at least, QFT has tempted mathematicians with ideas to pursue. In recent years, they’ve finally begun to understand some of the basic objects in QFT itself — abstracting them from the world of particle physics and turning them into mathematical objects in their own right.
Yet it’s still early days in the effort.
“We won’t know until we get there, but it’s certainly my expectation that we’re just seeing the tip of the iceberg,” said Greg Moore, a physicist at Rutgers University (US). “If mathematicians really understood [QFT], that would lead to profound advances in mathematics.”
Fields Forever
It’s common to think of the universe as being built from fundamental particles: electrons, quarks, photons and the like. But physics long ago moved beyond this view. Instead of particles, physicists now talk about things called “quantum fields” as the real warp and woof of reality.
These fields stretch across the space-time of the universe. They come in many varieties and fluctuate like a rolling ocean. As the fields ripple and interact with each other, particles emerge out of them and then vanish back into them, like the fleeting crests of a wave.
“Particles are not objects that are there forever,” said Tong. “It’s a dance of fields.”
To understand quantum fields, it’s easiest to start with an ordinary, or classical, field. Imagine, for example, measuring the temperature at every point on Earth’s surface. Combining the infinitely many points at which you can make these measurements forms a geometric object, called a field, that packages together all this temperature information.
In general, fields emerge whenever you have some quantity that can be measured uniquely at infinitely fine resolution across a space.
“You’re sort of able to ask independent questions about each point of space-time, like, what’s the electric field here versus over there,” said Davide Gaiotto, a physicist at the Perimeter Institute for Theoretical Physics (CA).
Quantum fields come about when you’re observing quantum phenomena, like the energy of an electron, at every point in space and time. But quantum fields are fundamentally different from classical ones.
While the temperature at a point on Earth is what it is, regardless of whether you measure it, electrons have no definite position until the moment you observe them. Prior to that, their positions can only be described probabilistically, by assigning values to every point in a quantum field that captures the likelihood you’ll find an electron there versus somewhere else. Prior to observation, electrons essentially exist nowhere — and everywhere.
“Most things in physics aren’t just objects; they’re something that lives in every point in space and time,” said Dijkgraaf.
A quantum field theory comes with a set of rules called correlation functions that explain how measurements at one point in a field relate to — or correlate with — measurements taken at another point.
Each quantum field theory describes physics in a specific number of dimensions. Two-dimensional quantum field theories are often useful for describing the behavior of materials, like insulators; six-dimensional quantum field theories are especially relevant to string theory; and four-dimensional quantum field theories describe physics in our actual four-dimensional universe. The Standard Model is one of these; it’s the single most important quantum field theory because it’s the one that best describes the universe.
There are 12 known fundamental particles that make up the universe. Each has its own unique quantum field. To these 12 particle fields the Standard Model adds four force fields, representing the four fundamental forces: gravity, electromagnetism, the strong nuclear force and the weak nuclear force.
It combines these 16 fields in a single equation that describes how they interact with each other. Through these interactions, fundamental particles are understood as fluctuations of their respective quantum fields, and the physical world emerges before our eyes.
It might sound strange, but physicists realized in the 1930s that physics based on fields rather than particles resolved some of their most pressing inconsistencies, ranging from issues regarding causality to the fact that particles don’t live forever. It also explained what otherwise appeared to be an improbable consistency in the physical world.
“All particles of the same type everywhere in the universe are the same,” said Tong. “If we go to the Large Hadron Collider and make a freshly minted proton, it’s exactly the same as one that’s been traveling for 10 billion years.
That deserves some explanation.” QFT provides it: All protons are just fluctuations in the same underlying proton field (or, if you could look more closely, the underlying quark fields).
But the explanatory power of QFT comes at a high mathematical cost.
“Quantum field theories are by far the most complicated objects in mathematics, to the point where mathematicians have no idea how to make sense of them,” said Tong. “Quantum field theory is mathematics that has not yet been invented by mathematicians.”
Too Much Infinity
What makes it so complicated for mathematicians? In a word, infinity.
When you measure a quantum field at a point, the result isn’t a few numbers like coordinates and temperature. Instead, it’s a matrix, which is an array of numbers. And not just any matrix — a big one, called an operator, with infinitely many columns and rows. This reflects how a quantum field envelops all the possibilities of a particle emerging from the field.
“There are infinitely many positions that a particle can have, and this leads to the fact that the matrix that describes the measurement of position, of momentum, also has to be infinite-dimensional,” said Kasia Rejzner of the University of York (UK).
And when theories produce infinities, it calls their physical relevance into question, because infinity exists as a concept, not as anything experiments can ever measure. It also makes the theories hard to work with mathematically.
“We don’t like having a framework that spells out infinity. That’s why you start realizing you need a better mathematical understanding of what’s going on,” said Alejandra Castro, a physicist at the University of Amsterdam [Universiteit van Amsterdam] (NL).
The problems with infinity get worse when physicists start thinking about how two quantum fields interact, as they might, for instance, when particle collisions are modeled at the Large Hadron Collider outside Geneva. In classical mechanics this type of calculation is easy: To model what happens when two billiard balls collide, just use the numbers specifying the momentum of each ball at the point of collision.
When two quantum fields interact, you’d like to do a similar thing: multiply the infinite-dimensional operator for one field by the infinite-dimensional operator for the other at exactly the point in space-time where they meet. But this calculation — multiplying two infinite-dimensional objects that are infinitely close together — is difficult.
“This is where things go terribly wrong,” said Rejzner.
Smashing Success
Physicists and mathematicians can’t calculate using infinities, but they have developed workarounds — ways of approximating quantities that dodge the problem. These workarounds yield approximate predictions, which are good enough, because experiments aren’t infinitely precise either.
“We can do experiments and measure things to 13 decimal places and they agree to all 13 decimal places. It’s the most astonishing thing in all of science,” said Tong.
One workaround starts by imagining that you have a quantum field in which nothing is happening. In this setting — called a “free” theory because it’s free of interactions — you don’t have to worry about multiplying infinite-dimensional matrices because nothing’s in motion and nothing ever collides. It’s a situation that’s easy to describe in full mathematical detail, though that description isn’t worth a whole lot.
“It’s totally boring, because you’ve described a lonely field with nothing to interact with, so it’s a bit of an academic exercise,” said Rejzner.
But you can make it more interesting. Physicists dial up the interactions, trying to maintain mathematical control of the picture as they make the interactions stronger.
This approach is called perturbative QFT, in the sense that you allow for small changes, or perturbations, in a free field. You can apply the perturbative perspective to quantum field theories that are similar to a free theory. It’s also extremely useful for verifying experiments. “You get amazing accuracy, amazing experimental agreement,” said Rejzner.
But if you keep making the interactions stronger, the perturbative approach eventually overheats. Instead of producing increasingly accurate calculations that approach the real physical universe, it becomes less and less accurate. This suggests that while the perturbation method is a useful guide for experiments, ultimately it’s not the right way to try and describe the universe: It’s practically useful, but theoretically shaky.
“We do not know how to add everything up and get something sensible,” said Gaiotto.
Another approximation scheme tries to sneak up on a full-fledged quantum field theory by other means. In theory, a quantum field contains infinitely fine-grained information. To cook up these fields, physicists start with a grid, or lattice, and restrict measurements to places where the lines of the lattice cross each other. So instead of being able to measure the quantum field everywhere, at first you can only measure it at select places a fixed distance apart.
From there, physicists enhance the resolution of the lattice, drawing the threads closer together to create a finer and finer weave. As it tightens, the number of points at which you can take measurements increases, approaching the idealized notion of a field where you can take measurements everywhere.
“The distance between the points becomes very small, and such a thing becomes a continuous field,” said Seiberg. In mathematical terms, they say the continuum quantum field is the limit of the tightening lattice.
Mathematicians are accustomed to working with limits and know how to establish that certain ones really exist. For example, they’ve proved that the limit of the infinite sequence 1/2 + 1/4 +1/8 +1/16 … is 1. Physicists would like to prove that quantum fields are the limit of this lattice procedure. They just don’t know how.
“It’s not so clear how to take that limit and what it means mathematically,” said Moore.
Physicists don’t doubt that the tightening lattice is moving toward the idealized notion of a quantum field. The close fit between the predictions of QFT and experimental results strongly suggests that’s the case.
“There is no question that all these limits really exist, because the success of quantum field theory has been really stunning,” said Seiberg. But having strong evidence that something is correct and proving conclusively that it is are two different things.
It’s a degree of imprecision that’s out of step with the other great physical theories that QFT aspires to supersede. Isaac Newton’s laws of motion, quantum mechanics, Albert Einstein’s theories of special and general relativity — they’re all just pieces of the bigger story QFT wants to tell, but unlike QFT, they can all be written down in exact mathematical terms.
“Quantum field theory emerged as an almost universal language of physical phenomena but it’s in bad math shape,” said Dijkgraaf. And for some physicists, that’s a reason for pause.
“If the full house is resting on this core concept that itself isn’t understood in a mathematical way, why are you so confident this is describing the world? That sharpens the whole issue,” said Dijkgraaf.
Outside Agitator
Even in this incomplete state QFT has prompted a number of important mathematical discoveries. The general pattern of interaction has been that physicists using QFT stumble onto surprising calculations that mathematicians then try to explain.
“It’s an idea-generating machine,” said Tong.
At a basic level, physical phenomena have a tight relationship with geometry. To take a simple example, if you set a ball in motion on a smooth surface, its trajectory will illuminate the shortest path between any two points, a property known as a geodesic. In this way, physical phenomena can detect geometric features of a shape.
Now replace the billiard ball with an electron. The electron exists probabilistically everywhere on a surface. By studying the quantum field that captures those probabilities, you can learn something about the overall nature of that surface (or manifold, to use the mathematicians’ term), like how many holes it has. That’s a fundamental question that mathematicians working in geometry, and the related field of topology, want to answer.
“One particle even sitting there, doing nothing, will start to know about the topology of a manifold,” said Tong.
In the late 1970s, physicists and mathematicians began applying this perspective to solve basic questions in geometry. By the early 1990s, Seiberg and his collaborator Edward Witten figured out how to use it to create a new mathematical tool — now called the Seiberg-Witten invariants — that turns quantum phenomena into an index for purely mathematical traits of a shape: Count the number of times quantum particles behave in a certain way, and you’ve effectively counted the number of holes in a shape.
“Witten showed that quantum field theory gives completely unexpected but completely precise insights into geometrical questions, making intractable problems soluble,” said Graeme Segal, a mathematician at the University of Oxford (UK).
Another example of this exchange also occurred in the early 1990s, when physicists were doing calculations related to string theory. They performed them in two different geometric spaces based on fundamentally different mathematical rules and kept producing long sets of numbers that matched each other exactly. Mathematicians picked up the thread and elaborated it into a whole new field of inquiry, called mirror symmetry, that investigates the concurrence — and many others like it.
“Physics would come up with these amazing predictions, and mathematicians would try to prove them by our own means,” said Ben-Zvi. “The predictions were strange and wonderful, and they turned out to be pretty much always correct.”
But while QFT has been successful at generating leads for mathematics to follow its core ideas still exist almost entirely outside of mathematics. Quantum field theories are not objects that mathematicians understand well enough to use the way they can use polynomials, groups, manifolds and other pillars of the discipline (many of which also originated in physics).
For physicists, this distant relationship with math is a sign that there’s a lot more they need to understand about the theory they birthed. “Every other idea that’s been used in physics over the past centuries had its natural place in mathematics,” said Seiberg. “This is clearly not the case with quantum field theory.”
And for mathematicians, it seems as if the relationship between QFT and math should be deeper than the occasional interaction. That’s because quantum field theories contain many symmetries, or underlying structures, that dictate how points in different parts of a field relate to each other. These symmetries have a physical significance — they embody how quantities like energy are conserved as quantum fields evolve over time. But they’re also mathematically interesting objects in their own right.
“A mathematician might care about a certain symmetry, and we can put it in a physical context. It creates this beautiful bridge between these two fields,” said Castro.
Mathematicians already use symmetries and other aspects of geometry to investigate everything from solutions to different types of equations to the distribution of prime numbers. Often, geometry encodes answers to questions about numbers. QFT offers mathematicians a rich new type of geometric object to play with — if they can get their hands on it directly, there’s no telling what they’ll be able to do.
“We’re to some extent playing with QFT,” said Dan Freed, a mathematician at the University of Texas, Austin. “We’ve been using QFT as an outside stimulus, but it would be nice if it were an inside stimulus.”
Make Way for QFT
Mathematics does not admit new subjects lightly. Many basic concepts went through long trials before they settled into their proper, canonical places in the field.
Take the real numbers — all the infinitely many tick marks on the number line. It took math nearly 2,000 years of practice to agree on a way of defining them. Finally, in the 1850s, mathematicians settled on a precise three-word statement describing the real numbers as a “complete ordered field.” They’re complete because they contain no gaps, they’re ordered because there’s always a way of determining whether one real number is greater or less than another, and they form a “field,” which to mathematicians means they follow the rules of arithmetic.
“Those three words are historically hard fought,” said Freed.
In order to turn QFT into an inside stimulus — a tool they can use for their own purposes — mathematicians would like to give the same treatment to QFT they gave to the real numbers: a sharp list of characteristics that any specific quantum field theory needs to satisfy.
A lot of the work of translating parts of QFT into mathematics has come from a mathematician named Kevin Costello at the Perimeter Institute. In 2016 he coauthored a textbook that puts perturbative QFT on firm mathematical footing, including formalizing how to work with the infinite quantities that crop up as you increase the number of interactions. The work follows an earlier effort from the 2000s called algebraic quantum field theory that sought similar ends, and which Rejzner reviewed in a 2016 book. So now, while perturbative QFT still doesn’t really describe the universe, mathematicians know how to deal with the physically non-sensical infinities it produces.
“His contributions are extremely ingenious and insightful. He put [perturbative] theory in a nice new framework that is suitable for rigorous mathematics,” said Moore.
Costello explains he wrote the book out of a desire to make perturbative quantum field theory more coherent. “I just found certain physicists’ methods unmotivated and ad hoc. I wanted something more self-contained that a mathematician could go work with,” he said.
By specifying exactly how perturbation theory works, Costello has created a basis upon which physicists and mathematicians can construct novel quantum field theories that satisfy the dictates of his perturbation approach. It’s been quickly embraced by others in the field.
“He certainly has a lot of young people working in that framework. [His book] has had its influence,” said Freed.
Costello has also been working on defining just what a quantum field theory is. In stripped-down form, a quantum field theory requires a geometric space in which you can make observations at every point, combined with correlation functions that express how observations at different points relate to each other. Costello’s work describes the properties a collection of correlation functions needs to have in order to serve as a workable basis for a quantum field theory.
The most familiar quantum field theories, like the Standard Model, contain additional features that may not be present in all quantum field theories. Quantum field theories that lack these features likely describe other, still undiscovered properties that could help physicists explain physical phenomena the Standard Model can’t account for. If your idea of a quantum field theory is fixed too closely to the versions we already know about, you’ll have a hard time even envisioning the other, necessary possibilities.
“There is a big lamppost under which you can find theories of fields [like the Standard Model], and around it is a big darkness of [quantum field theories] we don’t know how to define, but we know they’re there,” said Gaiotto.
Costello has illuminated some of that dark space with his definitions of quantum fields. From these definitions, he’s discovered two surprising new quantum field theories. Neither describes our four-dimensional universe, but they do satisfy the core demands of a geometric space equipped with correlation functions. Their discovery through pure thought is similar to how the first shapes you might discover are ones present in the physical world, but once you have a general definition of a shape, you can think your way to examples with no physical relevance at all.
And if mathematics can determine the full space of possibilities for quantum field theories — all the many different possibilities for satisfying a general definition involving correlation functions — physicists can use that to find their way to the specific theories that explain the important physical questions they care most about.
“I want to know the space of all QFTs because I want to know what quantum gravity is,” said Castro.
A Multi-Generational Challenge
There’s a long way to go. So far, all of the quantum field theories that have been described in full mathematical terms rely on various simplifications, which make them easier to work with mathematically.
One way to simplify the problem, going back decades, is to study simpler two-dimensional QFTs rather than four-dimensional ones. A team in France recently nailed down all the mathematical details of a prominent two-dimensional QFT.
Other simplifications assume quantum fields are symmetrical in ways that don’t match physical reality, but that make them more tractable from a mathematical perspective. These include “supersymmetric” and “topological” QFTs.
The next, and much more difficult, step will be to remove the crutches and provide a mathematical description of a quantum field theory that better suits the physical world physicists most want to describe: the four-dimensional, continuous universe in which all interactions are possible at once.
“This is [a] very embarrassing thing that we don’t have a single quantum field theory we can describe in four dimensions, nonperturbatively,” said Rejzner. “It’s a hard problem, and apparently it needs more than one or two generations of mathematicians and physicists to solve it.”
But that doesn’t stop mathematicians and physicists from eyeing it greedily. For mathematicians, QFT is as rich a type of object as they could hope for. Defining the characteristic properties shared by all quantum field theories will almost certainly require merging two of the pillars of mathematics: analysis, which explains how to control infinities, and geometry, which provides a language for talking about symmetry.
“It’s a fascinating problem just in math itself, because it combines two great ideas,” said Dijkgraaf.
If mathematicians can understand QFT, there’s no telling what mathematical discoveries await in its unlocking. Mathematicians defined the characteristic properties of other objects, like manifolds and groups, long ago, and those objects now permeate virtually every corner of mathematics. When they were first defined, it would have been impossible to anticipate all their mathematical ramifications. QFT holds at least as much promise for math.
“I like to say the physicists don’t necessarily know everything, but the physics does,” said Ben-Zvi. “If you ask it the right questions, it already has the phenomena mathematicians are looking for.”
And for physicists, a complete mathematical description of QFT is the flip side of their field’s overriding goal: a complete description of physical reality.
“I feel there is one intellectual structure that covers all of it, and maybe it will encompass all of physics,” said Seiberg.
Now mathematicians just have to uncover it.
See the full article here .
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• richardmitnick 10:54 am on July 27, 2019 Permalink | Reply
Tags: "Ask Ethan: Can We Really Get A Universe From Nothing?", , , , , Because dark energy is a property of space itself when the Universe expands the dark energy density must remain constant., , , , , Galaxies that are gravitationally bound will merge together into groups and clusters while the unbound groups and clusters will accelerate away from one another., , , Negative gravity?, , Quantum field theory
From Ethan Siegel: “Ask Ethan: Can We Really Get A Universe From Nothing?”
From Ethan Siegel
July 27, 2019
Our entire cosmic history is theoretically well-understood in terms of the frameworks and rules that govern it. It’s only by observationally confirming and revealing various stages in our Universe’s past that must have occurred, like when the first stars and galaxies formed, and how the Universe expanded over time, that we can truly come to understand what makes up our Universe and how it expands and gravitates in a quantitative fashion. The relic signatures imprinted on our Universe from an inflationary state before the hot Big Bang give us a unique way to test our cosmic history, subject to the same fundamental limitations that all frameworks possess. (NICOLE RAGER FULLER / NATIONAL SCIENCE FOUNDATION)
And does it require the idea of ‘negative gravity’ in order to work?
The biggest question that we’re even capable of asking, with our present knowledge and understanding of the Universe, is where did everything we can observe come from? If it came from some sort of pre-existing state, we’ll want to know exactly what that state was like and how our Universe came from it. If it emerged out of nothingness, we’d want to know how we went from nothing to the entire Universe, and what if anything caused it. At least, that’s what our Patreon supporter Charles Buchanan wants to know, asking:
“One concept bothers me. Perhaps you can help. I see it in used many places, but never really explained. “A universe from Nothing” and the concept of negative gravity. As I learned my Newtonian physics, you could put the zero point of the gravitational potential anywhere, only differences mattered. However Newtonian physics never deals with situations where matter is created… Can you help solidify this for me, preferably on [a] conceptual level, maybe with a little calculation detail?”
Gravitation might seem like a straightforward force, but an incredible number of aspects are anything but intuitive. Let’s take a deeper look.
Countless scientific tests of Einstein’s general theory of relativity have been performed, subjecting the idea to some of the most stringent constraints ever obtained by humanity. Einstein’s first solution was for the weak-field limit around a single mass, like the Sun; he applied these results to our Solar System with dramatic success. We can view this orbit as Earth (or any planet) being in free-fall around the Sun, traveling in a straight-line path in its own frame of reference. All masses and all sources of energy contribute to the curvature of spacetime. (LIGO SCIENTIFIC COLLABORATION / T. PYLE / CALTECH / MIT)
MIT /Caltech Advanced aLigo
VIRGO Gravitational Wave interferometer, near Pisa, Italy
Caltech/MIT Advanced aLigo Hanford, WA, USA installation
Caltech/MIT Advanced aLigo detector installation Livingston, LA, USA
LSC LIGO Scientific Collaboration
Cornell SXS, the Simulating eXtreme Spacetimes (SXS) project
Gravitational waves. Credit: MPI for Gravitational Physics/W.Benger
Gravity is talking. Lisa will listen. Dialogos of Eide
ESA/eLISA the future of gravitational wave research
Localizations of gravitational-wave signals detected by LIGO in 2015 (GW150914, LVT151012, GW151226, GW170104), more recently, by the LIGO-Virgo network (GW170814, GW170817). After Virgo came online in August 2018
Skymap showing how adding Virgo to LIGO helps in reducing the size of the source-likely region in the sky. (Credit: Giuseppe Greco (Virgo Urbino group)
If you have two point masses located some distance apart in your Universe, they’ll experience an attractive force that compels them to gravitate towards one another. But this attractive force that you perceive, in the context of relativity, comes with two caveats.
The first caveat is simple and straightforward: these two masses will experience an acceleration towards one another, but whether they wind up moving closer to one another or not is entirely dependent on how the space between them evolves. Unlike in Newtonian gravity, where space is a fixed quantity and only the masses within that space can evolve, everything is changeable in General Relativity. Not only does matter and energy move and accelerate due to gravitation, but the very fabric of space itself can expand, contract, or otherwise flow. All masses still move through space, but space itself is no longer stationary.
The ‘raisin bread’ model of the expanding Universe, where relative distances increase as the space (dough) expands. The farther away any two raisin are from one another, the greater the observed redshift will be by time the light is received. The redshift-distance relation predicted by the expanding Universe is borne out in observations, and has been consistent with what’s been known going all the way back to the 1920s. (NASA / WMAP SCIENCE TEAM)
NASA/WMAP 2001 to 2010
The second caveat is that the two masses you’re considering, even if you’re extremely careful about accounting for what’s in your Universe, are most likely not the only forms of energy around. There are bound to be other masses in the form of normal matter, dark matter, and neutrinos. There’s the presence of radiation, from both electromagnetic and gravitational waves. There’s even dark energy: a type of energy inherent to the fabric of space itself.
Now, here’s a scenario that might exemplify where your intuition leads you astray: what happens if these masses, for the volume they occupy, have less total energy than the average energy density of the surrounding space?
The gravitational attraction (blue) of overdense regions and the relative repulsion (red) of the underdense regions, as they act on the Milky Way. Even though gravity is always attractive, there is an average amount of attraction throughout the Universe, and regions with lower energy densities than that will experience (and cause) an effective repulsion with respect to the average. (YEHUDA HOFFMAN, DANIEL POMARÈDE, R. BRENT TULLY, AND HÉLÈNE COURTOIS, NATURE ASTRONOMY 1, 0036 (2017))
You can imagine three different scenarios:
1.The first mass has a below-average energy density while the second has an above-average value.
2.The first mass has an above-average energy density while the second has a below-average value.
3.Both the first and second masses have a below-average energy density compared to the rest of space.
In the first two scenarios, the above-average mass will begin growing as it pulls on the matter/energy all around it, while the below-average mass will start shrinking, as it’s less able to hold onto its own mass in the face of its surroundings. These two masses will effectively repel one another; even though gravitation is always attractive, the intervening matter is preferentially attracted to the heavier-than-average mass. This causes the lower-mass object to act like it’s both repelling and being repelled by the heavier-mass object, the same way a balloon held underwater will still be attracted to Earth’s center, but will be forced away from it owing to the (buoyant) effects of the water.
The Earth’s crust is thinnest over the ocean and thickest over mountains and plateaus, as the principle of buoyancy dictates and as gravitational experiments confirm. Just as a balloon submerged in water will accelerate away from the center of the Earth, a region with below-average energy density will accelerate away from an overdense region, as average-density regions will be more preferentially attracted to the overdense region than the underdense region will. (USGS)
So what’s going to happen if you have two regions of space with below-average densities, surrounded by regions of just average density? They’ll both shrink, giving up their remaining matter to the denser regions around them. But as far as motions go, they’ll accelerate towards one another, with exactly the same magnitude they’d accelerate at if they were both overdense regions that exceeded the average density by equivalent amounts.
You might be wondering why it’s important to think about these concerns when talking about a Universe from nothing. After all, if your Universe is full of matter and energy, it’s pretty hard to understand how that’s relevant to making sense of the concept of something coming from nothing. But just as our intuition can lead us astray when thinking about matter and energy on the spacetime playing field of General Relativity, it’s a comparable situation when we think about nothingness.
A representation of flat, empty space with no matter, energy or curvature of any type. With the exception of small quantum fluctuations, space in an inflationary Universe becomes incredibly flat like this, except in a 3D grid rather than a 2D sheet. Space is stretched flat, and particles are rapidly driven away. (AMBER STUVER / LIVING LIGO)
You very likely think about nothingness as a philosopher would: the complete absence of everything. Zero matter, zero energy, an absolutely zero value for all the quantum fields in the Universe, etc. You think of space that’s completely flat, with nothing around to cause its curvature anywhere.
If you think this way, you’re not alone: there are many different ways to conceive of “nothing.” You might even be tempted to take away space, time, and the laws of physics themselves, too. The problem, if you start doing that, is that you lose your ability to predict anything at all. The type of nothingness you’re thinking about, in this context, is what we call unphysical.
If we want to think about nothing in a physical sense, you have to keep certain things. You need spacetime and the laws of physics, for example; you cannot have a Universe without them.
A visualization of QCD illustrates how particle/antiparticle pairs pop out of the quantum vacuum for very small amounts of time as a consequence of Heisenberg uncertainty.
The quantum vacuum is interesting because it demands that empty space itself isn’t so empty, but is filled with all the particles, antiparticles and fields in various states that are demanded by the quantum field theory that describes our Universe. Put this all together, and you find that empty space has a zero-point energy that’s actually greater than zero. (DEREK B. LEINWEBER)
But here’s the kicker: if you have spacetime and the laws of physics, then by definition you have quantum fields permeating the Universe everywhere you go. You have a fundamental “jitter” to the energy inherent to space, due to the quantum nature of the Universe. (And the Heisenberg uncertainty principle, which is unavoidable.)
Put these ingredients together — because you can’t have a physically sensible “nothing” without them — and you’ll find that space itself doesn’t have zero energy inherent to it, but energy with a finite, non-zero value. Just as there’s a finite zero-point energy (that’s greater than zero) for an electron bound to an atom, the same is true for space itself. Empty space, even with zero curvature, even devoid of particles and external fields, still has a finite energy density to it.
The four possible fates of the Universe with only matter, radiation, curvature and a cosmological constant allowed. The top three possibilities are for a Universe whose fate is determined by the balance of matter/radiation with spatial curvature alone; the bottom one includes dark energy. Only the bottom “fate” aligns with the evidence. (E. SIEGEL / BEYOND THE GALAXY)
From the perspective of quantum field theory, this is conceptualized as the zero-point energy of the quantum vacuum: the lowest-energy state of empty space. In the framework of General Relativity, however, it appears in a different sense: as the value of a cosmological constant, which itself is the energy of empty space, independent of curvature or any other form of energy density.
Although we do not know how to calculate the value of this energy density from first principles, we can calculate the effects it has on the expanding Universe. As your Universe expands, every form of energy that exists within it contributes to not only how your Universe expands, but how that expansion rate changes over time. From multiple independent lines of evidence — including the Universe’s large-scale structure, the cosmic microwave background, and distant supernovae — we have been able to determine how much energy is inherent to space itself.
Constraints on dark energy from three independent sources: supernovae, the CMB (cosmic microwave background) and BAO (which is a wiggly feature seen in the correlations of large-scale structure). Note that even without supernovae, we’d need dark energy for certain, and also that there are uncertainties and degeneracies between the amount of dark matter and dark energy that we’d need to accurately describe our Universe. (SUPERNOVA COSMOLOGY PROJECT, AMANULLAH, ET AL., AP.J. (2010))
This form of energy is what we presently call dark energy, and it’s responsible for the observed accelerated expansion of the Universe. Although it’s been a part of our conceptions of reality for more than two decades now, we don’t fully understand its true nature. All we can say is that when we measure the expansion rate of the Universe, our observations are consistent with dark energy being a cosmological constant with a specific magnitude, and not with any of the alternatives that evolve significantly over cosmic time.
Because dark energy causes distant galaxies to appear to recede from one another more and more quickly as time goes on — since the space between those galaxies is expanding — it’s often called negative gravity. This is not only highly informal, but incorrect. Gravity is only positive, never negative. But even positive gravity, as we saw earlier, can have effects that look very much like negative repulsion.
Dark Energy Survey
Dark Energy Camera [DECam], built at FNAL
Timeline of the Inflationary Universe WMAP
The Dark Energy Survey (DES) is an international, collaborative effort to map hundreds of millions of galaxies, detect thousands of supernovae, and find patterns of cosmic structure that will reveal the nature of the mysterious dark energy that is accelerating the expansion of our Universe. DES began searching the Southern skies on August 31, 2013.
According to Einstein’s theory of General Relativity, gravity should lead to a slowing of the cosmic expansion. Yet, in 1998, two teams of astronomers studying distant supernovae made the remarkable discovery that the expansion of the universe is speeding up. To explain cosmic acceleration, cosmologists are faced with two possibilities: either 70% of the universe exists in an exotic form, now called dark energy, that exhibits a gravitational force opposite to the attractive gravity of ordinary matter, or General Relativity must be replaced by a new theory of gravity on cosmic scales.
DES is designed to probe the origin of the accelerating universe and help uncover the nature of dark energy by measuring the 14-billion-year history of cosmic expansion with high precision. More than 400 scientists from over 25 institutions in the United States, Spain, the United Kingdom, Brazil, Germany, Switzerland, and Australia are working on the project. The collaboration built and is using an extremely sensitive 570-Megapixel digital camera, DECam, mounted on the Blanco 4-meter telescope at Cerro Tololo Inter-American Observatory, high in the Chilean Andes, to carry out the project.
Over six years (2013-2019), the DES collaboration used 758 nights of observation to carry out a deep, wide-area survey to record information from 300 million galaxies that are billions of light-years from Earth. The survey imaged 5000 square degrees of the southern sky in five optical filters to obtain detailed information about each galaxy. A fraction of the survey time is used to observe smaller patches of sky roughly once a week to discover and study thousands of supernovae and other astrophysical transients.
How energy density changes over time in a Universe dominated by matter (top), radiation (middle), and a cosmological constant (bottom). Note that dark energy doesn’t change in density as the Universe expands, which is why it comes to dominate the Universe at late times. (E. SIEGEL)
If there were greater amounts of dark energy present within our spatially flat Universe, the expansion rate would be greater. But this is true for all forms of energy in a spatially flat Universe: dark energy is no exception. The only different between dark energy and the more commonly encountered forms of energy, like matter and radiation, is that as the Universe expands, the densities of matter and radiation decrease.
But because dark energy is a property of space itself, when the Universe expands, the dark energy density must remain constant. As time goes on, galaxies that are gravitationally bound will merge together into groups and clusters, while the unbound groups and clusters will accelerate away from one another. That’s the ultimate fate of the Universe if dark energy is real.
Laniakea supercluster. From Nature The Laniakea supercluster of galaxies R. Brent Tully, Hélène Courtois, Yehuda Hoffman & Daniel Pomarède at http://www.nature.com/nature/journal/v513/n7516/full/nature13674.html. Milky Way is the red dot.
So why do we say we have a Universe that came from nothing? Because the value of dark energy may have been much higher in the distant past: before the hot Big Bang. A Universe with a very large amount of dark energy in it will behave identically to a Universe undergoing cosmic inflation. In order for inflation to end, that energy has to get converted into matter and radiation. The evidence strongly points to that happening some 13.8 billion years ago.
When it did, though, a small amount of dark energy remained behind. Why? Because the zero-point energy of the quantum fields in our Universe isn’t zero, but a finite, greater-than-zero value. Our intuition may not be reliable when we consider the physical concepts of nothing and negative/positive gravity, but that’s why we have science. When we do it right, we wind up with physical theories that accurately describe the Universe we measure and observe.
See the full article here .
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• richardmitnick 8:41 am on June 5, 2019 Permalink | Reply
Tags: "Stanford joins collaboration to explore 'ultra-quantum matter'", , , Quantum field theory, , The Simons Collaboration on Ultra-Quantum Matter
From Stanford University: “Stanford joins collaboration to explore ‘ultra-quantum matter'”
Stanford University Name
From Stanford University
June 3, 2019
Ker Than
The Simons Collaboration on Ultra-Quantum Matter brings together physicists from 12 institutions to “understand, classify and realize” new forms of ultra-quantum matter in the lab.
Stanford physicist Shamit Kachru is a member of a new collaboration that aims to unravel the mystery of entangled quantum matter — macroscopic assemblages of atoms and electrons that seem to share the same seemingly telepathic link as entangled subatomic particles.
The Simons Collaboration on Ultra-Quantum Matter is funded by the Simons Foundation and led by Harvard physics Professor Ashvin Vishwanath. It is part of the Simons Collaborations in Mathematics and Physical Sciences program, which aims to “stimulate progress on fundamental scientific questions of major importance in mathematics, theoretical physics and theoretical computer science.” The Simons Collaboration on Ultra-Quantum Matter will be one of 12 such collaborations ranging across these fields.
Ultra-quantum matter, or UQM, exhibit non-intuitive quantum properties that were once thought to arise only in very small systems. One key property is “non-local entanglement,” in which two physically separated groups of atoms can share joint properties, so that measuring one affects the measurement outcome of the other. UQM should exhibit entirely new physical properties, a better understanding of which could lead to new types of quantum information storage systems and quantum materials.
The Simons Collaboration on Ultra-Quantum Matter brings together physicists from 12 institutions to “understand, classify and realize” new forms of ultra-quantum matter in the lab. To achieve this, the collaboration includes physicists working in different domains, including condensed matter and high energy theorists, as well as atomic and quantum information experts. Kachru’s own background is in string theory, theoretical cosmology, and condensed matter physics.
A confluence of factors makes this a particularly exciting time to study UQM, said Kachru, who is the Wells Family Director of the Stanford Institute for Theoretical Physics (SITP) and the chair of the physics department.
“Many of the cutting-edge questions in quantum field theory now seem to involve highly quantum condensed matter systems,” Kachru said. “These systems are often best studied using elegant and clean mathematical techniques, and there is a promise of genuine contact between high level theory and experiment. I can’t imagine better people to teach me about issues and opportunities here than the collaboration members, who are leading experts in all aspects of UQM.”
Kachru also looks forward to working again with former Stanford graduate student and collaboration member, John McGreevy, who was Kachru’s first PhD advisee and is now a professor of physics at the University of California, San Diego.
Ultra-Quantum Matter is an $8M four-year award funded by the Simons Foundation and renewable for three additional years. It will support researchers from the following institutions: Caltech, Harvard, the Institute for Advanced Study, MIT, Stanford, University of California Santa Barbara, University of California San Diego, the University of Chicago, the University of Colorado Boulder, the University of Innsbruck, University of Maryland and University of Washington.
A UQM meeting of the new collaboration is scheduled to take place at Stanford in May of 2020.
See the full article here .
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Stanford University campus. No image credit
Stanford University
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• richardmitnick 7:14 am on March 20, 2018 Permalink | Reply
Tags: , , , , Cosmological-constant problem, , , In 1998 astronomers discovered that the expansion of the cosmos is in fact gradually accelerating, , , Quantum field theory, , Saul Perlmutter UC Berkeley Nobel laureate, , Why Does the Universe Need to Be So Empty?, Zero-point energy of the field
From The Atlantic Magazine and Quanta: “Why Does the Universe Need to Be So Empty?”
Quanta Magazine
Quanta Magazine
Atlantic Magazine
The Atlantic Magazine
Mar 19, 2018
Natalie Wolchover
Physicists have long grappled with the perplexingly small weight of empty space.
The controversial idea that our universe is just a random bubble in an endless, frothing multiverse arises logically from nature’s most innocuous-seeming feature: empty space. Specifically, the seed of the multiverse hypothesis is the inexplicably tiny amount of energy infused in empty space—energy known as the vacuum energy, dark energy, or the cosmological constant. Each cubic meter of empty space contains only enough of this energy to light a light bulb for 11 trillionths of a second. “The bone in our throat,” as the Nobel laureate Steven Weinberg once put it [http://hetdex.org/dark_energy.html
], is that the vacuum ought to be at least a trillion trillion trillion trillion trillion times more energetic, because of all the matter and force fields coursing through it.
Somehow the effects of all these fields on the vacuum almost equalize, producing placid stillness. Why is empty space so empty?
While we don’t know the answer to this question—the infamous “cosmological-constant problem”—the extreme vacuity of our vacuum appears necessary for our existence. In a universe imbued with even slightly more of this gravitationally repulsive energy, space would expand too quickly for structures like galaxies, planets, or people to form. This fine-tuned situation suggests that there might be a huge number of universes, all with different doses of vacuum energy, and that we happen to inhabit an extraordinarily low-energy universe because we couldn’t possibly find ourselves anywhere else.
Some scientists bristle at the tautology of “anthropic reasoning” and dislike the multiverse for being untestable. Even those open to the multiverse idea would love to have alternative solutions to the cosmological constant problem to explore. But so far it has proved nearly impossible to solve without a multiverse. “The problem of dark energy [is] so thorny, so difficult, that people have not got one or two solutions,” says Raman Sundrum, a theoretical physicist at the University of Maryland.
To understand why, consider what the vacuum energy actually is. Albert Einstein’s general theory of relativity says that matter and energy tell space-time how to curve, and space-time curvature tells matter and energy how to move. An automatic feature of the equations is that space-time can possess its own energy—the constant amount that remains when nothing else is there, which Einstein dubbed the cosmological constant. For decades, cosmologists assumed its value was exactly zero, given the universe’s reasonably steady rate of expansion, and they wondered why. But then, in 1998, astronomers discovered that the expansion of the cosmos is in fact gradually accelerating, implying the presence of a repulsive energy permeating space. Dubbed dark energy by the astronomers, it’s almost certainly equivalent to Einstein’s cosmological constant. Its presence causes the cosmos to expand ever more quickly, since, as it expands, new space forms, and the total amount of repulsive energy in the cosmos increases.
However, the inferred density of this vacuum energy contradicts what quantum-field theory, the language of particle physics, has to say about empty space. A quantum field is empty when there are no particle excitations rippling through it. But because of the uncertainty principle in quantum physics, the state of a quantum field is never certain, so its energy can never be exactly zero. Think of a quantum field as consisting of little springs at each point in space. The springs are always wiggling, because they’re only ever within some uncertain range of their most relaxed length. They’re always a bit too compressed or stretched, and therefore always in motion, possessing energy. This is called the zero-point energy of the field. Force fields have positive zero-point energies while matter fields have negative ones, and these energies add to and subtract from the total energy of the vacuum.
The total vacuum energy should roughly equal the largest of these contributing factors. (Say you receive a gift of $10,000; even after spending $100, or finding $3 in the couch, you’ll still have about $10,000.) Yet the observed rate of cosmic expansion indicates that its value is between 60 and 120 orders of magnitude smaller than some of the zero-point energy contributions to it, as if all the different positive and negative terms have somehow canceled out. Coming up with a physical mechanism for this equalization is extremely difficult for two main reasons.
First, the vacuum energy’s only effect is gravitational, and so dialing it down would seem to require a gravitational mechanism. But in the universe’s first few moments, when such a mechanism might have operated, the universe was so physically small that its total vacuum energy was negligible compared to the amount of matter and radiation. The gravitational effect of the vacuum energy would have been completely dwarfed by the gravity of everything else. “This is one of the greatest difficulties in solving the cosmological-constant problem,” the physicist Raphael Bousso wrote in 2007. A gravitational feedback mechanism precisely adjusting the vacuum energy amid the conditions of the early universe, he said, “can be roughly compared to an airplane following a prescribed flight path to atomic precision, in a storm.”
Compounding the difficulty, quantum-field theory calculations indicate that the vacuum energy would have shifted in value in response to phase changes in the cooling universe shortly after the Big Bang. This raises the question of whether the hypothetical mechanism that equalized the vacuum energy kicked in before or after these shifts took place. And how could the mechanism know how big their effects would be, to compensate for them?
So far, these obstacles have thwarted attempts to explain the tiny weight of empty space without resorting to a multiverse lottery. But recently, some researchers have been exploring one possible avenue: If the universe did not bang into existence, but bounced instead, following an earlier contraction phase, then the contracting universe in the distant past would have been huge and dominated by vacuum energy. Perhaps some gravitational mechanism could have acted on the plentiful vacuum energy then, diluting it in a natural way over time. This idea motivated the physicists Peter Graham, David Kaplan, and Surjeet Rajendran to discover a new cosmic bounce model, though they’ve yet to show how the vacuum dilution in the contracting universe might have worked.
In an email, Bousso called their approach “a very worthy attempt” and “an informed and honest struggle with a significant problem.” But he added that huge gaps in the model remain, and “the technical obstacles to filling in these gaps and making it work are significant. The construction is already a Rube Goldberg machine, and it will at best get even more convoluted by the time these gaps are filled.” He and other multiverse adherents see their answer as simpler by comparison.
See the full article here .
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• richardmitnick 4:49 pm on July 1, 2017 Permalink | Reply
Tags: , , Quantum field theory
From PBS: Quantum Field Theory
Quantum Field Theory
Watch for Don Lincoln of FNAL
Watch, enjoy learn.
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• richardmitnick 4:58 pm on August 1, 2016 Permalink | Reply
Tags: , , Quantum field theory
From PPPL via Princeton Journal Watch: “PPPL researchers combine quantum mechanics and Einstein’s theory of special relativity to clear up puzzles in plasma physics (Phys. Rev. A)”
Princeton University
Princeton University
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Princeton Plasma Physics Laboratory
August 1, 2016
John Greenwald
Quantum field theory
Combining physics techniques
Standard formulas give inconsistent answers
See the full article here .
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be32fd43599a845c | Causal Determinism
First published Thu Jan 23, 2003; substantive revision Thu Jan 21, 2016
1. Introduction
In most of what follows, I will speak simply of determinism, rather than of causal determinism. This follows recent philosophical practice of sharply distinguishing views and theories of what causation is from any conclusions about the success or failure of determinism (cf. Earman, 1986; an exception is Mellor 1994). For the most part this disengagement of the two concepts is appropriate. But as we will see later, the notion of cause/effect is not so easily disengaged from much of what matters to us about determinism.
Traditionally determinism has been given various, usually imprecise definitions. This is only problematic if one is investigating determinism in a specific, well-defined theoretical context; but it is important to avoid certain major errors of definition. In order to get started we can begin with a loose and (nearly) all-encompassing definition as follows:
Determinism: The world is governed by (or is under the sway of) determinism if and only if, given a specified way things are at a time t, the way things go thereafter is fixed as a matter of natural law.
The italicized phrases are elements that require further explanation and investigation, in order for us to gain a clear understanding of the concept of determinism.
The roots of the notion of determinism surely lie in a very common philosophical idea: the idea that everything can, in principle, be explained, or that everything that is, has a sufficient reason for being and being as it is, and not otherwise. In other words, the roots of determinism lie in what Leibniz named the Principle of Sufficient Reason. But since precise physical theories began to be formulated with apparently deterministic character, the notion has become separable from these roots. Philosophers of science are frequently interested in the determinism or indeterminism of various theories, without necessarily starting from a view about Leibniz' Principle.
Since the first clear articulations of the concept, there has been a tendency among philosophers to believe in the truth of some sort of determinist doctrine. There has also been a tendency, however, to confuse determinism proper with two related notions: predictability and fate.
Fatalism is the thesis that all events (or in some versions, at least some events) are destined to occur no matter what we do. The source of the guarantee that those events will happen is located in the will of the gods, or their divine foreknowledge, or some intrinsic teleological aspect of the universe, rather than in the unfolding of events under the sway of natural laws or cause-effect relations. Fatalism is therefore clearly separable from determinism, at least to the extent that one can disentangle mystical forces and gods' wills and foreknowledge (about specific matters) from the notion of natural/causal law. Not every metaphysical picture makes this disentanglement possible, of course. But as a general matter, we can imagine that certain things are fated to happen, without this being the result of deterministic natural laws alone; and we can imagine the world being governed by deterministic laws, without anything at all being fated to occur (perhaps because there are no gods, nor mystical/teleological forces deserving the titles fate or destiny, and in particular no intentional determination of the “initial conditions” of the world). In a looser sense, however, it is true that under the assumption of determinism, one might say that given the way things have gone in the past, all future events that will in fact happen are already destined to occur.
Prediction and determinism are also easy to disentangle, barring certain strong theological commitments. As the following famous expression of determinism by Laplace shows, however, the two are also easy to commingle:
In this century, Karl Popper (1982) defined determinism in terms of predictability also, in his book The Open Universe.
Laplace probably had God in mind as the powerful intelligence to whose gaze the whole future is open. If not, he should have: 19th and 20th century mathematical studies showed convincingly that neither a finite, nor an infinite but embedded-in-the-world intelligence can have the computing power necessary to predict the actual future, in any world remotely like ours. But even if our aim is only to predict a well-defined subsystem of the world, for a limited period of time, this may be impossible for any reasonable finite agent embedded in the world, as many studies of chaos (sensitive dependence on initial conditions) show. Conversely, certain parts of the world could be highly predictable, in some senses, without the world being deterministic. When it comes to predictability of future events by humans or other finite agents in the world, then, predictability and determinism are simply not logically connected at all.
The equation of “determinism”with “predictability” is therefore a façon de parler that at best makes vivid what is at stake in determinism: our fears about our own status as free agents in the world. In Laplace's story, a sufficiently bright demon who knew how things stood in the world 100 years before my birth could predict every action, every emotion, every belief in the course of my life. Were she then to watch me live through it, she might smile condescendingly, as one who watches a marionette dance to the tugs of strings that it knows nothing about. We can't stand the thought that we are (in some sense) marionettes. Nor does it matter whether any demon (or even God) can, or cares to, actually predict what we will do: the existence of the strings of physical necessity, linked to far-past states of the world and determining our current every move, is what alarms us. Whether such alarm is actually warranted is a question well outside the scope of this article (see Hoefer (2002a), Ismael (2016) and the entries on free will and incompatibilist theories of freedom). But a clear understanding of what determinism is, and how we might be able to decide its truth or falsity, is surely a useful starting point for any attempt to grapple with this issue. We return to the issue of freedom in section 6, Determinism and Human Action, below.
2. Conceptual Issues in Determinism
Recall that we loosely defined causal determinism as follows, with terms in need of clarification italicized:
2.1 The World
Why should we start so globally, speaking of the world, with all its myriad events, as deterministic? One might have thought that a focus on individual events is more appropriate: an event E is causally determined if and only if there exists a set of prior events {A, B, C …} that constitute a (jointly) sufficient cause of E. Then if all—or even just most—events E that are our human actions are causally determined, the problem that matters to us, namely the challenge to free will, is in force. Nothing so global as states of the whole world need be invoked, nor even a complete determinism that claims all events to be causally determined.
For a variety of reasons this approach is fraught with problems, and the reasons explain why philosophers of science mostly prefer to drop the word “causal” from their discussions of determinism. Generally, as John Earman quipped (1986), to go this route is to “… seek to explain a vague concept—determinism—in terms of a truly obscure one—causation.” More specifically, neither philosophers' nor laymen's conceptions of events have any correlate in any modern physical theory.[1] The same goes for the notions of cause and sufficient cause. A further problem is posed by the fact that, as is now widely recognized, a set of events {A, B, C …} can only be genuinely sufficient to produce an effect-event if the set includes an open-ended ceteris paribus clause excluding the presence of potential disruptors that could intervene to prevent E. For example, the start of a football game on TV on a normal Saturday afternoon may be sufficient ceteris paribus to launch Ted toward the fridge to grab a beer; but not if a million-ton asteroid is approaching his house at .75c from a few thousand miles away, nor if his phone is about to ring with news of a tragic nature, …, and so on. Bertrand Russell famously argued against the notion of cause along these lines (and others) in 1912, and the situation has not changed. By trying to define causal determination in terms of a set of prior sufficient conditions, we inevitably fall into the mess of an open-ended list of negative conditions required to achieve the desired sufficiency.
Moreover, thinking about how such determination relates to free action, a further problem arises. If the ceteris paribus clause is open-ended, who is to say that it should not include the negation of a potential disruptor corresponding to my freely deciding not to go get the beer? If it does, then we are left saying “When A, B, C, … Ted will then go to the fridge for a beer, unless D or E or F or … or Ted decides not to do so.” The marionette strings of a “sufficient cause” begin to look rather tenuous.
They are also too short. For the typical set of prior events that can (intuitively, plausibly) be thought to be a sufficient cause of a human action may be so close in time and space to the agent, as to not look like a threat to freedom so much as like enabling conditions. If Ted is propelled to the fridge by {seeing the game's on; desiring to repeat the satisfactory experience of other Saturdays; feeling a bit thirsty; etc}, such things look more like good reasons to have decided to get a beer, not like external physical events far beyond Ted's control. Compare this with the claim that {state of the world in 1900; laws of nature} entail Ted's going to get the beer: the difference is dramatic. So we have a number of good reasons for sticking to the formulations of determinism that arise most naturally out of physics. And this means that we are not looking at how a specific event of ordinary talk is determined by previous events; we are looking at how everything that happens is determined by what has gone before. The state of the world in 1900 only entails that Ted grabs a beer from the fridge by way of entailing the entire physical state of affairs at the later time.
2.2 The way things are at a time t
The typical explication of determinism fastens on the state of the (whole) world at a particular time (or instant), for a variety of reasons. We will briefly explain some of them. Why take the state of the whole world, rather than some (perhaps very large) region, as our starting point? One might, intuitively, think that it would be enough to give the complete state of things on Earth, say, or perhaps in the whole solar system, at t, to fix what happens thereafter (for a time at least). But notice that all sorts of influences from outside the solar system come in at the speed of light, and they may have important effects. Suppose Mary looks up at the sky on a clear night, and a particularly bright blue star catches her eye; she thinks “What a lovely star; I think I'll stay outside a bit longer and enjoy the view.” The state of the solar system one month ago did not fix that that blue light from Sirius would arrive and strike Mary's retina; it arrived into the solar system only a day ago, let's say. So evidently, for Mary's actions (and hence, all physical events generally) to be fixed by the state of things a month ago, that state will have to be fixed over a much larger spatial region than just the solar system. (If no physical influences can go faster than light, then the state of things must be given over a spherical volume of space 1 light-month in radius.)
But in making vivid the “threat” of determinism, we often want to fasten on the idea of the entire future of the world as being determined. No matter what the “speed limit” on physical influences is, if we want the entire future of the world to be determined, then we will have to fix the state of things over all of space, so as not to miss out something that could later come in “from outside” to spoil things. In the time of Laplace, of course, there was no known speed limit to the propagation of physical things such as light-rays. In principle light could travel at any arbitrarily high speed, and some thinkers did suppose that it was transmitted “instantaneously.” The same went for the force of gravity. In such a world, evidently, one has to fix the state of things over the whole of the world at a time t, in order for events to be strictly determined, by the laws of nature, for any amount of time thereafter.
In all this, we have been presupposing the common-sense Newtonian framework of space and time, in which the world-at-a-time is an objective and meaningful notion. Below when we discuss determinism in relativistic theories we will revisit this assumption.
2.3 Thereafter
For a wide class of physical theories (i.e., proposed sets of laws of nature), if they can be viewed as deterministic at all, they can be viewed as bi-directionally deterministic. That is, a specification of the state of the world at a time t, along with the laws, determines not only how things go after t, but also how things go before t. Philosophers, while not exactly unaware of this symmetry, tend to ignore it when thinking of the bearing of determinism on the free will issue. The reason for this is that we tend to think of the past (and hence, states of the world in the past) as done, over, fixed and beyond our control. Forward-looking determinism then entails that these past states—beyond our control, perhaps occurring long before humans even existed—determine everything we do in our lives. It then seems a mere curious fact that it is equally true that the state of the world now determines everything that happened in the past. We have an ingrained habit of taking the direction of both causation and explanation as being past—-present, even when discussing physical theories free of any such asymmetry. We will return to this point shortly.
Another point to notice here is that the notion of things being determined thereafter is usually taken in an unlimited sense—i.e., determination of all future events, no matter how remote in time. But conceptually speaking, the world could be only imperfectly deterministic: things could be determined only, say, for a thousand years or so from any given starting state of the world. For example, suppose that near-perfect determinism were regularly (but infrequently) interrupted by spontaneous particle creation events, which occur only once every thousand years in a thousand-light-year-radius volume of space. This unrealistic example shows how determinism could be strictly false, and yet the world be deterministic enough for our concerns about free action to be unchanged.
2.4 Laws of nature
In the loose statement of determinism we are working from, metaphors such as “govern” and “under the sway of” are used to indicate the strong force being attributed to the laws of nature. Part of understanding determinism—and especially, whether and why it is metaphysically important—is getting clear about the status of the presumed laws of nature.
In the physical sciences, the assumption that there are fundamental, exceptionless laws of nature, and that they have some strong sort of modal force, usually goes unquestioned. Indeed, talk of laws “governing” and so on is so commonplace that it takes an effort of will to see it as metaphorical. We can characterize the usual assumptions about laws in this way: the laws of nature are assumed to be pushy explainers. They make things happen in certain ways , and by having this power, their existence lets us explain why things happen in certain ways. (For a defense of this perspective on laws, see Maudlin (2007)). Laws, we might say, are implicitly thought of as the cause of everything that happens. If the laws governing our world are deterministic, then in principle everything that happens can be explained as following from states of the world at earlier times. (Again, we note that even though the entailment typically works in the future→past direction also, we have trouble thinking of this as a legitimate explanatory entailment. In this respect also, we see that laws of nature are being implicitly treated as the causes of what happens: causation, intuitively, can only go past→future.)
Interestingly, philosophers tend to acknowledge the apparent threat determinism poses to free will, even when they explicitly reject the view that laws are pushy explainers. Earman (1986), for example, advocates a theory of laws of nature that takes them to be simply the best system of regularities that systematizes all the events in universal history. This is the Best Systems Analysis (BSA), with roots in the work of Hume, Mill and Ramsey, and most recently refined and defended by David Lewis (1973, 1994) and by Earman (1984, 1986). (cf. entry on laws of nature). Yet he ends his comprehensive Primer on Determinism with a discussion of the free will problem, taking it as a still-important and unresolved issue. Prima facie this is quite puzzling, for the BSA is founded on the idea that the laws of nature are ontologically derivative, not primary; it is the events of universal history, as brute facts, that make the laws be what they are, and not vice-versa. Taking this idea seriously, the actions of every human agent in history are simply a part of the universe-wide pattern of events that determines what the laws are for this world. It is then hard to see how the most elegant summary of this pattern, the BSA laws, can be thought of as determiners of human actions. The determination or constraint relations, it would seem, can go one way or the other, not both.
On second thought however it is not so surprising that broadly Humean philosophers such as Ayer, Earman, Lewis and others still see a potential problem for freedom posed by determinism. For even if human actions are part of what makes the laws be what they are, this does not mean that we automatically have freedom of the kind we think we have, particularly freedom to have done otherwise given certain past states of affairs. It is one thing to say that everything occurring in and around my body, and everything everywhere else, conforms to Maxwell's equations and thus the Maxwell equations are genuine exceptionless regularities, and that because they in addition are simple and strong, they turn out to be laws. It is quite another thing to add: thus, I might have chosen to do otherwise at certain points in my life, and if I had, then Maxwell's equations would not have been laws. One might try to defend this claim—unpalatable as it seems intuitively, to ascribe ourselves law-breaking power—but it does not follow directly from a Humean approach to laws of nature. Instead, on such views that deny laws most of their pushiness and explanatory force, questions about determinism and human freedom simply need to be approached afresh.
A second important genre of theories of laws of nature holds that the laws are in some sense necessary. For any such approach, laws are just the sort of pushy explainers that are assumed in the traditional language of physical scientists and free will theorists. But a third and growing class of philosophers holds that (universal, exceptionless, true) laws of nature simply do not exist. Among those who hold this are influential philosophers such as Nancy Cartwright, Bas van Fraassen, and John Dupré. For these philosophers, there is a simple consequence: determinism is a false doctrine. As with the Humean view, this does not mean that concerns about human free action are automatically resolved; instead, they must be addressed afresh in the light of whatever account of physical nature without laws is put forward. See Dupré (2001) for one such discussion.
2.5 Fixed
We can now put our—still vague—pieces together. Determinism requires a world that (a) has a well-defined state or description, at any given time, and (b) laws of nature that are true at all places and times. If we have all these, then if (a) and (b) together logically entail the state of the world at all other times (or, at least, all times later than that given in (a)), the world is deterministic. Logical entailment, in a sense broad enough to encompass mathematical consequence, is the modality behind the determination in “determinism.”
3. The Epistemology of Determinism
How could we ever decide whether our world is deterministic or not? Given that some philosophers and some physicists have held firm views—with many prominent examples on each side—one would think that it should be at least a clearly decidable question. Unfortunately, even this much is not clear, and the epistemology of determinism turns out to be a thorny and multi-faceted issue.
3.1 Laws again
As we saw above, for determinism to be true there have to be some laws of nature. Most philosophers and scientists since the 17th century have indeed thought that there are. But in the face of more recent skepticism, how can it be proven that there are? And if this hurdle can be overcome, don't we have to know, with certainty, precisely what the laws of our world are, in order to tackle the question of determinism's truth or falsity?
The first hurdle can perhaps be overcome by a combination of metaphysical argument and appeal to knowledge we already have of the physical world. Philosophers are currently pursuing this issue actively, in large part due to the efforts of the anti-laws minority. The debate has been most recently framed by Cartwright in The Dappled World (Cartwright 1999) in terms psychologically advantageous to her anti-laws cause. Those who believe in the existence of traditional, universal laws of nature are fundamentalists; those who disbelieve are pluralists. This terminology seems to be becoming standard (see Belot 2001), so the first task in the epistemology of determinism is for fundamentalists to establish the reality of laws of nature (see Hoefer 2002b).
Even if the first hurdle can be overcome, the second, namely establishing precisely what the actual laws are, may seem daunting indeed. In a sense, what we are asking for is precisely what 19th and 20th century physicists sometimes set as their goal: the Final Theory of Everything. But perhaps, as Newton said of establishing the solar system's absolute motion, “the thing is not altogether desperate.” Many physicists in the past 60 years or so have been convinced of determinism's falsity, because they were convinced that (a) whatever the Final Theory is, it will be some recognizable variant of the family of quantum mechanical theories; and (b) all quantum mechanical theories are non-deterministic. Both (a) and (b) are highly debatable, but the point is that one can see how arguments in favor of these positions might be mounted. The same was true in the 19th century, when theorists might have argued that (a) whatever the Final Theory is, it will involve only continuous fluids and solids governed by partial differential equations; and (b) all such theories are deterministic. (Here, (b) is almost certainly false; see Earman (1986),ch. XI). Even if we now are not, we may in future be in a position to mount a credible argument for or against determinism on the grounds of features we think we know the Final Theory must have.
3.2 Experience
Determinism could perhaps also receive direct support—confirmation in the sense of probability-raising, not proof—from experience and experiment. For theories (i.e., potential laws of nature) of the sort we are used to in physics, it is typically the case that if they are deterministic, then to the extent that one can perfectly isolate a system and repeatedly impose identical starting conditions, the subsequent behavior of the systems should also be identical. And in broad terms, this is the case in many domains we are familiar with. Your computer starts up every time you turn it on, and (if you have not changed any files, have no anti-virus software, re-set the date to the same time before shutting down, and so on …) always in exactly the same way, with the same speed and resulting state (until the hard drive fails). The light comes on exactly 32 µsec after the switch closes (until the day the bulb fails). These cases of repeated, reliable behavior obviously require some serious ceteris paribus clauses, are never perfectly identical, and always subject to catastrophic failure at some point. But we tend to think that for the small deviations, probably there are explanations for them in terms of different starting conditions or failed isolation, and for the catastrophic failures, definitely there are explanations in terms of different conditions.
There have even been studies of paradigmatically “chancy” phenomena such as coin-flipping, which show that if starting conditions can be precisely controlled and outside interferences excluded, identical behavior results (see Diaconis, Holmes & Montgomery 2004). Most of these bits of evidence for determinism no longer seem to cut much ice, however, because of faith in quantum mechanics and its indeterminism. Indeterminist physicists and philosophers are ready to acknowledge that macroscopic repeatability is usually obtainable, where phenomena are so large-scale that quantum stochasticity gets washed out. But they would maintain that this repeatability is not to be found in experiments at the microscopic level, and also that at least some failures of repeatability (in your hard drive, or coin-flipping experiments) are genuinely due to quantum indeterminism, not just failures to isolate properly or establish identical initial conditions.
If quantum theories were unquestionably indeterministic, and deterministic theories guaranteed repeatability of a strong form, there could conceivably be further experimental input on the question of determinism's truth or falsity. Unfortunately, the existence of Bohmian quantum theories casts strong doubt on the former point, while chaos theory casts strong doubt on the latter. More will be said about each of these complications below.
3.3 Determinism and Chaos
If the world were governed by strictly deterministic laws, might it still look as though indeterminism reigns? This is one of the difficult questions that chaos theory raises for the epistemology of determinism.
A deterministic chaotic system has, roughly speaking, two salient features: (i) the evolution of the system over a long time period effectively mimics a random or stochastic process—it lacks predictability or computability in some appropriate sense; (ii) two systems with nearly identical initial states will have radically divergent future developments, within a finite (and typically, short) timespan. We will use “randomness” to denote the first feature, and “sensitive dependence on initial conditions” (SDIC) for the latter. Definitions of chaos may focus on either or both of these properties; Batterman (1993) argues that only (ii) provides an appropriate basis for defining chaotic systems.
A simple and very important example of a chaotic system in both randomness and SDIC terms is the Newtonian dynamics of a pool table with a convex obstacle (or obstacles) (Sinai 1970 and others). See Figure 1.
Billiard table with convex obstacle
Figure 1: Billiard table with convex obstacle
The usual idealizing assumptions are made: no friction, perfectly elastic collisions, no outside influences. The ball's trajectory is determined by its initial position and direction of motion. If we imagine a slightly different initial direction, the trajectory will at first be only slightly different. And collisions with the straight walls will not tend to increase very rapidly the difference between trajectories. But collisions with the convex object will have the effect of amplifying the differences. After several collisions with the convex body or bodies, trajectories that started out very close to one another will have become wildly different—SDIC.
In the example of the billiard table, we know that we are starting out with a Newtonian deterministic system—that is how the idealized example is defined. But chaotic dynamical systems come in a great variety of types: discrete and continuous, 2-dimensional, 3-dimensional and higher, particle-based and fluid-flow-based, and so on. Mathematically, we may suppose all of these systems share SDIC. But generally they will also display properties such as unpredictability, non-computability, Kolmogorov-random behavior, and so on—at least when looked at in the right way, or at the right level of detail. This leads to the following epistemic difficulty: if, in nature, we find a type of system that displays some or all of these latter properties, how can we decide which of the following two hypotheses is true?
1. The system is governed by genuinely stochastic, indeterministic laws (or by no laws at all), i.e., its apparent randomness is in fact real randomness.
2. The system is governed by underlying deterministic laws, but is chaotic.
In other words, once one appreciates the varieties of chaotic dynamical systems that exist, mathematically speaking, it starts to look difficult—maybe impossible—for us to ever decide whether apparently random behavior in nature arises from genuine stochasticity, or rather from deterministic chaos. Patrick Suppes (1993, 1996) argues, on the basis of theorems proven by Ornstein (1974 and later) that “There are processes which can equally well be analyzed as deterministic systems of classical mechanics or as indeterministic semi-Markov processes, no matter how many observations are made.” And he concludes that “Deterministic metaphysicians can comfortably hold to their view knowing they cannot be empirically refuted, but so can indeterministic ones as well.” (Suppes 1993, p. 254) For more recent works exploring the extent to which deterministic and indeterministic model systems may be regarded as empirically indistinguishable, see Werndl (2016) and references therein.
There is certainly an interesting problem area here for the epistemology of determinism, but it must be handled with care. It may well be true that there are some deterministic dynamical systems that, when viewed properly, display behavior indistinguishable from that of a genuinely stochastic process. For example, using the billiard table above, if one divides its surface into quadrants and looks at which quadrant the ball is in at 30-second intervals, the resulting sequence is no doubt highly random. But this does not mean that the same system, when viewed in a different way (perhaps at a higher degree of precision) does not cease to look random and instead betray its deterministic nature. If we partition our billiard table into squares 2 centimeters a side and look at which quadrant the ball is in at .1 second intervals, the resulting sequence will be far from random. And finally, of course, if we simply look at the billiard table with our eyes, and see it as a billiard table, there is no obvious way at all to maintain that it may be a truly random process rather than a deterministic dynamical system. (See Winnie (1996) for a nice technical and philosophical discussion of these issues. Winnie explicates Ornstein's and others' results in some detail, and disputes Suppes' philosophical conclusions.)
The dynamical systems usually studied under the label of “chaos” are usually either purely abstract, mathematical systems, or classical Newtonian systems. It is natural to wonder whether chaotic behavior carries over into the realm of systems governed by quantum mechanics as well. Interestingly, it is much harder to find natural correlates of classical chaotic behavior in true quantum systems (see Gutzwiller 1990). Some, at least, of the interpretive difficulties of quantum mechanics would have to be resolved before a meaningful assessment of chaos in quantum mechanics could be achieved. For example, SDIC is hard to find in the Schrödinger evolution of a wavefunction for a system with finite degrees of freedom; but in Bohmian quantum mechanics it is handled quite easily on the basis of particle trajectories (see Dürr, Goldstein and Zhangì 1992).
The popularization of chaos theory in the relatively recent past perhaps made it seem self-evident that nature is full of genuinely chaotic systems. In fact, it is far from self-evident that such systems exist, other than in an approximate sense. Nevertheless, the mathematical exploration of chaos in dynamical systems helps us to understand some of the pitfalls that may attend our efforts to know whether our world is genuinely deterministic or not.
3.4 Metaphysical arguments
Let us suppose that we shall never have the Final Theory of Everything before us—at least in our lifetime—and that we also remain unclear (on physical/experimental grounds) as to whether that Final Theory will be of a type that can or cannot be deterministic. Is there nothing left that could sway our belief toward or against determinism? There is, of course: metaphysical argument. Metaphysical arguments on this issue are not currently very popular. But philosophical fashions change at least twice a century, and grand systemic metaphysics of the Leibnizian sort might one day come back into favor. Conversely, the anti-systemic, anti-fundamentalist metaphysics propounded by Cartwright (1999) might also come to predominate. As likely as not, for the foreseeable future metaphysical argument may be just as good a basis on which to discuss determinism's prospects as any arguments from mathematics or physics.
4. The Status of Determinism in Physical Theories
John Earman's Primer on Determinism (1986) remains the richest storehouse of information on the truth or falsity of determinism in various physical theories, from classical mechanics to quantum mechanics and general relativity. (See also his recent update on the subject, “Aspects of Determinism in Modern Physics” (2007)). Here I will give only a brief discussion of some key issues, referring the reader to Earman (1986) and other resources for more detail. Figuring out whether well-established theories are deterministic or not (or to what extent, if they fall only a bit short) does not do much to help us know whether our world is really governed by deterministic laws; all our current best theories, including General Relativity and the Standard Model of particle physics, are too flawed and ill-understood to be mistaken for anything close to a Final Theory. Nevertheless, as Earman stressed, the exploration is very valuable because of the way it enriches our understanding of the richness and complexity of determinism.
4.1 Classical mechanics
Despite the common belief that classical mechanics (the theory that inspired Laplace in his articulation of determinism) is perfectly deterministic, in fact the theory is rife with possibilities for determinism to break down. One class of problems arises due to the absence of an upper bound on the velocities of moving objects. Below we see the trajectory of an object that is accelerated unboundedly, its velocity becoming in effect infinite in a finite time. See Figure 2:
object accelerates to reach infinity
Figure 2: An object accelerates so as to reach spatial infinity in a finite time
By the time t = t*, the object has literally disappeared from the world—its world-line never reaches the t = t* surface. (Never mind how the object gets accelerated in this way; there are mechanisms that are perfectly consistent with classical mechanics that can do the job. In fact, Xia (1992) showed that such acceleration can be accomplished by gravitational forces from only 5 finite objects, without collisions. No mechanism is shown in these diagrams.) This “escape to infinity,” while disturbing, does not yet look like a violation of determinism. But now recall that classical mechanics is time-symmetric: any model has a time-inverse, which is also a consistent model of the theory. The time-inverse of our escaping body is playfully called a “space invader.”
space invader comes from infinity
Figure 3: A ‘space invader’ comes in from spatial infinity
Clearly, a world with a space invader does fail to be deterministic. Before t = t*, there was nothing in the state of things to enable the prediction of the appearance of the invader at t = t* +.[2] One might think that the infinity of space is to blame for this strange behavior, but this is not obviously correct. In finite, “rolled-up” or cylindrical versions of Newtonian space-time space-invader trajectories can be constructed, though whether a “reasonable” mechanism to power them exists is not clear.[3]
A second class of determinism-breaking models can be constructed on the basis of collision phenomena. The first problem is that of multiple-particle collisions for which Newtonian particle mechanics simply does not have a prescription for what happens. (Consider three identical point-particles approaching each other at 120 degree angles and colliding simultaneously. That they bounce back along their approach trajectories is possible; but it is equally possible for them to bounce in other directions (again with 120 degree angles between their paths), so long as momentum conservation is respected.)
Moreover, there is a burgeoning literature of physical or quasi-physical systems, usually set in the context of classical physics, that carry out supertasks (see Earman and Norton (1998) and the entry on supertasks for a review). Frequently, the puzzle presented is to decide, on the basis of the well-defined behavior before time t = a, what state the system will be in at t = a itself. A failure of CM to dictate a well-defined result can then be seen as a failure of determinism.
In supertasks, one frequently encounters infinite numbers of particles, infinite (or unbounded) mass densities, and other dubious infinitary phenomena. Coupled with some of the other breakdowns of determinism in CM, one begins to get a sense that most, if not all, breakdowns of determinism rely on some combination of the following set of (physically) dubious mathematical notions: {infinite space; unbounded velocity; continuity; point-particles; singular fields}. The trouble is, it is difficult to imagine any recognizable physics (much less CM) that eschews everything in the set.
Norton's dome
Figure 4: A ball may spontaneously start sliding down this dome, with no violation of Newton's laws. (Reproduced courtesy of John D. Norton and Philosopher's Imprint)
Finally, an elegant example of apparent violation of determinism in classical physics has been created by John Norton (2003). As illustrated in Figure 4, imagine a ball sitting at the apex of a frictionless dome whose equation is specified as a function of radial distance from the apex point. This rest-state is our initial condition for the system; what should its future behavior be? Clearly one solution is for the ball to remain at rest at the apex indefinitely.
But curiously, this is not the only solution under standard Newtonian laws. The ball may also start into motion sliding down the dome—at any moment in time, and in any radial direction. This example displays “uncaused motion” without, Norton argues, any violation of Newton's laws, including the First Law. And it does not, unlike some supertask examples, require an infinity of particles. Still, many philosophers are uncomfortable with the moral Norton draws from his dome example, and point out reasons for questioning the dome's status as a Newtonian system (see e.g. Malament (2007)).
4.2 Special Relativistic physics
Two features of special relativistic physics make it perhaps the most hospitable environment for determinism of any major theoretical context: the fact that no process or signal can travel faster than the speed of light, and the static, unchanging spacetime structure. The former feature, including a prohibition against tachyons (hypothetical particles travelling faster than light)[4]), rules out space invaders and other unbounded-velocity systems. The latter feature makes the space-time itself nice and stable and non-singular—unlike the dynamic space-time of General Relativity, as we shall see below. For source-free electromagnetic fields in special-relativistic space-time, a nice form of Laplacean determinism is provable. Unfortunately, interesting physics needs more than source-free electromagnetic fields. Earman (1986) ch. IV surveys in depth the pitfalls for determinism that arise once things are allowed to get more interesting (e.g. by the addition of particles interacting gravitationally).
4.3 General Relativity (GTR)
Defining an appropriate form of determinism for the context of general relativistic physics is extremely difficult, due to both foundational interpretive issues and the plethora of weirdly-shaped space-time models allowed by the theory's field equations. The simplest way of treating the issue of determinism in GTR would be to state flatly: determinism fails, frequently, and in some of the most interesting models. Here we will briefly describe some of the most important challenges that arise for determinism, directing the reader yet again to Earman (1986), and also Earman (1995) for more depth.
4.3.1 Determinism and manifold points
In GTR, we specify a model of the universe by giving a triple of three mathematical objects, <M, g,T>. M represents a continuous “manifold”: that means a sort of unstructured space (-time), made up of individual points and having smoothness or continuity, dimensionality (usually, 4-dimensional), and global topology, but no further structure. What is the further structure a space-time needs? Typically, at least, we expect the time-direction to be distinguished from space-directions; and we expect there to be well-defined distances between distinct points; and also a determinate geometry (making certain continuous paths in M be straight lines, etc.). All of this extra structure is coded into g, the metric field. So M and g together represent space-time. T represents the matter and energy content distributed around in space-time (if any, of course).
For mathematical reasons not relevant here, it turns out to be possible to take a given model spacetime and perform a mathematical operation called a “hole diffeomorphism” h* on it; the diffeomorphism's effect is to shift around the matter content T and the metric g relative to the continuous manifold M.[5] If the diffeomorphism is chosen appropriately, it can move around T and g after a certain time t = 0, but leave everything alone before that time. Thus, the new model represents the matter content (now h* T) and the metric (h*g) as differently located relative to the points of M making up space-time. Yet, the new model is also a perfectly valid model of the theory. This looks on the face of it like a form of indeterminism: GTR's equations do not specify how things will be distributed in space-time in the future, even when the past before a given time t is held fixed. See Figure 5:
Holediffeomorphismshifts contents of spacetime
Figure 5: “Hole” diffeomorphism shifts contents of spacetime
Usually the shift is confined to a finite region called the hole (for historical reasons). Then it is easy to see that the state of the world at time t = 0 (and all the history that came before) does not suffice to fix whether the future will be that of our first model, or its shifted counterpart in which events inside the hole are different.
This is a form of indeterminism first highlighted by Earman and Norton (1987) as an interpretive philosophical difficulty for realism about GTR's description of the world, especially the point manifold M. They showed that realism about the manifold as a part of the furniture of the universe (which they called “manifold substantivalism”) commits us to an automatic indeterminism in GTR (as described above), and they argued that this is unacceptable. (See the hole argument and Hoefer (1996) for one response on behalf of the space-time realist, and discussion of other responses.) For now, we will simply note that this indeterminism, unlike most others we are discussing in this section, is empirically undetectable: our two models <M, g, T> and the shifted model <M, h*g, h*T> are empirically indistinguishable.
4.3.2 Singularities
The separation of space-time structures into manifold and metric (or connection) facilitates mathematical clarity in many ways, but also opens up Pandora's box when it comes to determinism. The indeterminism of the Earman and Norton hole argument is only the tip of the iceberg; singularities make up much of the rest of the berg. In general terms, a singularity can be thought of as a “place where things go bad” in one way or another in the space-time model. For example, near the center of a Schwarzschild black hole, curvature increases without bound, and at the center itself it is undefined, which means that Einstein's equations cannot be said to hold, which means (arguably) that this point does not exist as a part of the space-time at all! Some specific examples are clear, but giving a general definition of a singularity, like defining determinism itself in GTR, is a vexed issue (see Earman (1995) for an extended treatment; Callender and Hoefer (2001) gives a brief overview). We will not attempt here to catalog the various definitions and types of singularity.
Different types of singularity bring different types of threat to determinism. In the case of ordinary black holes, mentioned above, all is well outside the so- called “event horizon”, which is the spherical surface defining the black hole: once a body or light signal passes through the event horizon to the interior region of the black hole, it can never escape again. Generally, no violation of determinism looms outside the event horizon; but what about inside? Some black hole models have so-called “Cauchy horizons” inside the event horizon, i.e., surfaces beyond which determinism breaks down.
Another way for a model spacetime to be singular is to have points or regions go missing, in some cases by simple excision. Perhaps the most dramatic form of this involves taking a nice model with a space-like surface t = E (i.e., a well-defined part of the space-time that can be considered “the state state of the world at time E”), and cutting out and throwing away this surface and all points temporally later. The resulting spacetime satisfies Einstein's equations; but, unfortunately for any inhabitants, the universe comes to a sudden and unpredictable end at time E. This is too trivial a move to be considered a real threat to determinism in GTR; we can impose a reasonable requirement that space-time not “run out” in this way without some physical reason (the spacetime should be “maximally extended”). For discussion of precise versions of such a requirement, and whether they succeed in eliminating unwanted singularities, see Earman (1995, chapter 2).
The most problematic kinds of singularities, in terms of determinism, are naked singularities (singularities not hidden behind an event horizon). When a singularity forms from gravitational collapse, the usual model of such a process involves the formation of an event horizon (i.e. a black hole). A universe with an ordinary black hole has a singularity, but as noted above, (outside the event horizon at least) nothing unpredictable happens as a result. A naked singularity, by contrast, has no such protective barrier. In much the way that anything can disappear by falling into an excised-region singularity, or appear out of a white hole (white holes themselves are, in fact, technically naked singularities), there is the worry that anything at all could pop out of a naked singularity, without warning (hence, violating determinism en passant). While most white hole models have Cauchy surfaces and are thus arguably deterministic, other naked singularity models lack this property. Physicists disturbed by the unpredictable potentialities of such singularities have worked to try to prove various cosmic censorship hypotheses that show—under (hopefully) plausible physical assumptions—that such things do not arise by stellar collapse in GTR (and hence are not liable to come into existence in our world). To date no very general and convincing forms of the hypothesis have been proven, so the prospects for determinism in GTR as a mathematical theory do not look terribly good.
4.4 Quantum mechanics
As indicated above, QM is widely thought to be a strongly non-deterministic theory. Popular belief (even among most physicists) holds that phenomena such as radioactive decay, photon emission and absorption, and many others are such that only a probabilistic description of them can be given. The theory does not say what happens in a given case, but only says what the probabilities of various results are. So, for example, according to QM the fullest description possible of a radium atom (or a chunk of radium, for that matter), does not suffice to determine when a given atom will decay, nor how many atoms in the chunk will have decayed at any given time. The theory gives only the probabilities for a decay (or a number of decays) to happen within a given span of time. Einstein and others perhaps thought that this was a defect of the theory that should eventually be removed, by a supplemental hidden variable theory[6] that restores determinism; but subsequent work showed that no such hidden variables account could exist. At the microscopic level the world is ultimately mysterious and chancy.
So goes the story; but like much popular wisdom, it is partly mistaken and/or misleading. Ironically, quantum mechanics is one of the best prospects for a genuinely deterministic theory in modern times! Everything hinges on what interpretational and philosophical decisions one adopts. The fundamental law at the heart of non-relativistic QM is the Schrödinger equation. The evolution of a wavefunction describing a physical system under this equation is normally taken to be perfectly deterministic.[7] If one adopts an interpretation of QM according to which that's it—i.e., nothing ever interrupts Schrödinger evolution, and the wavefunctions governed by the equation tell the complete physical story—then quantum mechanics is a perfectly deterministic theory. There are several interpretations that physicists and philosophers have given of QM which go this way. (See the entry on quantum mechanics.)
More commonly—and this is part of the basis for the popular wisdom—physicists have resolved the quantum measurement problem by postulating that some process of “collapse of the wavefunction” occurs during measurements or observations that interrupts Schrödinger evolution. The collapse process is usually postulated to be indeterministic, with probabilities for various outcomes, via Born's rule, calculable on the basis of a system's wavefunction. The once-standard Copenhagen interpretation of QM posits such a collapse. It has the virtue of solving certain problems such as the infamous Schrödinger's cat paradox, but few philosophers or physicists can take it very seriously unless they are instrumentalists about the theory. The reason is simple: the collapse process is not physically well-defined, is characterised in terms of an anthropomorphic notion (measurement)and feels too ad hoc to be a fundamental part of nature's laws.[8]
In 1952 David Bohm created an alternative interpretation of non relativistic QM—perhaps better thought of as an alternative theory—that realizes Einstein's dream of a hidden variable theory, restoring determinism and definiteness to micro-reality. In Bohmian quantum mechanics, unlike other interpretations, it is postulated that all particles have, at all times, a definite position and velocity. In addition to the Schrödinger equation, Bohm posited a guidance equation that determines, on the basis of the system's wavefunction and particles' initial positions and velocities, what their future positions and velocities should be. As much as any classical theory of point particles moving under force fields, then, Bohm's theory is deterministic. Amazingly, he was also able to show that, as long as the statistical distribution of initial positions and velocities of particles are chosen so as to meet a “quantum equilibrium” condition, his theory is empirically equivalent to standard Copenhagen QM. In one sense this is a philosopher's nightmare: with genuine empirical equivalence as strong as Bohm obtained, it seems experimental evidence can never tell us which description of reality is correct. (Fortunately, we can safely assume that neither is perfectly correct, and hope that our Final Theory has no such empirically equivalent rivals.) In other senses, the Bohm theory is a philosopher's dream come true, eliminating much (but not all) of the weirdness of standard QM and restoring determinism to the physics of atoms and photons. The interested reader can find out more from the link above, and references therein.
This small survey of determinism's status in some prominent physical theories, as indicated above, does not really tell us anything about whether determinism is true of our world. Instead, it raises a couple of further disturbing possibilities for the time when we do have the Final Theory before us (if such time ever comes): first, we may have difficulty establishing whether the Final Theory is deterministic or not—depending on whether the theory comes loaded with unsolved interpretational or mathematical puzzles. Second, we may have reason to worry that the Final Theory, if indeterministic, has an empirically equivalent yet deterministic rival (as illustrated by Bohmian quantum mechanics.)
5. Chance and Determinism
Some philosophers maintain that if determinism holds in our world, then there are no objective chances in our world. And often the word ‘chance’ here is taken to be synonymous with 'probability', so these philosophers maintain that there are no non-trivial objective probabilities for events in our world. (The caveat “non-trivial” is added here because on some accounts, under determinism, all future events that actually happen have probability, conditional on past history, equal to 1, and future events that do not happen have probability equal to zero. Non-trivial probabilities are probabilities strictly between zero and one.) Conversely, it is often held, if there are laws of nature that are irreducibly probabilistic, determinism must be false. (Some philosophers would go on to add that such irreducibly probabilistic laws are the basis of whatever genuine objective chances obtain in our world.)
The discussion of quantum mechanics in section 4 shows that it may be difficult to know whether a physical theory postulates genuinely irreducible probabilistic laws or not. If a Bohmian version of QM is correct, then the probabilities dictated by the Born rule are not irreducible. If that is the case, should we say that the probabilities dictated by quantum mechanics are not objective? Or should we say that we need to distinguish ‘chance’ and ‘probabillity’ after all—and hold that not all objective probabilities should be thought of as objective chances? The first option may seem hard to swallow, given the many-decimal-place accuracy with which such probability-based quantities as half-lives and cross-sections can be reliably predicted and verified experimentally with QM.
Whether objective chance and determinism are really incompatible or not may depend on what view of the nature of laws is adopted. On a “pushy explainers” view of laws such as that defended by Maudlin (2007), probabilistic laws are interpreted as irreducible dynamical transition-chances between allowed physical states, and the incompatibility of such laws with determinism is immediate. But what should a defender of a Humean view of laws, such as the BSA theory (section 2.4 above), say about probabilistic laws? The first thing that needs to be done is explain how probabilistic laws can fit into the BSA account at all, and this requires modification or expansion of the view, since as first presented the only candidates for laws of nature are true universal generalizations. If ‘probability’ were a univocal, clearly understood notion then this might be simple: We allow universal generalizations whose logical form is something like: “Whenever conditions Y obtain, Pr(A) = x”. But it is not at all clear how the meaning of ‘Pr’ should be understood in such a generalization; and it is even less clear what features the Humean pattern of actual events must have, for such a generalization to be held true. (See the entry on interpretations of probability and Lewis (1994).)
Humeans about laws believe that what laws there are is a matter of what patterns are there to be discerned in the overall mosaic of events that happen in the history of the world. It seems plausible enough that the patterns to be discerned may include not only strict associations (whenever X, Y), but also stable statistical associations. If the laws of nature can include either sort of association, a natural question to ask seems to be: why can't there be non-probabilistic laws strong enough to ensure determinism, and on top of them, probabilistic laws as well? If a Humean wanted to capture the laws not only of fundamental theories, but also non-fundamental branches of physics such as (classical) statistical mechanics, such a peaceful coexistence of deterministic laws plus further probabilistic laws would seem to be desirable. Loewer (2004) and Frigg & Hoefer (2015) offer forms of this peaceful coexistence that can be achieved within Lewis' version of the BSA account of laws.
6. Determinism and Human Action
In the introduction, we noted the threat that determinism seems to pose to human free agency. It is hard to see how, if the state of the world 1000 years ago fixes everything I do during my life, I can meaningfully say that I am a free agent, the author of my own actions, which I could have freely chosen to perform differently. After all, I have neither the power to change the laws of nature, nor to change the past! So in what sense can I attribute freedom of choice to myself?
Philosophers have not lacked ingenuity in devising answers to this question. There is a long tradition of compatibilists arguing that freedom is fully compatible with physical determinism; a prominent recent defender is John Fischer (1994, 2012). Hume went so far as to argue that determinism is a necessary condition for freedom—or at least, he argued that some causality principle along the lines of “same cause, same effect” is required. There have been equally numerous and vigorous responses by those who are not convinced. Can a clear understanding of what determinism is, and how it tends to succeed or fail in real physical theories, shed any light on the controversy?
Physics, particularly 20th century physics, does have one lesson to impart to the free will debate; a lesson about the relationship between time and determinism. Recall that we noticed that the fundamental theories we are familiar with, if they are deterministic at all, are time-symmetrically deterministic. That is, earlier states of the world can be seen as fixing all later states; but equally, later states can be seen as fixing all earlier states. We tend to focus only on the former relationship, but we are not led to do so by the theories themselves.
Nor does 20th (21st) -century physics countenance the idea that there is anything ontologically special about the past, as opposed to the present and the future. In fact, it fails to use these categories in any respect, and teaches that in some senses they are probably illusory.[9] So there is no support in physics for the idea that the past is “fixed” in some way that the present and future are not, or that it has some ontological power to constrain our actions that the present and future do not have. It is not hard to uncover the reasons why we naturally do tend to think of the past as special, and assume that both physical causation and physical explanation work only in the past present/future direction (see the entry on thermodynamic asymmetry in time). But these pragmatic matters have nothing to do with fundamental determinism. If we shake loose from the tendency to see the past as special, when it comes to the relationships of determination, it may prove possible to think of a deterministic world as one in which each part bears a determining—or partial-determining—relation to other parts, but in which no particular part (region of space-time, event or set of events, ...) has a special, privileged determining role that undercuts the others. Hoefer (2002a) and Ismael (2016) use such considerations to argue in a novel way for the compatiblity of determinism with human free agency.
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• van Fraassen, B., 1989, Laws and Symmetry, Oxford: Clarendon Press.
• Van Kampen, N. G., 1991, “Determinism and Predictability,” Synthese, 89: 273–281.
• Werndl, C. 2016, The Oxford Handbook of Philosophy of Science. Oxford: Oxford University Press. Online at, December 2015.
• Winnie, J. A., 1996, “Deterministic Chaos and the Nature of Chance,” in The Cosmos of Science—Essays of Exploration, J. Earman and J. Norton (eds.), Pittsburgh: University of Pitsburgh Press, pp. 299–324.
• Xia, Z., 1992, “The existence of noncollision singularities in newtonian systems,” Annals of Mathematics, 135: 411–468.
The author would like to acknowledge the invaluable help of John Norton in the preparation of this entry. Thanks also to A. Ilhamy Amiry for bringing to my attention some errors in an earlier version of this entry.
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New scientific highlights- by MUST PIs Chergui and Richardson
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Ursula Keller
January 2013
Prof. Ursula Keller was awarded a 2012 ERC Advanced Grant for the Attoclock project: Clocking fundamental attosecond electron dynamics.
The attoclock is a powerful, new, and unconventional tool to study fundamental attosecond dynamics on an atomic scale. Prof. Ursula Keller established its potential by using the first attoclock to measure the tunneling delay time in laser-induced ionization of helium and argon atoms. Building on these first proof-of-principle measurements, she proposed to amplify and expand this tool concept to explore the following key questions: How fast can light liberate electrons from a single atom, a single molecule, or a solid-state system? Related are more questions: How fast can an electron tunnel through a potential barrier? How fast is a multi-photon absorption process? How fast is single-photon photoemission?
Many of these questions will undoubtedly spark more questions – revealing deeper and more detailed insights on the dynamics of some of the most fundamental and relevant optoelectronic processes. Theory has failed to offer definitive answers, while – in most cases - simulations based on the exact time-dependent Schrödinger equation have not been possible. Instead, approximations and simpler models are used to capture the essential physics. Such semi-classical models potentially will help to understand attosecond energy and charge transport in larger molecular systems. Indeed the attoclock provides a unique tool to explore different semi-classical models, and resolve the question whether electron tunneling through an energetically forbidden region takes a finite time or is. The tunnelling process, charge transfer, and energy transport all play key roles in electronics, energy conversion, chemical and biological reactions, and fundamental processes important for improved information, health, and energy technologies. Prof. Keller believes the attoclock can help refine and resolve key models for many of these important underlying attosecond processes.
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99e752ae924ca3f0 | IA Scholar Query: Finite Versus Infinite Neural Networks: an Empirical Study. https://scholar.archive.org/ Internet Archive Scholar query results feed en info@archive.org Sun, 19 Jun 2022 00:00:00 GMT fatcat-scholar https://scholar.archive.org/help 1440 Foundations for Meaning and Understanding in Human-centric AI https://scholar.archive.org/work/joonfnadkrf2pab5rsubz2f7sq MUHAI is a European consortium funded by the EU Pathfinder program that studies how it is possible to build AI systems that rest on meaning and understanding. We call this kind of AI meaningful AI in contrast to AI that rests exclusively on the use of statistically acquired pattern recognition and pattern completion. Because meaning and understanding are rather vague and overloaded notions there is no obvious research path to achieve it. The consortium has therefore set up a task early on in the project to explore how understanding is being discussed and treated in other human-centred research fields, more specifically in social brain science, social psychology, linguistics, semiotics, economics, social history and medicine. Our explorations have yielded a wealth of insights: about understanding in general and the role of narratives in this process, about possible applications of meaningful AI in a diverse set of human-centred fields, and about the technology gaps that need to be plugged to achieve meaningful AI. This volume summarizes the outcome of these consultations. It has three main parts: I. A general introduction, II. A series of chapters reporting on what understanding means in various human-centered research fields other than AI, and III. A short conclusion identifying key research topics for meaning-based human-centric AI. Steels, Luc (ed.) work_joonfnadkrf2pab5rsubz2f7sq Sun, 19 Jun 2022 00:00:00 GMT Joint high-dimensional soft bit estimation and quantization using deep learning https://scholar.archive.org/work/fxtjnn2cqncvthtjya2cj2b4qa AbstractForward error correction using soft probability estimates is a central component in modern digital communication receivers and impacts end-to-end system performance. In this work, we introduce EQ-Net: a deep learning approach for joint soft bit estimation (E) and quantization (Q) in high-dimensional multiple-input multiple-output (MIMO) systems. We propose a two-stage algorithm that uses soft bit quantization as pretraining for estimation and is motivated by a theoretical analysis of soft bit representation sizes in MIMO channels. Our experiments demonstrate that a single deep learning model achieves competitive results on both tasks when compared to previous methods, with gains in quantization efficiency as high as $$20\%$$ 20 % and reduced estimation latency by at least $$21\%$$ 21 % compared to other deep learning approaches that achieve the same end-to-end performance. We also demonstrate that the quantization approach is feasible in single-user MIMO scenarios of up to $$64 \times 64$$ 64 × 64 and can be used with different soft bit estimation algorithms than the ones during training. We investigate the robustness of the proposed approach and demonstrate that the model is robust to distributional shifts when used for soft bit quantization and is competitive with state-of-the-art deep learning approaches when faced with channel estimation errors in soft bit estimation. Marius Arvinte, Sriram Vishwanath, Ahmed H. Tewfik, Jonathan I. Tamir work_fxtjnn2cqncvthtjya2cj2b4qa Mon, 13 Jun 2022 00:00:00 GMT Minun https://scholar.archive.org/work/buhdseulezadbjv4rmloolf3oi Entity Matching (EM) is an important problem in data integration and cleaning. More recently, deep learning techniques, especially pre-trained language models, have been integrated into EM applications and achieved promising results. Unfortunately, the significant performance gain comes with the loss of explainability and transparency, deterring EM from the requirement of responsible data management. To address this issue, recent studies extended explainable AI techniques to explain black-box EM models. However, these solutions have the major drawbacks that (i) their explanations do not capture the unique semantics characteristics of the EM problem; and (ii) they fail to provide an objective method to quantitatively evaluate the provided explanations. In this paper, we propose Minun, a model-agnostic method to generate explanations for EM solutions. We utilize counterfactual examples generated from an EM customized search space as the explanations and develop two search algorithms to efficiently find such results. We also come up with a novel evaluation framework based on a student-teacher paradigm. The framework enables the evaluation of explanations of diverse formats by capturing the performance gain of a "student" model at simulating the target "teacher" model when explanations are given as side input. We conduct an extensive set of experiments on explaining state-of-the-art deep EM models on popular EM benchmark datasets. The results demonstrate that Minun significantly outperforms popular explainable AI methods such as LIME and SHAP on both explanation quality and scalability. Jin Wang, Yuliang Li work_buhdseulezadbjv4rmloolf3oi Sun, 12 Jun 2022 00:00:00 GMT An Improved MobileNet Network with Wavelet Energy and Global Average Pooling for Rotating Machinery Fault Diagnosis https://scholar.archive.org/work/5aofja4tcjcdngi6g6nsciz6oe In recent years, neural networks have shown good performance in terms of accuracy and efficiency. However, along with the continuous improvement in diagnostic accuracy, the number of parameters in the network is increasing and the models can often only be run in servers with high computing power. Embedded devices are widely used in on-site monitoring and fault diagnosis. However, due to the limitation of hardware resources, it is difficult to effectively deploy complex models trained by deep learning, which limits the application of deep learning methods in engineering practice. To address this problem, this article carries out research on network lightweight and performance optimization based on the MobileNet network. The network structure is modified to make it directly suitable for one-dimensional signal processing. The wavelet convolution is introduced into the convolution structure to enhance the feature extraction ability and robustness of the model. The excessive number of network parameters is a challenge for the deployment of networks and also for the running performance problems. This article analyzes the influence of the full connection layer size on the total network. A network parameter reduction method is proposed based on GAP to reduce the network parameters. Experiments on gears and bearings show that the proposed method can achieve more than 97% classification accuracy under the strong noise interference of −6 dB, showing good anti-noise performance. In terms of performance, the network proposed in this article has only one-tenth of the number of parameters and one-third of the running time of standard networks. The method proposed in this article provides a good reference for the deployment of deep learning intelligent diagnosis methods in embedded node systems. Fu Zhu, Chang Liu, Jianwei Yang, Sen Wang work_5aofja4tcjcdngi6g6nsciz6oe Sat, 11 Jun 2022 00:00:00 GMT Investigation of chemical reactivity by machine-learning techniques https://scholar.archive.org/work/nihx2cehlba6pffgth4oaackg4 The concepts of potential energy surface (PES) and molecular geometry, defined within the Born-Oppenheimer (BO) approximation, are essential for computational chemistry. The PES is a multi-dimensional function of atomic coordinates and can be obtained by the solution of the electronic Schrödinger equation (SE). While estimating individual points on the PES by first-principles methods, such as density functional theory (DFT), for even moderately sized molecular and material systems is computationally expensive, approximate methods allow for simulations of large systems over long time scales. Machine-learned interatomic potentials (MLIPs) have been gaining in importance since, once trained, they hold the promise to be as accurate as the reference ab-initio electronic structure method while having an efficiency on par with empirical force fields. The derivation of a molecular representation is crucial for designing sample-efficient and accurate MLIPs, irrespective of the employed machine learning (ML) algorithm. Here, a novel molecular fingerprint referred to as Gaussian moment (GM) representation is developed. The GM representation is atom-centered, includes both structural and alchemical information of the local atomic neighborhood, and accounts for all essential invariances (translations, rotations, and permutations of like atoms). It is defined by pairwise atomic distance vectors and its runtime and memory complexity scale linearly with the number of atoms in the local neighborhood. Combined with atomistic neural networks (NNs), GM results in the Gaussian moment neural network (GM-NN) approach, which enables the generation of MLIPs with accuracy and efficiency similar to or better than other established ML models. The GM-NN source code is available free of charge from gitlab.com/zaverkin_v/gmnn. Another intriguing aspect of MLIPs is the generation of highly informative training data sets and consequently, uniformly accurate machine-learned PESs, by applying active learning (AL) strategies. The fundamental quantity of AL is the query strategy -an algorithmic criterion for deciding whether a given configuration should be included in the training set or not. This criterion is defined here by employing the uncertainty estimate derived in the optimal experimental design (OED) framework. The proposed AL scheme allows for a more efficient estimation of the uncertainty of atomistic NNs. Thus, it allows for a more efficient I generation of transferable and uniformly accurate potentials by selecting the most informative or extrapolative configurations. Aside from the conventional MLIPs, which typically aim to predict scalar energies, a methodology for learning the relationship between a structure and the respective tensorial property by atom-centered NNs has been proposed. To learn tensorial properties, specifically, the zero-field splitting (ZFS) tensors, the output of an NN is re-weighted by a tensor that satisfies the symmetry of the former. It has been shown that the proposed methodology can achieve high accuracy and has excellent generalization capability for out-of-sample configurations. Thus, it has been used to study the structural dependence of the ZFS tensor. Moreover, it has been demonstrated that complex processes such as spin-phonon relaxation can be investigated by employing machine-learned surrogate models. Finally, the developed ML approaches have been used to study various surface processes in interstellar environments. Specifically, the adsorption and desorption dynamics of N and H 2 on different surfaces have been investigated, providing binding energies, sticking coefficients, and desorption temperatures. The diffusion of a nitrogen atom on the surface of amorphous solid water (ASW) at low temperatures has drawn particular attention. The study requires long time scales, short time steps in direct molecular dynamics (MD), and a very accurate PES. It has been achieved by combining MLIP driven MD simulations, free energy sampling using well-tempered metadynamics, and kinetic Monte Carlo (kMC) simulations based on the minima and saddle points on the free-energy surface (FES). The study revealed that N atoms, as a paradigmatic case for light and weakly bound adsorbates, can hardly diffuse on bare ASW at 10 K. Surface coverage may change that considerably, increasing the effective diffusion coefficient over 9-12 orders of magnitude. II Zusammenfassung Die Konzepte der Potentialenergiefläche (PES, engl. für Potential Energy Surface) und der Molekülgeometrie sind in der Born-Oppenheimer-Näherung definiert und bilden eine Grundlage für die computergestützte Chemie. Die PES ist eine mehrdimensionale Funktion der Atomkoordinaten und kann durch die Lösung der elektronischen Schrödingergleichung erhalten werden. Die Berechnung einzelner Punkte auf der PES via First-Principles-Methoden, wie z. B. die Dichtefunktionaltheorie (DFT), wird bereits für Molekül-und Materialsysteme mittlerer Größe sehr rechenintensiv. Auf der anderen Seite ermöglichen Näherungsverfahren atomistische Simulationen großer Systeme über lange Zeitskalen. Mit ihrer, zur entsprechenden abinitio Referenzmethode ähnlichen, Präzision gewinnen die maschinell erlernten interatomaren Potentiale (MLIP, engl. für Machine Learned Interatomic Potential) an Bedeutung. Ein weiterer Vorteil ist die Recheneffizienz, vergleichbar zu empirischen Kraftfeldern. Die Herleitung einer molekularen Repräsentation ist entscheidend für die Entwicklung von einem dateneffizienten und genauen MLIP und ist unabhängig vom maschinellen Lernverfahren. In dieser Arbeit wird eine alternative Methode entwickelt, die im Folgenden als Gauß Moment (GM) Darstellung bezeichnet wird. Die GM-Darstellung ist auf einem Atom zentriert, enthält sowohl strukturelle als auch chemische Informationen der lokalen atomaren Umgebung und berücksichtigt alle wichtigen Invarianzen (Translationen, Rotationen und Permutationen von gleichartigen Atomen). Sie wird ausschließlich durch Abstandsvektoren zwischen benachbarten Atomen definiert. Außerdem skaliert die GM linear mit der Atomanzahl in der lokalen atomaren Umgebung. Kombiniert mit atomistischen neuronalen Netzen (NNs) ergibt sich der Ansatz des Gauß Moment Neuronalen Netzwerkes (GM-NN). Dieser ermöglicht die Erzeugung von maschinell erlernten (ML, engl. für Machine Learning) Potentialen, die im Vergleich zu etablierten ML-Modellen vergleichbar oder besser in puncto Präzision und Recheneffizienz sind. Der GM-NN-Quellcode ist unter gitlab.com/zaverkin_v/gmnn frei verfügbar. Ein weiterer wichtiger Aspekt von MLIPs ist die Generierung von hochinformativen Trainingsdatensätzen und damit gleichmäßig genauen ML-PESs. Dies kann durch Anwendung von Methoden des aktiven Lernens (AL) erreicht werden. Der Hauptbestandteil jeder AL-Methode III ist ein algorithmisches Kriterium für die Entscheidung, ob eine gegebene Konfiguration in den Trainingsdatensatz aufgenommen wird oder nicht. Ein solches Kriterium wird hier auf Basis der Unsicherheitsschätzung im Rahmen der optimalen Versuchsplanung (OED, engl. für Optimal Experimental Design) definiert. Der entwickelte AL-Algorithmus ermöglicht eine zeiteffizientere Schätzung der Unsicherheit atomistischer NNs. Durch die Auswahl der informativsten bzw. extrapolativsten Konfigurationen aus einem Trainingsdatensatz können übertragbare und gleichmäßig akkurate ML-Potentiale effizient erzeugt werden. Neben den konventionellen MLIPs, die typischerweise skalare Energien vorhersagen, wurde hier eine Methode zum Erlernen der tensoriellen Molekül-und Materialeigenschaften durch atomzentrierte NNs eingeführt. Um die entsprechenden Eigenschaften, insbesondere den Tensor der Nullfeldaufspaltung (ZFS, engl. für Zero-Field Splitting), zu modellieren, wird die Ausgabe eines NN durch einen weiteren Tensor neu gewichtet. Dieser erfüllt die Symmetrie der zu modellierenden Eigenschaft. Die entwickelte Methode bietet eine hohe Genauigkeit und besitzt außerdem eine ausgezeichnete Generalisierungsfähigkeit auf Konfigurationen, die während des Trainings nicht benutzt wurden. Konkret wurde die Methode für die Erforschung der Abhängigkeit des ZFS-Tensors von der Molekülstruktur benutzt. Darüber hinaus konnte die Möglichkeit zur Untersuchung komplexer Prozesse, z. B. der Spin-Phonon-Relaxation, durch den Einsatz von ML-Modellen gezeigt werden. Schließlich wurde eine Vielzahl von Oberflächenprozessen in interstellarer Umgebung untersucht, um die entwickelten ML-Methoden anwendungsbezogen zu nutzen. Insbesondere wurde die Adsorptions-und Desorptionsdynamik von N und H 2 auf verschiedenen Oberflächen simuliert sowie die Bindungsenergien, Adsorptionskoeffizienten und Desorptionstemperaturen berechnet. Ein besonderes Augenmerk wurde auf die Diffusion eines Stickstoffatoms auf amorphen Eisoberflächen bei niedrigen Temperaturen gelegt. Die entsprechende Studie erfordert lange Zeitskalen, kurze Zeitschritte in der direkten Moleküldynamik (MD) und eine hohe Genauigkeit der PES. Dies wurde durch die Kombination von MD-Simulationen auf einem MLIP, dem Sampling der Freie-Energie-Fläche (FES, engl. für Free-Energy Surface) mit der Methode der wohltemperierten Metadynamik und kinetischen Monte-Carlo-Simulationen (kMC) erreicht. Dabei wurden die Minima und Sattelpunkte auf der FES für die entsprechenden kMC-Simulationen verwendet. Das Resultat zeigte, dass N-Atome als paradigmatischer Fall für leichte und schwach gebundene Adsorbate auf den unkontaminierten, amorphen Eisoberflächen bei 10 K kaum diffundieren. Darüber hinaus konnte gezeigt werden, dass die Präsenz von anderen, inerten Atomen oder Molekülen den effektiven Diffusionskoeffizienten über neun bis zwölf Größenordnungen beeinflusst. IV Peer-reviewed publications This cumulative dissertation summarizes results that have been published in [1]: V. Zaverkin and J. Kästner: Gaussian Moments as Physically Inspired Molecular Descriptors for Accurate and Scalable Machine Learning Potentials. Viktor Zaverkin, Universität Stuttgart work_nihx2cehlba6pffgth4oaackg4 Fri, 10 Jun 2022 00:00:00 GMT Efficient instance and hypothesis space revision in Meta-Interpretive Learning https://scholar.archive.org/work/ijut72m36veyjnbn4mhddaiw6m Inductive Logic Programming (ILP) is a form of Machine Learning. The goal of ILP is to induce hypotheses, as logic programs, that generalise training examples. ILP is characterised by a high expressivity, generalisation ability and interpretability. Meta-Interpretive Learning (MIL) is a state-of-the-art sub-field of ILP. However, current MIL approaches have limited efficiency: the sample and learning complexity respectively are polynomial and exponential in the number of clauses. My thesis is that improvements over the sample and learning complexity can be achieved in MIL through instance and hypothesis space revision. Specifically, we investigate 1) methods that revise the instance space, 2) methods that revise the hypothesis space and 3) methods that revise both the instance and the hypothesis spaces for achieving more efficient MIL. First, we introduce a method for building training sets with active learning in Bayesian MIL. Instances are selected maximising the entropy. We demonstrate this method can reduce the sample complexity and supports efficient learning of agent strategies. Second, we introduce a new method for revising the MIL hypothesis space with predicate invention. Our method generates predicates bottom-up from the background knowledge related to the training examples. We demonstrate this method is complete and can reduce the learning and sample complexity. Finally, we introduce a new MIL system called MIGO for learning optimal two-player game strategies. MIGO learns from playing: its training sets are built from the sequence of actions it chooses. Moreover, MIGO revises its hypothesis space with Dependent Learning: it first solves simpler tasks and can reuse any learned solution for solving more complex tasks. We demonstrate MIGO significantly outperforms both classical and deep reinforcement learning. The methods presented in this thesis open exciting perspectives for efficiently learning theories with MIL in a wide range of applications including robotics, modelling of agent strategies and game [...] Céline Hocquette, Stephen Muggleton, Engineering And Physical Sciences Research Council (EPSRC) work_ijut72m36veyjnbn4mhddaiw6m Fri, 10 Jun 2022 00:00:00 GMT 77777777777777777777 https://scholar.archive.org/work/yr4dd2wzeneq5eydistmrwda2q Once the record has been published, you can no longer change the files in the record, Kjk work_yr4dd2wzeneq5eydistmrwda2q Tue, 07 Jun 2022 00:00:00 GMT 5 Tools for Systems Engineering https://scholar.archive.org/work/u73rz3eeg5c7tippt6sdzfay3e The development of integrated chemical processes in liquid multiphase systems requires extensive knowledge about the reaction kinetics in the different phase systems, the thermodynamics of the phase systems that govern the phase separation, and the distribution of the reactants of the products and the catalyst in the different phases, as well as, e.g., the mass transfer coefficients and separation efficiencies. The methods for acquiring this deep knowledge and the results for different prototypical reactions were described in detail in the previous chapters. This step involves large amounts of experimental work, as ab initio predictions of the yield, the speed of reactions in complex media, and of the phase separation and distribution are not possible yet. Based on experimental investigations, detailed mathematical models of the kinetics and the phase separation can be developed, which help to guide and to speed up the experimental work to determine optimal phase systems and conditions for the reaction and separation steps. This combination of experimental work and mathematical modeling was also discussed in the previous chapters and successful examples were presented that highlight the potential of model-guided experimental investigations and homogeneously catalyzed reactions in multiphase systems. Generally speaking, the design of chemical production processes consists of narrowing down the range of alternatives and, at the same time, removing uncertainty about the expected performance as well as about the best choice of the operating conditions, equipment parameters, etc. The search space comprises a huge number of possible alternatives, starting from the possible raw materials, over catalysts and ligands, solvent systems, types of equipment, to the sizing of the equipment, the ratios of the feed streams, residence times, temperatures, pressures, etc. It is not possible to deal with all these alternatives and their parameterization simultaneously. Therefore the design process proceeds in stages where some decisions are fixed sequentially (but may be revised if problems at subsequent stages are detected). In the beginning, the main goal is to single out promising options based on a preliminary evaluation of their potential, which necessarily has to be done with incomplete knowledge or under uncertainty. Traditionally, this step is very much based on the experience of the developers, gained in previous investigations. The goal is an economically viable, if possible economically optimal process that meets the sustainability criteria, as well as possible. A third, also very relevant criterion in the initial phase is the minimization of risk, i.e., to ensure that the product specifications are met and the economic viability is maintained under uncertainties about the precise properties of the raw materials, with limited Open Access. Sebastian Engell work_u73rz3eeg5c7tippt6sdzfay3e Tue, 07 Jun 2022 00:00:00 GMT Stochastic modelling and statistical inference for electricity prices, wind energy production and wind speed https://scholar.archive.org/work/o3jxdkcctjfjtjuupxojvnxima Although wind energy helps us slow down the increase of global temperatures, its weather-dependence and unpredictability make it risky to invest in. In this thesis we apply statistical and mathematical tools to enable energy providers to accurately plan such investments. In the first part we want to understand the impact of wind energy on electricity prices. We extend an existing multifactor model of electricity spot prices by including stochastic volatility as well as the information about wind energy production. Empirical studies indicate that these additions improve the model fit. We also model wind-related variables directly, using Brownian semistationary processes with generalised hyperbolic marginals. Finally, we introduce a joint model of prices and wind energy production suitable for quantifying the risk faced by energy distributors. The second goal is to produce accurate short-term wind speed forecasts based on historical data instead of computationally expensive physical models. We achieve this by splitting the wind speed into two horizontal components and modelling them with Brownian semistationary processes with a novel triple-scale kernel. We develop efficient estimation and forecasting procedures. Empirical studies show that such modelling choices result in good forecasting performance. Paulina A. Rowińska, Almut Veraart, Engineering And Physical Sciences Research Council (EPSRC), EDF Energy (Firm) work_o3jxdkcctjfjtjuupxojvnxima Mon, 06 Jun 2022 00:00:00 GMT Resource-Constrained Learning and Inference for Visual Perception https://scholar.archive.org/work/urx3hnzcbrbcdmqqhsnirfbxpi We have witnessed rapid advancement across major computer vision benchmarks over the past years. However, the top solutions' hidden computation cost prevents them from being practically deployable. For example, training large models until convergence may be prohibitively expensive in practice, and autonomous driving or augmented reality may require a reaction time that rivals that of humans, typically 200 milliseconds for visual stimuli. Clearly, vision algorithms need to be adjusted or redesigned when meeting resource constraints. This thesis argues that we should embrace resource constraints into the first principles of algorithm designs. We support this thesis with principled evaluation frameworks and novel constraintaware solutions for both the cases of training and inference of computer vision tasks. For evaluation frameworks, we first introduce a formal setting for studying training under the non-asymptotic, resource-constrained regime, i.e., budgeted training. Next,we propose streaming accuracy to evaluate latency and accuracy coherently with a single metric for real-time online perception. More broadly, building upon this metric, we introduce a meta-benchmark that systematically converts any single-frame task into a streaming perception task. For constraint-aware solutions, we propose a budget-aware learning rate schedule for budgeted training, and dynamic scheduling and asynchronous forecasting for streaming perception. We also propose task-specific solutions, including foveated image magnification and progressive knowledge distillation for 2D object detection, multi-range pyramids for 3D object detection, and future object detection with backcasting for end-to-end detection, tracking and forecasting. We conclude the thesis with discussions on future work. We plan to extend streaming perception to include long-term forecasting, generalize our foveated image magnification to arbitrary spatial image understanding tasks, and explore multi-sensor fusion for long-range 3D detection. Mengtian Li work_urx3hnzcbrbcdmqqhsnirfbxpi Mon, 06 Jun 2022 00:00:00 GMT Amodal Visual Scene Representations With and Without Geometry https://scholar.archive.org/work/xlf37hif2ng47a4bukr5nzhejq Most computer vision models in deployment today describe the pixels of images. This does not suffice, because images are only projections of the scene in front of the camera. In this thesis we build representations that attempt to describe the scene itself. We call these representations "amodal" (i.e., without modality), emphasizing the fact that they describe elements of the scene for which we have no sensory input. We present two methods for amodal visual scene representation. The first focuses on modelling space, and proposes geometry-based methods for lifting images into 3D maps, where the objects are complete, despite partial occlusions in the imagery. We show that this representation allows for self-supervised learning from multi-view data, and yields state-of-the- art results as a perception system for autonomous vehicles, where the goal is to estimate a "bird's eye view" semantic map from multiple sensors. The second method focuses on modelling time, and proposes geometry-free methods for tracking image elements through partial and full occlusions across a video. Using learned temporal priors and within inference optimization, we show that our model can track points across outperform flow-based and feature-matching methods on fine-grained multi-frame correspondence tasks. Adam Harley work_xlf37hif2ng47a4bukr5nzhejq Mon, 06 Jun 2022 00:00:00 GMT Simulation-Based Inference for Whole-Brain Network Modeling of Epilepsy using Deep Neural Density Estimators https://scholar.archive.org/work/5qdgu2ffb5hznpr4luu7em77ym Whole-brain network modeling of epilepsy is a data-driven approach that combines personalized anatomical information with dynamical models of abnormal brain activity to generate spatio-temporal seizure patterns as observed in brain imaging signals. Such a parametric simulator is equipped with a stochastic generative process, which itself provides the basis for inference and prediction of the local and global brain dynamics affected by disorders. However, the calculation of likelihood function at whole-brain scale is often intractable. Thus, likelihood-free inference algorithms are required to efficiently estimate the parameters pertaining to the hypothetical areas in the brain, ideally including the uncertainty. In this detailed study, we present simulation-based inference for the virtual epileptic patient (SBI-VEP) model, which only requires forward simulations, enabling us to amortize posterior inference on parameters from low-dimensional data features representing whole-brain epileptic patterns. We use state-of-the-art deep learning algorithms for conditional density estimation to retrieve the statistical relationships between parameters and observations through a sequence of invertible transformations. This approach enables us to readily predict seizure dynamics from new input data. We show that the SBI-VEP is able to accurately estimate the posterior distribution of parameters linked to the extent of the epileptogenic and propagation zones in the brain from the sparse observations of intracranial EEG signals. The presented Bayesian methodology can deal with non-linear latent dynamics and parameter degeneracy, paving the way for reliable prediction of neurological disorders from neuroimaging modalities, which can be crucial for planning intervention strategies. Meysam Hashemi, Anirudh Nihalani Vattikonda, Jayant Jha, Viktor Sip, Marmaduke M Woodman, Fabrice Bartolomei, Viktor Jirsa work_5qdgu2ffb5hznpr4luu7em77ym Fri, 03 Jun 2022 00:00:00 GMT Dynamic Privacy Budget Allocation Improves Data Efficiency of Differentially Private Gradient Descent https://scholar.archive.org/work/rcxdipdctfglhialemyhonh74i Protecting privacy in learning while maintaining the model performance has become increasingly critical in many applications that involve sensitive data. A popular private learning framework is differentially private learning composed of many privatized gradient iterations by noising and clipping. Under the privacy constraint, it has been shown that the dynamic policies could improve the final iterate loss, namely the quality of published models. In this talk, we will introduce these dynamic techniques for learning rate, batch size, noise magnitude and gradient clipping. Also, we discuss how the dynamic policy could change the convergence bounds which further provides insight of the impact of dynamic methods. Junyuan Hong and Zhangyang Wang and Jiayu Zhou work_rcxdipdctfglhialemyhonh74i Thu, 02 Jun 2022 00:00:00 GMT Metrizing Fairness https://scholar.archive.org/work/zlfiy4wxmfdobgmob5i325eq4a We study supervised learning problems for predicting properties of individuals who belong to one of two demographic groups, and we seek predictors that are fair according to statistical parity. This means that the distributions of the predictions within the two groups should be close with respect to the Kolmogorov distance, and fairness is achieved by penalizing the dissimilarity of these two distributions in the objective function of the learning problem. In this paper, we showcase conceptual and computational benefits of measuring unfairness with integral probability metrics (IPMs) other than the Kolmogorov distance. Conceptually, we show that the generator of any IPM can be interpreted as a family of utility functions and that unfairness with respect to this IPM arises if individuals in the two demographic groups have diverging expected utilities. We also prove that the unfairness-regularized prediction loss admits unbiased gradient estimators if unfairness is measured by the squared ℒ^2-distance or by a squared maximum mean discrepancy. In this case, the fair learning problem is susceptible to efficient stochastic gradient descent (SGD) algorithms. Numerical experiments on real data show that these SGD algorithms outperform state-of-the-art methods for fair learning in that they achieve superior accuracy-unfairness trade-offs – sometimes orders of magnitude faster. Finally, we identify conditions under which statistical parity can improve prediction accuracy. Yves Rychener, Bahar Taskesen, Daniel Kuhn work_zlfiy4wxmfdobgmob5i325eq4a Thu, 02 Jun 2022 00:00:00 GMT Improving Transparency and Intelligibility of Multi-Objective Probabilistic Planning https://scholar.archive.org/work/az7okfddjjbpjfi5fe35pmdz6e Sequential decision-making problems with multiple objectives are natural to many application domains of AI-enabled systems. As these systems are increasingly used to work with people or to make decisions that impact people, it is important that their reasoning is intelligible to the end-users and stakeholders, to foster trust and effective human-agent collaborations. However, understanding the reasoning behind solving sequential decision problems is difficult for end-users even when white-box decision models such as Markov decision processes (MDPs) are used. Such intelligibility challenge is due to the combinatorial explosion of possible strategies for solving long-horizon problems. The multi-objective optimization aspect further complicates the problem as different objectives may conflict and reasoning about tradeoffs is required. These complexities pose a barrier for end-users to know whether the agent has made the right decisions for a given context, and may prohibit them from intervening if the agent was wrong. The goal of this thesis is to develop an explainability framework that enables the agent making sequential decisions to communicate its goals and rationale for its behavior to the end-users. We present an explainable planning framework for MDP, particularly to support problem domains with multiple optimization objectives. We propose consequence oriented contrastive explanations, in which an argument for an agent's policy is in terms of its expected consequences on the task objectives, put in context of the selected viable alternatives to demonstrate the optimization and tradeoff reasoning of the agent. Our modeling framework supports reward decomposition, and augments MDP representation to ground the components of the reward or cost function in the domain-level concepts and semantics, to facilitate explanation generation. Our explanation generation method computes policy-level contrastive foils that describe the inflection points in the agent's decision making in terms of optimization and tradeoff reas [...] Roykrong Sukkerd work_az7okfddjjbpjfi5fe35pmdz6e Thu, 02 Jun 2022 00:00:00 GMT Evaluating the Stability of Numerical Schemes for Fluid Solvers in Game Technology https://scholar.archive.org/work/4sp3mkgztndmhctlkhm73x6seq A variety of numerical techniques have been explored to solve the shallow water equations in real-time water simulations for computer graphics applications. However, determining the stability of a numerical algorithm is a complex and involved task when a coupled set of nonlinear partial differential equations need to be solved. This paper proposes a novel and simple technique to compare the relative empirical stability of finite difference (or any grid-based scheme) algorithms by solving the inviscid Burgers' equation to analyse their respective breaking times. To exemplify the method to evaluate numerical stability, a range of finite difference schemes is considered. The technique is effective at evaluating the relative stability of the considered schemes and demonstrates that the conservative schemes have superior stability. Craig R. Stark, Declan A. Diver, Peican Zhu work_4sp3mkgztndmhctlkhm73x6seq Thu, 02 Jun 2022 00:00:00 GMT The effective noise of Stochastic Gradient Descent https://scholar.archive.org/work/e2p3z4zp4rgkjcg6m7sdlhgqf4 Stochastic Gradient Descent (SGD) is the workhorse algorithm of deep learning technology. At each step of the training phase, a mini batch of samples is drawn from the training dataset and the weights of the neural network are adjusted according to the performance on this specific subset of examples. The mini-batch sampling procedure introduces a stochastic dynamics to the gradient descent, with a non-trivial state-dependent noise. We characterize the stochasticity of SGD and a recently-introduced variant, persistent SGD, in a prototypical neural network model. In the under-parametrized regime, where the final training error is positive, the SGD dynamics reaches a stationary state and we define an effective temperature from the fluctuation-dissipation theorem, computed from dynamical mean-field theory. We use the effective temperature to quantify the magnitude of the SGD noise as a function of the problem parameters. In the over-parametrized regime, where the training error vanishes, we measure the noise magnitude of SGD by computing the average distance between two replicas of the system with the same initialization and two different realizations of SGD noise. We find that the two noise measures behave similarly as a function of the problem parameters. Moreover, we observe that noisier algorithms lead to wider decision boundaries of the corresponding constraint satisfaction problem. Francesca Mignacco, Pierfrancesco Urbani work_e2p3z4zp4rgkjcg6m7sdlhgqf4 Wed, 01 Jun 2022 00:00:00 GMT Some Thoughts on Official Statistics and its Future (with discussion) https://scholar.archive.org/work/o6q55ed3dreevgsmhzucbs7caq In this article, we share some reflections on the state of statistical science and its evolution in the production systems of official statistics. We first try to make a synthesis of the evolution of statistical thinking. We then examine the evolution of practices in official statistics, which had to face very early on a diversification of sou rces: first with the use of censuses, then sample surveys and finally administrative files. At each stage, a profound revision of methods was necessary. We show that since the middle of the 20th century, one of the major challenges of statistics has been to produce estimates from a variety of sources. To do this, a large number of methods have been proposed which are based on very different f oundations. The term "big data" encompasses a set of sources and new statistical methods. We first examine the potential of valorization of big data in official statistics. Some applications such as image analysis for agricultural prediction are very old and will be further developed. However, we report our skepticism towards web-scrapping methods. Then we examine the use of new deep learning methods. With access to more and more sources, the great challenge will remain the valorization and harmonization of these sources. Yves Tillé, Marc Debusschere, Henri Luomaranta, Martin Axelson, Eva Elvers, Anders Holmberg, Richard Valliant work_o6q55ed3dreevgsmhzucbs7caq Wed, 01 Jun 2022 00:00:00 GMT Likelihood-Free Inference with Generative Neural Networks via Scoring Rule Minimization https://scholar.archive.org/work/d7hbolchlrdyrbky4svh2xkp7u Bayesian Likelihood-Free Inference methods yield posterior approximations for simulator models with intractable likelihood. Recently, many works trained neural networks to approximate either the intractable likelihood or the posterior directly. Most proposals use normalizing flows, namely neural networks parametrizing invertible maps used to transform samples from an underlying base measure; the probability density of the transformed samples is then accessible and the normalizing flow can be trained via maximum likelihood on simulated parameter-observation pairs. A recent work [Ramesh et al., 2022] approximated instead the posterior with generative networks, which drop the invertibility requirement and are thus a more flexible class of distributions scaling to high-dimensional and structured data. However, generative networks only allow sampling from the parametrized distribution; for this reason, Ramesh et al. [2022] follows the common solution of adversarial training, where the generative network plays a min-max game against a "critic" network. This procedure is unstable and can lead to a learned distribution underestimating the uncertainty - in extreme cases collapsing to a single point. Here, we propose to approximate the posterior with generative networks trained by Scoring Rule minimization, an overlooked adversarial-free method enabling smooth training and better uncertainty quantification. In simulation studies, the Scoring Rule approach yields better performances with shorter training time with respect to the adversarial framework. Lorenzo Pacchiardi, Ritabrata Dutta work_d7hbolchlrdyrbky4svh2xkp7u Tue, 31 May 2022 00:00:00 GMT 55555555555555 https://scholar.archive.org/work/vao7vkpr3jev3hnj5g2tmsc4ze This book aims to provide a theoretically oriented introduction to the scientifific study of human episodic memory—memory for events experienced in a spe KMJ work_vao7vkpr3jev3hnj5g2tmsc4ze Tue, 31 May 2022 00:00:00 GMT |
f591c804bc781826 | Ghost universes kill Schrödinger's quantum cat
Quantum weirdness is a sign of many ordinary but invisible universes jostling to share the same space as ours, according to a bold new idea
Physics 5 November 2014
New Scientist Default Image
Not as strange as it seems
(Image: ESO)
THE wave function has collapsed – permanently. A new approach to quantum mechanics eliminates some of its most famous oddities, including the concept of quantum objects being both a wave and a particle, and existing in multiple states at once.
In short, the approach removes the wave function and demotes the equation that describes it. In its place are a huge but finite number of ordinary, parallel worlds, whose jostling explains the weird effects normally ascribed to quantum mechanics.
Quantum theory was dreamed up to describe the strange behaviour of particles like atoms and electrons. For nearly a century, physicists have explained the peculiarities of their quantum properties – such as wave-particle duality and indeterminism – by invoking an entity called the wave function, which exists in a superposition of all possible states at once right up until someone observes it, at which point it is said to “collapse” into a single state.
Physicist Erwin Schrödinger famously illustrated this idea by imagining a cat in a box that is both dead and alive until someone opens the box to check on it. The probability that the cat will survive is given by the Schrödinger equation, which describes all the possible states that the wave function can take.
The Schrödinger equation predicts the outcomes of experiments perfectly. But many physicists are uncomfortable with seeing the wave function as a fundamental aspect of reality, preferring to treat its companion equation as a calculating device and seeking a deeper theory to explain what is really going on.
“You can’t think of the wave function as a real thing,” says Howard Wiseman of Griffith University in Queensland, Australia. But if the wave function is not real, what is?
Now, Wiseman and colleagues have come up with an answer. Our universe, they claim, shares space with a large number of other universes, each of which follows the classical, Newtonian laws of physics. In this view, particles in our universe feel a subtle push from corresponding particles in all the other universes. Everything we think of as quantum weirdness is the result of these worlds bumping into each other (Physical Review X,
“One way to think about it is that they coexist in the same space as our universe, like ghost universes,” Wiseman says. These other worlds are mostly invisible because they only interact with ours under very strict conditions, and only in very minute ways, he says, via a force acting between similar particles in different universes. But this interaction could be enough to explain quantum mechanics.
“One way to think about it is the other universes coexist in the same space as ours, like ghosts”
To demonstrate that the idea has legs, Wiseman and his team showed mathematically that the many interacting worlds theory can explain specific effects. “The first thing you have to do is show that it can reproduce results, because quantum mechanics has been tested to incredible accuracy,” Wiseman says. “If you can’t actually reproduce quantum mechanics, you’re sunk from the beginning.”
They chose a classic test called the double-slit experiment, which is usually read as evidence that photons act like both a wave and a particle. The effects of phantom photons in as few as 41 other worlds could give qualitatively the same result, the team says (see “Phantom photons“).
They show that the theory can account for other effects as well, including the stability of matter: other worlds pressing in on our own stop electrons from falling into the nucleus of atoms, as they would in a purely Newtonian world.
“If those interactions were turned off, or if there was only one world, then all such effects would vanish, and Newtonian physics would reign supreme,” says Wiseman’s colleague Michael Hall, who is also at Griffith.
The approach is still in its infancy, and the theorists have a lot of work to do to flesh it out. For example, they haven’t modelled how a finite number of worlds can explain entanglement, a long-distance relationship between quantum particles that Einstein called “spooky action at a distance”. But it could work if the number of worlds is infinite.
Many worlds
The many interacting worlds idea echoes an earlier many-worlds interpretation of quantum mechanics, thought up by theorist Hugh Everett in the 1950s, in which the universe splits into pairs of parallel universes every time a wave function collapses. The cat is both dead and alive, according to Everett – it just depends which universe you look in.
But there are some important differences. The Everettian many-worlds interpretation treats the wave function as a fundamental part of reality, for one. And once two worlds split, they almost never interact with each other.
In the new theory, the parallel worlds have always been there, and interact via repulsive forces between corresponding particles as soon as they diverge a tiny bit.
The approach has some advantages over the standard many-worlds interpretation, says Lev Vaidman at Tel Aviv University in Israel, who has worked extensively on the Everettian approach. For example, the many-worlds view struggles to explain probability in a world where everything that could possibly happen does happen. With many interacting worlds, probability falls easily out of the mathematics.
The idea also raises the exciting possibility of actually doing experiments to see evidence of these other worlds – and maybe even of communicating with our twins there.
“The idea raises the exciting possibility of doing experiments to see evidence of other worlds”
The Schrödinger equation only predicts quantum behaviour exactly if there are an infinite number of worlds, Wiseman says. If the number of worlds is finite, then the Schrödinger equation is just an approximation. With sufficiently careful experiments, we could see its predictions differ from what experiments measure.
“This leads to the amazing possibility that an experiment could determine how many worlds there are,” says Eric Cavalcanti at the University of Sydney in Australia.
Another possible probe into the other worlds could be via the proposed force that acts between them. Wiseman says that force may be able to help explain outstanding mysteries like gravity. He even says it’s not crazy to imagine that if the theory is right, but the force is just a little bit different to what he’s proposed, then communication between worlds could be possible.
That’s still speculative, though. For now, other physicists find the proposal interesting, although it’s not clear that it will actually replace the wave function.
“It is the concept of the quantum wave function that resolved the paradoxes of classical mechanics one century ago,” Vaidman says. “So I am not sympathetic to the current attempt to replace the quantum wave function with something else.” Still, he says, the theory could be interesting if theorists can make the maths simpler and more elegant.
Even if the theory has nothing to do with the way the world really is, it could prove a useful tool for simplifying quantum predictions.
“Once you have six or seven quantum particles interacting, the Schrödinger equation is far too hard to solve, even approximately,” Hall says. “With our theory, we just have to worry about where the particles are in each world, and calculate that inter-world force between them.”
Cavalcanti agrees, noting that many other quantum interpretations have led to new technology, despite nobody agreeing on whether they are correct. “The no-cloning theorem, quantum cryptography, quantum teleportation, all have roots at least partly in these kinds of foundational questions,” he says. “It is not inconceivable therefore that an approach like this may be useful in such a way. But it is too early to tell.”
Phantom photons
In the “many interacting worlds” theory, nearly identical particles in parallel universes bump into each other to create all the weird quantum effects we observe in our world (see main story). To show this idea works, Howard Wiseman at Griffith University in Australia and colleagues demonstrated mathematically that the theory predicts the results of the classic double-slit experiment.
In this experiment, single photons are fired one at a time at a phosphorus screen that detects them. Between the source and the screen is a black material that absorbs photons, but with two slits in it.
If the photons were merely particles, you would expect them to end up evenly spread over the screen with time. But they don’t. Instead, they pile up in stripes, making a characteristic pattern on the screen.
This is exactly what would happen if they were passing through both slits as a wave, and the two wavefronts interfered with one another, so the experiment is usually taken as evidence that photons can be both a particle and a wave.
With many interacting worlds, each photon exerts a force on all the others, nudging the photons in our world on a specific, slightly different trajectory. Wiseman and colleagues showed that with just 41 worlds, you can produce the same pattern as in the experiment.
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a877d05ad078724e | Empiricism, Materialism, Physicalism avoiding Solipsism
Great post on Reductionism over at Emil’s blog.
Once you choose to accept materialism a lot of consequences follow.
Empiricism and Physicalist Monism – How To Do It
Ideas, Concepts, Thoughts – Physical Instantiation In Brains
[This is part of a set: Thinking][This is part of a set: Consciousness]
Abstract ideas, concepts, thoughts, occur in human brains. But how are they instantiated in those brains? Physically.
There are patterns of matter and energy in the universe, sometimes called ‘fractures in the continuum’, or ‘lack of conformity’. In informational terms there are distinctions – distinct data patterns. These are synonymous to all intents and purposes, though some philosophers may object to this – but then I think if they object to this they’ve got bigger problems with solipsism anyway. Certainly from an inductive point of view this acknowledgement of the correspondence between reality and the patterns or distinctions in it is sufficient.
On this basis, everything is essentially data – including human brains. The change in human brains that occurs when thoughts flit through them or when they remember something is merely brain matter changing state, changing pattern. Conversely, everything is also material – including data, by virtue of the fact that it consists of the organisation of matter into distinct patterns, whether that’s a configuration of electrons in the capacitive element of a logic transistor, or the configuration of synapses in a human brain.
Even when we think in our minds of abstract data existing in some Platonic plane, that very idea itself has an existence in the formation of matter in the brain. The odd thing to grasp with this is that we have this abstract notion that there is nothing abstract, it’s all real, except the abstraction itself, which doesn’t have some separate reality independent of physical reality.
I think it important to note that all ideas, such as ‘idea’, ‘concept’, ‘abstract’, along with religious ideas like ‘soul’, ‘God’, are all inventions of the human mind – as is ‘mind’ of course, so I should really say, inventions of the human brain. No science has ever discovered the existence of a material object, or any trace of energy, or anything else, that is a ‘soul’, or an ‘idea’, or a ‘concept’, other than their physical instantiation as patterns in matter/energy.
So that when philosophers talk about these as if they have some existence, it’s pure invention with no verification through evidence. What we do find are patterns in matter which are used to represent these, which then invokes something in the brain.
Representation = Physical Implementation.
So, the word ‘concept’ itself invokes the concept of ‘concept’ in my brain as I read it. But given that this is happening in a material brain then there is little more to expect other than the word on the screen has triggered a corresponding pattern in the brain: word on screen, light to eye, retina activity, complex neuronal activity, triggered concept recognition.
This is why I think that even when talking about human ‘knowledge’ in the brain we are better sticking to terms like data, or information. This view also unifies the idea of knowledge as data within human brains, and outside them, on paper, in books and databases, and even unifies the idea with the material world.
Data = Physical Distinction
I accept that as a matter of convenience we will want to differentiate between the places where this data/matter resides. So, on some occasions we’ll talk about ‘the body of human knowledge’ when we mean the accumulation of all of what has at sometimes been in some human brains and has been translated into common media, such as books. On other occasions we’ll talk of how a person ‘knows some proposition to be true’, when we are talking about their commitment to the correspondence of the proposition to some relating thing or event in the world outside the human head. But when looking at this in the whole, and at the same time looking for how all this ‘knowledge’ exists in some detailed but unified way, it’s easier to talk about information, data, matter.
Let’s compare software. A piece of software is only ever an abstraction in a human mind. There is nothing you can touch that is a Microsoft Word program. When you buy it on disk you are actually taking with you a disk with some pattern on it. Look at the pattern on the disk and you see pits in a CD. You do not see nebulous software. When you install it onto a PC there is real physical energy transfer, from the CD reader, through the system, into magnetic patterns on the hard drive. Other than wear and tear and any decay, loss of fidelity on the disk through laser action is entirely incidental – the disk pattern largely remains. Software has not been transferred. It has been copied – re-represented. When it’s loaded into PC memory and run, it’s just bit states in the memory. Programs are data; data is information; information is distinction in physical state.
Abstractions, ideas, concepts, are our software. They don’t exist in any physical sense other than they are patterns. They are patterns in the brain, no matter how permanent, like long term memory, or how transient, like short term memory, or even non-memorised flashes across areas of the working brain.
Take a concept, any concept. Can you hold one? Or are they fleeting brain content? If I have the concept of a car, and I draw that car on paper, and show that paper to someone, and they recognise the pattern as representing a car, their brain will likely construct, immediately, a concept of a car. At no time did that concept exist on the paper. Only a representation of it existed. If the other person did not share the concept of car, had they never seen one (our classical ‘jungle native’, ignorant of all technology), then, they would only see lines on the paper – and might even mistake the paper for some kind of leaf or some object they are familiar with. The lines in which we see a car would not invoke the concept of a car in anyone ignorant of the human technology.
An example used by Sam Harris is language. When I hear English spoken it triggers patterns in my brain. My brain recognises the words and converts them into brain patterns that emerge into consciousness as concepts. This is to a great extent unconscious, thanks to my having learned English from childhood. I have limited experience of other languages. If I listen to a French speaker speaking quickly I may pick up only a portion of the content, and may miss some key words so that I get the story completely wrong. I know some French but I’m not fluent. My brain is not attuned to the sound patterns of quickly spoken French. If I listen to Korean it will be pure noise. I don’t know that I know any Korean. Just as someone who has no experience or knowledge of cars would not recognise a line drawing of a car, so my brain does not pick anything useful out of Korean. It’s noise.
Information theory relies on distinction for any information at all. In our physical universe distinction amounts to different states of matter/energy; and dynamic states at that. The whole point of the heat death of the universe is the complete and utter loss of distinction. Our very existence relies on distinction in states of matter. Our brains undergo dynamic changes to the matter of which it is constituted to form distinct states.
Is it surprising that thoughts, concepts, ideas, only came into being along with our evolved brains, and even more so when our brains acquired language? But, you might ask, what about the thoughts of God? Well, so far, all the evidence points to God coming into existence, as a concept, along with the development of human brains. I don’t know of any encoded record of God being present along with any fossils. Our first notions of gods appear with the early artifacts of creatures that were already human.
Epistemology is a problem for philosophy. Knowledge doesn’t have a satisfactory water tight definition that gets us anywhere. Far simpler to accept the information theory use of knowledge which is more about the correspondence between what we have in our heads and the material experience it represents. The problem is that we are inundated with continuous experiences from our first conception, though cognitive experiences await some rudimentary brain development in the fetus. By the time we’re old enough to think consciously about ideas like ‘concept’, ‘knowledge’ and other ‘abstract’ ideas, our brains are already full of them. This leaves us with the impression that they have some sort of abstract life of their own, but they don’t. They exist as brain states, and changing states: behaviour.
I find it odd that anti-physicalists want to use the insubstantial ephemeral nature of ‘ideas’, ‘concepts’, as evidence of a real and active ‘mind’ that is distinct from the brain. To my physical brain, my mind, the very nebulous nature of ‘concepts’ and ‘ideas’ is evidence of their non-existence in any independent reality, and better as evidence of their existence only in the brain.
Physicalism and Conciousness
See section 2 on Conciousness, and in particular the Mary problem.
As Colin McGinn has stated, “Consciousness defies explanation in [compositional, spatial] terms. Consciousness does not seem to be made up out of smaller spatial processes…. Our faculties bias us towards understanding matter in motion, but it is precisely this kind of understanding that is inapplicable to the mind-body problem.”
Nonsense. What is computer software? Can you explain it? How can you copy it without creating new matter or energy? It’s information, that’s why. Our thoughts are information, the product of physicalism and caused by it. Nothing inherently mysterious, though it might appear so to the human mind that is actually experiencing it. The mind-body duality dilema that people struggle with is analogous to an optical illusion – e.g. the hollow mask that appears solid, or the wire cube that flips orientation – as with these it’s difficult to think in our mind of both states simultaneously. We can flip states, but we can’t ‘see’ or imagine both simultaneously. In a similar way we can (almost) imagine computer software as information, but have greater difficulty imagining this condition when applying it to our own thoughts. It becomes even more confusing, and more like the attempt to simultaneously ‘see’ both states of an optical illusion, when we try an imagine what’s happening when we think about what we are thinking now in the first person; and some explanations of conciousness and dualism confuse the issue by trying to do this.
Did Mary (see site) learn something new about pain? Yes. She physically experienced (both in terms of physical neurological responses and informational interpretation) the real pain for which she had only previously had a physical neurological model. Her model has simply been updated with real first hand experiential data, when previously the only experiential data she had was neurological mapping of things she had already experienced. In practice of course this ‘schrodingers’s cat’ type of thought experiment is limited. The definition of the experiment is incorrect. Pain is simply a more intense stimulus of corresponding stimuli – presumably Mary hadn’t been denide the sense of touch, otherwise she would have had difficulty relating to much of the theoretical information she had read in the first place. What sort of human would have emerged from the room if that had been the case. It’s a hypethetical case where the accuracy of the perceived consequences are dubious, to the extent that the conclusion does not necessarily follow. Mary can’t even pick up the bowling ball if she’s been deprived of the appropriate senses!
This is metaphysical mumbo-jumbo. “compositional, spatial analysis of the intrinsic nature of an event” – does this actually mean anything? These arguments are often dressed up in these phrases that some researcher has latched onto or invented to describe some concept that is difficult to understand – fair enough. But then the problem is that these phrases are used in ways that make it difficult to grasp what is being said.
“…can he (physicalist) at least provide a plausible explanation of how it came about that the universe contains occurrences such as experiences of pain and pleasure? We doubt it.”
Why, when it has expressly been given? The dualist is confusing a simple causal relationship between an excessive physical stimulus and the informational model that the receiving organism experiences as a result, as a separate entity.
How does a human feel pain? A cat? A worm? A bacterium? A cell? A complex molecule? A grain of sand? Physicaly, they don’t, they simply react – either extremely passivily according to relatively simple laws of physics for a grain of sand, or in more complex physical/chemical ways for a molecule, or in increasingly more complex chemical/exlectrical/biological/neurological ways for higher organisms.
Being organsims with a complex nervous system that includes the brain we have adapted ourselves to the interpretation of our environment. One of our interpretations is to feel/think/experience our environment in terms of our own experiences. The more animate and the more similar to us other entities are, the more easly we make this mapping – we anthropomorphise or personify. We do this with ourselves and our ‘thoughts’ to the greatest degree. Some of us even have to create, or imagine, or to model non-existant entities using the same principle – demons, faires, ghosts, gods, etc. Sometimes our brains get it wrong – they extrapolate (a very valuable tool used in the prediction process) – they extrapolate too much, they become gullible, seeing optical illusions, even delusions.
“What, then, is the theistic alternative? Theism begins by acknowledging that experiences of pleasure and pain and choices are events that occur in subjects which refer to themselves by the first-person pronoun ‘I.'”
Do some of the lower organsims not feel pain? If they do, do they refer to themselves in the first person? Again, when is this magical dualism switched on – just humans, apes, …? Be careful, else you’ll be dragging up biblical nonsense again.
“As the theist René Descartes wrote…(quotes Descartes)…”
The dualist is here acknowledging the simplicity of the mind in one respect, but denying it from the physicalist respect, which itself is very simple.
Decartes: “I cannot distinguish in myself any parts” – could that be because there is nothing to distinguish? Is Decartes referring to the distinction between mind and body, or the distinction between parts of his thoughts? Is he struggling to identify his thoughts as distinct physical entities? Maybe he’s struggling because they don’t exist as such. When my computer is running some software I can see the results on screen, I can imaging the electrons moving at amazing speeds around the silicon based microscopic circuitry, and I can imaging the source code I have written if it’s my program that’s running – but can I imaging the actual ‘software’ itself as a physical entity? No more than I can be self aware and imagine my own thoughts as something distict from my physicality.
I can certainly imagine what the dualists are describing. I can imaging some ghostly substance that might be my soul, spirit, thoughts – but that’s all it is, an imagined concept. I have no reason to think it exists. When movies portray a dead soul rising out of a body – is that what we really think is happeng in some invisible dimension? Of course not (or maybe you do). But there is no evidence to support that imagining, that concept. I can imagine flying pigs, with little wings – do they exist? Because I can imagine something doesn’t mean it exists.
I can imagine God, angels – all with typically anthropomorphised representations. If God really exists with some of the real properties he’s supposed to have, such as omniscience, can I imagine that? Only in a limited way, as I imagine the mathematical concept of infinity – something bigger than anything, but to which if I add more it is the same thing? Does that sound a little like the ontological argument for God? Figments of our limited imaginations!
In postulating the concept of dualism we are using a limited capacity tool (the mind) to grasp something of itself that is merely apparent. We accept illusions, hoaxes, some delusions, for what they are – the mind not presenting a sufficiently good approximation of the external physical reality – but then for no apparent reason than the mystery of not underestaning something, we invent dualism, supernatural external agents, theism. Figments of our limited imaginations.
Why is it so difficult to see that the alternative – the physical causal relationship between neurological activity and the resulting mental models?
Don’t be fooled by the apparent complexity. How can this proposed simple process take part in this argument, including those parts of the process that produce the written (typed) work above (whether you think its good or not it’s still apparently complex). But, just as the many many simple little steps of evolution have produced us, so the many many simple little processes in this organism have produced this. If I had omnisciently and omnipotently flashed out all this text instantly, in zero time, then we might be closer to the realisation of what God is. But I didn’t. Every impulse to my fingers to type, every nuerologocal action that contributes, is very very simple – they are simply working very fast and in great numbers. The sophisticaion comes from the co-ordination. But co-ordinated lesser orgaisms that are independent to some extent also produce similarly amazing results. Bees building honey combs, ants foreging for food – they are all sophisticated co-ordinated processes where the individual elements are all amazingly simple whan compared with the result.
We are at the top of the chain, as far as we know, in this evolutionary scale, so we find it difficult to imagine anything that might be more complex than ourselves that is not some ultimate God.
Dualism, as with God, is a failed attempt to come to terms with the complex. We can imagine the simple. We can imagine somethings more complex. But eventually, as complexity increases we lose touch and make a giant leap to something bigger, but conceptually easier to identify – even if not easier to understand.
In maths, imagine a simple sum: 1 + 1 = 2. Now imagine some complex formula – say some series using powers and factorials – still with me? Now try some complex differential equations – still here? Now Schrödinger equation… – have you seen them and do you understand them? By now some, if not most of us (including me) has lost track of these equations – they are more complex than I am familar with. I can imagine some vague representation on a physicists blackboard, employing symbols I’m not familar with – it’s all Greek to me. Now, let’s imagine infinity – got that?
I bet more people with upper high school and graduate level maths find it easier to grasp the notion of infinity than they do some complex expression representing something in physics. It’s quite straight forward to imagine clearly some simpler things, and relatively easy to grasp something of the notion of a concept that is very extensive, in size, number, power, infomational capacity, than it is to imagine some things that are just more complex than we are used to. It’s easier to imagine God as represented by some very vague notions of extreme extension to simpler human properties, than it is to imagine in detail more complex processes or organisms than those with which we are currently familar.
Dualism is similar to some extent. We find it difficult to imagine where the boundary lies – or how the continuum flows – from the physical bodies that we have come to be familiar with and the thoughts that we are also familiar with. Because we can’t imagine this we invent a separation – dualism. It’s a failure of our current capacity to understand.
So, are physicalists so advanced that they can conceive of it, while the poor dumb dualists can’t? No, of course not. What is most likely at work here is an ingrained view that’s difficult to shake off. I would guess, though I have nothing to support this, that all physicalists have had dualist interpretations at one time – simply because it is easier to imagine.
This is an imagination gap. If the gap is narrow we can build a bridge easily. If the gap is wide we prefer to fly across, skipping whatever is missing. Go from what we are familiar with to some extreme concept based on the familar properties. It’s difficult to imagine what we don’t know. This imagination gap should be familar to most students, particularly the more advanced your studies*. You can read the fear of the apparent consequences in the writings of theists. We are dealing with a ‘duality of the gaps’ that is similar to the ‘God of the gaps’.
“we are not arguing that there is some gap in an otherwise seamless naturalist view of reality”
Oh yes you are.
“This is an argument from the fundamental character of reality and what kinds of things exist (purposes, feelings…”
Yes, purpose and feelings exist, but not as some distinct dualist entity. They are properties of the organism that is experiencing. Particularly feelings and emotions – simple hormonal biological chemical electrical reactions. ‘Purpose’ is apparent, not real in the sense that is independent free-will.
The only dualism I see in all this is that in the mind of the dualist. On the one had an imagination failure in not seeing the continuum and inclusiveness of physicalism that encompases conciousness, and on the other, the runaway imagination that goes in leaps and bounds from missing data regarding conciousness, to mind-body dualism, on to basic theism, and then on to all the wild imaginings of heaven, hell, saints, miracles, etc.
*I remember very clearly the earliest experience of this, on a very limited scale. In primary school I could do ‘short-division’ but I couldn’t fathom out ‘long-division’ – it was very frustrating, and even frightening – I feared I was really dumb!. Then a neigbour’s son, a year older than me, spent some time going through examples. I remember very clearly when the penny dropped. A spiritual revalation? Later, at university I struggled with some concepts of advanced chemistry – it was an electronics course and I naively hadn’t expected to be learning chemistry and I’d skipped chemistry at highschool, so I was ill equiped for some of this stuff. I remember the anguish in class, seeing all the other students nodding knowingly while I was thinking “what the hell is he talking about”. Recognising the response I went off to the library and made sure I caught up. Never be afraid of what you don’t know! If you need to know it, put in sufficient effort so that your brain and its neurological patterns become famialar with it – eventually you’ll see the light – alleluiah! |
010c5f3325d5790c | Researchers enrich silver chemistry
Researchers from the Moscow Institute of Physics and Technology have teamed up with colleagues in Russia and Saudi Arabia and proposed an efficient method for obtaining fundamental data necessary for understanding chemical and physical processes involving substances in the gaseous state. The proposed numerical protocol predicts the thermal effect of gas-phase formation of silver compounds and their absolute entropy. This includes first-ever such data for over 90 compounds. Published in the journal Inorganic Chemistry, the findings are important for the practical applications of substances containing silver: in water and wound disinfection, photography, rainmaking via cloud seeding, etc.
The team derived the precise values of two key characteristics — the enthalpy of formation and the entropy — of numerous silver compounds. The enthalpy (from Greek “thalpein,” meaning “to heat”) of a system describes its state in terms of the energy of the constituent particles, pressure, and volume. According to Hess’ law, by multiplying stoichiometric coefficients and the difference between the formation enthalpies of the reactants and those of the products, one gets the amount of heat generated or consumed in a chemical reaction. Entropy is a measure of how disordered a system is. The second law of thermodynamics states that a system can spontaneously adopt a less organized state, so entropy grows with time.
Knowing the values of enthalpy and entropy is crucial for predicting whether a reaction will ever occur at given conditions. These characteristics also indicate how reaction yield and selectivity — the ratio between products — vary with temperature and pressure, allowing for optimization. The findings enable researchers to make predictions concerning chemical processes occurring in the gas phase. The data will also help manage the processes involved in thin film and pure sample deposition from the gas phase.
There are basically two ways for determining enthalpy and entropy changes: either through complex and costly experiments or by using the data from reference books and doing some arithmetic based on Hess’ law.
“The choice seems to be obvious, more so considering that you cannot experimentally measure the heat of some reactions,” said Yury Minenkov, senior researcher at the Laboratory of Supercomputing Methods in Condensed Matter Physics. “For example, incomplete graphite combustion always yields both carbon monoxide and carbon dioxide. So even by measuring the thermal effect of the reaction we could not determine carbon monoxide formation enthalpy.
“But the computational approach faces some problems,” Minenkov went on. “First, the enthalpies of formation and entropies are not known for every compound. Second, even if the data are available, no one can guarantee their accuracy. The values vary widely between reference books. At times, the measurement errors may be quite large.”
Luckily, quantum chemistry helps obtain the entropy and, to some extent, the enthalpy data. Each constituent molecule of a gaseous substance can be viewed as a system of positively charged nuclei and negatively charged electrons. Researchers can then apply electronic structure calculation methods to solve the molecular Schrödinger equation. This reveals the total electronic energy of the molecule, its wave function, and the spatial configuration of nuclei — that is, its 3D geometric structure. Physicists can then calculate the entropy and enthalpy of an ideal gas composed of such molecules. The entropy values obtained in this way are then introduced into reference books and used in thermodynamic calculations.
The problem with enthalpy is that, not being a fundamental value at this point, it significantly depends on the chosen method for Schrödinger equation calculation.
Figure 1. Simplified diagram illustrating formation enthalpy calculation for silver compounds. The compound depicted is silver sulfate, Ag₂SO₄. Credit: @tsarcyanide/MIPT Press Office
Atomization reactions are usually employed to calculate the enthalpy of formation. In such reactions, the compound of interest breaks down into individual atoms. For example, silver sulfide — Ag₂S — yields one sulfur and two silver atoms. Since the enthalpies of formation of atomic substances are well-known and reported in reference books, it is possible to calculate the enthalpy of formation of the initial substance — in this case, silver sulfide — by finding the enthalpy change in the reaction via quantum chemistry methods.
However, when molecules composed of many atoms undergo atomization, this affects the electronic structure to such an extent that enthalpy, too, is significantly changed. The currently available methods of theoretical chemistry cannot account for these effects with enough accuracy.
The team of researchers from MIPT, the Frumkin Institute of Physical Chemistry and Electrochemistry of the Russian Academy of Sciences, Ivanovo State University of Chemistry and Technology, and Saudi Arabia’s King Abdullah University of Science and Technology has published a series of papers proposing a way to calculate the thermodynamic characteristics of organic and inorganic compounds with more accuracy.
In the case of silver sulfide, the researchers found its enthalpy of formation from the reaction with hydrochloric acid, which yields silver chloride and hydrogen sulfide (fig. 2). Since the number of bonds in the top row is the same as in the bottom row, the change in energy can be calculated with the least error.
Figure 2. A diagram illustrating the chemical reaction between one silver sulfide (Ag₂S) and two hydrochloric acid (HCl) molecules, producing two molecules of silver chloride (AgCl) and one of hydrogen sulfide (H₂S). Credit: @tsarcyanide/MIPT Press Office
The heats of formation for silver chloride, hydrogen sulfide, and hydrochloric acid are known with a high accuracy, and computer modeling reveals the thermal effect of the reaction. From these data, one can derive the heat of formation of silver sulfide using Hess’ law.
“The classic coupled-cluster approach, known as CCSD(T), is the gold standard for solving the electronic Schrödinger equation in modern quantum chemistry,” Minenkov explained. “We replaced it with the local version called DLPNO-CCSD(T), which was developed not long ago at Max Planck Institute. This shrank the required computing power by an order of magnitude. Under the conventional CCSD(T), the computation time varies with the size N of the molecule as N⁷, so it is not an option for large molecules. The local version is much less time- and resource consuming.”
The team began by checking how well the results of their quantum chemical calculations agreed with the experimental thermodynamic and structural data. Reference books reported the values for 10 silver compounds, and they were a good match. Having thus ensured that their model is adequate, the researchers calculated the thermodynamic characteristics for 90 silver compounds missing in the books. The data are of use to both scientists working in the field of silver chemistry and for testing and calibrating new methods of electronic structure calculation.
Originally published at
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b539bc3cfa175691 | The bottom quark mass from sum rules
[0.2cm] at next-to-next-to-leading order
M. Beneke Theory Division, CERN, CH-1211 Geneva 23, Switzerland
A. Signer Department of Physics, University of Durham,
Durham DH1 3LE, England
(June, 22, 1999)
We determine the bottom quark mass and the quark mass in the potential subtraction scheme from moments of the production cross section and from the mass of the Upsilon 1S state at next-to-next-to-leading order in a reorganized perturbative expansion that sums Coulomb exchange to all orders. We find GeV and GeV for the potential-subtracted mass at the scale GeV, adopting a conservative error estimate.
Introduction. Accurate determinations of the bottom quark mass in perturbative QCD usually rely on properties of the spectrum of Upsilon mesons and production near threshold. Since already for the state the momentum scale GeV and energy scale
In this case the characteristic momentum and energy scales are replaced by and , respectively. The requirement of perturbativity puts an upper limit on the admissible values of . On the other hand, only the resonance contribution to the sum rule is experimentally well known, and needs to be taken large enough to reduce the error from the continuum. When , the perturbative expansion of in the strong coupling breaks down, because there exist terms of the form in any order of perturbation theory. This suggests a summation of the perturbative expansion to all orders in which is treated as order 1 [2].
In this letter we analyse the sum rule (1) at next-to-next-to-leading order (NNLO) in this resummed perturbative expansion. [A preliminary analysis was presented in Ref. [3].] The resummed perturbative cross section is computed at NNLO using recent 2-loop results on the Coulomb potential [4] and the vertex [5, 6, 7] and non-relativistic effective field theory in dimensional regularization as described in [3, 8, 9]. Rather than determining the quark pole mass, as has usually been done, we apply the potential subtraction (PS) scheme and determine the PS mass [10] from the sum rule. We expect perturbative corrections in this and related schemes to be smaller than in the on-shell scheme [10, 11]. We then convert the extracted PS mass to the mass, thus by-passing the infrared sensitivity problem of the on-shell scheme [12], and yet implementing the resummation necessary in the non-relativistic kinematics enforced by taking large moments. Other NNLO analyses of the sum rule have already appeared [13, 14, 15, 16]. Nevertheless, we think that an independent analysis, together with a critical discussion of the quark mass error, is still useful. We also perform a complementary analysis and determine the quark mass directly from the mass of the state. This has been done previously in a NNLO analysis presented in [17], which, however, concentrated on the quark pole mass, as did [13, 14].
Experimental moments. We first evaluate the integrals (1) by expressing the cross section in terms of the six resonances and the open continuum. The masses and leptonic widths of the resonances are taken from [18]. Very little information exists on the continuum above [19]. We parametrize the continuum by setting . With this crude parametrization the experimental error on the determination of is MeV for , and small compared to the theoretical error for interesting moments with -. Some experimental moments are shown in Table 1. For - about 70%-85% of the experimental moment comes from the resonance.
Table 1: The experimental moments.
Theoretical moments. The theoretical moments are computed by first matching QCD to non-relativistic QCD. In a second step this theory is matched to a non-local Schrödinger field theory, in which pairs propagate through the Coulomb Green function. We then solve the Schrödinger equation to NNLO. We refer to [3, 8, 9] for some details of the method; further useful information can be found in [13, 14, 15]. The result for the cross section to NNLO, still in the on-shell scheme, is expressed as
where , , , and is the pole mass. The functions contain bound-state poles that correspond to the resonances. We obtained these functions analytically. After integrating numerically over according to (1), these functions sum all terms of the form to all orders. Writing in the form of (2) implies that we expand the bound-state pole -functions around the leading-order pole position. Expanding the bound-state pole -functions rather than leaving them unexpanded is motivated by the fact that the sum rule relies on global duality. Using dispersion relations, the moments can be expressed in terms of derivatives of the vacuum polarization as indicated in (1), which makes no reference to individual resonances. Computing these derivatives in resummed perturbation theory to NNLO implies that we expand the resonance -functions in the expression for .222 For very large one should keep the -functions unexpanded, because the effective smearing interval in becomes smaller than the perturbative correction to the bound-state pole position. But for such large one has to rely on local duality and the sum rules suffers from non-perturbative uncertainties as we discuss further below.
Before integrating over , we convert the expression for from the on-shell to the potential subtraction scheme. The pole mass is eliminated using the relation [10]
where is the Coulomb potential in momentum space. Explicit expressions for can be found in [10]. Note that is proportional to a subtraction scale , which should not exceed the characteristic scale of the moments . We insert (3) into (2) and expand the small correction terms involving . However, the term is not expanded when is replaced in , or , because counts as being of the same order as . The result is an expression of the same form as (2), but with as input parameter. As mentioned in the introduction we expect the expansion (2) in this new variable to be more convergent, and hence the PS mass can be determined with smaller error than the pole mass. The PS mass depends on and we choose GeV as our default. The PS masses for different are connected by a renormalization group equation that follows directly from the definition (3).
The dominant theoretical uncertainty arises from the residual dependence of the theoretical moments on the renormalization scale of the strong coupling in the scheme. We now discuss the choice and variation of this scale and the choice of moments that go into our analysis.
As indicated in (2) explicit logarithms of always come as . When , the integral (1) falls exponentially as , so that the characteristic energy scale is . This determines the parametric form of the renormalization scale to be . This, however, is not strictly true, because the moments also contain parts in which gluons carry momentum of order and momentum of order . In the renormalization-group-improved treatment (see [3, 9]) the hard scale enters as the starting point of the renormalization group evolution of the Wilson coefficient functions. The dependence on is negligible, of the order of MeV on the output for , compared to the dependence on the scale , which determines the endpoint of the renormalization-group evolution. It is therefore not considered further. Gluons with three-momentum of order enter only at order . This leaves us with the scale above and we adopt as the most ‘natural’ scale.
One may object that the form of the logarithm gives the natural scale only parametrically, but that the scale is arbitrary up to a multiplicative factor, since the physical scale in the scheme corresponds to a different scale in another scheme, for instance MS. We can address this question by searching for constants that appear systematically in conjunction with the logarithm . While this is complicated for the full cross section, it is easily done for the bound state energies that correspond to the resonances. We find that for the th energy level the analogous logarithm always appears in the combination , where . Since , this suggests – if anything –, that the physical scale is even smaller than what we inferred from the logarithm alone.
The useful moments are restricted from below by the uncertainty in the experimental value of the moment. If we aim at an error of about MeV in the determination of the quark mass, we need . There is also a technical restriction, which could be overcome. Our expression for sums all terms of the form , but it does not make use of the exact fixed-order coefficients at order , which are known [20], because terms of relative order or smaller are dropped. This could be compensated for by matching the resummed result and the fixed-order result. However, we find that for this matching correction is small, as can be seen from Table 2.
It is advantageous to take large moments, because large moments are more sensitive to , while the experimental error does not increase, see Table 1. An upper limit arises, because the characteristic scales must remain perturbative. As concerns , the requirement does not seem to pose a serious restriction. In practice, we find that the theoretical prediction becomes unstable already when is smaller than 1.5-2.0GeV; requiring to be larger than this is restrictive, if we also allow for a variation of about . A more serious constraint arises from the scale , which enters the NNLO calculation implicitly. At NNNLO there is a contribution to the moment that scales as . When , we should count as order 1. In this case, we have an uncontrolled non-perturbative contribution to the moments that is formally of NNLO.333It is worth noting that the scales and do not really approach 0 as , but freeze at values of order and , respectively. While this is of interest for a very heavy quark, it is of little practical relevance to quarks. We therefore require . In the literature larger moments are often used. The justification for this is that the gluon condensate contribution to the moments, which represents the leading non-perturbative power correction, is small even for moments much larger than 10. However, the operator product expansion in local operators is itself only valid when and so the estimate is not rigorous. It may, however, indicate that ultrasoft contributions from the scale are smaller than what we would estimate on parametric grounds.
For reference we give some selected moments in Table 2 in the on-shell and PS scheme. The table also quantifies the importance of resumming corrections and the error incurred by not including the exact fixed-order coefficients at order . Resummation is crucial even for , because the contribution from the bound-state poles, which does not exist in the NNLO fixed-order approximation, is large. On the other hand, already for the fixed-order moments (FO1) are well approximated by the leading three terms in their large- expansion (FO2).
Table 2: Selected moments (recall the normalization of the moments in (1)) in the on-shell and PS scheme (square brackets) for and GeV (on-shell scheme) and GeV (PS scheme). The renormalization scale is taken to be , where or . ‘Res.’ refers to the resummed cross section with LO, NLO, NNLO in the sense of (2). ‘FO1’ refers to the fixed-order result without resummation, including terms of order (LO, NLO, NNLO). ‘FO2’ refers to the fixed-order result, dropping terms with relative suppression or more. The effect of resummation is given by the difference of ‘Res.’ and ‘FO2’. The correction due to matching to the fixed-order result is roughly given by ‘FO1’ minus ‘FO2’.
Figure 1: The value of obtained from the 10th moment as a function of the renormalization scale in NLO and NNLO and for . The dark region specifies the variation due to the experimental error on the moment. The middle line marks the scale , the two outer lines determine the scale variation from which the theoretical error is computed.
Numerical analysis. For a given the theoretical moments are functions of , which we would like to determine; the strong coupling , for which we use together with 3-loop evolution; and the renormalization scale , which is our (rough) handle to estimate the uncertainty due to the NNLO approximation of the moment calculation. We first compute, for given and (and ), the values of , for which the theoretical moment lies within the experimental range. For the result is shown in Fig. 1. It is evident that the resulting varies significantly as function of the scale at which the sum rule is evaluated. Furthermore, there is no overlap between the range of masses that is obtained from the NNLO and the NLO sum rule for any reasonable range of . The same conclusion is obtained for or . We also determined from simultaneously by minimizing a with equal weights. The value we obtain from this procedure differs by no more than MeV from that obtained from single moments, when is varied between and GeV, reflecting that the -dependence of the theoretical moments is completely correlated. The same analysis in the on-shell scheme results in an identical qualitative picture; however, the scale-dependence is even larger in the on-shell scheme.
As explained above, we take as our default choice of scale. We would then follow common practice and estimate a theoretical error by varying the scale between one half and twice this value. But from Fig. 1 we observe that the theoretical prediction becomes unstable (compare the behaviour of the NLO and NNLO results) for scales below GeV and one may argue that varying the scale into this region does not provide a reliable error estimate. We therefore compute the theoretical error from a variation between GeV and . It is clear that the error so estimated is rather sensitive to the lower scale cut-off. Taking and adding the error from and the experimental moments, we obtain
If the scale is varied down to GeV, the scale error decreases (increases) to MeV. In comparison, a NLO analysis of the sum rule would return the central value GeV with a smaller scale uncertainty (see Fig. 1). The large difference with the NNLO result casts doubt on the convergence of successive perturbative approximations. The origin of this difference and the origin of the large scale dependence will become clear below.
The PS mass is a useful parameter (replacing the pole mass) for short-distance observables involving quarks close to their mass shell. For high energy processes, we would like to convert the PS mass to the definition. Call the quark mass at the renormalization scale and the coefficient at order that relates the pole mass to . From (3) we obtain the relation
where we defined . An NLO analysis of the sum rule determines the PS mass with a parametric accuracy of order . This is most easily seen by noting that an NLO calculation of the resummed cross section determines the (perturbative) masses to order . To determine the mass with the same parametric accuracy implies that one should use (5) at order . At present (5) is known only to third order, combining the result of [21] for and the one for from [10].
To obtain an order-of-magnitude estimate of the missing fourth-order term, we estimate the coefficient in the ‘large-’ approximation [22, 23]. This gives and for and GeV.444For comparison, note that the ‘large-’ approximation for results in 2140.36 rather than the exact value of 1870.54. For one obtains 6526.91 rather than the ‘exact’ value 6144(128). (The brackets specify the error on the ‘exact’ result, see [21].) refers to the number of light-quark flavours. Although the individual coefficients are large, there are large cancellations in the combination that enters (5), which reflect the infrared cancellation that motivated the introduction of the potential subtraction [10]. With these numbers, given , we estimate that the term reduces by MeV. (For comparison, the term provides a MeV reduction.) We therefore assume an additional MeV correction in the relation between and beyond the third-order formula. This results in the mass555 We compute by solving (5) exactly for a given PS mass, rather then inverting (5) perturbatively to order . The second procedure would result in a central value that is negligibly different by MeV from the one given.
The dependence on nearly cancels out and ‘conv.’ refers to the conversion from the PS to the scheme just discussed. We have repeated the analysis with GeV and GeV for the subtraction scale of the PS mass. Converting to , we find agreement with (6) within MeV.
Origin of the large scale dependence. The scale uncertainty in (4) is only about 30% smaller than the uncertainty we would have found in the on-shell scheme. To understand the origin of this marginal improvement, we consider a truncated sum rule, in which both experimental and theoretical moments are given only in terms of the first resonance. This is actually not a bad approximation to the full sum rule and allows us to discuss the origin of scale dependence in a transparent form. In this approximation, performing in addition a non-relativistic approximation to the -integration measure in (1), we can write the sum rule as
where and are the leptonic width and mass of the state computed to NNLO. In the on-shell scheme, the series expansions for the two quantities read666The mass and leptonic width have been obtained to NNLO in [17, 15] and [15], respectively. Our analytic expressions for an arbitrary state coincide with those previous results, provided we neglect the renormalization-group improvement for the leptonic width. At present only the logarithms from the renormalization of the external current are taken into account.
where , and and the second line is given for such that (for GeV), in which case . Neither of the series is converging and, because of the large power of of the NNLO term, the scale dependence is huge at small scales, where varies fast. This is seen from Fig. 2, which shows the scale dependence of the left-hand side and right-hand side of (7) separately.
Figure 2: The left-hand side (lhs, short-dashed line) and right-hand side (rhs, dash-dotted and solid lines) of (7) as a function of the scale for , GeV (on-shell scheme, dash-dotted) and GeV (PS scheme, solid). The figure clearly shows the reduction of the scale dependence in the PS scheme for the predicted mass. The scale dependence of the width (short-dashed line) is identical in the on-shell and PS schemes.
Going from the on-shell to the PS scheme improves the convergence and scale dependence of the predicted mass as seen from the solid line in Fig. 2, but has little effect on the series expansion (9) for the width. Hence the large scale uncertainty in (4) can be traced to the poor control over the perturbative expansion for the leptonic width that controls the over-all normalization of the theoretical moments.
Constraints from . The fact that the mass determination from the sum rule is dependent on the theoretical prediction of the leptonic width, and limited in accuracy for this reason, suggests that we consider determining directly from the spectrum. Non-perturbative corrections to the masses grow rapidly for higher radial excitations and preclude using any state other than the ground state. The problem with this method is that even for the state it is difficult to estimate the non-perturbative correction reliably. The perturbative expression for the mass in the on-shell scheme is given by (8) above.
If , the non-perturbative correction to the mass can be computed in terms of vacuum condensates of local operators. The leading contribution is [24]
where is the gluon condensate. The actual magnitude of is rather uncertain. If we choose the ‘natural’ scale GeV, we obtain MeV. However, as noted earlier, the logarithm that determines this scale appears together with constants that tend to make the effective scale lower. Furthermore, it may be argued that the coupling should be taken as the coefficient of in the Coulomb potential rather than in the scheme. This coupling is larger than the coupling. Both effects can decrease substantially.
Since the inequality that justifies the operator product expansion (OPE) does not hold, we should consider the subsequent term in the OPE to judge whether the expansion converges. Using the result of [25], we find that the contribution from dimension-6 operators could be anything between a fraction of and twice , where the large uncertainty stems from the poorly known dimension-6 condensates and the ambiguity in the value of .777Ref. [25] concludes that the OPE appears to be convergent, because the minimal value of the strong coupling in the denominator of (10) assumed there is larger than the minimal value allowed in our estimate. This puts the convergence of the OPE in question. We therefore consider (10) as an order-of-magnitude estimate of the non-perturbative correction and treat it as a theoretical error rather than adding it to (8). In our opinion, assigning an error of MeV to from this source is conservative.
We proceed to determine from using (8) converted to the PS scheme. This renders the series (8) convergent and leads to a very small scale uncertainty of the extracted value of , as anticipated from the solid curve in Fig. 2. Varying from to GeV, we obtain
which is consistent with (4). In this case a NLO analysis would return the central value GeV, which suggests that the corresponding small value in the sum rule analysis is an anomaly related to the behaviour of the series for the leptonic width. From (11) we obtain the mass
The central value varies by only about MeV, when is varied between and GeV. Contrary to the sum rule determination, the error is dominated by the non-perturbative contribution to the mass. This leaves room for improving upon the error, if some quantitative insight into the non-perturbative contribution could be obtained.
Comparison with previous results. We compare the bottom quark mass obtained in this work with the results of earlier NNLO analyses of the sum rule and the mass. Our comments will be restricted to those analyses that quote a result for the quark mass [13, 14, 15, 16, 17, 26]. With the exception of [17], which obtains from and, therefore, should be compared with (12), all other NNLO analyses use the sum rule (1) and should be compared with (6).
The value given in (12) is significantly smaller than GeV, obtained by [17]. This difference is explained by the fact that [17] first uses the on-shell scheme to extract the pole mass and then uses the 2-loop truncation of (5) [with ] to obtain . However, contrary to the PS scheme with GeV, the 3-loop and 4-loop terms are large in the on-shell scheme; at least the 3-loop term888Since the large infrared contribution in the 4-loop term cancels against an NNNLO contribution to the mass, it can be argued that only the 3-loop term is to be used. This is different from the PS scheme, where no systematically large coefficients appear. Compare (5) [with ] and the discussion in [10, 27] regarding combining different powers of to make infrared cancellations manifest. has to be included when the pole mass is determined from the NNLO formula for the mass. Estimating the terms missing in [17] in the large- limit, and subtracting them from GeV, we find that the result of [17] becomes (roughly) consistent with ours. The error estimate of [17] is, however, less conservative than ours.
A related difficulty concerns the comparison with the value GeV quoted in [13]. While apparently consistent with the one obtained in this work, it is obtained via a 2-loop relation from the quark pole mass, which in turn is determined from the sum rule. If we add the 3-loop and/or 4-loop term, the result of [13] would be about MeV lower than ours. This difference is a reflection of the fact that the pole mass quoted in [13] is roughly MeV lower than the one we would have obtained had we chosen to determine it. This difference in turn can be traced to the use of a high renormalization scale for the evaluation of the sum rule, cf. Fig. 1. In our opinion, the choice of such a high scale is not well motivated. We also think that it is mandatory to use intermediate mass definitions such as the PS mass to determine reliably. Otherwise large perturbative coefficients make it impossible to disentangle true theoretical errors from correlated and spurious ones caused by those large perturbative coefficients.
The most recent analysis by the authors of [26] determines the mass directly from the sum rule and gives GeV from moments with -. No resummation is performed, because it is assumed, incorrectly, that this is unnecessary in the scheme. However, for high moments the Coulomb interaction must be treated non-perturbatively, and a resummation has to be done, irrespective of the mass renormalization convention. The scheme actually makes the expansion worse, because the expansion contains terms of order in addition to . To avoid such terms, one has to use an intermediate convention, such as the potential subtraction scheme, and then relate this convention to the scheme in a second step. Because of this theoretical shortcoming, the result of [26] cannot be compared with (6).
The other papers quoted above perform a NNLO resummation as in this work. Differences arise either in the representation of the NNLO-resummed moments or in the analysis and error evaluation strategy. Both [15] (MY) and [16] (Hoang) also use intermediate mass subtractions, different from the PS scheme, but conceptually similar to it, before converting these intermediate masses to the scheme.999As the analysis in [16] supersedes [14], we do not discuss [14] in detail.
The differences in the theoretical representation of the moments are the following:
• MY and Hoang use a factorization scheme different from dimensional regularization. Since the final result is physical, this is a technical difference that should bear no consequences on the final result.
• Hoang applies the non-relativistic approximation also to the -integration in (1), while MY and the present work obtain the resummed cross section analytically and then integrate it numerically according to (1) or after an equivalent contour deformation into the complex -plane. The difference is negligible.
• MY and Hoang have taken the short-distance coefficient as an over-all factor, while we have multiplied it out to NNLO. Keeping it as an over-all factor is problematic, because this results in a spurious factorization scheme and scale dependence, which is not small as can be seen from Table 3 of [16]. Since both short-distance and long-distance contributions are computed perturbatively, the factorization scale is a purely technical construct and no dependence on it should be left in the result. One motivation for writing the short-distance coefficient as an over-all factor is that the scales in the coupling constant are different in the long- and short-distance parts. However, this effect, related to logarithms of , can be treated consistently only in the context of a full renormalization group treatment. This has been done in the present work (see [3, 9]), but not in [15, 16]. As a consequence there is no analogue of in the present approach, while the role of in [15, 16] is taken by the starting scale for the renormalization group evolution. As mentioned earlier, this dependence is negligible in our representation of the moments.
• Hoang and this work expand the bound-state -functions for reasons discussed earlier; MY keep them unexpanded. This increases the theoretical moments significantly at NNLO. Not expanding the bound state -functions would increase the value quoted in (4) and (6) by almost MeV.
• MY and Hoang use a 2-loop formula to obtain from their intermediate mass. As explained above, a NNLO analysis of the sum rule determines the PS (or related) masses with accuracy and to fully exploit this accuracy, the 4-loop relation between the PS and mass should be used. We estimated that the 3- and 4-loop terms decrease by MeV and this additional shift has been incorporated in (6). Employing a less accurate relation entails a corresponding loss in parametric accuracy of , although this is a numerically small effect, if our estimate is correct.
As in this work, MY obtain their result from an analysis of single moments (checking consistency between a set of moments), although larger moments - are used, which could be considered problematic. They use the so-called kinetic mass as intermediate mass definition. The analogue of in (3), needed for a NNLO sum rule analysis, is not yet known in this scheme; MY estimate it in the large- limit, an additional assumption we had to use only for the 4-loop term when relating (4) to (6), but not to extract the PS mass in the first place. MY vary the renormalization scale from GeV to GeV, while we would argue that, for the high moments used in [15], the scale should be chosen lower. If we repeat our analysis with the same assumptions as those of MY, we reproduce their error estimate for the kinetic mass, which is smaller than the more conservative procedure that leads to (4).
Hoang uses an analysis and error estimate that is different from the single-moment analysis performed by MY and in this work. Simplifying somewhat, Hoang fits the quark mass from the linear combination of moments, where the coefficients are determined by the covariance matrix of the experimental input data such as the measured leptonic widths of the six resonances. This linear combination (which entails a cancellation of one part in 4000) turns out to be very insensitive to the renormalization scale , yet retaining a large sensitivity to . Hoang then scans the theoretical parameter space and finds an error of only MeV for the so-called 1S-mass, compared to the error of (4). We have repeated our analysis for this linear combination and obtain GeV in this way, where the quoted error is due to variation of the renormalization scale only. The result is consistent with (4), but the error is much smaller. The central value is MeV higher than the value reported in Sect. 6 of [16]. Such differences can be explained by different implementations of the NNLO result, as discussed above.
Several circumstances make us suspicious that the theoretical error is underestimated by Hoang’s procedure. For example, if we increase the error of the leptonic width of the by a factor of 10, or if we increase the error on the measured mass of the to MeV, which is still small compared to the expected theoretical error, the procedure chooses a linear combination that exhibits less stability in , and has no or two solutions for for some ranges of , even though our experimental knowledge of the width or mass should have no bearing on the theoretical error estimate. This remark may not be considered as a serious objection, because we could abandon the way the linear combination is chosen in [16] and optimize it deliberately. However, even for the original linear combination, there is a second solution GeV in addition to GeV, because the linear combination is no longer a monotonic function of . The criterium of renormalization scale stability does not exclude obtaining solutions that differ by more than the error estimated from the -dependence. The problem is compounded by the observation that the stability under variations of the renormalization scale, and hence the small error obtained by Hoang, crucially depends on the assumption that the four moments are combined at the same value of the renormalization scale . This is a serious assumption, in particular as the natural scale of the moments is . If we combine the moments at their natural rather than at equal scales, the stability is lost. To be fair, we should mention that Hoang’s analysis is more involved than analysing a single linear combination, although the covariance matrix is such that it does indeed give most weight to a single one. Nevertheless, we think that the simplified discussion above emphasizes the problem with estimating a theoretical error in the way done in [16].
The final results by MY (GeV) and by Hoang (GeV) agree with (6) within the quoted errors. However, if the MeV shift were applied to those results, there is actually a discrepancy of about MeV in the central value. This could be a consequence of the different representations of the moments as discussed above. However, we find it difficult to reconcile a significantly smaller than GeV with the analysis of [see (12)], unless there is indeed a large positive non-perturbative contribution to .
Summary. We determined the bottom quark mass in the scheme and the potential subtraction (PS) scheme [10] at next-to-next-to-leading order from sum rules for the cross section and the mass of the state. The results are in excellent agreement with each other as summarized by (4), (6), (11) and (12). There is no systematic procedure to combine the two results. On the one hand, the two determinations are not independent, because some theoretical input is common to both. On the other hand, the dominant source of theoretical error is different. We therefore combine the two determinations to yield the PS mass
and the mass (at the scale of the mass)
The calculations that go into these results imply partial resummations of the QCD perturbative expansion to all orders. An important point is that a NNLO resummation allows us to determine the quark masses with a parametric accuracy of order , i.e. the residual error scales formally as . In the case of the mass this requires that one controls the four-loop relation to the PS mass. We estimated the 4-loop term, which is not yet known exactly and found that it should be very small.
Unfortunately, the sum rule analysis yields a much less precise determination of the bottom quark mass than what might have been expected with NNLO accuracy. We identify as the reason for this the bad behaviour of the perturbative expansion for the leptonic width of the resonances. The same is true for the mass in the on-shell scheme. However, in this case it is understood that the large coefficients are unphysical and can be removed by a suitable mass subtraction procedure. If a similar mechanism underlied the expansion for the leptonic width, the error on the bottom quark mass could be reduced. In the absence of any understanding of this point, we have adopted a more conservative error estimate than in previous works [13, 14, 15, 16], mainly because of a more generous variation of the renormalization scale. Eq. (14) is in agreement within errors, but larger than the quark mass values quoted there, but is about MeV smaller than the one in [17]. We argued that the result of [17] should be corrected for the large 3-loop term in the relation between the pole mass and the mass.
Eq. (14) is also in good agreement with found in [28]. This work uses the meson mass, a lattice calculation of the (properly defined) binding energy of the meson in the unquenched, two-flavour approximation to heavy quark effective theory, and a two-loop perturbative matching to the scheme. To our knowledge, this is the only other NNLO determination of the mass besides the sum rule calculations mentioned above (which, in fact, are NLO as far as is concerned). However, because of the heavy quark limit, there are corrections, which remain to be estimated. Finally, the result is also in agreement with earlier, parametrically less accurate determinations, as for example in [29].
Acknowledgements. We thank G. Buchalla, A.H. Hoang and A. Pineda for useful discussions and comments on the manuscript. We also thank M. Steinhauser for providing us with the numerical code that produced the fixed-order moments (FO1) in Table 2. This work was supported in part by the EU Fourth Framework Programme ‘Training and Mobility of Researchers’, Network ‘Quantum Chromodynamics and the Deep Structure of Elementary Particles’, contract FMRX-CT98-0194 (DG 12 - MIHT).
For everything else, email us at [email protected]. |
82ef3d9ce255566b | Applied Bohmian Mechanics : From Nanoscale Systems to Cosmology book cover
2nd Edition
Applied Bohmian Mechanics
From Nanoscale Systems to Cosmology
ISBN 9789814800105
Published May 16, 2019 by Jenny Stanford Publishing
700 Pages
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Book Description
Most textbooks explain quantum mechanics as a story where each step follows naturally from the one preceding it. However, the development of quantum mechanics was exactly the opposite. It was a zigzag route, full of personal disputes where scientists were forced to abandon well-established classical concepts and to explore new and imaginative pathways. Some of the explored routes were successful in providing new mathematical formalisms capable of predicting experiments at the atomic scale. However, even such successful routes were painful enough, so that relevant scientists like Albert Einstein and Erwin Schrödinger decided not to support them.
In this book, the authors demonstrate the huge practical utility of another of these routes in explaining quantum phenomena in many different research fields. Bohmian mechanics, the formulation of the quantum theory pioneered by Louis de Broglie and David Bohm, offers an alternative mathematical formulation of quantum phenomena in terms of quantum trajectories. Novel computational tools to explore physical scenarios that are currently computationally inaccessible, such as many-particle solutions of the Schrödinger equation, can be developed from it.
Table of Contents
1. Overview of Bohmian Mechanics
X. Oriols and J. Mompart
2. Hydrogen Photoionization with Strong Lasers
A. Benseny et al.
3. Atomtronics. Coherent Control Of Atomic Flow via Adiabatic Passage
A. Benseny et al.
4. Bohmian Pathways into Chemistry
Á. S. Sanz
5. Adaptive Quantum Monte Carlo Approach States for High-Dimensional Systems
E. R. Bittner, et al.
6. Nanoelectronics. Quantum Electron Transport
E. Colomés et al.
7. Beyond the Eikonal Approximation in Classical Optics and Quantum Physics
A. Orefice, R. Giovanelli, and D. Ditto
8. Relativistic Quantum Mechanics and Quantum Field Theory
H. Nikolic
9. Quantum Accelerating Universe
P. F. González-Díaz and A. Rozas-Fernández
10. Bohmian Quantum Gravity and Cosmology
N. Pinto-Neto and and W. Struyve
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Xavier Oriols is an associate professor at the UAB. His research interests range from quantum foundations to practical engineering of electron devices. He is the author or coauthor of more than 140 papers and has developed the quantum electron transport simulator, named BITLLES, based on Bohmian mechanics.
Jordi Mompart, after a postdoctoral stay at the Leibniz Universität Hannover, he became an associate professor at the UAB. |
270a35c8ca806383 | Quantum-to-Classical Transition
The descriptions of the quantum realm and the macroscopic classical world differ significantly not only in their mathematical formulations but also in their foundational concepts and philosophical consequences. When and how physical systems stop to behave quantumly and begin to behave classically is still heavily debated in the physics community and subject to theoretical and experimental research.
Conceptually different from the decoherence program, in 2007 we introduced a novel theoretical approach to macroscopic realism and classical physics from within quantum theory [1]. It focuses on the limits of observability of quantum effects for macroscopic objects, i.e., on the required precision of our measurement apparatuses such that quantum phenomena can still be observed. First, we demonstrated that for unrestricted measurement accuracy a violation of macrorealism (i.e. as quantified by a violation of the Leggett-Garg inequalities) is possible for arbitrarily large systems. Then we show for certain time evolutions under the restriction of coarse-grained measurements not only macrorealism but even the classical Newtonian laws emerge out of the Schrödinger equation and the projection postulate. This resolves the apparent impossibility of how classical realism and deterministic laws can emerge out of fundamentally random quantum events.
Finally, we demonstrate that there exist “non-classical dynamics” that enable a violation of macroscopic realism even under classical coarse-grained measurements [2]. The question why we then normally do not see such violations arises again. We suggested that the reason for this is that non-classical time evolutions are of high computational complexity. Figuratively, this means that if nature spontaneously “chooses” a time evolution, it is much more likely that a low complex, i.e. a classical, time evolution is realized and thus our every-day world appears classical under coarse-grained measurements.
In Ref. [3] we introduced a necessary condition for violation of macrorealism, which is called “no-signalling in time” and is analogous to the no-signalling condition in the case of Bell’s inequality tests. Most importantly, it can be violated in situations where no violation of the Leggett-Garg inequalities is possible.
(a) The probability for the outcome of a spin component measurement is given by a Gaussian distribution, which can be seen under the „magnifying glass“ of sharp measurements. (b,c) If the measurement resolution is poor, the sharply peaked Gaussian cannot be distinguished anymore from the delta function, and one arrives at deterministic predictions.
[1] J. Kofler and Č. Brukner, Classical World Arising out of Quantum Physics under the Restriction of Coarse-grained Measurements, Phys. Rev. Lett. 99, 180403 (2007).
[2] J. Kofler and Č. Brukner, Conditions for Quantum Violation of Macroscopic Realism, Phys. Rev. Lett. 101, 090403 (2008).
[3] J. Kofler and Č. Brukner, Condition for Macroscopic Realism beyond the Leggett-Garg Inequalities, Phys. Rev. A 87, 052115 (2013).
Additional reading:
Feature of New Scientist (17 March 2007), M. Chown, The illusion of reality in a quantum world, with cover page: Reality is an illusion.
Nature News (22 Nov. 2007): P. Ball, Schrödinger’s kittens enter the classical world.
Seed Magazine cover story (June 2008): J. Roebke, The reality tests.
PhysOrg.com (12 Nov. 2007): M. Marquit, Do classical laws arise from quantum laws? |
6c595cedf3c822cc | Science, Mathematics, And Sufism
I know a professor of theoretical physics, with whom I’ve had many interesting discussions over the years. (Disclosure: I came to Sufism via science.) I wanted to do an interview on the topics we covered with someone who, like me, had progressed from science to Sufism. For those who are the least bit interested in science, physics, and mathematics, the article below will, I believe, prove quite rewarding. The language is simple and no higher mathematics is involved, except only briefly.
Of the “99 Beautiful Names of God,” one is al-Muhsi (The Reckoner, Appraiser, or Accountant): The One who possesses all quantitative knowledge, who comprehends everything, small or great, who knows the number of every single thing in existence. In Arabic, the root HSY connotes “to number, count, reckon, compute,” “to collect in an aggregate by numbering,” “to register or record something,” “to take an account of something.” I conclude that a more concise rendition in English would be: God the Mathematician.
Of course, another of God’s Beautiful Names, the Omniscient (al-Alim), is all-inclusive, so that God’s Knowledge (ilm) encompasses mathematics, physics, and biology alike. But “the Mathematician” makes it more explicit.
In fact, quantity (miqdar) and destiny (qadar) both derive from the root QDR, and thus are inseparably intertwined.
On May 18, 2014, I recorded a lively conversation with my friend, who wishes to remain anonymous. Highlights from that discussion follow. Text in bold, in brackets, and below graphics belongs to me.
The incredibly sophisticated nanotech machine designs within a single cell. Watch it and weep. (Go to “Settings” and select 480 for best view.) Then ask yourself: can this be the outcome of any random collocation of atoms? When a cell dies, it has precisely the same components. Why then do they lie motionless in the case of a dead cell?
So… Where shall we start?
Well… They say, “When a person comes of age, s/he becomes responsible” [religion-wise]. Why? Because a person can comprehend the existence of God by reason alone. The mind is enough to know that God exists.newton1
A flower, a bit of soil, a car. Can these nice things have come about by themselves? We’re talking about initial creation, of course. Once the mechanism is in place, after it becomes self-reproducing, things are easier.
Order, disorder. What I’ve seen in life is, unless it’s cultivated, nothing tends to improvement. If something has a chance of going wrong, it will.
That’s Murphy’s Law.
But there’s such an established order that you don’t have to be a professor, you could be a mountain peasant. When you look around, you see this exists. Your child is born. If you leave it alone, it won’t grow up, the child will die. You have to show it exceptional care. There is no need for intelligence to know that a child has parents. That is, you already know it has parents. And this child that is the universe has a parent too, it has an Owner, a Creator. You go to the moon, you find a color television there. Would anyone in their right mind say, “This TV was formed spontaneously out of the ground”? This is absurd.
But they do say that. It’s called evolution by random mutation.
What do they take refuge in? They take refuge in time. But the law of entropy tells us the exact opposite. Time is more of a negative factor in these matters. Time is something that degenerates, unless there is a driving force supporting the process.
They say that radiation causes the mutations, but in all the examples I know of, radiation has a deleterious effect on living tissue.
Radiation is one of the causes of cancer. “A drowning man will grasp at any straw.” That’s why a child is responsible upon reaching eighteen years of age. Because the child is no longer a child, s/he can analyze and see certain things. As a result, I think the intellect alone is sufficient to comprehend God. Prophethood and so on are something else. They’re more specialized matters.
Now science has a dead-end of this sort. They used to define the law of entropy as: “Left by themselves, systems tend to disorder.” Now they’ve changed this, they’ve removed the word “disorder.” They’re trying to abstract entropy away from disorder, they’re trying not to use the words “entropy” and “disorder” together. This is in the newer textbooks. Because otherwise, you ask: “how did this order come about?” Now they write entropy as an equation, they don’t mention disorder. Physicists have tried to circumvent this, to find a solution to the question of entropy, and have wound up nowhere.
That’s the second law of thermodynamics, isn’t it?
Yes. And a peasant doesn’t call this entropy, but he says, “If you don’t tend your garden, you’ll get weeds.” If you were to bring together all the ingredients of a cell and shake them up, the probability that something will come of that is inconceivably less than 1 divided by 10130, which is already a vanishingly small number. That is, it’s zero. For all practical purposes, this means zero. [See Appendix A. We’re talking about the first living, self-replicating cell.]
But people usually miss the really important point here. If the probability of something occurring randomly is zero, then the probability that it did not occur by chance is a certainty. 1 – 10-130 = 1 – 0 = 1. Now they don’t emphasize that, of course!
Mind-blowing Animations of Molecular Machines inside Your Body [TED]. To claim that all the intricate mechanisms and processes of life could have arisen from inert matter by blind chance, given no matter how many billions of years, is not just an insult to God’s intelligence, but also to our own. It is to elevate the “intelligence” that can emerge from chance to the level of God’s, to impute the highest IQ to random events. Is that anything other than “chance-olatry”—the worship of chance?
And if you say it will form into a cell if shaken for umpteen billion years, that’s an untestable hypothesis, and hence not science. Actually, quite to the contrary, entropy militates that not long afterwards, you’ll have a homogeneous mixture, and it’ll stay that way. Try it with two or three different powders or differently colored liquids, and you’ll see. Shaking more vigorously, adding more energy, doesn’t change the result.
So time is no solution, either. On the contrary, time has an adverse effect. Hence, a mind that can’t perceive this shouldn’t be considered responsible. Because from the point of view of religion, there’s no responsibility when there’s a problem with the intellect. A sacred verse says, “God casts defilement on those who don’t use their reason” (10:100). So you have to use your intellect. There are so many verses that say “men possessed of minds,” “do you not reflect?” But we use our mind for other things. We know very well how to use it for diabolical stuff.
What do scientists do when they’re desperate? They resort to time. Whereas entropy tells us the exact opposite. So a cause is an unavoidable problem. What do you do to get rid of it? You say there was a big bang, and before the big bang there was something else, and before that… you look for a way to wiggle out. Even if you didn’t know about the big bang, I think one ought to know that this can’t be of itself when one beholds this order. One has to see. This is insight.
[For more on this see the Appendix B, taken from another discussion.]
planthoppergears planthoppergear-jumping
Interacting Gears Synchronize Propulsive Leg Movements in a Jumping Insect (Science, 13 September 2013; click on picture at right to view animation (size: 3 MB)). Gear technology designed into legs (and hence the genes and DNA) of young planthoppers. The mechanical gear was invented around 300 B.C. by humans. For millions of years, a 3-millimeter long hopping insect known as Issus coleoptratus has had intermeshing gears on its legs with 10 to 12 tapered teeth, each about 80 micrometers (or 80 millionths of a meter) wide. The gears enable the creature to jump straight. The teeth even have filleted curves at the base, a design also used in human-made mechanical gears since it reduces wear over time.
Right: screw-and-nut system in hip joint of the weevil Trigonopterus oblongus. The screw thread is half a millimeter in size. Weevils, of which there are 50 thousand species, are a kind of beetle, and have been around for 100 million years. These are examples of God’s handiwork in His aspect of Engineer.
You pose a problem in mathematics. One person sees the solution in a second, another sees it in an hour, a third doesn’t see it at all. I think this is like that, with the difference that psychology plays no role in a mathematical problem. Psychology does have a role when you look at nature and infer God. The way you were raised, what your parents taught you, what you received from your surroundings, can prevent you at that point. Because there’s a phenomenon called hypnotism, and this is a form of hypnosis.
I hypnotize someone, I plant the suggestion: “when you wake up, you won’t see that phone.” After they wake up, I ask for the phone. They just can’t find the phone. These experiments have been performed. And human beings are hypnotized like that, only they’re not aware of it. So that person can’t ever find God, because they’ve been hypnotized since childhood.
They’ve been conditioned.
Conditioning takes time. The Prophet said, “Every child is born a Moslem, their parents turn them into something else.” How do they do that? Just so, by conditioning.
So the intellect is very important. But intellect is not enough by itself. Until about the year 1700, we talked trusting our intellect. Science didn’t advance much. We talked for thousands of years. Physics was like history, like geography. Everybody was a physicist.
How did this change? With Newton. Prior to the twentieth century, there are three great scientists: Newton, Galileo, and Maxwell. Maxwell isn’t emphasized that much, but he did something of paramount importance. He’s the one who solidified the mathematization of physics. Newton started the mathematics.
He laid down the “method of fluxions” (differential calculus)…
He introduced mathematics to mechanics. Galileo emphasized the importance of experiment. But Maxwell is the person who wrote down all electromagnetic phenomena in the form of differential equations. So there’s a solid mathematization there. And at that point, a discrepancy in the equations presented itself: a conceptual discrepancy. Maxwell resolved the discrepancy according to his own lights, he balanced the equations by adding another term. That’s when it emerged mathematically that electromagnetic waves exist. And so, we actually owe the foundations of our present technology to Maxwell. The mathematization there is as significant as Newton’s.
The physicists of his time objected. One of the protesters was Faraday. Maxwell mathematized Faraday’s Law, as well. Faraday’s objection at that time was: By itself, mathematics does not include any laws of physics. In other words he’s saying, you’re doing this, but you’re doing it in vain. He objects, he says it won’t contribute much to physics. But Maxwell mathematizes these laws.
Now this is very important in present-day physics. You pose a problem, you build a mathematical model of it. Writing the math is a skill all its own. Maxwell did this, and then the objections ceased. When Newton did it, they said, “You’ve done this, but physics has become a specialized science.” We were all physicists before that. You’ve done this math, but it’s a specialized field. So you’ve reduced it to a very small scope, they said, and the objections continued. The principle of gravitation, for example: you say it’s mathematical, but you don’t explain how it occurs. But after Maxwell, because there was that prediction, the objections ceased.
Hence, mathematics accomplishes a very great thing. Looking at it from the viewpoint of classical mechanics, Galileo says, “Let’s check it with experiment.” The mathematical mind may be beautiful, but it’s not everything. The superiority of the mathematical mind to other kinds of mind is that it is a very concrete form of mind. For instance, there’s water vapor and then there’s ice. But the second is concrete. Water vapor exists, too, but it’s not as tangible as ice.
Now you have your way of thinking, I have mine, she has her own. And the logic of each of us has internal weaknesses which we can’t perceive. But mathematics prevents that. Mathematics has become concrete, that is, it has been tested, formulated, thought through by thousands of people. When you apply mathematics, you’re automatically freed of the weaknesses, the fallacies of your personal logic. So mathematics is a more concrete form of logic, of the mind. feynmanI’m saying this in terms of its application to physics. Otherwise, there are fields where it can’t be applied. It can’t be applied that much to psychology. I don’t know to what extent it will prove applicable to neuroscience, to modeling the brain.
But much that is useful has come of this. We know the seven planets, it was thanks to mathematics that the existence of the eighth planet was proved. Mathematics predicts. You do the calculations, they don’t agree. The coordinates don’t match, they diverge. Either our model is wrong, or something else is afoot that we don’t know about. What is required for this to occur? You say, there has to be a planet of this mass in such-and-such a position. They say, look at this point on this day, at this hour, and you’ll see a planet. That’s how the eighth planet was first sighted. Two astronomers, one French and the other British, are involved. Lo and behold, on that day at that hour at that point, a planet [Neptune] is observed.
Now this invalidates Faraday’s claim. He was saying that mathematics could not make physical predictions on its own. What did it do? It predicted. That is, mathematics is usually regarded as a tool. But it’s slowly going beyond being a tool. It’s becoming a means of discovery. It’s becoming something of a trailblazer, a pioneer. A tool is a thing that helps you do something, it’s passed beyond that.
And the same with the ninth planet, too. This time, perturbations in the orbit of the eighth planet led to the discovery of the ninth [Pluto]. But the ninth planet was discovered with more difficulty. And then it was demoted from the status of being a planet. They call them “dwarf planets.” Because of the tenth planet, the ninth was demoted.
Now, back to Maxwell: he says there’s a discrepancy, a mathematical, a logical discrepancy. As he gets rid of that, he finds a wave equation there. Hence he says, electromagnetic waves exist. He calculates their velocity, it turns out to be the speed of light. Therefore, says he, light is an electromagnetic wave. And these are all things that were subsequently verified experimentally. Hertz, Marconi… The basis of today’s technology and communications lies there. This is one of the major breakthroughs.
What did mathematics do? It paved the way for something. It led to a new discovery. After being confirmed by experiment, of course. In physics, one should never forget that principle of Galileo.
mathpauldirac1Examples of this abound. We now come to quantum mechanics. For instance, in quantum mechanics, Dirac’s equation. Dirac’s equation renders quantum mechanics and relativity compatible with each other. The solutions of this equation are more accurate than those of the Schrödinger equation. But here, too, there is a discrepancy, just as there was in the case of electromagnetics. Then Dirac says, there has to be a particle with the same mass as an electron, but with opposite charge. Within a year or two, the positron is discovered. It was so unexpected that the discoverer was awarded the Nobel Prize. Now what has mathematics done? It has again led to a new discovery, it has served as the means to finding a new physical entity. Again, it has passed beyond being a mere tool. And there are many more examples like this.
Now, physicists are amazed by this. Eugene Wigner wrote an article on “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.”
Stephen Hawking has a saying like that, too. He asks: “What breathes fire into the equations?”
This confusion arises from the assumption that the system excludes the God concept. Otherwise, they wouldn’t be amazed. Because such precision… everywhere there is a logic, a mind, an Infinite Mind at work. Entropy is the cause of our amazement: how can such order exist? The assumptions are wrong. These phenomena clearly tell us that these things can’t happen by themselves, there is an Infinite Mind here.
And as far as physicists are concerned, this is a real conversation-stopper. From here, scientists and philosophers go on to other things. They say [with mathematician David Hilbert]: “Mathematics is a game.” Well, if it is a game, how come it’s so effective in physics? Mathematics is real. But there is no mathematics in nature.
The numbers 2, 3, … don’t exist as objects in nature.
You infer these yourself. For instance, half-integers. Irrational numbers. Rational numbers. Complex numbers. These are entirely constructs of the mind. For example, complex numbers were invented completely independently of physics, so that certain mathematical equations could have a solution. And what do we find, centuries after they were invented? Without complex numbers, quantum mechanics cannot be formulated. There are four or five formulations of quantum mechanics, all of them require complex numbers. There’s just no way to avoid them.
Isn’t it the same with electricity?
No. Complex numbers provide simplicity there. But you can do the calculations without resorting to complex numbers at all. Here, on the other hand, you can’t do anything without complex numbers. You don’t have that luxury.
And here again, the question arises: Weren’t numbers a construct of the mind? Why are mind and nature such an inseparable whole? These are presumably surprising questions for physicists. Also, there is intellect there, but not every intellect. That’s why Galileo is so important. You have to test it against nature, to check whether that intellect is there or not.
For instance, there are four kinds of what are called “division algebras”: real numbers, complex numbers, quaternions and octonions. If a number has an inverse, it’s part of a division algebra. As you move from the first to the last, you lose a property at each stage. Real numbers have the property of ordering: for instance, 5 is greater than 3. With complex numbers, you can no longer say which is greater, 3 + 5i or 5 – 6i. With quaternions, you lose the property of commutativity, and with octonions, you also lose the property of associativity.
Now real numbers and complex numbers are used in nature, but quaternions and octonions are not. A group of physicists tried to formulate quantum mechanics in terms of quaternions, and nothing came of it. And the same holds for octonions. So that’s why experimentation is so important: you have to check the applicability of your mathematics to reality.
In conclusion, the effectiveness of mathematics is unreasonable only if you exclude God. If you include that concept, then it becomes eminently reasonable.
Now Plato says that mathematics has a reality independent of us. He says we access it by extensions of the mind, and project it on the physical world. That’s why it’s called a Platonic reality. And the same with love: you love another, that person doesn’t know anything about it, it’s all in the lover’s mind. That’s why that love is Platonic love.
But this Platonic reality is a peculiar kind of reality. Where would physics be without mathematics? We would still be talking. We would be in the situation that existed prior to 1600-1700. There would still be a physics, crude, experimental, somewhat like meteorology. In meteorology you make forecasts. But is it like that now? I launch a rocket, thanks to my calculations I know where it’s going to fall, down to the centimeter. With our calculations, we can predict the exact time and duration of a solar or lunar eclipse that will happen 100 years from now down to the second. Now these are not trivial things. Mathematics equates with the mind, an intelligence that pervades the entire universe.
Now we have trouble admitting this. So we don’t want to see or hear certain things. The question of entropy remains unresolved. The formation of the first living cell remains unresolved. It cannot be resolved, because there’s the law of entropy. Those experiments have been performed, that organic soup has been made. Stanley Miller did one experiment, Sidney Fox did another. You place the gases you imagine composed the atmosphere at that time, you give the electric current, that corresponds to lightning strokes. You get amino acids. Amino acids are the building blocks of proteins, so you conclude that life emerged from there.
But it’s not merely a giant step, it’s an impossible step, from amino acids to proteins, if you’re going by chance.
OK, how are these organized? Sidney Fox did that experiment. Nothing came of it. By that time, ten years had passed. And nothing would come of it if they were to remain there for ten million years more, because there’s the law of entropy. We say that given time, we’ll solve this. And that’s just kicking the can down the road.
Now, why is mathematics so effective? Because nature is the product of a mind. There’s an Infinite Mind in the universe, a Mind that beggars our minds, that makes us look like mongoloids. Moreover, that Mind also has to possess infinite power, in order to enforce those laws all across the universe, from the macrocosmos down to the microcosmos at every level.
Take a single cell, a single human, a single life form. There’s a phenomenal mechanism there, there’s a monumental set of laws. We’ve understood little bits and pieces of these, that is, what we understand doesn’t amount to much. And that, we understand by isolating. For example, we understand an atom, we try to understand a hydrogen atom.
We act from the principle of linear superposition. We dismantle things like a clock and assume that like a clock, they’ll work in the same way when they’re reassembled.
Of course, because our approach is atomistic. We haven’t seen any other kind, we don’t know. And we can’t wrap our minds around it, because it’s nothing comprehensible. Now a holistic approach, that’s something else. It’s the outcome of a different state of consciousness. Since we’re in atomistic states of consciousness, our minds too are atomistic. If we had holistic states of consciousness, perhaps we would have holistic minds. There are people with holistic consciousness. We don’t always understand what they say, because they’re talking from a different state of consciousness. A butterfly has a consciousness of its own, a mind of its own. A human has a consciousness of his own, a mind of his own. It’s like that, that is. There’s a relationship between consciousness and mind.
You always say that “Quantum physics is holistic”…
Not many people realize this. Before Newton, mathematics is at the level of arithmetic. Until quantum mechanics, in classical physics, we understand events atomistically, that is, we understand them one at a time. We draw diagrams, those diagrams have correlates. The resultant of two forces, and so on.
In quantum mechanics, the dose of mathematics is stepped up even more. But our understanding diminishes. We have difficulty in comprehending the phenomena. In classical physics, we thought we understood the phenomena. We could take events on a piecemeal basis. In quantum mechanics, there’s a helium atom, it has 2 electrons and a nucleus, the nucleus has 2 protons and 2 neutrons. But we deal with it as a system. When we speak of the energy level of the helium atom, we don’t mean the energy level of the electron, the nucleus, or the proton, we consider the energy level of the system. The phenomenon is approached as a whole.
What happens then? We can’t draw a diagram. The diagrams we draw are abstract. Hence, they have no pictorial representation. Pictures are out. So, three stages: first, arithmetic. Next, a physics at the level of calculus. Third, again physics at the level of calculus, but depiction is lost. Because our assumptions changed. We approached the phenomenon holistically.
Why did we do that? Not because we wanted to. We were forced to do so. In order to make sense of the experiments. We can’t comprehend the results of experiments. The experiment is there, but its results don’t make any sense. We had to derive this formulation in spite of ourselves. The experiments forced this on us. And what is essential in physics is the experiment.
Then we sat down and thought about what it was we had discovered. We had found something holistic.
How about a definition of “holistic,” while we’re at it?
First, let’s clarify what we mean by “atomistic.” Let’s say there’s an event in the solar system. We take the sun separately, the moon separately, this planet separately. Then we do our calculations. Each component has an identity of its own. The values of every component are important. Now, for example, the helium atom, the hydrogen atom, the individual states of protons, of electrons, are no longer of importance. We’re looking at it as a system, that is, as a whole. That’s what “holistic” is. In other words, not to go from the parts to the whole, but to deal only and directly with the whole.
To make a jump, could we deal with the universe in the same way?
The wave function of the universe. There have been studies like that. [Everett-Wheeler-Graham (EWG), “The Many-Worlds Interpretation of Quantum Mechanics.”] Here’s what this means: let there be a wave function, let all that can be known in the universe be in that wave function. And in the representation of the hydrogen atom, there’s all the information related to the system.
Now this is a significant jump. First, it places us in a more helpless situation. It’s like Gödel’s theorems in mathematics. What do Gödel’s theorems do? They undermine the foundations of mathematics, they make it more insecure. We used to be determinists, we used to know everything. Now, we don’t know everything. We don’t know what we’re going to find when we conduct an experiment. We can only say, you’ll find this with this probability and that with that probability. And I don’t know how correct that is, because in order to say that with certainty, you’d have to conduct an infinite number of experiments. Only the menu I’m offering you is definite. But I can’t tell you which item you’ll discover. Because this is a holistic matter, there’s an indeterminacy there. There’s always this in holistic things: a lack of certainty. We can’t understand it, but in the end, we can know the energy levels. And we can do this with great accuracy. We can observe them in experiments. And this has been a very great success.
[Quantum electrodynamics, or QED, has been tested to an accuracy of one part in 100 billion (more recently, in 2006, eight parts in a trillion). The famous American physicist Richard Feynman compared this degree of accuracy to mathematically calculating the distance between New York and Los Angeles to within a hair’s breadth. In other words, this is equivalent to predicting the width of North America with the precision of plus or minus one human hair.]
There’s no such thing in classical physics. But actually, there’s a parallel between classical physics and quantum mechanics. Classical mechanics has four or five different formalisms, quantum mechanics has four or five different formalisms. This is not valid for every formalism. For example, the Poisson bracket formalism of classical mechanics is almost the same as the formalism of quantum mechanics, with one difference. The general appearance of the equations is the same.
To me, this looks like the following: in the Koran, they say Ibn Abbas gave a verse’s hidden meaning by interpreting it differently. That’s not what you understand when you read the verse. And I say, that’s what the equation states, but you have to take it as a commutator. That is, there can be different approaches like that in reading the book of nature. There’s actually a one-to-one correspondence, so you penetrate to a deeper level of meaning.
But you can’t logically prove one from the other. That is, you can’t prove the equations of quantum mechanics starting from the equations of classical mechanics. You see the similarity, but there’s no direct proof.
That sounds like pattern recognition, doesn’t it? That is, there’s a form-al similarity.
It’s not just a morphological similarity. For instance, the values of the commutators are identical. So it’s not only a matter of form. Give me the Poisson bracket of anything, I’ll write down its quantum mechanical equivalent for you. This goes beyond form-al. I know the Poisson bracket of a hydrogen atom, of a harmonic oscillator, I can write down the corresponding equation in quantum mechanics, because of this similarity. And the results are phenomenal. This is a different meaning of “a book with twin verses” [the Koran], that is, they have dual meanings. [The book of the universe is here being compared to the Koran.]
Taking the meaning of “verse” (ayah) as “sign” here…
Of course, not as words, but as God’s universe, God’s signs. That is, there’s a signifier in everything. In fact, there are even deeper meanings, and that happens in quantum field theory. Then you give a slightly different meaning.
Now there are operators, and the things they operate on. If you assume commutation relations in the operated (operand), it becomes quantum field theory, that yields even more accurate results. In other words, there are nested meanings. Maybe that’s the case with everything, I don’t know. I’m saying this in terms of physics. But mathematics has an extraordinary role in our discovery of these.
From the viewpoint of physics, however, not every mathematics is always useful. If the assumptions are valid, if you base your mathematics on those, the result is sensational. If the assumptions are wrong, nothing will come of it even if the math is correct.
thought-of-god-ramanujan1That is, mathematics is actually a kind of gardening. Seed, cultivation, result. If the seed is the seed of a thorn, no matter how well you cultivate, you won’t get apples from it. Your seed simply has to be the right seed. And that seed is your assumptions. Why, for example, can’t we reach a result in the case of entropy? We can’t sow the right seed there, due to psychological reasons. That’s our problem. So we continue to be surprised. “The unreasonable effectiveness of mathematics” is not unreasonable at all. Why should you be surprised about the mind of God? [Nothing lies beyond its ken.]
You mean it’s not so hard to pass from science to religion?
You can pass to religion from anything, even from art. Perhaps you’ve heard of the joke: “I used to believe in no God, until I saw her. That’s when my opinion changed.” That is, such beauty can’t be accidental. This art can’t happen of itself. This rose doesn’t grow of itself. This scent doesn’t emerge by itself. This beauty, this intricate design, can’t exist of itself. You don’t have to be a physicist to understand this. Take any phenomenon. After you see the balance, the beauty there, you’ll say, this can’t happen on its own.
Of course, there’s the matter of faith here. Anything can be a cause of faith. But there’s also the verse: “Nobody can have faith unless God desires it” (10:100). Some come to faith easily, others just can’t. But if there has to be an occasion for it, it doesn’t have to be mathematics or physics. But mathematics and physics make it crystal clear. So does medicine. A doctor. If the diagnosis is wrong, you can’t heal no matter what the therapy is, right? But for the diagnosis to be correct, you have to have a firm grasp of the processes. And you have to know that nothing is accidental, you have to know the mechanisms, to be able to reach the right diagnosis.
Feynman explains all this elegantly. There were two objections against Newton: 1. You mathematized physics, you made it specialized. 2. You didn’t explain how gravitation occurs, you called it “action at a distance.” This is magic, and it has to remain so. The sun attracts the earth. How does it do this? The mechanism isn’t described.
This was also Einstein, Podolsky and Rosen’s (EPR) objection to quantum theory. Einstein opposed quantum entanglement on the grounds that it was “spooky action at a distance” (spukhafte Fernwirkungen).
It was everyone’s objection. Einstein turns gravitation into the curvature of spacetime, that has problems of its own. For two hundred years, people tried to devise a mechanism for it. There isn’t any. According to Feynman, there’s no difference between saying that gravitation attracts and that “the angel of gravitation” performs the attraction, because we don’t know what it is.
For example, how does a proton attract an electron? Via an “electric field.” These are just words. Are these empty concepts, or can they be filled with meaning? That’s what we have to look at. Except that in quantum field theory, there’s an exchange of photons. We call them “virtual photons.” This tosses a photon to that, and vice versa. That’s how attraction occurs. Mathematically, many nice things have emerged from this. In the weak interaction (weak nuclear force), there is an exchange of W and Z bosons instead of photons. And in the strong interaction (strong nuclear force), gluons are exchanged. All these are by analogy.
But there’s nothing there. There is no impressive prediction. Those in the know don’t say it out loud, but they know and feel it in their hearts. Because the assumptions are wrong, nothing comes of it. It’s the same in every science. Science is an activity performed by humans, and human beings have egos.
How did you pass from science to religion?
From the intellect rather than from science. But science refines this further. You see the accuracy more clearly. Let’s say that a human with a mind, anyone intelligent enough, can comprehend that all this can’t happen by itself when s/he looks at these relationships, this order, this art. But when you go deeper into the relationships, you discover how finely tuned, how delicate, how highly ordered the relationships are, with such great precision, and that cements it. That’s the real contribution of science.
For instance, a doctor. When a doctor goes into that, s/he begins to see things on more of a micro level. They see much deeper than you or I do. So what happens? That cements it. And the same in other places, as well. For instance, if the distance between the sun and the earth were not what it is, there would be no life on earth. There are a thousand things like that. These things wouldn’t be if the ratio between gravitation and the electromagnetic force were not what it is. You perceive that so many coincidences just can’t coincide by themselves.
I actually found God before science, but science riveted it. For example, human beings, couples. There’s a man, a woman. God created them compatible with each other. From that union, a child is born. He gave affection so that that child could live, He created that environment. The male seed, the female seed, there’s an extraordinary design. At that stage, there’s no need to be a physicist to see this. What’s really important here is the patent. Once the factory is in place and working, things are a bit easier. I built up the shop, I left it to my child and went off. The child’s task is a bit easier. Forming it is more difficult. But if the children can’t take it forward, it’ll degenerate and get closed. “It happened by itself.” If so, why can’t I take it forth? Why can’t the child, who took it over in a ready state, take it forward? Therefore, it didn’t happen of itself.
Now, this logic is all very clear, very simple. But you won’t see it if you don’t want to. That’s the real issue. One has to be blind to not see it. Or you have to have grown up blinded. For me, it’s impossible not to see.
Now, it’s possible to pass to the concept of God from science or art or something else. But how do we go from the God concept to religion?
There, a vehicle is needed. The mind sees: OK, there’s something here. Why is experiment important in physics? The mind can’t solve everything. Reason has to be tested against reality. Experience is more important. Sometimes we know something from experience, we construct its reasoning later.
To understand religion completely, experience is very important. The phenomenon of prophethood. You can’t understand that with physics, with mathematics. The phenomenon of sainthood, you can’t understand it with the intellect. Our Prophet inspired such a sense of trust in everyone, but in spite of that, not everyone believed in him. Either that, or you have to be able to reach great conclusions from small experiences you live. You saw something in your dream, the next day it took place, it came true. This happened once, twice, three times, … There’s no place for this in science. Well then, hold on, friend, there’s something here that eludes your intellect.
Now of course, this gives way to listening, to heeding. Why don’t literate people take religion seriously? Because they trust their own mind and do not listen. They don’t listen, they don’t feel the need to listen. First, they were raised that way. Second, they haven’t had experiences like that to astound them. Even if they have, they feel the immediate need to rationalize it. They bypass it. Otherwise, if only they were to start researching, the place to be reached is clear. There’s a world you don’t know, a whole range of experiences you don’t know about. It’s all here. We call it the world of light. There’s the Realm of Power (Jabarut), the Stage of Nondetermination (La ta’ayyun), right? If you ask when that was, they say it’s all simultaneous. That is, they’re all here, and they’re here according to the level of consciousness you’re in.
You mean they’re not in any temporal sequence.
They’re not. Not anywhere else either, they’re actually here [and now]. That doesn’t mean nobody sees them. And that’s our main error. For example, I study mathematics, but I don’t understand it. That doesn’t mean nobody understands it. Or, there’s going to be an earthquake, a dog hears it, I can’t hear it. In other words, there are things I can’t perceive. For example, elephants can hear a sound from a distance of ten kilometers. Its ears are designed that way. Its trunk is designed to emit that sound. That is, both its transmitter and its receiver are suited to the task. My ears and mouth haven’t been designed for that. So the sizes and frequencies tally. Because its wavelength is greater, its frequency is lower. I would be wrong to claim it doesn’t exist.
This has also been said of vision: of the electromagnetic spectrum, we see only a tiny sliver.
Of course, of course. Now they can photograph the same place in every spectrum. crabnebulaThis is used in science, it’s even used in daily life. A thing that can’t be seen at one frequency can be seen at another. Why didn’t this exist before? It wasn’t done until now because we said, this can’t be. In the infrared spectrum, you see something there that you don’t normally see. So we shouldn’t trust our own perceptions too much, just as we shouldn’t trust our intellect too much.
This is also an ego problem. The stronger your sense of self, the more heedless you are, the more you trust yourself. And the greatest catastrophes occur because of that. It’s also true in daily life: you trust yourself too much, your company folds. And such like.
Either you have to have nonordinary experiences, or you have to have experienced people by your side. They explain certain things to us. But of course, in order to understand these events, holistic concepts are needed. This makes comprehension even more difficult.
Do we need to think holistically in order to understand religion?
Religion [Islam] has its own kind of classical mechanics, that’s the Divine Law. It has its quantum mechanics, that’s Paths and Schools. For example, religion tells us, “Do this and this,” “Don’t do that and that.” These are things at the atomistic level. You have to do them yourself, you’re not exonerated if someone else does them. To understand other concepts, holistic things enter: “He who kills one person, kills entire humankind. He who saves one person, saves entire humankind” (5:32). Or, “Don’t gossip, you’ll put that person’s spirit in pain.” You find that you are no longer yourself, everything is interlocked, everything is connected with everything else.
Holistic concepts are less well-understood, more delicate things. One reads them in one way, another in another. Like in classical physics versus quantum physics. The second taxes you from a holistic viewpoint, you understand with difficulty unless you’re used to it in terms of experience. If not, you shouldn’t deny, you shouldn’t take risks. That’s what the great Sufi saint Ibn Arabi says: “Even if you don’t believe, don’t deny.” Don’t say, How can this be? “What is in the universe, that is in man.” Don’t say this is impossible. You don’t have that, but don’t say nobody can have it, don’t take that risk.
Now this is entirely holistic. Everything is in the human being. “In man there’s a mountain,” as the Master said. Well, I see no such thing? I can’t reconcile a mountain with a human being. Neither my intellect nor my spiritual condition are up to the task. I can’t understand quantum mechanics, either. Nothing in a high school student is ready for quantum mechanics. And those who understand aren’t entirely there either, but at least we agree that there’s truth in it. There’s a similar situation here. You can’t explain everything to everyone, because they won’t understand. Plus, maybe there’s nothing to be understood, only something to be experienced.
Mind alone is not sufficient to discover religion. The mind that comprehends the existence of God is responsible religiously. In order to go beyond that, you need an extra grace from God. Belief in God is a must. For that, the mind is enough. But believing in religion, believing in the Prophet, is a grace from God. There’s a verse to the effect: Noah says, “I’m telling you these things, but they’re no use if God doesn’t wish it.” As Joseph’s brothers are going on their second visit to him, their father Jacob says, “Enter through separate gates. But if God doesn’t desire it, it won’t work.” No matter what you do, it’ll make no difference.
Now we don’t understand this. We don’t understand the will of God. The Master once said, “God scattered a light. It struck some and didn’t strike others.” We don’t know the reason why. In particular, faith in the Prophet rests with God. That is, it’s a very special grace, believing in him is very difficult. Because when you say “God,” you bow to a superior authority. But the Prophet? “Well, he’s human and so am I.” There, the ego enters at once. “He could only have been an ordinary man. The conditions then were such-and-such, he said this, he administered, he was wild,” in the end there’s nothing there. “There was a clever man,” you say. And with that, you miss a lot. You need a special favor to believe that our Prophet was very special, that he was very different, that he was “a mercy to the worlds.” There’s no other way. Or else, God has to have given you the aptitude to derive great conclusions from small experiences. Then it’s possible.
The Master riveted this. I reached that faith only with difficulty: the Prophet is a prophet. But the Master riveted down that faith in place. Our Prophet is very special.
Now, this is very hard to believe. He is the best locus of manifestation the world has ever known. To believe like that is very difficult. Why is that true? Because all the Names of God were manifested in him. There’s no need for someone else. Why is there no need for another Book? Everything is in it [the Koran], even if we can’t understand this. So it’s not necessary. Whereas with the others, it wasn’t like that.
Now it’s hard to accept it like this for our mind-dominated human beings. The ego is strong. Even at birth, children are princes or queens. Those egos won’t bend when they grow up. Here, you need to bend. You need to believe that God gave a mind-boggling boon to someone other than you. But I’m the king… In the language of his state, he says, “If He were to give it to someone, He’d give it to me, I’m king.”
But God favors some human beings. Now, we look at the Koran. What’s there that’s bad about it? It says: “Do good, don’t do evil, don’t harm your neighbor, don’t charge interest on money, don’t be a burden to others.” It counsels all that is good. It says, “Don’t hurt anyone.”
It also defines what is good. It says “This is good, do it, that is bad, don’t do it.” Otherwise, goodness is a relative thing. Thieves think what they’re doing is good.
And that is like abandoning your mind to mathematics. Before Newton, everyone had intelligence. They still do, but everyone does things according to their own lights.
In science, you receive guidance from mathematics, in religion you receive guidance from the Koran.
You have to have a reference. Otherwise, everyone has their own reference point. Take morality. Everyone’s ethics is good from their own standpoint. Why are saints necessary? They hold a mirror to you. They show you yourself, they make you know yourself. Otherwise, nobody is aware of themselves. The Master shows you your error with extraordinary finesse. These things are entirely beyond the ken of contemporary human beings, even conceptually. They can’t even conceive of them, they can’t even conceive what they’re missing. These university professors, these people who think they’re clever, they don’t even know what they’re missing. Meeting the Master, I regard as God’s grace. There’s no other explanation. That is, the mind is at sea here. Everybody’s smart. Many university professors are more intelligent than I am. So this can’t be solely a matter of intelligence, there’s something else. I’m not smarter than they are just because I was graced with the presence of the Master.
I realized that the world is not as I thought it was. This left me shaken. From that I passed on to other things. I already had faith in God, I believed in the Prophet, too. Scientists need experiences that will stagger them, experiences that will shake their belief that they know everything. That’s the only way. Because these are matters of consciousness. In its essence, religion has to do with consciousness. You have to observe changes in your consciousness. You’ll realize then that things are different. There are different states of consciousness: your present state of consciousness, there’s hypnosis, there are different levels in hypnosis, there’s the consciousness of sleep, there’s dream consciousness, there’s lucid dream consciousness. Each is different than the other. And there are who-knows-what-other states of consciousness that I don’t know about.
Would you define religion as consciousness alteration?
Here’s how I view religion: religion is the process of becoming worthy of God by changing one’s morality. But as you alter your ethics, that has an impact on your consciousness. That’s of secondary importance. Being moral is more important than being in a different state of consciousness. The person whose ethics, whose character traits, are closer to the Prophet’s, that person is the winner. This is the primary criterion that I’ve come to understand in the long run.
Morality is very important. For example, we read in the Koran: “I chose him for Myself.” This is about Moses: “I chose you for Myself.” And the same for Abraham: “God chose Abraham as His friend.” Many of Abraham’s morals, character traits, are recounted in the Koran: “Abraham was of mild-mannered mien.” It also tells what God looks at: “He looks at your heart.” “God loves these, God does not love those,” right? “God does not love misers,” “God loves the generous,” God has given all the codes.
Those things all pertain to morality. It doesn’t say, “God loves those who go to Mars in one leap.” It doesn’t say, “God loves those who do Spacefolding.” Nor does it mean that God doesn’t love those who do Spacefolding, but it’s important only in the second-third-fourth degree. It’s not important if it’s not there. The Koran states very clearly: “God loves these, God does not love those.” If we were to list these, that’s where religion is.
Because this is a matter of love. The heart of religion is love. Justice, that’s the Divine Law. Conscience, that’s the Paths. Love is the Reality. [The reference here is to the Master’s pamphlet: “The Secret That is Love.”] The main task is love. In other words, He created human beings out of love. That’s how I understand it. He loves human beings very much.
The Master stated that clearly: “God loved human beings very much.” (Teachings of A Perfect Master, p. 56.)
The “Secret of Islam” is Love, nothing else. But if I remain at the level of a dog or some other animal, how is God going to love me? That is, religion is more a matter of changing one’s state of morality than of changing one’s state of consciousness. The focus is always on ethics.
After the New Age philosophies, this all became: “Let’s change our state of consciousness.” But without a change in one’s state of morality, a permanent change in one’s state of consciousness can’t be obtained. You go up in a helicopter, five minutes later it comes down when it runs out of gas.
For example, let’s get top grades in the exam. How? Let’s cheat. But the means are more important than the ends. To obtain those credentials legitimately. This is actually stated very clearly in the books of great Sufis. For instance, in the “Holy Bestowal” [by Abdulqader Geylani]. Worshipers: worship is very important. Scientists/scholars: knowledge is very important. The wise: the secret and maybe the state of consciousness are very important. But most important of all is the love of God.
Then, the question becomes: “How can we attain that love?” And that’s not possible except by ethics, and that’s a very hard thing to do. If only our ethics were beautified by our saying so, my ethics would have improved long ago. No, that happens by suffering. By suffering hardships. It’s not easy for a rock to become earth. It happens in time, by suffering hardships. It happens by paying careful attention to principles. It happens by paying careful attention to the Prohibited and the Permitted.
Religion is a matter of ethics, a matter of becoming worthy of God by this means. First things first. That’s what God wants. He says, “First fix your ethics, then come to Me.”
Intelligence is also important in these matters. “Who has no mind has no religion.” There’s a Tradition of the Prophet. Someone said: “My friend is highly moral.” The Prophet asked: “How is his intelligence?” “Not that much.” “Then he can’t progress very far.” On the other hand, if you’re not straight inwardly, the more intelligent you are, the more harmful you are.
But the Master posits courtesy. Why? Because courtesy is actually morality. Courtesy is the refined form of morality. If you want the Owner, you have to fix your ethics.
At first, I didn’t understand that. I’m reading the Koran, it says “those who want Paradise,” but it also says “those who want God.” So there is such a concept as desiring God. What is this? It’s in the Koran. So some people desire God more than Paradise. [The Turkish Sufi poet] Yunus Emre said that, and the expression is in the verses of the Koran. But it’s hard to discern it there.
He sang, “I need You and You alone.”
It’s been said, “When God is present, neither heaven nor hell exist,” right? That is something amazing. Because we want to re-establish our severed link with God [re-ligio]. That’s our real quest. Heaven and hell pale in comparison. When you’re dealing with God, everything pales in comparison.
Of course. Compared to infinity, every finite thing is zero.dyson1
It’s like this in our lives, too. How so? When our friends come visiting, we prepare a treat. But our friends don’t come for that bounty, they come for a reunion. The reunion is the important thing, not bounties or Paradise. Now suppose that some come for the food. Well, let them! Let no one remain hungry. But the main point is not the bounty.
Paradise is a boon, a wonderful boon. But in the end, it’s a boon. The phenomenon of Union is very different. What’s important for us is Union, just as it is for God.
I see this in Sufi writings. What God desires is Union. God created human beings for Himself. And He said: “Fix your ethics, and come.” There’s something that will put “blessings such as no eye has seen and no ear has heard” to shame. That must be what they mean by “the Truth of Certainty.” You reach the highest level of proximity. Beyond “the Knowledge of Certainty” and “the Eye of Certainty.” That’s how we see the Master, he’s at the level of the Truth of Certainty.
We’re going to perform the Prayer, we’re going to Fast. But what does the Master say? “Even if your head doesn’t rise from prostration, it won’t happen without these.”
So it’s a matter of ethics. Actually, this is religion: religion is the task of making yourself worthy of God. Can we achieve that? That’s another matter entirely. But that’s the purpose. We don’t know if we can go to Mars, but that’s our calling: to go to Mars. It’s not a matter of knowledge, of consciousness. You can have those too, but there’s a ranking in terms of importance.
The important thing is to display praiseworthy conduct. A man rescues a kitten from the rain, that night he dreams that the Prophet is stroking his beard. So it pleased him. And what’s pleasing to the Prophet is pleasing to God as well. He couldn’t have dreamt that if he had spent that whole night in worship. Let him worship, by all means, but the thing is beauteous conduct. That is, God’s pleasure, something that pleases Him.
broccoli1Mathematics is important because it represents the mind. Physics plus mathematics proves God’s existence. For it is by mathematics that we best analyze nature. The root of the matter is there. Nothing is accidental. Everything is calculated, programmed, precise. And this is a very clear indicator of God’s existence.
If the seed is right, it will yield results. God attaches great importance to the intellect. If you have no mind, you’re not responsible. Because you can deduce the existence of God based purely on reason. If you accept the Prophet too, that’s awesome. And mathematics is important because it has become a means of discovery. But if your assumptions are wrong, mathematics won’t help you. If they’re correct, unexpected things can emerge from that. The mind, mathematics, and experiment have brought us to a place in three hundred years that we hadn’t been able to reach in the previous three thousand. It’s magnificent.
Great scientists, and Dirac is one of them, have arrived at the point that from now on, we need to study consciousness. We don’t know how to study it yet. The Sufi masters have been studying it for centuries.
So where Dirac ends, the Sufi masters begin.
Dirac arrived at that point. So did [Roger] Penrose. And that’s where everyone will arrive at, sooner or later. That’s the point where the masters enter the loop. And then, you have to understand the importance of religion better. You have to perceive that religion is important, that morality is important, that things are not as you imagine them, that the intellect alone is not sufficient, in order to come to that door.
A small protein may typically contain 100 amino acids, each with 20 varieties. For example, the protein histone-4 has a chain of 102 amino acids. The probability of even one small enzyme/protein molecule of 100 amino acids being arranged randomly in a useful (and hence, necessarily specific) sequence would be 1 part in 20100 = ~10130. For comparison, there are ~1080 protons in the entire universe. Even the smallest catalytically active protein molecules of the living cell consist of at least a hundred amino acid residues, and they thus already possess more than 10130 sequence alternatives. Getting a useful configuration of amino acids from the zillions of useless combinations is an exercise in futility. A primitive organism has about the same chance of arising by pure chance as a general textbook of biochemistry has of arising by the random mixing of a sufficient number of letters. And the moment you say that non-chance events are involved, such as the folding and fitting of molecules, you fall outside the field of randomness. You implicitly admit the presence of order.
It appears that some people lack an adequate understanding of either the mathematical law of large numbers, or the physical law of entropy, or both. The law of large numbers (LLN) solidifies the expected probability or improbability of an event. If an event is improbable to begin with, an extremely large number of trials will only certify that improbability.
Actually, the two are linked: “The law of the increase of entropy is guaranteed by the law of large numbers… order is an exception in a world of chance” (Hans Reichenbach, p. 54-55), and the LLN is at the core of the second law of thermodynamics.
It would be unfair to one of the great names in quantum physics, Erwin Schrödinger, if we were to neglect mention here of his monograph, What Is Life? (1944). There, he explicitly associated life with negative entropy, or “negentropy” for short. This also ties in with Information Theory: information is a measure of order, entropy is a measure of disorder, so information is the negative of entropy.
The “randomists”—that’s what I call people who try to explain the origin and development of life by random events occurring over eons—claim that there are highly improbable events which nevertheless occur every once in a while. For instance, winning the lottery is a highly improbable event, yet somebody does win the lottery. And getting a royal flush in a card game is an extremely improbable event, yet it does happen every now and then. Starting from such examples, they argue that highly improbable events can become possible, probable, and even actual, given billions of years.
First, I should perhaps clarify that I’m not opposed to evolution as such. There’s the fossil record and all that. Natural selection exists. Mutations are a fact of life. What I’m against is supposing that extremely highly ordered phenomena, such as we witness everywhere in life, can be the outcome of chance events. Order does not arise spontaneously out of disorder.
[To be more explicit: directed evolution is a possibility, random evolution is not. Nature cannot produce blueprints that have not been encoded into it.]
Now like I said, the reason for this can’t be found in logic. Rather, it’s psychological. Those who make this claim, the “randomists” as you’ve called them, are in a hypnotic state that makes you Godproof. They don’t want to see. These people who impute the most important things to chance: observe them and you’ll see, in their own lives they leave nothing to chance. Because deep in their hearts, they know that chance alone won’t get you there.
The lottery is designed so that at least one person will win. And you need not one, but a run of at least a thousand consecutive royal flushes to even begin to approximate the complexity of life processes. You know Murphy’s law. It says: “If anything can go wrong, it will.” This is actually the law of entropy. And you need, not only intelligence, but also will, to counteract this.
Consider a TV set. One component in the wrong place, and the device won’t work. Now put all the components of a TV set in a sack and start shaking. Do you actually expect that after a sufficient number of shakes, they will all fall into the right place and the TV will assemble itself? First you need a plan, a blueprint. For that, intelligence is needed. And then, you need an iron will and constant, diligent supervision at every step of the way, to ensure that the thing actually gets done. Otherwise, it’s hopeless. Without that, everything tends to disorder, as anyone who’s ever accomplished anything knows firsthand.
Let’s say you’re a Martian, and you see the Mars Rover moving about doing things. There’s no human being around, there’s nothing around, and yet it’s doing those things. It seems to be doing everything by itself, but it’s not. Someone has built it and is guiding it from millions of miles away. A chick lives and dies, but someone has to have programmed it, to have arranged it that way. We now have pilotless planes, but they were planned and developed over time. It didn’t happen all of a sudden.
That reminds me of what a friend once said about the “infinite monkey theorem,” as it’s called. There’s even a jingle about it, which I can’t resist quoting here:babassoon
There once was a brassy baboon
Who used to breathe down a bassoon
He said: “It appears,
in millions of years,
I’m certain to hit on a tune.”
In its simplest form, the infinite monkey theorem states that a monkey randomly punching at the keys of a typewriter (or keyboard) will, given infinite time, type out the complete works of William Shakespeare, without a single error, punctuation marks included. This is one of the arguments set forth to support the idea of evolution by random mutation.
Now this friend was a doctor, and he said this when he was a medical student, when they were studying the intricate workings of the human body. He said: “OK, I’ll accept that a monkey can actually do that, given infinite time. What I cannot accept is that this human body, with its millions of processes going on simultaneously, can ever be the work of chance.”
How do those who deny the lack of randomness do so? They defer to infinite time. Because you can’t test it. Or they invoke higher dimensions. You can’t test that, either. Or they call it a “quantum jump” [punctuated equilibrium]. That is, they throw the issue into untestable territory.
Feynman’s principles here are great. I like his approach. He says no theory is right. For it to be correct, it has to pass an infinite series of experiments. A theory passes an experiment, that means it has passed that experiment, it has not yet been falsified. [The concept of falsifiability was developed by philosopher of science Karl Popper.]
Today, there’s the situation that when a theory doesn’t conform with experimental facts, you go back and mathematically tweak the theory until it does, and hence you remove the possibility of falsifying it. And that’s an illusion.
There’s a couplet by the famous Turkish Sufi poet, Niyazi Misri, that expresses all this in a nutshell:
Nothing is more apparent than God
He is hidden only to the eyeless.
3 comments on “Science, Mathematics, And Sufism
1. Dear Imran Khan,
You have asked:
>how do you find the two related, I mean physics to Sufism.
The two are related through quantum mechanics. Not through its mathematics, but through the interpretation of that mathematics. Of course there have been various interpretations of QM, but one thing that is not in doubt is that QM is “holistic.” In the words of physicist David Bohm, it treats the world as an “undivided whole.” In the interview, it is said that it treats its scope of investigation as a “system.” A collection of fifty atoms or particles is not treated as some kind of sum of fifty separate atoms or particles, but as a single, indivisible system. For this reason, it is difficult to understand, because pictorial representation is not possible. In fact, the observer/subject and observed/object themselves constitute a single whole.
Now Sufism, too, is holistic. In the Koran it says: “Who kills one innocent person (is like one who) has killed all humankind” (5:32). It treats all humanity as a single entity. This is a holistic worldview. And it has been articulated by the famous Sufi Ibn Arabi in particular. Sometimes he sounds as if he is talking about quantum physics. Though not widely known, Sufism’s and Ibn Arabi’s affinity with quantum physics has been noted by various researchers. Google “Ibn Arabi quantum physics” and you will find various examples of this.
NOTE: Modern quantum field theory conceives of physical phenomena as fluctuations of the underlying quantum vacuum. A 2015 Physics Today article described the quantum vacuum as “a turbulent sea, roiling with waves…” This has its exact counterpart in Sufism, which hundreds of years ago conceived of phenomena as waves on the surface of a sea.
“The best credo of all times is that of modern physics — that everything is an unbroken, undivided wholeness.”
—Pir Vilayat Inayat Khan, echoing Ibn Arabi’s famous doctrine of the Unity of Being (wahdat al-wujûd).
—Erwin Schrödinger
2. Rukhsan ul Haq on said:
Dear Henry Bayman
I read your articles with a lot of interest and they always give joyful insights into the wisdom of Islam and what I like about them is the modern langauge based mostly on physics. I am a theoretical physicist myself so they appeal to me in that vein as well…
With lots of love
Bangalore India
3. Rukhsan ul Haq on said:
I have the privilege to have known you through articles and books available from your website. I feel blessed to have the opportunity to cherish the wisdom you share with us and which you have inherited directly from a Sufi master in Turkey. I am a theoretical physicist by profession and a Sufi at heart. So there is no wonder that your articles and writings resonate with me because I see that you present Sufi wisdom in a scientific idiom… I will always behold you with love in my heart.
With best wishes and regards
Rukhsan ul Haq
Bangalore India |
cae656edfa5506bb | Top and bottom envelope functions for a modulated sine wave.
In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes.[1] The envelope thus generalizes the concept of a constant amplitude. The figure illustrates a modulated sine wave varying between an upper and a lower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable.
Example: beating wavesEdit
A modulated wave resulting from adding two sine waves of nearly identical wavelength and frequency.
A common situation resulting in an envelope function in both space x and time t is the superposition of two waves of almost the same wavelength and frequency:[2]
which uses the trigonometric formula for the addition of two sine waves, and the approximation Δλ ≪ λ:
Here the modulation wavelength λmod is given by:[2][3]
The modulation wavelength is double that of the envelope itself because each half-wavelength of the modulating cosine wave governs both positive and negative values of the modulated sine wave. Likewise the beat frequency is that of the envelope, twice that of the modulating wave, or 2Δf.[4]
If this wave is a sound wave, the ear hears the frequency associated with f and the amplitude of this sound varies with the beat frequency.[4]
Phase and group velocityEdit
The argument of the sinusoids above apart from a factor 2π are:
with subscripts C and E referring to the carrier and the envelope. The same amplitude F of the wave results from the same values of ξC and ξE, each of which may itself return to the same value over different but properly related choices of x and t. This invariance means that one can trace these waveforms in space to find the speed of a position of fixed amplitude as it propagates in time; for the argument of the carrier wave to stay the same, the condition is:
which shows to keep a constant amplitude the distance Δx is related to the time interval Δt by the so-called phase velocity vp
On the other hand, the same considerations show the envelope propagates at the so-called group velocity vg:[5]
A more common expression for the group velocity is obtained by introducing the wavevector k:
We notice that for small changes Δλ, the magnitude of the corresponding small change in wavevector, say Δk, is:
so the group velocity can be rewritten as:
where ω is the frequency in radians/s: ω = 2πf. In all media, frequency and wavevector are related by a dispersion relation, ω = ω(k), and the group velocity can be written:
Dispersion relation ω=ω(k) for some waves corresponding to lattice vibrations in GaAs.[6]
In a medium such as classical vacuum the dispersion relation for electromagnetic waves is:
where c0 is the speed of light in classical vacuum. For this case, the phase and group velocities both are c0.
In so-called dispersive media the dispersion relation can be a complicated function of wavevector, and the phase and group velocities are not the same. For example, for several types of waves exhibited by atomic vibrations (phonons) in GaAs, the dispersion relations are shown in the figure for various directions of wavevector k. In the general case, the phase and group velocities may have different directions.[7]
Example: envelope function approximationEdit
Electron probabilities in lowest two quantum states of a 160Ǻ GaAs quantum well in a GaAs-GaAlAs heterostructure as calculated from envelope functions.[8]
In condensed matter physics an energy eigenfunction for a mobile charge carrier in a crystal can be expressed as a Bloch wave:
where n is the index for the band (for example, conduction or valence band) r is a spatial location, and k is a wavevector. The exponential is a sinusoidally varying function corresponding to a slowly varying envelope modulating the rapidly varying part of the wavefunction un,k describing the behavior of the wavefunction close to the cores of the atoms of the lattice. The envelope is restricted to k-values within a range limited by the Brillouin zone of the crystal, and that limits how rapidly it can vary with location r.
In determining the behavior of the carriers using quantum mechanics, the envelope approximation usually is used in which the Schrödinger equation is simplified to refer only to the behavior of the envelope, and boundary conditions are applied to the envelope function directly, rather than to the complete wavefunction.[9] For example, the wavefunction of a carrier trapped near an impurity is governed by an envelope function F that governs a superposition of Bloch functions:
where the Fourier components of the envelope F(k) are found from the approximate Schrödinger equation.[10] In some applications, the periodic part uk is replaced by its value near the band edge, say k=k0, and then:[9]
Example: diffraction patternsEdit
Diffraction patterns from multiple slits have envelopes determined by the single slit diffraction pattern. For a single slit the pattern is given by:[11]
where α is the diffraction angle, d is the slit width, and λ is the wavelength. For multiple slits, the pattern is [11]
where q is the number of slits, and g is the grating constant. The first factor, the single-slit result I1, modulates the more rapidly varying second factor that depends upon the number of slits and their spacing.
See alsoEdit
1. ^ C. Richard Johnson, Jr; William A. Sethares; Andrew G. Klein (2011). "Figure C.1: The envelope of a function outlines its extremes in a smooth manner". Software Receiver Design: Build Your Own Digital Communication System in Five Easy Steps. Cambridge University Press. p. 417. ISBN 0521189446.
2. ^ a b Blair Kinsman (2002). Wind Waves: Their Generation and Propagation on the Ocean Surface (Reprint of Prentice-Hall 1965 ed.). Courier Dover Publications. p. 186. ISBN 0486495116.
3. ^ Mark W. Denny (1993). Air and Water: The Biology and Physics of Life's Media. Princeton University Press. p. 289. ISBN 0691025185.
4. ^ a b Paul Allen Tipler; Gene Mosca (2008). Physics for Scientists and Engineers, Volume 1 (6th ed.). Macmillan. p. 538. ISBN 142920124X.
5. ^ Peter W. Milonni; Joseph H. Eberly (2010). "§8.3 Group velocity". Laser Physics (2nd ed.). John Wiley & Sons. p. 336. ISBN 0470387718.
6. ^ Peter Y. Yu; Manuel Cardona (2010). "Fig. 3.2: Phonon dispersion curves in GaAs along high-symmetry axes". Fundamentals of Semiconductors: Physics and Materials Properties (4th ed.). Springer. p. 111. ISBN 3642007090.
7. ^ V. Cerveny; Vlastislav Červený (2005). "§2.2.9 Relation between the phase and group velocity vectors". Seismic Ray Theory. Cambridge University Press. p. 35. ISBN 0521018226.
8. ^ G Bastard; JA Brum; R Ferreira (1991). "Figure 10 in Electronic States in Semiconductor Heterostructures". In Henry Ehrenreich; David Turnbull (eds.). Solid state physics: Semiconductor Heterostructures and Nanostructures. p. 259. ISBN 0126077444.
9. ^ a b Christian Schüller (2006). "§2.4.1 Envelope function approximation (EFA)". Inelastic Light Scattering of Semiconductor Nanostructures: Fundamentals And Recent Advances. Springer. p. 22. ISBN 3540365257.
10. ^ For example, see Marco Fanciulli (2009). "§1.1 Envelope function approximation". Electron Spin Resonance and Related Phenomena in Low-Dimensional Structures. Springer. pp. 224 ff. ISBN 354079364X.
11. ^ a b Kordt Griepenkerl (2002). "Intensity distribution for diffraction by a slit and Intensity pattern for diffraction by a grating". In John W Harris; Walter Benenson; Horst Stöcker; Holger Lutz (eds.). Handbook of physics. Springer. pp. 306 ff. ISBN 0387952691.
This article incorporates material from the Citizendium article "Envelope function", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL. |
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学术地点 物电学院A栋335学术报告厅 主讲人 Stephen J. Pennycook，Andrew Wee，Karl-Hei
讲座时间 2019年7月5日9:00-12:00 14:30-18:00
主题：International Workshop on Nanomaterials and Nanodevices
时间：2019年7月5日9:00-12:00 14:30-18:00
Designing Energy Materials via Atomic-resolution Microscopy and Spectroscopy
主讲人：Stephen J. Pennycook，National University of Singapore
时 间：2019年7月5日9:10-9:40
讲座摘要：In recent years, the sensitivity of the electron microscope for imaging and spectroscopy has dramatically improved due to aberration correction, greatly assisting the correlation of atomic-scale structure and bonding to materials’properties. Trial and error materials’ development is increasingly being replaced by atomic scale engineering, informed by the powerful combination of microscopy and theoretical calculations.
In catalysis for example, it has become almost routine to image single atoms and probe their coordination by spectroscopy, greatly aiding the development of so-called single atom catalysts (SACs). Their unique coordination can impart exceptional activity and selectivity, and much effort is ongoing to replace platinum group metals by cheaper, earth abundant metals such as cobalt or nickel. One example is the synthesis of graphene-supported cobalt SACs with a tunable high loading using atomic layer deposition that show exceptional activity and selectivity for the hydrogenation of nitroarenes to azoxy aromatic compounds. Single Co atoms are visible in the Z-contrast image, and electron energy loss (EEL) spectra from them show that whenever Co is detected, so also is O; theory then shows that these proximal O atoms expose partially-filled Co-d orbitals, resulting in the excellent catalytic activity. Another example of a cobalt SAC uses porous nitrogen-doped carbon nanoflake arrays as support. These SACs show a lower oxygen evolution reaction (OER) overpotential and higher oxygen reduction reaction (ORR) saturation current than Co nanoparticle catalysts, showing that Co metal clusters are actually redundant for both the OER and ORR reactions. The well-dispersed Co single atoms are the active sites, attached to the carbon network through N−Co bonding. The electrocatalyst was used as the air cathode in a solid-state Zn−air battery, achieving good cycling stability (2500 min, 125 cycles) and a high open circuit potential (1.411 V).
Single atom sensitivity is also important for developing thermoelectric materials. Whereas nanostructuring has been well appreciated, recently the key role of interstitials and interstitial clusters on thermal and electrical transport properties has also been elucidated. In piezoelectrics, gradual atomic-scale polarization rotation among co-existing phases has been recently found in lead-free piezoelectrics, a feature that seems common to all high-performance piezoelectric systems at phase boundaries.
The Organic-2D Heterointerface
主讲人：Andrew Wee，National University of Singapore
时 间：2019年7月5日9:40-10:10
讲座摘要：For more than a decade after the discovery of the unique physical properties of graphene, two-dimensional (2D) materials have been attracting the attention of the nanoscale research community. 2D materials can be stacked on top of the other or be interfaced with organic molecules, and this leads to a paradigm shift in the way nanoscale heterostructures can be artificially fabricated.
In this talk, I will introduce our work on the use of high resolution scanning tunneling microscopy/spectroscopy (STM/STS) to study the atomic structure and local electronic properties of 2D graphene and transition metal dichalcogenides (TMDs) monolayers, e.g. MoS2, WSe2, MoSe2. We show that the electronic bandgaps can be tuned by strain at grain boundaries and dislocations. Using monolayers of adsorbed organic molecules, we demonstrate the surface transfer doping of epitaxial graphene and TMDs. We also discuss the fabrication and electronic properties of a lateral doped/intrinsic heterojunction in 2D WSe2, partially covered with a molecular acceptor C60F48 (Fig1). Using PTCDA as a prototype semiconductor organic molecule, we show that a monolayer TMD can effectively screen an organic-inorganic heterointerface. Recent results of DAP on MoSe2 will also be introduced. The use of organic-2D hybrid heterointerfaces is a promising approach to manipulate the electronic properties for flexible and wearable applications.
On the density of racemic and homochiral crystals:Wallach, Liebisch and Sommerfeld in Göttingen
主讲人：Karl-Heinz Ernst，EMPA, Swiss Federal Laboratories for Materials Testing & Research, Switzerland
时 间：2019年7月5日10:30-11:00
讲座摘要：Alfred Werner (1866–1919) is the undisputed founder of coordination chemistry, but many years passed before his stereochemical insights were accepted. Only after he proved conclusively that metal complexes can be chiral did his model become accepted and earn him the nickname “Inorganic Kekulé” and the Nobel Prize in Chemistry in 1913. But it took more than ten years from the time he predicted chirality in coordination compounds for his group to succeed in separating enantiomers. During the 1980s, reports appeared stating that some of the compounds originally prepared by one of Werner’s students, Edith Humphrey, resolve spontaneously into the enantiomers during crystallization. This led to the claim that Werner could have proven his theory much earlier if he had only tested a single crystal for optical activity. However, our re-examination of the original samples, which are stored in the Werner collection at the University of Zurich, and perusal of the corresponding doctoral theses of Werner’s students, reveals new aspects of conglomerate crystallization in the old samples.
The first comparison of densities of heterochiral crystals with their homochiral counterparts was given by Otto Wallach (Nobel Prize in Chemistry 1910) in an account on carvone bromide crystals in 1895 in Liebigs Annalen der Chemie. Although the well-known mineralogist Theodor Liebisch, professor in Göttingen from 1887 to 1908, wrote the last four pages of that Annalen paper, his colleague from chemistry, Wallach served as sole author. The tedious density measurements in Liebsch’s laboratory were performed by one of the fathers of theoretical physics, Arnold Sommerfeld! We discuss whether Wallach or Liebisch had the idea of a comparative study of crystal densities of racemates and their homochiral analogues and who of the two should be credited.
Tip and local environment induced manipulation of molecular properties
主讲人：Uta Schlickum，Institute for Applied Physics, Technical University Braunschweig, Germany
时 间：2019年7月5日11:00-11:30
讲座摘要：Functionalities of organic molecules on surfaces can be manipulated locally using a scanning tunneling microscope or globally by altering the properties of the local environment. In this talk I will demonstrate these capabilities on the example of a single molecular switch and the intercalation of graphene patches to tailor the electronic surface properties.
In the first example we demonstrate a perfect bipolar switch exploiting a bistability of molecular conformations stabilized by different charge transfers between molecule and metal substrate. Single molecular switches can be addressed individually [1] but also entire ensembles can be switched using the scanning tunneling microscopy tip as a local stimulus [2]. In the latter case hot charge carrier injection into a surface state and the long mean free path of this specific state allows switching for distances of the order of 100 nm.
In the second example we describe a new way to create h-BN/carbon nanostructures. We can alter the electronic and structural properties of hexagonal boron nitride (h-BN) on Rh(111) by the controlled intercalation of carbon between the h-BN and the Rh(111) surface. The carbon atoms – natural impurities in Rh bulk crystals – diffuse to the surface during the h-BN growth and segregate at the surface during cooling. Due to the specific h-BN/Rh(111) interaction, resulting in a strong corrugation of the h-BN superstructure, hexagonal carbon rings are formed situated at specific sites under the h-BN layer. This intercalated carbon rings lead to a modified appearance of the Moiré pattern as well as to altered electronic properties. The observed work-function variations could be shown to affect the local reactivity of the surface by modified preferred adsorption positions of organic molecules.
Low-dimensional Metal Halide Perovskites for Integrated Photonics
主讲人：Anlian Pan，Hunan University, China
时 间：2019年7月5日11:30-12:00
讲座摘要：Low-dimensional metal halide perovskites (PVK) have attracted enormous attentions due to their superior optical and electronic properties, holding promise for integrated laser and photodetector applications.
High-quality low-dimensional PVK nanostructures with natural optical cavities, are suitable for laser applications. Herein, we systhesised CsPbX3 nanorods through a vapor method, then we achieved tunable lasing[1] and investigated the underlying mechanism. Achieving large-area integration of PVKs is of great significance, highly aligned CsPbBr3 nanowire arrays were successfully grown on annealed M-plane sapphire, showing excellent photodetecting performance and ideal for constructing laser arrays[4]. The instability characteristic of PVK has limited the device fabrication, we constructed devices through directly growing CsPbBr3 nanoplates on ITO electrodes, achieving electroluminescence and visulization of carrier transport.
High-quality PVK films are excellent platforms for integrated photodetection. Flexible photodetector arrays were patterned on CH3NH3PbI3-xClx film, demonstrating real-time photosensing. Combining PVK film with erbium silicate nanosheet is a nice solution to achieve high-performance near-infrared photoresponse. In addition to the film structure, 2D ultrathin PVKs with strong quantum confinement have attracted booming attention. TMDcs were found to be ideal substrates for growing few-nanometer-thick PVK, the obtained ultrathin PVK/TMDc heterostructures show outstanding photodetection performance. The above results suggest that high-quality PVKs may open up new opportunities for various applications in high-performance integrated lasers and optoelectronics. More details about our recent works on PVKs can be found in our review paper.
Programmed assembly of terpyridine derivatives into porous, on-surface networks
主讲人：Thomas Jung，Paul Scherrer Institut,，Switzerland
时 间：2019年7月5日14:30-15:00
讲座摘要：Metal organic frameworks and metal organic networks comprise tunable systems made from chemical linkers with different coordinating metals. We here present the complex behavior of symmetric and slightly asymmetric terpyrimidine building blocks in their van der Waals assembly and in their coordination with Cu. V-shaped terpyridine building blocks self-assemble into hydrogen-bonded domains and upon addition of copper atoms undergo metallation with concomitant transformation into a coordination network. Interestingly multiple, energetically similar, structural motifs are observed in both hydrogen-bonded and adatom-coordinated networks and provide insight into the structure function property relation of these tunable 2D and 3D architectures.
Functional Large Scale, Single Layer Hexagonal Boron Nitride
主讲人：Thomas Greber，University of Zürich, Switzerland
时 间：2019年7月5日15:00-15:30
讲座摘要：Two-dimensional (2D) van der Waals materials may be stacked layer by layer and thus allow for the realization of unprecedented properties of condensed matter systems. This perspective relies on the availability of inert 2D materials, where boron nitride is expected to play the first violin.
I will report on our recent progress on the exfoliation of centimeter sized single orientation, single layer boron nitride from its metal growth substrate . To demonstrate the quality of the material on a large scale, it was employed as a packing-layer to protect a germanium wafer from oxidation in air at high temperature. A second set of experiments involved the nanoscale engineering of the h-BN layer with the “can opener effect” prior to the transfer. This allowed the realization of boron nitride membranes with 2 nm voids, across which we measured ion transport in aqueous solutions.
The new BN exfoliation process involves in the first step the application of tetraoctylammonium (TOA) from a water free electrochemical reaction with the h-BN/Rh(111) substrate before the standard hydrogen bubbling. With high-resolution x-ray photoelectron spectroscopy, atomic force microscopy and density functional theory we identify the proximity of the metal substrate to enable covalent functionalization of h-BN with TOA constituents.
Tuning the band structures of graphene nanoribbons via functional group edge modification
主讲人：Jincheng Li，Nanoscience Cooperative Research Center, Spain
时 间：2019年7月5日15:30-16:00
讲座摘要：The tunable electronic structure of Graphene Nanoribbons (GNRs) has provoked great interest due to potential applications in electronic devices as molecular diodes or transistors or as interconnectors or electrodes. On-surface synthesis strategy has been developed to fabricate GNRs with atomic precision [1]. The high precision in the bottom-up synthesis allows to tune their electronic structure via width control, edge topology or chemical doping.
A common strategy for chemical doping of GNRs is through the substitution of carbon atoms by heteroatoms in the organic precursor [1]. However, the on-surface synthesis strategy provides further tuning flexibility, such as the addition of functional groups to the GNR structure. In this talk, I will show you that the functional groups attached to the backbone of GNRs can effectively dope the GNRs. Furthermore, the n-doping or p-doping of GNRs can be precisely controlled through different functional groups. By means of Scanning Tunneling Spectroscopy and Density Functional Theory, I will show you that how the nitrile (CN) functional groups n-dope the GNRs [2], while the amino (NH2) functional group p-dopes the GNRs. Interestingly, the amino (NH2) functional groups can turn narrow chiral GNRs from semiconducting to metallic by doping.
Tuning charge and spin interactions at hybrid organic/metal and organic/topological insulator interfaces
主讲人：Aitor Mugarza Ezpeleta，Catalan Institute of Nanoscience and Nanotechnology, Spain
时 间：2019年7月5日16:30-17:00
讲座摘要：Interfacing materials with different functionalities is an efficient way to manipulate their respective properties and promote the emergence of novel phenomena. Controlling interfacial interactions is however a complicated task in most cases. In that respect, the tunability offered by ligand chemistry in organic materials is an interesting asset that can be exploited at hybrid interfaces. Here I will present two examples where the molecular strategy is employed to tune the interactions of localized transition metal ions with underlying spin-degenerated electrons in non-magnetic metals, and with spin-textured electrons in topological insulators. In both cases, we obtain a comprehensive picture of the phenomenology by combining scanning tunnelling microscopy/spectroscopy, atomic manipulation, X-ray absorption and photoelectron spectroscopy, and ab-initio calculations.
On our systematic study of transition metal phthalocyanines on noble metals we show how the molecular charge redistribution can either quench or enhance the molecular magnetic moment depending on the relative ligand/ion interaction strengths, and how the molecular spin and charge can be manipulated by doping them with alkali atoms one by one. This tunability will be employed to study different intramolecular and molecule-metal spin interactions.
For molecular films on topological insulators, the tunability of ligands is exploited to tune the interaction of Co ions with the underlying topological surface state (TSS), going from the strongly interacting regime where the TSS is quenched in the first quintuple layer , to the weakly interacting regime where both the TSS and the Co magnetic moment are preserved.
Tomonaga Luttinger Liquid Hosted by Line Defects in 2D Semiconductors
主讲人：Matthias Batzill，University of South Florida, USA
时 间：2019年7月5日17:00-17:30
讲座摘要：Mirror twin grain boundaries (MTBs) in MoSe2 or MoTe2 are well-ordered 1D defects, that exhibit metallic properties. The presence of such 1D-metals that are decoupled from the semiconducting host material opens the prospect of studying truly 1D electron systems. We show that dense networks of Mo-rich MTBs can be synthesized by simple incorporation of excess Mo atoms into the lattice structure of Mo-dichalcogenides. The densely packed MTBS enable us to characterize their 1D electronic properties by ARPES . These measurements show signature of spin charge separation consistent with Tomonaga Luttinger Liquid theory and thus prove the true 1D nature of these electron systems.
Structural and Electronic Properties of Germanene
主讲人：Lijie Zhang，School of Physics and Electronics, Hunan University
时 间：2019年7月5日17:30-18:00
讲座摘要：Germanene, the germanium analogue of graphene, is in many aspects very similar to graphene, but in contrast to the planar graphene lattice, the germanene honeycomb lattice is slightly buckled and composed of two vertically displaced sub-lattices . Frist principles total energy calculations have revealed that freestanding germanene is a two-dimensional Dirac fermion system, i.e. the electrons behave as massless relativistic particles that are described by the Dirac equation, i.e. the relativistic variant of the Schrödinger equation. Recently, it has been shown that germanene can be synthesized on various substrates, including MoS2 and Ge2Pt . As predicted germanene’s honeycomb lattice is indeed buckled and the experimentally measured density of states exhibits a V-shape, which is one of the hall marks of a two-dimensional Dirac system. Spatial maps of the Dirac point of germanene synthesized on MoS2 reveal the presence of charge puddles, which are induced by charged defects of the underlying substrate. |
668aeccb6c2806ab | Cover Petrov Yu. I. Delusions and Errors in Fundamental Concepts of Physics. Пер. с рус.
Id: 189300
Delusions and Errors in Fundamental Concepts of Physics.
Пер. с рус.
URSS. 504 pp. (English). ISBN 978-5-396-00812-0.
This book revealed and demonstrated latent or evident errors in the mathematical constructions of the general and special theories of relativity, quantum mechanics, and surface tensions in condensed bodies. This required examination of a wide range of questions concerning the essence of magnetism, effects of relative motion, and wave-corpuscular dualism of particles. Einstein’s errors in deriving the Lorentz transformations and estimating the secular ...(More)precession of planetary perihelion were brought to light. The general relativity theory was shown to be incapable in principle to solve the problem of displacement of the perihelion, for example, of Mercury. The Lagrange formalism was found to be inapplicable to the system of charges moving in magnetic field. As a corollary, many equations in the Dirac and Landau quantum theories which incorporate magnetic field become fictitious. A simple solution of the problems in question is offered by the model of “blinking particles”. The absence of the Laplace compression pressure in small particles was established. The fundamental equation for the jump of the chemical potential on the surface of small particles was obtained, from which the Kelvin and Thomson formulae, as well as the Wulff rule, follow at once. The book is intended for a broad circle of readers with an interest in the fundamental problems of physics.
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Publishers’ Note8
Comments on “Delusions and Errors in Fundamental Concepts of Physics” by Yu. I. Petrov9
Chapter 1. Space. Forces. Fields29
1.1. General concepts of space and forces29
1.1.1. Heinrich Hertz’s views on the principles of mechanics29
1.1.2. Formation of the concept of force and rigid hidden constraints in Hertz’s mechanics35
1.1.3. Dimensionality and metric of space36
1.1.4. Concept of Riemann geometry38
1.1.5. Conception of tensors40
1.1.6. On the principle of relativity47
1.1.7. What general relativity proposes53
1.2. Dynamics of particles in potential field according to modified Hertz mechanics72
1.3. On application of some operators of vector analysis77
1.3.1. Potential Field77
1.3.2. Curl field81
1.3.3. Contrast between potential and curl fields85
1.4. Gravitational and Coulomb potential fields92
1.4.1. Gravitational field93
1.4.2. Electrostatic field95
1.5. Motion of charged particles in magnetic fields96
1.6. Conclusion97
Chapter 2. Electrodynamics101
2.1. On magnetism101
2.2. Magnetic field of a direct current109
2.3. Ampère’s formula121
2.4. Interaction between moving charges124
2.5. Maxwell equations128
2.6. Electromagnetic field energy148
2.7. On inapplicability of Lagrange formalism to magnetic phenomena157
2.7.1. Is the concept of magnetism acceptable to classical physics?157
2.7.2. Analytical mechanics and magnetic interactions158
2.7.3. Van Leeuwen’s theorem166
2.7.4. Failure of attempts to describe motion of a charge in a magnetic field by analytical mechanics168
2.8. Collapse of Larmor theorem173
2.8.1. Zeeman effect and Larmor frequency173
2.8.2. Failure of various proofs of Larmor theorem174
2.8.3. Unreality of Larmor precession179
Chapter 3. Relative motion effects187
3.1. Time and relativity187
3.2. Lorentz transformations190
3.3. Model of blinking particles195
3.4. Doppler effect201
3.4.1. Relative motion of emitter and observer in vacuum202
3.4.2. Change of light frequency and wavelength during relative motion of emitter and observer in vacuum203
3.4.3. Light propagation in moving medium206
3.4.4. Illusory character of Lorentz transformations208
3.5. New view of the effects of relative motion. Relativity theory: unperceived delusion210
3.6. Behaviour of photons in force fields219
3.6.1. Results of general relativity theory219
3.6.2. What the model of blinking particles offers232
3.7. On secular precession of planetary perihelion236
3.7.1. Approach of general relativity theory236
3.7.2. Einstein’s errors238
3.7.3. Inability of general relativity to solve the problem246
3.7.4. A new description of planetary perihelion precession248
3.8. Comparison of the theory with experimental data250
3.8.1. On testing Lorentz transformations250
3.8.2. Change of emission frequency in force fields277
Chapter 4. Wave-corpuscular dualism of particles283
4.1. Introduction283
4.2. Material blinking particles286
4.3. Quantization of blackbody radiation. Corpuscular aspect of radiation290
4.4. Wave aspect of radiation296
4.5. Motion of electrons in atom298
4.5.1. Stationary and non-stationary orbits298
4.5.2. Bohr’s model of atom300
4.5.3. Effect of magnetic field on orbital motion of electrons302
4.5.4. Quantum numbers, selection rules and Zeeman effect. Spin of electron308
4.5.5. Angular momentum of photons318
4.6. Fundamentals of quantum mechanics323
4.6.1. Schrödinger’s wave equation and Heisenberg’s uncertainty principle323
4.6.2. Hamiltonian of a charge moving in a magnetic field345
4.7. Matrix quantum mechanics348
4.7.1. Definition of vectors and matrices349
4.7.2. Linear operators and their matrix representation351
4.7.3. Eigenvectors and eigenvalues351
4.7.4. Types of operators352
4.7.5. Physics and operators352
4.7.6. Commutation relations involving energy354
4.7.7. Angular momentum356
4.7.8. Spin matrices357
4.7.9. Eigenvalues of angular momentum operator358
4.7.10. Dirac’s theory359
Chapter 5. Metaphysical nature of mathematical foundations of quantum mechanics363
5.1. Introduction363
5.2. Derivation of Schrödinger equation365
5.3. Schrödinger equation as a Sturm—Liouville problem366
5.4. Spatial harmonic functions369
5.5. Important solutions of Schrödinger equation371
5.5.1. Harmonic oscillator371
5.5.2. Rotator373
5.5.3. Motion of an electron in a Coulomb field374
5.6. Illusory nature of the mathematics of quantum mechanics376
Chapter 6. Radiation and matter383
6.1. Formation and development of optical laws. Wave-corpuscle confrontation383
6.2. Interaction of radiation with matter393
6.3. Some dead-end difficulties of the wave theory402
6.4. Radiation sources in optics405
6.5. Once again about aether and revision of physical concepts409
6.6. Incorrect Hamiltonian leads to collapse of quantum electrodynamics424
6.7. Is quantum electrodynamics confirmed experimentally?432
6.8. Model of blinking particles as synthesis of wave and corpuscular aspects443
6.8.1. Time as a measure of motion443
6.8.2. Are there waves in vacuum?447
Chapter 7. New view on surface tension451
7.1. Introduction451
7.2. Criticism of mechanical models of surface tension454
7.3. Statistical-mechanical interpretation of internal pressure in liquids459
7.4. Thermodynamics of small particles473
About the author
Petrov Yuri Ivanovich
Born in 1922. Received a mathematical physics degree in 1954 and a Ph.D. degree in mathematical physics in 1967. After graduating from secondary school in 1940 he had been serving for six years in the Red Army participating in the Great Patriotic War. Graduated with Honours from the Physics Department of M. V. Lomonosov Moscow State University in 1949. Since then he has been working at the N. N. Semenov Institute of Chemical Physics, RAS, currently as a Chief Scientific Officer. The main range of scientific interests includes physics of clusters and small particles of metals, their alloys and compounds. Author of 167 scientific papers, including 5 monographs and 3 inventions. |
48db0e18f0053f82 | Intramolecular halogen-halogen bonds?
By analysing the properties of the electron density in the structurally simple perhalogenated ethanes, X3C-CY3 (X, Y = F, Cl), a previously overlooked non-covalent attraction between halogens attached to opposite carbon atoms is found. Quantum chemical calculations extrapolated towards the full solution of the Schrödinger equation reveal the complex nature of the interaction. When at least one of the halogens is a chlorine, the strength of the interaction is comparable to that of hydrogen bonds. Further analysis shows that the bond character is quite different from standard non-covalent halogen bonds and hydrogen bonds; no bond critical points are found between the halogens, and the σ-holes of the halogens are not utilised for bonding. Thus, the nature of the intramolecular halogen⋯halogen bonding studied here appears to be of an unusually strong van der Waals type |
8194057f5eb6f70e | Saturday, 29 July 2017
This is an invitation to see an interpretation of Quantum Mechanics as a geometrical process of energy exchanges that forms what we see and feel as the passage of time! With classical physics representing processes over a period of time as in Newton’s differential equations.
Video explaining the continuum of time.
The Quantum wave particle function Ψ or probability function radiates out as an inverse sphere and is represent by Schrödinger equation with three quantum numbers representing three dimensional space.
In this theory the uncertainly that is formed by the Quantum Wave Particle Function is the same uncertainty we have with any future event at the smallest scale of the creative process.
This process is formed by the spontaneous absorption and emission of light forming the movement of positive and negative charge. With the outer convex surface of the sphere representing positive charge and the inner concaved surface representing negative charge.
Whenever the atoms bond and break there is an exchange of photon energy with the movement of charge and whenever objects touch it is charge that makes contact. Therefore this process forms the ever changing world of our everyday life that we measure as a period of time relative to the atoms of the periodic table.
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07701ea199338edc | Dictionary Definition
1 (theology) being determined in advance; especially the doctrine (usually associated with Calvin) that God has foreordained every event throughout eternity (including the final salvation of mankind) [syn: predestination, foreordination, preordination]
2 the act of determining or ordaining in advance what is to take place
User Contributed Dictionary
1. The act of determining beforehand.
2. Something that has been decided in advance.
Extensive Definition
Determinism (also called antiserendipity) is the philosophical proposition that every event, including human cognition and behaviour, decision and action, is causally determined by an unbroken chain of prior occurrences. With numerous historical debates, many varieties and philosophical positions on the subject of determinism exist from traditions throughout the world.
Philosophy of determinism
It is a popular misconception that determinism necessarily entails that humanity or individual humans have no influence on the future and its events (a position known as Fatalism); however, determinists believe that the level to which human beings have influence over their future is itself dependent on present and past. Causal determinism is associated with, and relies upon, the ideas of Materialism and Causality. Some of the philosophers who have dealt with this issue are Steven M. Cahn, Omar Khayyám, Thomas Hobbes, Baruch Spinoza, Gottfried Leibniz, David Hume, Baron d'Holbach (Paul Heinrich Dietrich), Pierre-Simon Laplace, Arthur Schopenhauer, William James, Friedrich Nietzsche and, more recently, John Searle, Ted Honderich, and Daniel Dennett.
Mecca Chiesa notes that the probabilistic or selectionistic determinism of B.F. Skinner comprised a wholly separate conception of determinism that was not mechanistic at all. A mechanistic determinism would assume that every event has an unbroken chain of prior occurrences, but a selectionistic or probabilistic model does not.
The nature of determinism
The exact meaning of the term determinism has historically been subject to several interpretations. Some, called Incompatibilists, view determinism and free will as mutually exclusive. The belief that free will is an illusion is known as Hard Determinism. Others, labeled Compatibilists, (or Soft Determinists) believe that the two ideas can be coherently reconciled. Incompatibilists who accept free will but reject determinism are called Libertarians — not to be confused with the political sense. Most of this disagreement is due to the fact that the definition of free will, like that of determinism, varies. Some feel it refers to the metaphysical truth of independent agency, whereas others simply define it as the feeling of agency that humans experience when they act.
Ted Honderich, in his book How Free Are You? - The Determinism Problem gives the following summary of the theory of determinism:
In its central part, determinism is the theory that our choices and decisions and what gives rise to them are effects. What the theory comes to therefore depends on what effects are taken to be... [I]t is effects that seem fundamental to the subject of determinism and how it affects our lives.
Varieties of determinism
Causal (or nomological) determinism is the thesis that future events are necessitated by past and present events combined with the laws of nature. Such determinism is sometimes illustrated by the thought experiment of Laplace's demon. Imagine an entity that knows all facts about the past and the present, and knows all natural laws that govern the universe. Such an entity might, under certain circumstances, be able to use this knowledge to foresee the future, down to the smallest detail. Simon-Pierre Laplace's determinist dogma (as described by Stephen Hawking) is generally referred to as "scientific determinism" and predicated on the supposition that all events have a cause and effect and the precise combination of events at a particular time engender a particular outcome. http://www.hawking.org.uk/lectures/dice.html. This causal determinism has a direct relationship with predictability. (Perfect) predictability implies strict determinism, but lack of predictability does not necessarily imply lack of determinism. Limitations on predictability could alternatively be caused by factors such as a lack of information or excessive complexity. An example of this could be found by looking at a bomb dropping from the air. Through mathematics, we can predict the time the bomb will take to reach the ground, and we also know what will happen once the bomb explodes. Any small errors in prediction might arise from our not measuring some factors, such as puffs of wind or variations in air temperature along the bomb's path.
Additionally, there is environmental determinism, also known as climatic or geographical determinism which holds the view that the physical environment, rather than social conditions, determines culture. Those who believe this view say that humans are strictly defined by stimulus-response (environment-behavior) and cannot deviate. Key proponents of this notion have included Ellen Churchill Semple, Ellsworth Huntington, Thomas Griffith Taylor and possibly Jared Diamond, although his status as an environmental determinist is debated.
Biological determinism is the idea that all behavior, belief, and desire are fixed by our genetic endowment. There are other theses on determinism, including cultural determinism and the narrower concept of psychological determinism. Combinations and syntheses of determinist theses, e.g. bio-environmental determinism, are even more common. Addiction Specialist Dr. Drew Pinski relates addiction to biological determinism: "Absolutely. It's a complex disorder, but it clearly has a genetic basis. In fact, in the definition of the disease, we consider genetics absolutely a crucial piece of the definition. So the definition as stated in a consensus conference that was published in the early '90s, it's a genetic disorder with a biological basis. The hallmark is the progressive use in the face of adverse consequence, and then finally denial."
Theological determinism is the thesis that there is a God who determines all that humans will do, either by knowing their actions in advance, via some form of omniscience or by decreeing their actions in advance. The problem of free will, in this context, is the problem of how our actions can be free, if there is a being who has determined them for us ahead of time.
Determinism with regard to Ethics
Some hold that, were determinism true, it would negate human morals and ethics. Counter to this argument, some would say that determinism is simply the sum of empirical scientific findings, making it devoid of subjectivism. Morals and Ethics do not hold the universal permanence that physical rules do (like magnetism polarity), but their very existence can also mean they were an inevitable product themselves. That, possibly through an extended period of social development, a confluence of events formed to generate the very idea of morals and ethics in our minds. In other words, all events that actually occur are unavoidable, proven by the fact that these events do, in fact, occur. The "chicken before the egg?" debate manifests again, here.
Determinism in Eastern tradition
The idea that the entire universe is a deterministic system has been articulated in both Eastern and non-Eastern religion, philosophy, and literature. Determinism has been expressed in the Buddhist doctrine of Dependent Origination, which states that every phenomenon is conditioned by, and depends on, the phenomena that it is not. A common teaching story, called Indra's Net, illustrates this point using a metaphor. A vast auditorium is decorated with mirrors and/or prisms hanging on strings of different lengths from an immense number of points on the ceiling. One flash of light is sufficient to light the entire display since light bounces and bends from hanging bauble to hanging bauble. Each bauble lights each and every other bauble. So, too, each of us is "lit" by each and every other entity in the Universe. In Buddhism, this teaching is used to demonstrate that to ascribe special value to any one thing is to ignore the interdependence of all things. Volitions of all sentient creatures determine the seeming reality in which we perceive ourself as living, rather than a mechanical universe determining the volitions which humans imagine themselves to be forming.
In the story of the Indra's Net, the light that streams back and forth throughout the display is the analogy of karma. (Note that in popular Western usage, the word "karma" often refers to the concept of past good or bad actions resulting in like consequences.) In the Eastern context "Karma" refers to an action, or, more specifically, to an intentional action, and the Buddhist theory holds that every karma (every intentional action) will bear karmic fruit (produce an effect somewhere down the line). Volitional acts drive the universe. The consequences of this view often confound our ordinary expectations.
A shifting flow of probabilities for futures lies at the heart of theories associated with the Yi Jing (or I Ching, the Book of Changes). Probabilities take the center of the stage away from things and people. A kind of "divine" volition sets the fundamental rules for the working out of probabilities in the universe, and human volitions are always a factor in the ways that humans can deal with the real world situations one encounters. If one's situation in life is surfing on a tsunami, one still has some range of choices even in that situation. One person might give up, and another person might choose to struggle and perhaps to survive. The Yi Jing mentality is much closer to the mentality of quantum physics than to that of classical physics, and also finds parallelism in voluntarist or Existentialist ideas of taking one's life as one's project.
The followers of the philosopher Mozi made some early discoveries in optics and other areas of physics, ideas that were consonant with deterministic ideas.
Determinism in Western tradition
In the West, the Ancient Greek atomists Leucippus and Democritus were the first to anticipate determinism when they theorized that all processes in the world were due to the mechanical interplay of atoms, but this theory did not gain much support at the time. Determinism in the West is often associated with Newtonian physics, which depicts the physical matter of the universe as operating according to a set of fixed, knowable laws. The "billiard ball" hypothesis, a product of Newtonian physics, argues that once the initial conditions of the universe have been established the rest of the history of the universe follows inevitably. If it were actually possible to have complete knowledge of physical matter and all of the laws governing that matter at any one time, then it would be theoretically possible to compute the time and place of every event that will ever occur (Laplace's demon). In this sense, the basic particles of the universe operate in the same fashion as the rolling balls on a billiard table, moving and striking each other in predictable ways to produce predictable results.
Whether or not it is all-encompassing in so doing, Newtonian mechanics deals only with caused events, e.g.: If an object begins in a known position and is hit dead on by an object with some known velocity, then it will be pushed straight toward another predictable point. If it goes somewhere else, the Newtonians argue, one must question one's measurements of the original position of the object, the exact direction of the striking object, gravitational or other fields that were inadvertently ignored, etc. Then, they maintain, repeated experiments and improvements in accuracy will always bring one's observations closer to the theoretically predicted results. When dealing with situations on an ordinary human scale, Newtonian physics has been so enormously successful that it has no competition. But it fails spectacularly as velocities become some substantial fraction of the speed of light and when interactions at the atomic scale are studied. Before the discovery of quantum effects and other challenges to Newtonian physics, "uncertainty" was always a term that applied to the accuracy of human knowledge about causes and effects, and not to the causes and effects themselves.
Minds and bodies
Some determinists argue that materialism does not present a complete understanding of the universe, because while it can describe determinate interactions among material things, it ignores the minds or souls of conscious beings.
A number of positions can be delineated:
1. Immaterial souls exist and exert a non-deterministic causal influence on bodies. (Traditional theistic free-will, interactionist dualism).
2. Immaterial souls exist, but are part of deterministic framework.
3. Immaterial souls exist, but exert no causal influence, free or determined (epiphenomenalism, occasionalism)
4. Immaterial souls do not exist — the mind-body problem has some other solution.
5. Immaterial souls are all that exist (Idealism).
Modern perspectives on determinism
Determinism and a first cause
Since the early twentieth century when astronomer Edwin Hubble first hypothesized that red shift shows the universe is expanding, prevailing scientific opinion has been that the current state of the universe is the result of a process described by the Big Bang. Many theists and deists claim that it therefore has a finite age, pointing out that something cannot come from nothing. The big bang does not describe from where the compressed universe came; instead it leaves the question open. Different astrophysicists hold different views about precisely how the universe originated (Cosmogony). The philosophical argument here would be that the big bang triggered every single action, and possibly mental thought, through the system of cause and effect.
Determinism and generative processes
In emergentist or generative philosophy of cognitive sciences and evolutionary psychology, free will does not exist. However an illusion of free will is experienced due to the generation of infinite behaviour from the interaction of finite-deterministic set of rules and parameters. Thus the unpredictability of the emerging behaviour from deterministic processes leads to a perception of free will, even though free will as an ontological entity does not exist. As an illustration, the strategy board-games chess and Go have rigorous rules in which no information (such as cards' face-values) is hidden from either player and no random events (such as dice-rolling) happen within the game. Yet, chess and especially Go with its extremely simple deterministic rules, can still have an extremely large number of unpredictable moves. By analogy, emergentists or generativists suggest that the experience of free will emerges from the interaction of finite rules and deterministic parameters that generate infinite and unpredictable behaviour. Yet, if all these events were accounted for, and there were a known way to evaluate these events, the seemingly unpredictable behaviour would become predictable.
Dynamical-evolutionary psychology, cellular automata and the generative sciences, model emergent processes of social behaviour on this philosophy, showing the experience of free will as essentially a gift of ignorance or as a product of incomplete information.
Determinism in mathematical models
Many mathematical models are deterministic. This is true of most models involving differential equations (notably, those measuring rate of change over time). Mathematical models that are not deterministic because they involve randomness are called stochastic. Because of sensitive dependence on initial conditions, some deterministic models may appear to behave non-deterministically; in such cases, a deterministic interpretation of the model may not be useful due to numerical instability and a finite amount of precision in measurement. Such considerations can motivate the consideration of a stochastic model when the underlying system is accurately modeled in the abstract by deterministic equations.
Arguments against determinism
Libertarianism is the belief that we have complete free will. Compatibilism is a mixture of Libertarianism and Determinism. The negation of determinism is sometimes called indeterminism.
Determinism, quantum mechanics and classical physics
Since the beginning of the 20th century, quantum mechanics has revealed previously concealed aspects of events. Newtonian physics, taken in isolation rather than as an approximation to quantum mechanics, depicts a universe in which objects move in perfectly determinative ways. At human scale levels of interaction, Newtonian mechanics gives predictions that in many areas check out as completely perfectible, to the accuracy of measurement. Poorly designed and fabricated guns and ammunition scatter their shots rather widely around the center of a target, and better guns produce tighter patterns. Absolute knowledge of the forces accelerating a bullet should produce absolutely reliable predictions of its path, or so was thought. However, knowledge is never absolute in practice and the equations of Newtonian mechanics can exhibit sensitive dependence on initial conditions, meaning small errors in knowledge of initial conditions can result in arbitrarily large deviations from predicted behavior.
At atomic scales the paths of objects can only be predicted in a probabilistic way. The paths may not be exactly specified in a full quantum description of the particles; "path" is a classical concept which quantum particles do not exactly possess. The probability arises from the measurement of the perceived path of the particle. In some cases, a quantum particle may trace an exact path, and the probability of finding the particles in that path is one. The quantum development is at least as predictable as the classical motion, but it describes wave functions that cannot be easily expressed in ordinary language. In double-slit experiments, light is fired singly through a double-slit apparatus at a distant screen and does not arrive at a single point, nor do the photons arrive in a scattered pattern analogous to bullets fired by a fixed gun at a distant target. Instead, the light arrives in varying concentrations at widely separated points, and the distribution of its collisions can be calculated reliably. In that sense the behavior of light in this apparatus is deterministic, but there is no way to predict where in the resulting interference pattern an individual photon will make its contribution (see Heisenberg Uncertainty Principle).
Some have argued that, in addition to the conditions humans can observe and the laws we can deduce, there are hidden factors or "hidden variables" that determine absolutely in which order photons reach the detector screen. They argue that the course of the universe is absolutely determined, but that humans are screened from knowledge of the determinative factors. So, they say, it only appears that things proceed in a merely probabilistically-determinative way. In actuality, they proceed in an absolutely deterministic way. Although matters are still subject to some measure of dispute, quantum mechanics makes statistical predictions which would be violated if some local hidden variables existed. There have been a number of experiments to verify those predictions, and so far they do not appear to be violated, though many physicists believe better experiments are needed to conclusively settle the question. (See Bell test experiments.) It is possible, however, to augment quantum mechanics with non-local hidden variables to achieve a deterministic theory that is in agreement with experiment. An example is the Bohm interpretation of quantum mechanics.
On the macro scale it can matter very much whether a bullet arrives at a certain point at a certain time, as snipers are well aware; there are analogous quantum events that have macro- as well as quantum-level consequences. It is easy to contrive situations in which the arrival of an electron at a screen at a certain point and time would trigger one event and its arrival at another point would trigger an entirely different event. (See Schrödinger's cat.)
Even before the laws of quantum mechanics were fully developed, the phenomenon of radioactivity posed a challenge to determinism. A gram of uranium-238, a commonly occurring radioactive substance, contains some 2.5 x 1021 atoms. By all tests known to science these atoms are identical and indistinguishable. Yet about 12600 times a second one of the atoms in that gram will decay, giving off an alpha particle. This decay does not depend on external stimulus and no extant theory of physics predicts when any given atom will decay, with realistically obtainable knowledge. The uranium found on earth is thought to have been synthesized during a supernova explosion that occurred roughly 5 billion years ago. For determinism to hold, every uranium atom must contain some internal "clock" that specifies the exact time it will decay. And somehow the laws of physics must specify exactly how those clocks were set as each uranium atom was formed during the supernova collapse.
Exposure to alpha radiation can cause cancer. For this to happen, at some point a specific alpha particle must alter some chemical reaction in a cell in a way that results in a mutation. Since molecules are in constant thermal motion, the exact timing of the radioactive decay that produced the fatal alpha particle matters. If probabilistically determined events do have an impact on the macro events -- such as when a person who could have been historically important dies in youth of a cancer caused by a random mutation -- then the course of history is not determined from the dawn of time.
The time dependent Schrödinger equation gives the first time derivative of the quantum state. That is, it explicitly and uniquely predicts the development of the wave function with time.
\ i\hbar\frac = - \frac \frac+V(x)\psi
So quantum mechanics is deterministic, provided that one accepts the wave function itself as reality (rather than as probability of classical coordinates). Since we have no practical way of knowing the exact magnitudes, and especially the phases, in a full quantum mechanical description of the causes of an observable event, this turns out to be philosophically similar to the "hidden variable" doctrine.
According to some, quantum mechanics is more strongly ordered than Classical Mechanics, because while Classical Mechanics is chaotic, quantum mechanics is not. For example, the classical problem of three bodies under a force such as gravity is not integrable, while the quantum mechanical three body problem is tractable and integrable, using the Faddeev Equations. That is, the quantum mechanical problem can always be solved to a given accuracy with a large enough computer of predetermined precision, while the classical problem may require arbitrarily high precision, depending on the details of the motion. This does not mean that quantum mechanics describes the world as more deterministic, unless one already considers the wave function to be the true reality. Even so, this does not get rid of the probabilities, because we can't do anything without using classical descriptions, but it assigns the probabilities to the classical approximation, rather than to the quantum reality.
Asserting that quantum mechanics is deterministic by treating the wave function itself as reality implies a single wave function for the entire universe, starting at the big bang. Such a "wave function of everything" would carry the probabilities of not just the world we know, but every other possible world that could have evolved from the big bang. For example, large voids in the distributions of galaxies are believed by many cosmologists to have originated in quantum fluctuations during the big bang. (See cosmic inflation and primordial fluctuations.) If so, the "wave function of everything" would carry the possibility that the region where our Milky Way galaxy is located could have been a void and the Earth never existed at all. (See large-scale structure of the cosmos.)
First cause
Intrinsic to the debate concerning determinism is the issue of first cause. Deism, a philosophy articulated in the seventeenth century, holds that the universe has been deterministic since creation, but ascribes the creation to a metaphysical God or first cause outside of the chain of determinism. God may have begun the process, Deism argues, but God has not influenced its evolution. This perspective illustrates a puzzle underlying any conception of determinism:
Assume: All events have causes, and their causes are all prior events. There is no cycle of events such that an event (possibly indirectly) causes itself.
The picture this gives us is that Event AN is preceded by AN-1, which is preceded by AN-2, and so forth.
Under these assumptions, two possibilities seem clear, and both of them question the validity of the original assumptions:
(1) There is an event A0 prior to which there was no other event that could serve as its cause.
(2) There is no event A0 prior to which there was no other event, which means that we are presented with an infinite series of causally related events, which is itself an event, and yet there is no cause for this infinite series of events.
Under this analysis the original assumption must have something wrong with it. It can be fixed by admitting one exception, a creation event (either the creation of the original event or events, or the creation of the infinite series of events) that is itself not a caused event in the sense of the word "caused" used in the formulation of the original assumption. Some agency, which many systems of thought call God, creates space, time, and the entities found in the universe by means of some process that is analogous to causation but is not causation as we know it. This solution to the original difficulty has led people to question whether there is any reason for there only being one divine quasi-causal act, whether there have not been a number of events that have occurred outside the ordinary sequence of events, events that may be called miracles. Another possibility is that the "last event" loops back to the "first event" causing an infinite loop. If you were to call the Big Bang the first event, you would see the end of the Universe as the "last event". In theory, the end of the Universe would be the cause of the beginning of the Universe. You would be left with an infinite loop of time with no real beginning or end. This theory eliminates the need for a first cause, but does not explain why there should be a loop in time.
Immanuel Kant carried forth this idea of Leibniz in his idea of transcendental relations, and as a result, this had profound effects on later philosophical attempts to sort these issues out. His most influential immediate successor, a strong critic whose ideas were yet strongly influenced by Kant, was Edmund Husserl, the developer of the school of philosophy called phenomenology. But the central concern of that school was to elucidate not physics but the grounding of information that physicists and others regard as empirical. In an indirect way, this train of investigation appears to have contributed much to the philosophy of science called logical positivism and particularly to the thought of members of the Vienna Circle, all of which have had much to say, at least indirectly, about ideas of determinism.
See also
• Albert Messiah, Quantum Mechanics, English translation by G. M. Temmer of Mécanique Quantique, 1966, John Wiley and Sons, vol. I, chapter IV, section III.
• Dennett D. (2003) Freedom Evolves. Viking Penguin, NY, USA.
• "Physics and the Real World" by George F. R. Ellis, Physics Today, July, 2005 — This article seems to make the common error of thinking quantum probability goes on in nature; but its explanation, in terms of homeostasis, of why life is understandable in terms so different from those of microscopic physics is relevant to the distinction between physical and moral determinism.
• Kenrick, D. T., Li, N. P., & Butner, J. (2003). Dynamical evolutionary psychology: Individual
decision rules and emergent social norms. Psychological Review, 110, 3–28
• Nowak A., Vallacher R.R., Tesser A., Borkowski W., (2000) Society of Self: The emergence of collective properties in self-structure. Psychological Review 107
• Epstein J.M. and Axtell R. (1996) Growing Artificial Societies — Social Science from the Bottom. Cambridge MA, MIT Press.
• Epstein J.M. (1999) Agent Based Models and Generative Social Science. Complexity, IV (5)
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Calculating d<p>/dt
1. Feb 2, 2014 #1
we were calculating $$\frac{d<p>}{dt}$$ in class and here are my class notes (sorry for the messiness):
why/how does the term circled in green go to zero??..
here is a separate note when i attempted the same problem..
where did i go wrong?
Last edited: Feb 2, 2014
2. jcsd
3. Feb 2, 2014 #2
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Have you heard about the abstract Hilbert-space formulation of quantum theory? Then I'd recommend to use the Heisenberg picture, where all the time dependence is at the operators that represent observables. For an observable [itex]A[/itex] that is not explicitly time dependent (as is the case for momentum in standard quantum mechanics), you have
[tex]\frac{\mathrm{d} \hat{A}}{\mathrm{d} t}=\frac{1}{\mathrm{i} \hbar} [\hat{A},\hat{H}].[/tex]
The state vectors are time independent. Thus you have
[tex]\frac{\mathrm{d}}{\mathrm{d} t} \langle A \rangle=\frac{\mathrm{d}}{\mathrm{d} t} \langle \psi|\hat{A}|\psi \rangle = \left \langle \psi \left | \frac{\mathrm{d} \hat{A}}{\mathrm{d} t} \right | \psi \right \rangle.[/tex]
So, finally you only need to evaluate the commutator for the time derivative of the observable's operator. The result is Ehrenfest's theorem.
Of course, all this is equivalent to the use of the Schrödinger equation and the scalar product in position representation, but it's somewhat easier to do :-).
4. Feb 2, 2014 #3
If iScience wants to continue in the Schrodinger picture, he can try integration by parts on the integral that he desires to equate to zero noting that ψ → 0 as x → [itex]\pm[/itex]∞.
Last edited: Feb 2, 2014
5. Feb 2, 2014 #4
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Because the term:
[itex] Ψ^{*} \frac{∂^{3}Ψ}{∂x^{3}} =\frac{∂}{∂x}(Ψ^{*} \frac{∂^{2}Ψ}{∂x^{2}})-\frac{∂Ψ^{*} }{∂x}\frac{∂^{2}Ψ}{∂x^{2}}=\frac{∂}{∂x}(Ψ^{*} \frac{∂^{2}Ψ}{∂x^{2}})-\frac{∂}{∂x}(\frac{∂Ψ^{*} }{∂x}\frac{∂Ψ}{∂x})+\frac{∂^{2}Ψ^{*} }{∂x^{2}}\frac{∂Ψ}{∂x}[/itex]
The last term is equal but oposite to your second term in green... the other two are total derivatives, which vanish at the integral's limits. |
f8332e372547afdf | Thursday, January 29, 2009
AWT and definition of intelligence
By AWT correct - i.e. physically relevant - definition of intelligence is rather important, as it can give us a clue about direction of psychological time arrow.
From certain perspective every free particle appears like quite intelligent "creature", because it can find the path of the optimal potential gradient unmistakably even inside of highly dimensional field where interactions of many particles overlaps mutually. Whereas single particle is rather "silly" and it can follow just a narrow density gradient, complex multidimensional fluctuations of Aether can follow a complex gradients and they can even avoid a wrong path or obstacles at certain extent. They're "farseeing" and "intelligent". Note that the traveling of particle along density gradient leads into gradual dissolving of it and "death". The same forces, which are keeping the particle in motion will lead to its gradual disintegration of it.
The ability of people to make correct decisions in such fuzzy environment is usually connected with social intelligence. We can say, motion of particle is fully driven by its "intuition". They can react fast in many time dimensions symmetrically (congruently), whereas their ability to interact with future (i.e. ability of predictions) still remains very low, accordingly to low (but nonzero) memory capacity of single gradient particle. Nested clusters of many particles are the more clever, the more hidden dimensions are formed by. Electrochemical waves of neural system activity should form a highly nested systems of energy density fluctuations.
Neverthelles, if we consider intelligence as "an ability to obtain new abilities", then the learning ability and memory capacity of single level density fluctuations still remains very low. Every particle has a surface gradient from perspective of single level of particle fluctuations, so it has an memory (compacted space-time dimensions) as well. Therefore for single object we can postulate the number of nested dimensions inside of object as a general criterion of intelligence. The highly compactified character of neuron network enables people to handle a deep level of mutual implications, i.e. manifolds of causual space defined by implication tensors of high order. Such definition remains symmetrical, i.e. invariant to both intuitive behaviour driven by parallel logics, both conscious behaviour, driven by sequential logics.
Every highly condensed system becomes chaotic, because intelligent activities of individual particles are temporal and they're compensating mutually here. By such way, the behavior of human civilization doesn't differ very much from behavior of dense gas, as we can see from history of wars and economical crisis, for instance. The ability of people to drive the evolution of their own society is still quite limited in general. We can consider such ability as a criterion of social self-awareness. The process of phase transition corresponds learning phase of multi-particle system.
Interesting point is, individual members of such systems may not be aware of incoming phase transition, because theirs space-time expands (the environment becomes more dense) together with these intelligent artifacts. At certain moment the environment becomes more conscious (i.e. negentropic), then the particle system formed by it and phase transition will occur. The well known superfluidity and superconductivity phenomena followed by formation of boson condensate can serve as a physical analogy of sectarian community formation, separated from the needs/feedback of rest of society. Members of community can be internally characterized by their high level of censorship (total reflection phenomena with respect to information spreading) and by superfluous homogeneity of individual stance distribution, followed by rigidity and fragility of their opinions (i.e. by duality of odd and even derivations in space and time) from outside perspective.
AWT explains, how even subtle forces of interests between individuals crowded around common targets cumulate under emergence of irrational behavior gradually. Because such environment becomes more dense, the space-time dilatation occurs here and everything vents OK from insintric perspective. As the result, nobody from sectarian community will realize, he just lost control over situation.
For example, people preparing LHC experiments cannot be accused from evil motives - they just want to do some interesting measurements on LHC, to finish their dissertations, make some money in attractive job, nurse children, learn French, and so on… Just innocent wishes all the time, am I right? But as a whole their community has omitted serious precautionary principles under hope, successful end justifies the means.
Particle model explains, how even subtle forces of interests between individuals crowded around common targets cumulate under emergence of irrational behavior gradually. For example, nobody of this community has taken care about difference in charged and neutral black holes in their ability to swallow surrounding matter. As a result, nobody of members of such community realizes consequence of his behavior until very end.
And this is quite silly and unscouscios behavior, indeed.
AWT and LHC safety risk
Tuesday, January 27, 2009
AWT and Bohmian mechanics
This post is a reaction to recent L. Motl's comments (1, 2, reactions) concerning the Bohm interpretation of quantum mechanics (QM), the concept of Louis de Broglie pilot wave in particular (implicate/explicate order is disputed here). Bohm's holistic approach (he was proponent of marxistic ideas) enabled him to see general consequences of this concept a way deeper, the aristocratic origin of de Broglie. It's not surprising, Bohm's interpretation has a firm place in AWT interpretations of various concepts, causual topology of implications and famous double slit experiment in particular. After all, we have a mechanical analogy of double slit experiment (DSE) presented already (videos), therefore it’s evident, QM can be interpreted by classical wave mechanics without problem..
Single-particle interference observed for macroscopic objects
AWT considers pilot wave as an analogy of Kelvin waves formed during object motion through particle environment. Original AWT explanation of double slit experiment is, every fast moving particle creates an undulations of vacuum foam around it by the same way, like fish flowing beneath water surface in analogy to de Broglie wave.
These undulations are oriented perpendicular to the particle motion direction and they can interfere with both slits, whenever particle passes through one of them. Aether foam gets more dense under shaking temporarily, thus mimicking mass/energy equivalence of relativity and probability density function of quantum mechanics at the same moment. The constructive interference makes a flabelliform paths of more dense vacuum foam, which the particle wave follows preferably, being focused by more dense environment, thus creating a interference patterns at the target.
By AWT the de Broglie wave or even quantum wave itself are real physical artifacts. The fact, they cannot be observed directly by the using of light wave follows from Bose statistics: the surface waves are penetrating mutually, so they cannot be observed mutually. But by Hardy's theorem weak (gravitational or photon coupling) measurement of object location without violating of uncertainty principle is possible. What we can observe is just a gravitational lensing effect of density gradients (as described by probability function), induced by these waves in vacuum foam by thickening effect during shaking.
Other thing is, whether pilot wave concept supplies a deeper insight or even other testable predictions, then for example time dependent Schrödinger equation does. By my opinion it doesn't, or it's even subset of information contained in classical QM formalism. This doesn't mean, in certain situations pilot wave formalism cannot supply an useful shortcut for formal solution (by the same way, like for example Bohr's atom model) - whereas in others cases it can become more difficult to apply, then other interpretations.
Monday, January 26, 2009
AWT and definition of observable reality
When comparing contemporary physical theories, a natural question can emerge immediately: if AWT is proclamativelly more general, then for example various quantum field or quantum gravity theories, shouldn't it lead to even more solutions, then these theories can supply? And if the vagueness is the main objection against these theories, why we should take care about AWT, after then?
The true is, AWT can lead into virtually infinite number of solutions, because even in quite limited particle system the number of possible states increases by extremely fast way. But AWT introduces a gradient driven reality concept, which is probability driven. Many results of particle-particle collisions aren't simply probable, because they're too rare. Therefore we can see only density gradients inside of dense particles system, not a particles or intermediate states as such. The concept of gradient driven reality is apparently anthropocentric, but it can be derived from AWT concept independently, because only artifacts, which were created by long term evolution of high number of mutations, i.e. by causal time events can interact with reality by gradient driven way.
The probability based approach based on particle statistics brings a rather strict restriction into number of possible solutions of every fuzzy theory. String theorists are aware of this opportunity, so they're trying to apply a statistical approach onto landscape of string theory predictions as well. But because the number of predictions of string theory (~10E+500) roughly corresponds the number of particles states inside of observable portion of Universe, then such approach is phenomenologically identical to AWT, if we simply omit whole intermediate step related to tedious string theory formalism (which is serving like random number generator only) - and if we apply Boltzmann statistics to these states directly.
By such way, the AWT wins over formal theories in simplicity (i.e. by Occam's razor criterion), just because it introduces a gradient driven definition of observable reality into physics, thus reducing the number of possible observable states in it: every object can be observed only and if only it contains some space-time gradient from sufficiently general perspective. For example, the (movement of) density gradients inside of condensing supercritical vapor can be observed, while the molecules (motion) itself not. The single Aether concept i.e. material conditional (antecedent) is sufficient for such decision, if we apply an observability criterion (consequent), thus introducing basic implication vector, which the AWT is based on: if Universe is formed by chaotic/particle environment, then every fluctuation evolved/emerged in it via (number of) causal events would see only the (same number of) causal gradients of it. (... and we can predict an appearance of this observable reality by unique way). By such way, we can always see exactly the part of Universe, which has served for our evolution (space-time emergence) and the observable scope of reality expands gradually. This is the way, how Bohm's implicate/explicate order may be undertood in context of AWT, because implication vector defines a time arrow of causual space-time curvature and subsequent compactification of it here.
The testability of AWT insintric perspective is provided by nonscalar implication vector, which is based on nonsingular (zero or infinite) order of axiomatic tensor. Outside of this perspective AWT remains tautology inherently, whis is given by fact, no assumption can consider itself, or less generally, that no object of observation can serve both like mean, both like subject of the same observation in the same time and space point. Aether concept itself remains a tautology, as it cannot be proven by observation and causual logic without violation of this logic in less or more distant perspective by the same way, like God concept.
It can be demonstrated easily, many conceptual problems of contemporary science simply follows from the fact, the scientists have no clue, what is observable and what not, because of lack of relevant definition of observable reality. By such way, many possible combinations would simply disappear from testable predictions, if we apply the gradient driven statistics or Lagrange/Hamilton mechanics, which is based on it. In particular, the misinterpretation of results of M-M experiment just follows from the fact, scientists didn't realize, the motion of environment isn't observable by waves of this environment. The refusal of deBroglie /Bohmian mechanics is misunderstanding of the same category: scientists didn't realize, deBroglie wave cannot be observable by light wave (so easily), being a wave of the same environment, so that the lack of experimental evidence of deBroglie wave cannot serve as an evidence against Bohmian mechanics.
AWT, emergence and Hardy's paradox
Recently, the fundamental experimental evidence of Hardy's paradox was given, which basically means, quantum mechanics isn't pure statistics based theory following Bell inequalities anymore. The non-formal understanding of this paradox is easy: if every combination of mutually commutable quantities cannot be measured with certainty, how can we be sure about it? Whether some combination exists, which violates such uncertainty? By such way, uncertainty principle of quantum mechanics violates itself on background, thus enabling so called "weak" measurements.
This was demonstrated recently for the case of entangled photon pairs - it can serve as an evidence, even the photons have a distinct "shape", which is the manifestation of the rest mass of photon. This is because the explicit formulation of quantum mechanics neglects the gravity phenomena and the rest mass concept on background: by Schrödinger equation every particle should dissolve into whole Universe gradually - which violates the everyday observations, indeed. Such behavior is effectively prohibited by acceleration following from omni-directional Universe expansion i.e. the gravity potential, so that every locatable particle has a nonzero surface curvature and its conditionally stable at the human scale. From nested character of Aether fluctuations follows, not only single level of "weak" measurement should be achievable here. After all, the fact we can interact with another people and object without complete entanglement can serve as an evidence, the "weak" observation is very common at the human scale.
By AWT every strictly causual theory violates itself in less or more distant perspective due the emergence phenomena. While the classical formulation of general relativity remains seemingly self-consistent (being strictly based on single causality arrow) - the deeper analysis reveals, derivation of Einstein field equations neglects the stress energy tensor contribution (Yilmaz, Heim, Bekenstein and others), which is the result of mass-energy equivalence. This approach makes relativity implicit and infinitely fractal theory by the same way, like the quantum mechanics (which is AdS/CFT dual theory). For example, gravitational lensing, multiple event horizons of charged black holes and/or dark matter phenomena can serve as an evidence of spontaneous symmetry breaking of time arrows and manifestation of quantum uncertainty and super-symmetry in relativity. This uncertainty leads into landscape of many solutions for every theory quantum field or quantum gravity theory, based on combination of mutually inconsistent (i.e. different) postulates.
Such behavior follows Gödel's incompleteness theorems, by which formal proof of rules valid for sufficiently large natural number sets becomes more difficult, then these rules itself - thus remaining unresolvable by their very nature. This is a consequence of emergence, which introduces a principal dispersion into observation of large causal objects and/or phenomena, which cannot be avoided, or such artifacts wouldn't observable anymore. By such way, every strictly formal (i.e. sequential logic based) proof of natural law becomes violated in less or more distant perspective and it follows "More is Different" theorem. AWT demonstrates, this emergence is followed by causal (i.e. transversal wave based) energy spreading through large system of scale invariant symmetry fluctuations (unparticles), which are behaving like soap foam with respect to light spreading and they enable to observe the universe (and all objects inside it) both from excentric, both from insintric perspective simultaneously. The mutual interference of these two perspectives leads to the quantization of observable reality, which is insintrically chaotic, exsintrically causal by its very nature.
In this connection it's useful (..and sometimes entertaining) to follow deductions of formally thinking theorists, like Lubos Motl, whose strictly formal thinking leads him to the deep contradiction/confrontation with common sense and occasionally the whole rest of world undeniably. It may appear somewhat paradoxical, just fanatic proponent of string theory - which has introduced the duality concept into physics - has so deep problem with dual/plural thinking. This paradox is still logical though, if we realize, how complex the string theory is and how strictly formal thinking it requires for its comprehension.
By such way, "emergence group" of dense Aether theory makes understanding of observable reality quite transparent and easy task at sufficiently general level. It still doesn't mean, here's not still a lotta things to understand at the deeper levels, dedicated to individual formal theories. |
db3aec69347e60ca | Friday, August 23, 2013
New Father-and-Son Quantum Text Book
Samarkand, Uzbekistan by Richard-Karl Karlovitch Zommer
Samarkand, one of the world's oldest inhabited cities, once prospered as a trading post on the Silk Road between China and Europe. During the Islamic Golden Age (750 AD -- 1258 AD) the city became a famous focus of Arab scholarship in astronomy, medicine and mathematics. In more modern times, there graduated from the State University of Samarkand a physicist Moses Fayngold, who with his son Vadim, also a physicist, has written a new text book on quantum mechanics, intended for advanced undergraduates and beginning graduate students. I found this book rich and unpredictable and, like the romantic Silk Road metropolis, offering something fresh and exotic around every corner.
Why does the world need yet another book about quantum mechanics? This question was raised by the father. "[The father], who by his own admission, used to think of himself as something of an expert in QM, was not initially impressed by the idea, citing a huge number of excellent contemporary presentations of the subject. Gradually, however, as he grew involved in discussing the issues brought up by his younger colleague, he found it hard to explain some of them even to himself. Moreover, to his surprise, in many instances he could not find satisfactory explanations even in those texts he had previously considered to contain authoritative accounts on the subject." (from the Preface).
Unlike most conventional quantum physics texts which merely explain things, this book also focuses on many of the loopholes, exceptions, imperfections, misunderstandings, man traps and pitfalls that exist in this complex field.
When you buy a new car, you will find an Owner's Manual in the glove compartment that tells you how to change the oil and how to replace the light bulbs. But if you are handy with tools you will also want to purchase the Mechanic's Manual to learn how to do things that only professionals should attempt. And, in particular, to learn things that YOU SHOULD NOT DO. (Never unscrew part A before releasing part B.)
This new quantum text book is the equivalent of a Mechanic's Manual that makes previous text books seem mere Owner's Manuals.
Most quantum text books tell you how to do things, but I have never run across a text book like Moses and Vadim's which tells you WHAT NOT TO DO. Over and over again in this text, I ran across comments to the effect that "The naive way to do this is B, but B will give you the wrong answer. Here's how to do things right." The authors seem to have anticipated many pitfalls that lie in wait for the quantum neophyte and have posted the appropriate warnings. My guess is that these pitfalls are those into which Moses and Vadim have themselves fallen. Niels Bohr once claimed that the definition of an "expert" in a field is a person who has made all the mistakes in that field. In this unusual book Moses and Vadim give you the advantage of that kind of street-smart expertise.
Their book begins by describing some major phenomena that classical physics could not explain (black-body radiation, photoelectric effect, low-temperature specific heats and atomic spectra), then show how one simple concept--the quantization of energy--could correctly reproduce these results.
Moses and Vadim then describe the origin of Louis DeBroglie's hypothesis--that matter possesses a wave-like nature whose wavelength DeBroglie could calculate. Altho this textbook confines itself to non-relativistic quantum mechanics, I was surprised (one surprise of many) to discover that DeBroglie's calculation was motivated by special relativity which means that his discovery is deeper than necessary and transcends its non-relativistic buddies such as the Schrödinger equation.
Using the DB hypothesis to physically justify energy quantization (similar to the way that resonance modes quantize the notes of stringed instruments), Moses and Vadim then use the Superposition Principle for waves to construct an "embryonic quantum mechanics" from which much more good physics can be derived without yet mentioning the Schrödinger Equation.
This book includes in-depth discussions (always accompanied by Moses and Vadim's dependable pitfall warning signs) of most of the conventional topics in quantum theory including Hilbert space, Dirac notation, angular momentum, scattering theory, band structure, quantum tunneling, density matrices, Kaon and neutrino oscillations, quantum entanglement, CHSH, POVMs, CNOT and XOR gates, the Bloch sphere, Zeno's paradox, Schrödinger's Cat, and much much more.
Moses and Vadim also introduce a novel topic they call "submissive quantum mechanics" in which they show how to manipulate potentials to create customized wave functions never before realized in nature--a useful skill that may prove profitable in the emerging field of nanotechnology.
Again and again while reading this book I got the feeling of a wise adviser at my side. The ratio of explanatory text to equations is large--resulting in a lucidity reminiscent of the classic Feynman Lectures as well as Quantum Theory by David Bohm.
Besides devising the shortest proof of Bell's theorem, Nick Herbert's main claim to physics fame is his FLASH (First Laser-Amplified Superluminal Hookup) proposal which purported to send signals faster-than-light using a "laser-like device" to clone single photons. The FLASH proposal was refuted by Wooters and Zurek who proved that "a single (unknown) photon cannot be cloned", a result which crucially limits what quantum computers can do--for instance, when quantum hard drives or quantum DVDs are built, the no-cloning theorem provides automatic copy protection courtesy of the laws of physics.
Naturally I was curious about how Moses and Vadim would deal with my FLASH proposal in their hyper-informative "Mechanic's Manual" style. In this I was not disappointed.
The authors agree that the W&Z "no perfect cloning of unknown states" proof definitively refutes my FLASH proposal. But what about "imperfect cloning"?, they ask. And what about the cloning of states that are not completely unknown but part of a small prearranged set of known states? Moses and Vadim carefully consider these loopholes (and a few more) to the standard FLASH refutation and definitively decide that FLASH won't work. But in the course of their detailed refutation the reader learns a lot about quantum cloning machines.
This book is a wonderful Mechanic's Manual crammed full of intimate details about the operation of one of the most elegant intellectual sports cars we possess--the theory of non-relativistic quantum mechanics. But in addition to this Mechanic's Manual, I urge you to also purchase an Owner's Manual of your choice, a book that you can use to solve everyday problems in simple ways. (My own favorite Owner's Manual is the classic text by Leonard Schiff from which I learned QM in those bygone days when the world's largest particle accelerator was the Berkeley Bevatron.)
But next to your trusted Owner's Manual, be sure to include this helpful Mechanic's Manual on your book shelf, both to deepen your knowledge of quantum mechanics and to help you avoid some of its more obvious pitfalls.
This book is perfect for those quantum mechanics who know how to fix Volkswagons and now want to go to work on Porsches.
New father-and-son quantum text book |
ebef8d7ed9563d79 | Wednesday, June 30, 2010
Déjà vu
Last week, on the workshop in Bonn, I was in for a nasty surprise. Sitting there, listening to one talk after the other about black holes, I saw pictures reappear that I had made. Four different pictures of mine, in four different talks. All without picture credits. When I told the speakers later that they've been using a picture that took me in some cases hours to make without even putting my name below it, they apologized. One shrugged shoulders and said "It came up in Google." I checked that, it did come up when doing a Google image source for "Black Hole Evaporation," the source being my home page. I'm not surprised by this, my homepage has always been well indexed by Google. Apparently I was expecting too much when thinking people could at least look at the front page and find my name.
I will admit that I am very dismayed by this. Yes, I too sometimes do use other people's figures and plots in my talks, but I usually add a source, if possible to find. It's more complicated with photos, who will typically appear in so many copies on some dozen websites that it's next to impossible to find out who originally took the photo. In any case, some of the pictures I saw reappearing in those talks I don't even hold the copyright on. They were published in one of my papers, and with that the copyright went to the publisher.
I don't mind at all if people use my pictures, otherwise I wouldn't upload them to my website. I receive the occasional email from somebody asking if they can use one or the other for a talk or a paper and I always say yes. (I once was asked for a picture to be reprinted in a popular science book, but when the publisher of my picture was asked for the reprint permission they said no for reasons I still don't understand.) But of course I do expect that people add at least my name below it. It has previously happened that I saw pictures of mine reappear, this one showing an evaporating black hole seems to be the favorite
but that workshop convinced me to add my name in a corner of all these pictures. Sure, one can cut it out, but it takes a deliberate effort.
This also reminds me that I once received a paper for peer review. It was written in dramatically bad English, then all of a sudden there were two paragraphs that weren't only readable but sounded eerily familiar. A quick check confirmed my suspicion that it was an introduction from one of my own papers. They had cited my paper somewhere, but it was by no means clear they had copied half a page from it. Again, my paper was published, the copyright was with the publisher. The paper I reviewed wasn't only badly written but also wrong, so it didn't get published. However, I later wrote to the authors making it very clear that this is not an appropriate way to cite. They either mark it as a quotation, or they rewrite it. They apologized and then rearranged a few words here and there. I know other people who have made exactly the same experience with one of their papers.
I find it very worrisome that more and more people make so unashamedly use of other's work without even thinking about it. My mother is a high school teacher and as a standard procedure she'll have to check every essay for whether it's been copied elsewhere. Evidently, there's still kids stupid enough to try nevertheless. I know these checks are being done in many other places too, there's even software for it so you don't have to Google every sentence manually. An extreme case that I know of was a PhD candidate who had copied together half of his thesis from other people's review articles, including equations, references and footnotes. He did cite the papers he used, but certainly didn't mark the "borrowed" pieces as quotations.
It is clear that when thousands of people write introductions to the same topic, then many of them will sound quite similar. I also understand that when you find a nice picture for your talk online it seems superfluous to spend time yourself on what Google gives it to you on a silver plate. Certainly you have better things to do than making a pictures for your talk, right? But what you're doing is simply using someone else's effort and selling them as your own. So next time, spend the three seconds and check whose homepage you've been downloading your pictures from.
And here's a recent copyright story that I found hilarious "Greek man sues Swedish firm over Turkish yoghurt pic"
"A Greek man has sued a dairy firm in southern Sweden after his picture ended up on a Turkish yoghurt product. The man whose picture adorns the Turkish yoghurt product, manufactured by Lindahls dairy in Jönköping, argues that the company does not have permission to use his image [...]
The man, who lives in Greece, was made aware of the use of his picture on the popular Swedish product when an acquaintance living in Stockholm recognized his bearded friend [...]
In his writ the man has underlined that he is not Turkish, he is Greek, and lives in Greece, and the use of his picture is thus misleading both for those who know him and for buyers of the product.
Lindahls dairy has expressed surprise at the writ and argues that the image was bought from a picture agency [...]"
Monday, June 28, 2010
The left-handed Piano
As a left-hander, I have an early hands-on experience with the concept of chirality, or handedness: It can be quite difficult to cut a piece of paper with the left hand using standard scissors; the blades usually do not close precisely, resulting in a frayed cut. And of course, scissors with modern, "ergonomically-formed" handles cannot be used with the left hand in the first place.
There is a small niche market for all kinds of chiral partners of standard right-handed everyday products and tools: left-handed scissors, left-handed can-openers, left-handed pencil sharpeners. However, I do not utilize any of them, and use standard instruments with the right hand instead.
Today, I heard on the radio about something really amazing on the market on left-handed products: There are left-handed pianos!
Invented by Geza Loso, musician, piano teacher, left-hander and father of three left-handed kids, they are exact mirror images of usual pianos, with the pitch rising from the right to the left. As Geza Loso explains on his website: For the first time left-handed people receive a real chance to learn how to play the piano on an adequate instrument. Left-handed people would basically use their right hand to accompany and the skilled hand to handle the main functions of a piano-play, to play the melody. This is very decisive for every artistic interpretation.
The left-handed piano will be distributed by the Leipzig piano-manufacturing company Blüthner. Chief executive Christian Blüthner doesn't expect a big commercial success, but thinks that the left-handed piano demonstrates his company's inventiveness. And I am wondering if my career with the piano may have have been longer than a couple of lessons if the instrument would have been left-handed.
Saturday, June 26, 2010
Hello from Bonn
Stefan and I, we are currently in Bonn for a workshop on "Black Holes in a Violent Universe." Bonn is the former German capital and a quite charming city, though not what you'd expect from a capital. So probably a good thing Berlin has taken over the burden. Germany is collectively in a good mood these days since the Germans won Friday's soccer game, and everybody is looking forward to Sunday's game.
We're staying in a small hotel near the river Rhine. Needless to say, our room is on the 4th floor without elevator. On the other hand, we have a small roof patio. And here's what we found looking out of the window on the side opposite the patio: A small staircase leading to a platform (the top of the downstairs windows) with railing. That little walkway ends then, leaving you with the only option of a 4 floors' jump down on the paved street. I was thinking it might be the emergency exit, but the evacuation plan on our door points another direction. So not sure what this is. An invitation for suicide? A diving platform in case the river floods?
My talk about the black hole information loss problem went very well (slides here). I wish you all a great weekend.
Thursday, June 24, 2010
Guestpost: Marcelo Gleiser
[A month ago, I was at a workshop at Perimeter Institute and I reported on a talk by Marcelo Gleiser. Marcelo's talk was very interesting and thought-stimulating. It touched upon very many different topics, from the process of knowledge discovery to the question of whether we should be searching for a fundamental theory of everything. In my post I expressed my opinion that of course believing in a theory of everything, if you take the name literally, is religion not science because if we had one we would never know if not one day we'd discover something that the theory would not explain. But the whole question of whether it exists is somewhat besides the point, the actual question (for me, the pragmatist) is what is a promising approach to take that will lead to progress.
Marcelo has now written a reply to some of the points that came up in my post and the comments, and to some other reactions that he got. This reply can also be found at his blog 13.7.]
To Unify Or Not To Unify: That Is (Not) The Question
My latest book, A Tear at the Edge of Creation, came out in the US early April. In it, I present a critique of some deeply ingrained ideas in physics. In particular, I examine the question of unification and the search for a theory of everything, arriving at conclusions that—judging from some of the reactions I’ve been getting in lectures and in various blogs around the world—are shocking to many people.
Of course, I welcome criticism and skepticism. We are used to this in scientific debates. What’s surprising to me, and perhaps alarming, is the speed with which superficial commentary in the blogosphere quickly escalates into complete misunderstanding of what it is that I am saying and why. So, I think the time is ripe for sketching a reply, even though the space here won’t do justice to the details of the argument. I do hope, however, that this will at least inspire critics and skeptics to actually read the book and judge for themselves and not through a few lines on a blog post.
Among other things, in the book I suggest that the notion of a final theory, that is, a theory that encompasses complete knowledge of how matter particles interact with one another, is impossible. First, note that “final theory” here deals only with fundamental particle physics. Any claim that physical theories could be complete in the sense of describing (and predicting) all natural phenomena, including why you’re reading this, shouldn’t be taken seriously.
First, we must consider if a complete theory of matter does exist. Second, assuming it does, if we can ever get to it. The first question is quite nebulous. We have no way of knowing if such a complete theory exists. We don’t even know what a “complete” theory is. You may believe it does and spend your life searching for it. That’s a personal choice. Or, like most physicists, you may believe this is nonsense, more metaphysics than physics. The second question, though, is tangible. Can humans achieve complete knowledge of the subatomic world?
To answer this question, we must look at how science actually works. In a post at her blog Back Reaction, physicist Sabine Hossenfelder expressed her surprise at my statement that it took me 15 years to figure out that the notion of a final theory is faulty. Sorry Sabine, I guess old habits are hard to break. At least, I did see the light in the end. Happily, she agreed with my basic argument, that since what we know of the world depends on our measurements of the world, we can never be sure that we arrived at a final explanation: as tools advance, there is always room for new discoveries. Knowledge is limited by discovery.
I go on to describe how the unifications that we have achieved so far, beautiful and enlightening as they are, are approximations and not “perfect” in any sense. The electroweak theory, a unification of the electromagnetic and the weak nuclear forces, is not a true unification but a mixing of the two interactions. Even electromagnetism, the paradigm of unification, only works flawlessly in the absence of sources. To be a truly perfect unification, objects called magnetic monopoles would have to exist. And even though they could still be found, their properties are clearly very different from the ubiquitous electric monopoles, e.g. point-like particles like electrons. We have partial unifications and we should keep on looking for more of them. This is the job of theoretical physicists. The mistake is made when symmetry, a very useful tool in physics, is taken as dogma.
I don’t agree with Sabine when she says that it doesn’t matter what you believe in as long as the search “helps you in your research.” I think beliefs are very important, and to a large extent drive what it is that we are searching and the cultural context in which research is undertaken. Wrong beliefs can have very negative consequences. And can keep us blind for a long time.
So, one of the points I make is that science is a construction that evolves in time to expand our body of knowledge through a combination of intuition and experimental consensus. There is no end point to it, no final truth to arrive at.
Now, here are some of the things that have been said about my arguments:
“Marcelo is disillusioned with unification; he has closed up his mind to string theory; he couldn’t find a Theory of Everything and now thinks no one can find one as well; he’s just frustrated; he doesn’t understand the role of symmetry in physics (!); he’s timing is bad because the LHC will be revealing new physics.” George Musser, at a Scientific American blog post wrote “My own reaction was that although it’s useful to caution against clinging to preconceived ideas about a final theory, Gleiser was too insistent on seeing the glass of physics as half-empty.” Musser goes on to say how much we do know about Nature and how much of that is due to the fact that simple laws govern natural phenomena.
It’s true that Musser (and Sabine) were basing their comments on a lecture I gave recently at the Perimeter Institute and not on my book (you can watch the video here). Even so, as I tried to make clear in my text, I would never put down the remarkable achievements of science and much less be foolish to say that there are no patterns and symmetries in Nature! After all, that is how science works, by searching for simplifying explanation of natural phenomena. Having the LHC turned on and able to probe physics at energies higher than ever before is a very exciting prospect.
The same general defensive zeitgeist was echoed by Neil Turok, the current director of the Perimeter Institute. We recently participated in a televised debate hosted by TV Ontario on Stephen Hawking’s ideas. We were a group of six physicists, hosted by Steve Paikin and had a great time. But at the end, when I made my arguments about final unification and the limits of knowledge, Turok accused me of pessimism!
If anything, my book is a celebration of the human mind and all that we have achieved in such a short time. The fact that I point out that science has limitations doesn’t detract from all of its achievements. Or from all that lies ahead.
I’m not disillusioned for not having found a TOE or for believing it doesn’t exist. I’m actually relieved!
The reactions that I have encountered only reinforce my point, that there is great confusion these days about the cultural role of science and scientists. Science is not a new form of religion, scientists are not holy men and women, and we don’t have or can have all the answers.
As I wrote in Tear at the Edge of Creation, “Human understanding of the world is forever a work in progress. That we have learned so much, speaks well of our creativity. That we want to know more, speaks well of our drive. That we think we can know all, speaks only of our folly.”
Hopefully, this acceptance of our perennial ignorance won’t be interpreted as an opening to religion and supernatural explanations. Let me make my position clear: behind our ignorance there is only the science we still don’t know.
Monday, June 21, 2010
Friday, June 18, 2010
The summer solstice is near and days here in Stockholm are getting longer and longer. The other day I woke up early and, looking out of the window, saw that it was dawning already. Or so I thought. The clock revealed that it wasn't the dawn I was seeing, but that the sun hadn't even set. My biorhythm seems to be a little confused these days.
Along with midsummer also the long awaited wedding of Sweden's Crown Princess Victoria is coming closer. Tomorrow Victoria will exchange I-do's in Stockholm Cathedral with her former personal trainer Daniel Westling. It's a giant marketing event: The Swedes have declared Stockholm's airport Arlanda the "Official Love Airport 2010" and the two weeks before the wedding we had to endure the "LOVE Stockholm 2010," a "two-week festival of love, right in the centre of Stockholm." You can buy postcards and posters of the happy couple in every supermarket here, together with loads of blue-yellow decorations. Busy cityworkers have planted yellow and blue flowers all over the place. Just the weather isn't really playing along, today it's rainy at 17° C.
My Swedish isn't good enough to actually understand the traffic report on the radio, but I understand as much as a long list of streets separated by stängt stängt stängt stängt (closed). I for certain will stay as far away as possible from the city center tomorrow. If your national TV station doesn't broadcast the event, you can follow the wedding ceremonies live tomorrow via SVT. I think it's great the two get married tomorrow because that way I was able to grab a slot for the laundry room on Saturday morning.
Next week, I'll be on a short trip to Bonn for a workshop on quantum black holes, where I'll give a talk about my paper with Lee on the black hole information loss. I wish you all a lovely weekend :-)
Thursday, June 17, 2010
Science Metrics
Nature has a very interesting News Feature on metrics for scientific achievement, titled Metrics: Do metrics matter? The use of scientific metrics is a recurring theme on this blog. I wrote about it most recently in my post Against Measure.
The main point of my criticism on science metrics is that they deviate researchers' interests. It is what I refer to as a deviation from primary goals to secondary criteria. Here, the primary goal is good research. The secondary criteria are some measures that for whatever reason are thought to be relevant quantifiers for the primary goal. The problem is that, even if the secondary criteria have initially had some relevance, their implementation inevitably affects researcher's own assessment of what success means and leads them to strive for the secondary criteria rather than the primary goal. With that, the secondary criteria become less and less useful since they are being pursued as an end in itself. Typical example: number of publications. In principle not a completely useless criterion to assess a researcher's productivity. But it becomes increasingly less useful the more tricks scientists pull to increase the number of publications instead of focusing on the quality of their research.
Note that for a deviation of interests to happen it is not necessary that the measures are actually used! It is only relevant that researchers believe they are used. It's a sociological effect. You can cause such believes by simply doing much talk about science metrics. The better known a measure is, the more likely people are to believe it has some relevance. It is a well known fact about human psychology that people pay attention to what they hear repeatedly.
Now Nature did a little poll asking readers how much they believe science metrics are used at their institution for various purposes. 150 readers responded; the results are available here. They then contacted scientists in administrative positions at nearly 30 research institutions around the world and asked them what metrics are being used, and how heavily they are relied on. In a nutshell the administrators claim that metrics are being used much less than scientists believe they are.
"The results suggest that there may be a disconnect between the way researchers and administrators see the value of metrics."
While this is an interesting suggestion, it is not much more than a suggestion. It is entirely unclear whether the sample of people who replied to the poll had a good overlap with the sample of administrators being asked. By such a small sample size the distribution of people in both groups over countries matters significantly. It remained unclear to me from the article whether in their contacting of institutes they have made sure that the representation of countries is the same as that of the poll's participants, and also if the distribution of research fields is the same. If not, the mismatch between the administration and the researchers might simply show national differences or differences between fields of research. Also, it is conceivable that people who filled out the questionnaire had some concerns about the topic to begin with, while this would not have been the case for people contacted. It did not become clear to me how the poll was publicized.
In any case, given what I said earlier, we should of course appreciate the suggestion of these results. Please do not believe that science metrics matter for your career!
Tuesday, June 15, 2010
Why do people get tattooed?
Saturday, June 12, 2010
Book review: From Eternity to Here by Sean Carroll
By Sean Carroll
Dutton Adult (January 7, 2010)
Most of you will know Sean Carroll, who blogs at Cosmic Variance. Sean is a Senior Research Associate at CalTech and his research focuses on cosmology, general relativity and the standard model, as well as extensions thereof. He has written a textbook on General Relativity, and the lecture notes that gave rise to the book are available online. I've met Sean a few times, he's an interesting person and gives great talks. Sean has a special interest in the arrow of time, and that is also the topic of book “From Eternity to Here.” The arrow of time is, in a nutshell, the question why the past is different from the present.
I bought the book for three reasons. One is that for many years I've been using the PDF version of his lecture notes as a handy quick reference when on travel and had a bad consciousness for never buying the book. The second one is that from reading Sean's blog I know he writes well. The third reason is that adding a second book to the order rendered delivery free.
“From Eternity to Here” is a very well written book that communicates a lot of science, both textbook science and contemporary science, while at the same time being amazingly accurate. The biggest part of the book - all but the last chapter - is dedicated to accurately framing the question. Why is it interesting to ask why the past was what it was? What exactly is it that we don't understand? How do we get a grip on the problem? For this, Sean covers first of all the second law of thermodynamics, then special relativity, general relativity, cosmology, quantum mechanics, black hole physics, and finally inflation and the multiverse. In the last chapter, he then discusses possible solutions to the question he has posed and puts forward his own solution as the most plausible one. Along the way he scratches on topics like the vacuum energy, structure formation, the AdS/CFT duality and magnetic monopoles.
Sean is very careful with distinguishing between established science and unconfirmed speculations. The only glitch is the section on the holographic principle where he fails to point out that there is no experimental evidence for such a feature of Nature to be true in all generality. I am somewhat sick of being misinterpreted on this point so let me be very clear here. All I am saying is that, absent experimental evidence, scientists should be very careful with what they put forward as a true description of Nature. Theoretical evidence can very easily be biased simply because a topic that attracts attention may mount one-sided “evidence.” This can never replace actual tests of a hypothesis. The holographic principle certainly does not rest on the same basis as ΛCDM or the Schrödinger equation and I wish its status had been framed more clearly. Anyway, Sean needs the holographic counting of degrees of freedom for the rest of his argument.
I was very pleased that Sean's explanations of physical concepts are not as superficial and vague as one frequently finds in popular science books. He does not shy away from the phase space, using logarithms, and discusses the amplitude of the wave function. The chapter on quantum mechanics however somewhat suffers from the overuse of cats and dogs. The book has plenty of footnotes with additional explanations, and offers many references so that the interested reader will easily be able to find the relevant keywords and dig deeper, should they wish so. On several occasions I took a note that Sean had forgotten to point out a specific assumption that entered his argument or left out some exceptions. In every single case, these points were later addressed, so I am left with nothing to complain about.
I personally don't have a large interest in the topic and don't care very much about the whole discussion. I think the question is ill-posed and when we have a better understanding of quantum gravity we'll see why. Sean's book didn't succeed in increasing my interest. Nevertheless, it was a pleasure to read. Sean has a good sense of humor, but doesn't overdo it. The story he tells is also well embedded into its scientific history and I learned a thing or two here that I hadn't known before. Both the historical and the philosophical aspects however play a secondary role and don't take over the scientific discussion. All together, the book is very well balanced and a recommendable read. It has something to offer for anybody who has an interest in modern cosmology and/or the arrow of time. I'd give this book 5 out of 5 stars.
From January through April, Sean offered a book club at his blog, each weak discussing another chapter. You might find this a useful addition to the book itself.
Wednesday, June 09, 2010
Perimeter Institute is looking for a Scientific IT specialist
Two years ago, I organized a conference on Science in the 21st Century, focusing on topics at the intersection on science, society and information technology. (I wrote about the conference here, a summary is here and a brief write-up of my own talk is here.) There are three aspects to the changes that the use of information technologies are bringing to science. One is the improving communication with the public - this blog is an example for such a change. The second one is that advances in hard- and software allow us to better understand the process of knowledge discovery and the dynamics of the scientific communities itself - the Maps of Science are an example for this. The third aspect, and probably the one most interesting for the scientist at work, is the development of new tools that support research and researchers in their every day work.
As I learned the other day, Perimeter Institute is now looking for a person who works at exactly this intersection. The job description reads as follows:
The Perimeter Institute for Theoretical Physics (PI) is looking for a Scientific IT specialist -- a creative individual with experience in both scientific research and information technology (IT). This is a new, hybrid, research/IT position within the Institute, dedicated to helping PI’s scientific staff make effective use of IT resources. It has two clear missions. First, to directly assist researchers in using known, available IT tools to do their research. Second, to uncover or develop cutting-edge IT resources, introduce and test them with PI researchers, and then share the things we create and discover with the worldwide scientific community.
By "tools", we mean almost anything. Coding techniques are an obvious example. Collaboration and communication technologies are another: tools for peer-to-peer interactions (such as skype), virtual whiteboards, video conferencing tools, platforms for running virtual conferences (that can do justice to talks in the mathematical sciences), and novel ways of presenting research results such as archives for recorded seminars, blogs, and wikis. Further examples include tools for helping researchers organize information (e.g., specialized search engines and filtering schemes), and end-user software that facilitates bread-and-butter scientific activities like writing papers collaboratively, preparing presentations, and organizing references.
We are seeking a person who brings an independent and ambitious vision that will help define this vision. The job is as yet quite malleable in its scope and duties! We're looking for someone who is inspired by the possibility that new IT tools can improve or perhaps even revolutionize the way that physics research is done, and someone who can take full advantage of a mandate to create and implement that vision.
Some Duties and Responsibilities:
- Participate in the creation of a high quality “standard" Researcher IT environment (desktop hardware, software set-up), built from a mix of open source software and popular commercial packages.
- Help with High Performance Computing demands.
- Maintain expert level knowledge in the use of the main packages used by Researchers, including Mathematica, Maple, LaTex, etc.
For the official job ad, go here.
[Via Rob Spekkens]. The deadline for applications is Friday, July 2, 2010. The Albert Einstein Institute in Potsdam meanwhile offers an almost identically sounding position. I've been told PI was first, but their posting is not dated.
I very much like this development. My requirements on IT staff these days are however very modest. I am happy when the printer spits out my paper without chewing up some pages or leaving them blank. My biggest wish would be not a virtual whiteboard but an actual whiteboard with a plugin to my computer so I could use the board for equations and figures during a skype call. The equations are usually cumbersome but still doable, in the worst case by typing them in LaTex into the chat interface. But diagrams are a disaster. Drawing with a mouse yields no sensible results and the drawing pads that I've tried weren't too convincing either, even neglecting the problem on how to incorporate them into the call. On occasion I've thus drawn on a paper and held it into the camera. This however only works for figures with few details and necessitates plenty of additional explanations.
What is the software or hardware you dream of for your research life?
Saturday, June 05, 2010
Diamonds in Earth Science
To clarify the situation, experiments would need to push above 120 Gigapascal and 2500 Kelvin. I [...] started laboratory experiments using diamond-anvil cell, in which samples of mantle-like materials are squeezed to high pressure between a couple of gem-quality natural diamonds (about two tenths of a carat in size) and then heated with a laser. Above 80 Gigapascal, even diamond—the hardest known material—starts to deform dramatically. To push pressure even higher, one needs to optimize the shape of the diamond anvils's tip so that the diamond will not break. My colleagues and I suffered numerous diamond failures, which cost not only research funds but sometimes our enthusiasm as well.
(From The Earth's Missing Ingredient)
But in the end, Kei Hirose and his group succeeded in subjecting a small sample of magnesium silicate to the pressure and temperature that prevails in the lower Earth's mantle, about 2700 kilometer below our feet.
Planet Earth has an onion-like structure, as has been revealed by the analysis of seismological data: There is a central core consisting mostly of iron, solid in the inner part, molten and liquid in the upper part. On top of this follows the mantle, which is made up of silicates, compounds of silicon oxides with magnesium and other metals. The solid crust on which we live is just a thin outer skin.
The lower part of the mantle down to the iron core was long thought to consist of MgSiO3 in a crystal structure called perovskite. However, seismological data also revealed that the part of the mantle just above the CMB (in earth science, that's the core-mantle boundary, not the cosmic microwave background... ) somehow is different from the rest of the mantle. This lower-mantle layer was dubbed D″ (D-double-prime, shown in the light shade in the figure), and it was unclear if the difference was by chemical composition or by crystal structure.
As Kei Hirose describes in the June 2010 issue of the Scientific American, his group started a series of experiments to study the properties of magnesium silicate at a pressure up to 130 Gigapascal (water pressure at an ocean depth of 1 kilometer is 0.01 GPa) and a temperature exceeding 2500 Kelvin ‒ the conditions expected for the D″ layer of the lower mantle.
To achieve such extreme conditions, one squeezes a tiny piece of magnesium silicate between the tips of two diamonds, and heats up the probe by a laser. The press used in such experiments is called "laser-heated diamond anvil cell".
The figure shows the core of a diamond anvil cell: The sample to be probed is fixed by a gasket between the tips of two diamonds. The diameter of the tips is about 0.1 millimeter, so applying a moderate force results in huge pressure.
Diamonds are used because of their hardness, but they have the additional bonus of being transparent. Hence, the probe can be observed, or irradiated by a laser for heating, or x-rayed for structure determination.
The diamonds are fixed in cylindrical steel mounts, but creating huge pressure does not require huge equipment: The whole device fits on a hand! (Photo from a SPring-8 press release about Kei Hirose's research.)
Actually, the force on the diamond tips is applied in such a device by tightening screws by hand.
In the experiment, the cell was mounted in a brilliant, thin beam of x-rays created by the SPring-8 synchrotron facility in Japan. This allows to monitor the crystal structure of the probe by observing the pattern of diffraction rings.
It was found that under the conditions of the D″ layer of the lower mantle, magnesium silicate forms a crystal structure unknown before for silicates, which was called "Post-Perovskite". The formation of post-perovskite in the lower mantle is a structural phase transition of the magnesium silicate, and this transition can explain the existence of a separate the D″ layer, and many of its peculiar features. It also facilitates heat exchange between core and mantle, which seems to have quite important implications for earth science.
And here is the heart of the experiment (from the "High pressure and high temperature experiments" site of the Maruyama & Hirose Laboratory at the Department of Earth and Planetary Sciences, Tokyo Institute of Technology) ‒ a diamond used in a diamond anvil pressure cell:
High-quality diamonds of this size cost about US $500 each.
Thursday, June 03, 2010
Impressions from the PI workshop on the Laws of Nature
As you know, 2 weeks ago I was at Perimeter Institute for the workshop on the Laws of Nature: Their Nature and Knowability. It was a very interesting event, bringing together physicists with philosophers, a mix that isn't always easy to deal with.
People (them)
On the list of participants, you'll find some well known names. Besides the usual suspects Julian Barbour and Lee Smolin, Paul Davies was there (though only for the first day), Anthony Aquirre (the event was sponsored by FQXi) and of course several people from PI and the University of Waterloo. In my previous post, I already wrote about Marcelo Gleiser's talk. Marcelo is from Brazil, and he is apparently well known there for his popular science books (which was confirmed by Christine in an earlier post.) I had frankly never heard of him before. I talked to him later over dinner, and he told me he writes for a group blog called 13.7 together with, among others, Stuart Kauffman who is also well known for his popular science books. (13.7 is the estimated age of the universe in billion years. What will they do if that number gets updated?)
Another interesting name on the list of participants is Roberto Unger, who is a well-known Brazilian politician and besides that a professor for law at Harvard Law School, and author of multiple books on social and political theory. He apparently has an interest not only in the laws of societies, but also in the laws of Nature*. And finally let me mention George Musser was also at the workshop. George writes for Scientific American and is author of The Complete Idiot’s Guide to String Theory. He turned out to be a very nice guy with the journalist's theme "I want to know more about that."
Talks (their)
Now let me say a word about the talks. First, and most important, all the talks were recorded and are available on PIRSA here. The talks on the first day were heavily philosophical. I will admit that I often have problems making sense of that. Not because I don't have an interest in philosophy, but because one frequently ends up arguing about the meaning of words which is, at the bottom of things, a consequence of lacking definitions and thus a waste of time. Yes, my apologies, I'm, duh, a theoretical physicist with some semesters maths on my CV. If I don't see a definition and an equation, I get lost easily. In some cases it seems the philosophers imply some specific meaning that they just never bother to explain. But in other cases they'll start arguing about it themselves, and that's when I usually zoom out wondering what's the point in arguing if they don't know what they're arguing about anyway.
The most interesting event on the first day was arguably Lee Smolin's and Roberto Unger's shared talk "Laws and Time in Cosmology". Let me add that I've heard Smolin talking about the "reality of time" several times and I still can't make sense of it. The problem I have is simply that I don't know what he's talking about. This recent talk didn't change anything about my confusion, but if you haven't heard it before, you might find it inspiring. Unger's talk is very impressive on the rhetorical side. Unfortunately, it made even less sense to me than Lee's talk. For all I can see, there's no tension neither between a block-universe and a notion of simultaneity nor between a block-universe and causality, as I think I heard Unger saying (thus my question in the end). Point is, I don't understand the problem they're attempting to address to begin with. I see no problem. As Barbara Streisand already told us "Life is a moment in space" and "In love there is no measure of time." Consequently, a universe where time is real must be loveless. I don't like that idea.
On that note, let me recommend Julian Barbour's talk "A case for geometry". Julian is a charming British guy and he has his own theory of a lovely, timeless universe. I don't buy a word of what he says, but his talk is very accessible and fun to listen to. It makes your head spin what he's saying, just try it out, it's very intriguing. I am curious to see how these ideas will develop, it seems to me they might be on the brink of actually making predictions. (A somewhat more detailed explanation of his ideas is here, audio becomes audible at 3:30 min.)
On the second day, we had several talks discussing concrete proposals for how one could think of the laws of Nature off the trodden path. You probably won't be surprised to hear that one of the suggestions is that of "Law without Law: Entropic Dynamics" by Ariel Caticha. It is not directly related to Erik Verlinde's entropic gravity, but certainly plays in the same corner of the room: exploiting the possibility that fundamentally all our dynamics is simply a consequence of the increase of entropy. Ariel's talk however isn't really recommendable, it sits on a funny edge between too many and too few details.
Another approach is Kevin Knuth's who put forward in his talk "The Role of Order in Natural Law" the idea that on the basis of all, there's order - in a well-defined mathematical sense. I can't avoid the impression though that even if this worked out to reproduce the standard model, it would merely be a reformulation. Kevin's talk was basically a summary of this recent paper. And Philip Goyal gave a very nice talk on "The common symmetries underlying quantum theory, probability theory, and number systems." I have a lot of sympathy for the attempt to reconstruct quantum theory, it's just that I don't understand why literally all the quantum foundations guys hang themselves up on the measurement process in quantum mechanics. For what I'm concerned, quantum field theory is the thing, and I'm still waiting for somebody to reconstruct the non-commutativity of annihilation and creation operators.
Finally, let me mention Kevin Kelley's talk "How does simplicity help science find true laws?" Kelley is a philosopher from Carnegie Mellon, and in his talk he explored whether it is possible to put Ockham's Razor on a rational basis. Unfortunately, while the theme could in principle have been very interesting, his talk is not particularly accessible. He assumed way too much knowledge from the audience. At least I get very easily frustrated when technical terms are dropped and procedures are mentioned without being explained, since it's not a field I work in. In any case, I'll spare you the time watching the full thing and just mention an interesting remark that came up in the discussion. Apparently there have been efforts to create a computer software that could simulate a "scientist," in this case for the example of trying to extract a theory from data of the motion of the planets. At least so far, such attempts failed (if anybody knows a reference, it would be highly appreciated.) So it seems, for the time being, scientists will not be replaced by computers.
At the end of the last day we had a discussion session, moderated by Steven Weinstein, wrapping up some of the topics that came up the previous days and some others. One of them is the question about the power of mathematics and if there are limits to what humans can grasp (a theme we have previously discussed here). For a fun anecdote making the point well, watch Steven at 1:13:50 min ("I remember distinctively being in a graduate quantum mechanics class by Bob Wald...") Of course Tegmark's mathematical universe made an appearance as well, another topic we have previously discussed on this blog. For what I am concerned, declaring that all is mathematics may be some sort of unification of the laws of Nature, alright, but it's eventually a completely useless unification. And that brings me to...
Thoughts (mine)
On several occasions at the workshop, I felt like the stereotypical physicist among philosophers, and it took me a while to figure out what I found lacking at this workshop. You could say I'm a very pragmatic person. There's even an ism that belongs to that! If you talk about reality and truth, I don't know what you mean, and I actually don't care. This is just words. I'll start caring if you tell me what it's good for. If you want to reformulate the laws of physics, fine, go ahead. But if you want me to spend time on it, you'll have to tell me what the advantage is. If there's two theories and they make the same predictions, that doesn't cause me headaches. For what I'm concerned, if they make the same predictions, they're the same theory.
What matters in the end about a law or a theory or a model is not whether it's philosophically appealing and not even if there's a rational process by which it's been selected (and btw, what means "rational" anyway), but simply whether it's useful. And usefulness is eventually a notion deeply connected to human societies and values. For that reason I think to understand the scientific method and its success one inevitably needs to take into account the dynamics of the communities and the embedding of scientific knowledge into our societies. (It should be clear that with usefulness I don't necessarily mean technical applications as I have recently expressed in this post.)
Leaving aside that I found this aspect entirely missing to the discussions about the process of science itself and its possible limitations, the workshop has given me a lot to think about. Having said that the pragmatist in me searches for the use in all that enters my ears, I nevertheless have enough fantasy to imagine that some of the themes discussed at the workshop will become central to shaping our thinking about the laws of Nature in the future and thus eventually prove their usefulness. It was a very stimulating meeting and the approaches that were presented are all as bold as courageous. It will be interesting to follow the progress of these thoughts.
*I once made an attempt to read one of Unger's books, What should the left propose? I had to look up every second word in a dictionary, and even that didn't always help. When I had, after an hour or so, roughly deciphered the meaning of a page it seemed to me one could have said the same in one simple sentence, avoiding 3 or more syllable words. I gave up on page 20. Sorry for being so incredibly unintellectual, but to me language is first and foremost a means of communication. If you want to be heard, you better use a code that the receiver can decipher. Friedrich Engels, for example, was an excellent writer...
Tuesday, June 01, 2010
Update on the ESQG 2010
• What to sacrifice?
• The Future of Particle Physics.
• Experiments and Thought Experiments
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cc3b41a719e68f35 | The wavefunction of a zero-mass particle
Post scriptum note added on 11 July 2016: This is one of the more speculative posts which led to my e-publication analyzing the wavefunction as an energy propagation. With the benefit of hindsight, I would recommend you to immediately the more recent exposé on the matter that is being presented here, which you can find by clicking on the provided link. In fact, I actually made some (small) mistakes when writing the post below.
Original post:
I hope you find the title intriguing. A zero-mass particle? So I am talking a photon, right? Well… Yes and no. Just read this post and, more importantly, think about this story for yourself. 🙂
One of my acquaintances is a retired nuclear physicist. We mail every now and then—but he has little or no time for my questions: he usually just tells me to keep studying. I once asked him why there is never any mention of the wavefunction of a photon in physics textbooks. He bluntly told me photons don’t have a wavefunction—not in the sense I was talking at least. Photons are associated with a traveling electric and a magnetic field vector. That’s it. Full stop. Photons do not have a ψ or φ function. [I am using ψ and φ to refer to position or momentum wavefunction. You know both are related: if we have one, we have the other.] But then I never give up, of course. I just can’t let go out of the idea of a photon wavefunction. The structural similarity in the propagation mechanism of the electric and magnetic field vectors E and B just looks too much like the quantum-mechanical wavefunction. So I kept trying and, while I don’t think I fully solved the riddle, I feel I understand it much better now. Let me show you the why and how.
I. An electromagnetic wave in free space is fully described by the following two equations:
1. B/∂t = –∇×E
2. E/∂t = c2∇×B
We’re making abstraction here of stationary charges, and we also do not consider any currents here, so no moving charges either. So I am omitting the ∇·E = ρ/ε0 equation (i.e. the first of the set of four equations), and I am also omitting the j0 in the second equation. So, for all practical purposes (i.e. for the purpose of this discussion), you should think of a space with no charges: ρ = 0 and = 0. It’s just a traveling electromagnetic wave. To make things even simpler, we’ll assume our time and distance units are chosen such that = 1, so the equations above reduce to:
1. B/∂t = –∇×E
2. E/∂t = ∇×B
Perfectly symmetrical! But note the minus sign in the first equation. As for the interpretation, I should refer you to previous posts but, briefly, the ∇× operator is the curl operator. It’s a vector operator: it describes the (infinitesimal) rotation of a (three-dimensional) vector field. We discussed heat flow a couple of times, or the flow of a moving liquid. So… Well… If the vector field represents the flow velocity of a moving fluid, then the curl is the circulation density of the fluid. The direction of the curl vector is the axis of rotation as determined by the ubiquitous right-hand rule, and its magnitude of the curl is the magnitude of rotation. OK. Next step.
II. For the wavefunction, we have Schrödinger’s equation, ∂ψ/∂t = i·(ħ/2m)·∇2ψ, which relates two complex-valued functions (∂ψ/∂t and ∇2ψ). Complex-valued functions consist of a real and an imaginary part, and you should be able to verify this equation is equivalent to the following set of two equations:
[Two complex numbers a + ib and c + id are equal if, and only if, their real and imaginary parts are the same. However, note the −i factor in the right-hand side of the equation, so we get: a + ib = −i·(c + id) = d −ic.] The Schrödinger equation above also assumes free space (i.e. zero potential energy: V = 0) but, in addition – see my previous post – they also assume a zero rest mass of the elementary particle (E0 = 0). So just assume E= V = 0 in de Broglie’s elementary ψ(θ) = ψ(x, t) = eiθ = a·e−i[(E+ p2/(2m) + V)·t − p∙x]/ħ wavefunction. So, in essence, we’re looking at the wavefunction of a massless particle here. Sounds like nonsense, doesn’t it? But… Well… That should be the wavefunction of a photon in free space then, right? 🙂
Maybe. Maybe not. Let’s go as far as we can.
The energy of a zero-mass particle
What m would we use for a photon? It’s rest mass is zero, but it’s got energy and, hence, an equivalent mass. That mass is given by the m = E/cmass-energy equivalence. We also know a photon has momentum, and it’s equal to its energy divided by c: p = m·c = E/c. [I know the notation is somewhat confusing: E is, obviously, not the magnitude of E here: it’s energy!] Both yield the same result. We get: m·c = E/c ⇔ m = E/c⇔ E = m·c2.
OK. Next step. Well… I’ve always been intrigued by the fact that the kinetic energy of a photon, using the E = m·v2/2 = E = m·c2/2 formula, is only half of its total energy E = m·c2. Half: 1/2. That 1/2 factor is intriguing. Where’s the rest of the energy? It’s really a contradiction: our photon has no rest mass, and there’s no potential here, but its total energy is still twice its kinetic energy. Quid?
There’s only one conclusion: just because of its sheer existence, it must have some hidden energy, and that hidden energy is also equal to E = m·c2/2, and so the kinetic and hidden energy add up to E = m·c2.
Huh? Hidden energy? I must be joking, right?
Well… No. No joke. I am tempted to call it the imaginary energy, because it’s linked to the imaginary part of the wavefunction—but then it’s everything but imaginary: it’s as real as the imaginary part of the wavefunction. [I know that sounds a bit nonsensical, but… Well… Think about it: it does make sense.]
Back to that factor 1/2. You may or may not remember it popped up when we were calculating the group and the phase velocity of the wavefunction respectively, again assuming zero rest mass, and zero potential. [Note that the rest mass term is mathematically equivalent to the potential term in both the wavefunction as well as in Schrödinger’s equation: (E0·t +V·t = (E+ V)·t, and V·ψ + E0·ψ = (V+E0)·ψ—obviously!]
In fact, let me quickly show you that calculation again: the de Broglie relations tell us that the k and the ω in the ei(kx − ωt) = cos(kx−ωt) + i∙sin(kx−ωt) wavefunction (i.e. the spatial and temporal frequency respectively) are equal to k = p/ħ, and ω = E/ħ. If we would now use the kinetic energy formula E = m·v2/2 – which we can also write as E = m·v·v/2 = p·v/2 = p·p/2m = p2/2m, with v = p/m the classical velocity of the elementary particle that Louis de Broglie was thinking of – then we can calculate the group velocity of our ei(kx − ωt) = cos(kx−ωt) + i∙sin(kx−ωt) as:
vg = ∂ω/∂k = ∂[E/ħ]/∂[p/ħ] = ∂E/∂p = ∂[p2/2m]/∂p = 2p/2m = p/m = v
[Don’t tell me I can’t treat m as a constant when calculating ∂ω/∂k: I can. Think about it.] Now the phase velocity. The phase velocity of our ei(kx − ωt) is only half of that. Again, we get that 1/2 factor:
vp = ω/k = (E/ħ)/(p/ħ) = E/p = (p2/2m)/p = p/2m = v/2
Strange, isn’t it? Why would we get a different value for the phase velocity here? It’s not like we have two different frequencies here, do we? You may also note that the phase velocity turns out to be smaller than the group velocity, which is quite exceptional as well! So what’s the matter?
Well… The answer is: we do seem to have two frequencies here while, at the same time, it’s just one wave. There is only one k and ω here but, as I mentioned a couple of times already, that ei(kx − ωt) wavefunction seems to give you two functions for the price of one—one real and one imaginary: ei(kx − ωt) = cos(kx−ωt) + i∙sin(kx−ωt). So are we adding waves, or are we not? It’s a deep question. In my previous post, I said we were adding separate waves, but now I am thinking: no. We’re not. That sine and cosine are part of one and the same whole. Indeed, the apparent contradiction (i.e. the different group and phase velocity) gets solved if we’d use the E = m∙v2 formula rather than the kinetic energy E = m∙v2/2. Indeed, assuming that E = m∙v2 formula also applies to our zero-mass particle (I mean zero rest mass, of course), and measuring time and distance in natural units (so c = 1), we have:
E = m∙c2 = m and p = m∙c2 = m, so we get: E = m = p
Waw! What a weird combination, isn’t it? But… Well… It’s OK. [You tell me why it wouldn’t be OK. It’s true we’re glossing over the dimensions here, but natural units are natural units, and so c = c2 = 1. So… Well… No worries!] The point is: that E = m = p equality yields extremely simple but also very sensible results. For the group velocity of our ei(kx − ωt) wavefunction, we get:
vg = ∂ω/∂k = ∂[E/ħ]/∂[p/ħ] = ∂E/∂p = ∂p/∂p = 1
So that’s the velocity of our zero-mass particle (c, i.e. the speed of light) expressed in natural units once more—just like what we found before. For the phase velocity, we get:
vp = ω/k = (E/ħ)/(p/ħ) = E/p = p/p = 1
Same result! No factor 1/2 here! Isn’t that great? My ‘hidden energy theory’ makes a lot of sense. 🙂 In fact, I had mentioned a couple of times already that the E = m∙v2 relation comes out of the de Broglie relations if we just multiply the two and use the v = λ relation:
2. v = λ ⇒ f·λ = v = E/p ⇔ E = v·p = v·(m·v) ⇒ E = m·v2
But so I had no good explanation for this. I have one now: the E = m·vis the correct energy formula for our zero-mass particle. 🙂
The quantization of energy and the zero-mass particle
Let’s now think about the quantization of energy. What’s the smallest value for E that we could possible think of? That’s h, isn’t it? That’s the energy of one cycle of an oscillation according to the Planck-Einstein relation (E = h·f). Well… Perhaps it’s ħ? Because… Well… We saw energy levels were separated by ħ, rather than h, when studying the blackbody radiation problem. So is it ħ = h/2π? Is the natural unit a radian (i.e. a unit distance), rather than a cycle?
Neither is natural, I’d say. We also have the Uncertainty Principle, which suggests the smallest possible energy value is ħ/2, because ΔxΔp = ΔtΔE = ħ/2.
Huh? What’s the logic here?
Well… I am not quite sure but my intuition tells me the quantum of energy must be related to the quantum of time, and the quantum of distance.
Huh? The quantum of time? The quantum of distance? What’s that? The Planck scale?
No. Or… Well… Let me correct that: not necessarily. I am just thinking in terms of logical concepts here. Logically, as we think of the smallest of smallest, then our time and distance variables must become count variables, so they can only take on some integer value n = 0, 1, 2 etcetera. So then we’re literally counting in time and/or distance units. So Δx and Δt are then equal to 1. Hence, Δp and ΔE are then equal to Δp = ΔE = ħ/2. Just think of the radian (i.e. the unit in which we measure θ) as measuring both time as well as distance. Makes sense, no?
No? Well… Sorry. I need to move on. So the smallest possible value for m = E = p would be ħ/2. Let’s substitute that in Schrödinger’s equation, or in that set of equations Re(∂ψ/∂t) = −(ħ/2m)·Im(∇2ψ) and Im(∂ψ/∂t) = (ħ/2m)·Re(∇2ψ). We get:
1. Re(∂ψ/∂t) = −(ħ/2m)·Im(∇2ψ) = −(2ħ/2ħ)·Im(∇2ψ) = −Im(∇2ψ)
2. Im(∂ψ/∂t) = (ħ/2m)·Re(∇2ψ) = (2ħ/2ħ)·Re(∇2ψ) = Re(∇2ψ)
Bingo! The Re(∂ψ/∂t) = −Im(∇2ψ) and Im(∂ψ/∂t) = Re(∇2ψ) equations were what I was looking for. Indeed, I wanted to find something that was structurally similar to the ∂B/∂t = –∇×and E/∂t = ∇×B equations—and something that was exactly similar: no coefficients in front or anything. 🙂
What about our wavefunction? Using the de Broglie relations once more (k = p/ħ, and ω = E/ħ), our ei(kx − ωt) = cos(kx−ωt) + i∙sin(kx−ωt) now becomes:
ei(kx − ωt) = ei(ħ·x/2 − ħ·t/2)/ħ = ei(x/2 − t/2) = cos[(x−t)/2] + i∙sin[(x−t)/2]
Hmm… Interesting! So we’ve got that 1/2 factor now in the argument of our wavefunction! I really feel I am close to squaring the circle here. 🙂 Indeed, it must be possible to relate the ∂B/∂t = –∇×E and ∂E/∂t = c2∇×B to the Re(∂ψ/∂t) = −Im(∇2ψ) and Im(∂ψ/∂t) = Re(∇2ψ) equations. I am sure it’s a complicated exercise. It’s likely to involve the formula for the Lorentz force, which says that the force on a unit charge is equal to E+v×B, with v the velocity of the charge. Why? Note the vector cross-product. Also note that ∂B/∂t and ∂E/∂t are vector-valued functions, not scalar-valued functions. Hence, in that sense, ∂B/∂t and ∂E/∂t and not like the Re(∂ψ/∂t) and/or Im(∂ψ/∂t) function. But… Well… For the rest, think of it: E and B are orthogonal vectors, and that’s how we usually interpret the real and imaginary part of a complex number as well: the real and imaginary axis are orthogonal too!
So I am almost there. Who can help me prove what I want to prove here? The two propagation mechanisms are the “same-same but different”, as they say in Asia. The difference between the two propagation mechanisms must also be related to that fundamental dichotomy in Nature: the distinction between bosons and fermions. Indeed, when combining two directional quantities (i.e. two vectors), we like to think there are four different ways of doing that, as shown below. However, when we’re only interested in the magnitude of the result (and not in its direction), then the first and third result below are really the same, as are the second and fourth combination. Now, we’ve got pretty much the same in quantum math: we can, in theory, combine complex-valued amplitudes in four different ways but, in practice, we only have two (rather than four) types of behavior only: photons versus bosons.
vector addition
Is our zero-mass particle just the electric field vector?
Let’s analyze that ei(x/2 − t/2) = cos[(x−t)/2] + i∙sin[(x−t)/2] wavefunction some more. It’s easy to represent it graphically. The following animation does the trick:
I am sure you’ve seen this animation before: it represents a circularly polarized electromagnetic wave… Well… Let me be precise: it presents the electric field vector (E) of such wave only. The B vector is not shown here, but you know where and what it is: orthogonal to the E vector, as shown below—for a linearly polarized wave.
Let’s think some more. What is that ei(x/2 − t/2) function? It’s subject to conceiving time and distance as countable variables, right? I am tempted to say: as discrete variables, but I won’t go that far—not now—because the countability may be related to a particular interpretation of quantum physics. So I need to think about that. In any case… The point is that x can only take on values like 0, 1, 2, etcetera. And the same goes for t. To make things easy, we’ll not consider negative values for x right now (and, obviously, not for t either). So we’ve got a infinite set of points like:
• ei(0/2 − 0/2) = cos(0) + i∙sin(0)
• ei(1/2 − 0/2) = cos(1/2) + i∙sin(1/2)
• ei(0/2 − 1/2) = cos(−1/2) + i∙sin(−1/2)
• ei(1/2 − 1/2) = cos(0) + i∙sin(0)
Now, I quickly opened Excel and calculated those cosine and sine values for x and t going from 0 to 14 below. It’s really easy. Just five minutes of work. You should do yourself as an exercise. The result is shown below. Both graphs connect 14×14 = 196 data points, but you can see what’s going on: this does effectively, represent the elementary wavefunction of a particle traveling in spacetime. In fact, you can see its speed is equal to 1, i.e. it effectively travels at the speed of light, as it should: the wave velocity is v = f·λ = (ω/2π)·(2π/k) = ω/k = (1/2)·(1/2) = 1. The amplitude of our wave doesn’t change along the x = t diagonal. As the Last Samurai puts it, just before he moves to the Other World: “Perfect! They are all perfect!” 🙂
graph imaginarygraph real
In fact, in case you wonder how the quantum vacuum could possibly look like, you should probably think of these discrete spacetime points, and some complex-valued wave that travels as it does in the illustration above.
Of course, that elementary wavefunction above does not localize our particle. For that, we’d have to add a potentially infinite number of such elementary wavefunctions, so we’d write the wavefunction as ∑ ajeiθj functions. [I use the symbol here for the subscript, rather than the more conventional i symbol for a subscript, so as to avoid confusion with the symbol used for the imaginary unit.] The acoefficients are the contribution that each of these elementary wavefunctions would make to the composite wave. What could they possibly be? Not sure. Let’s first look at the argument of our elementary component wavefunctions. We’d inject uncertainty in it. So we’d say that m = E = p is equal to
m = E = p = ħ/2 + j·ħ with j = 0, 1, 2,…
That amounts to writing: m = E = p = ħ/2, ħ, 3ħ/2, 2ħ, 5/2ħ, etcetera. Waw! That’s nice, isn’t it? My intuition tells me that our acoefficients will be smaller for higher j, so the aj(j) function would be some decreasing function. What shape? Not sure. Let’s first sum up our thoughts so far:
1. The elementary wavefunction of a zero-mass particle (again, I mean zero rest mass) in free space is associated with an energy that’s equal to ħ/2.
2. The zero-mass particle travels at the speed of light, obviously (because it has zero rest mass), and its kinetic energy is equal to E = m·v2/2 = m·c2/2.
3. However, its total energy is equal to E = m·v= m·c2: it has some hidden energy. Why? Just because it exists.
4. We may associate its kinetic energy with the real part of its wavefunction, and the hidden energy with its imaginary part. However, you should remember that the imaginary part of the wavefunction is as essential as its real part, so the hidden energy is equally real. 🙂
So… Well… Isn’t this just nice?
I think it is. Another obvious advantage of this way of looking at the elementary wavefunction is that – at first glance at least – it provides an intuitive understanding of why we need to take the (absolute) square of the wavefunction to find the probability of our particle being at some point in space and time. The energy of a wave is proportional to the square of its amplitude. Now, it is reasonable to assume the probability of finding our (point) particle would be proportional to the energy and, hence, to the square of the amplitude of the wavefunction, which is given by those aj(j) coefficients.
OK. You’re right. I am a bit too fast here. It’s a bit more complicated than that, of course. The argument of probability being proportional to energy being proportional to the square of the amplitude of the wavefunction only works for a single wave a·eiθ. The argument does not hold water for a sum of functions ∑ ajeiθj. Let’s write it all out. Taking our m = E = p = ħ/2 + j·ħ = ħ/2, ħ, 3ħ/2, 2ħ, 5/2ħ,… formula into account, this sum would look like:
a1ei(x − t)(1/2) + a2ei(x − t)(2/2) + a3ei(x − t)(3/2) + a4ei(x − t)(4/2) + …
But—Hey! We can write this as some power series, can’t we? We just need to add a0ei(x − t)(0/2) = a0, and then… Well… It’s not so easy, actually. Who can help me? I am trying to find something like this:
power series
Or… Well… Perhaps something like this:
power series 2
Whatever power series it is, we should be able to relate it to this one—I’d hope:
power series 3
Hmm… […] It looks like I’ll need to re-visit this, but I am sure it’s going to work out. Unfortunately, I’ve got no more time today, I’ll let you have some fun now with all of this. 🙂 By the way, note that the result of the first power series is only valid for |x| < 1. 🙂
Note 1: What we should also do now is to re-insert mass in the equations. That should not be too difficult. It’s consistent with classical theory: the total energy of some moving mass is E = m·c2, out of which m·v2/2 is the classical kinetic energy. All the rest – i.e. m·c2 − m·v2/2 – is potential energy, and so that includes the energy that’s ‘hidden’ in the imaginary part of the wavefunction. 🙂
Note 2: I really didn’t pay much attentions to dimensions when doing all of these manipulations above but… Well… I don’t think I did anything wrong. Just to give you some more feel for that wavefunction ei(kx − ωt), please do a dimensional analysis of its argument. I mean, k = p/ħ, and ω = E/ħ, so check the dimensions:
• Momentum is expressed in newton·second, and we divide it by the quantum of action, which is expressed in newton·meter·second. So we get something per meter. But then we multiply it with x, so we get a dimensionless number.
• The same is true for the ωt term. Energy is expressed in joule, i.e. newton·meter, and so we divide it by ħ once more, so we get something per second. But then we multiply it with t, so… Well… We do get a dimensionless number: a number that’s expressed in radians, to be precise. And so the radian does, indeed, integrate both the time as well as the distance dimension. 🙂
3 thoughts on “The wavefunction of a zero-mass particle
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eac4e0dd15fe3abf | Document Type
Date of Original Version
Pendlebury et al. [Phys. Rev. A 70, 032102 (2004)] were the first to investigate the role of geometric phases in searches for an electric dipole moment (EDM) of elementary particles based on Ramsey-separated oscillatory field magnetic resonance with trapped ultracold neutrons and comagnetometer atoms. Their work was based on the Bloch equation and later work using the density matrix corroborated the results and extended the scope to describe the dynamics of spins in general fields and in bounded geometries. We solve the Schrödinger equation directly for cylindrical trap geometry and obtain a full description of EDM-relevant spin behavior in general fields, including the short-time transients and vertical spin oscillation in the entire range of particle velocities. We apply this method to general macroscopic fields and to the field of a microscopic magnetic dipole.
PACS numbers: 28.20.-v, 14.20.Dh, 21.10.Tg |
f78e220a6f9298fb | The world as emergent from pure entropy
2017-12-28T14:36:49Z (GMT) by Alexandre Harvey-tremblay
<div>We propose a meta-logical framework to understand the world by an ensemble of theorems rather than by a set of axioms. We prove that the theorems of the ensemble must have *feasible* proofs and must recover *universality*. The ensemble is axiomatized when it is constructed as a partition function, in which case its axioms are, up to an error rate, the leading bits of Omega (the halting probability of a prefix-free universal Turing machine). The partition function augments the standard construction of $\Omega$ with knowledge of the size of the proof of each theorems. With this knowledge, it is able to decide *feasible mathematics*.</div><div><br></div><div>As a consequence of the axiomatization, the ensemble additionally adopts the mathematical structure of an ensemble of statistical physics; it is from this context that the laws of physics are derived. The Lagrange multipliers of the partition function are the fundamental Planck units and the background, a thermal space-time, emerges as a consequence of the limits applicable to the conjugate pairs. The background obeys the relations of special and general relativity, dark energy, the arrow of time, the Schrödinger equation, the Dirac equation and it embeds the holographic principle. In this context, the limits of feasible mathematics are mathematically the same as the laws of physics.</div><div><br></div><div>The framework is so fundamental that informational equivalents to length, time and mass (assumed as axioms in most physical theories) are here formally derivable. Furthermore, it can prove that no alternative framework can contain fewer bits of axioms than it contains (thus it is necessarily the simplest theory). Furthermore, it can prove that, for all worlds amenable to this framework, the laws of physics will be the same (hence there can be no alternatives).</div><div><br></div><div>Thus, the framework is a possible candidate for a final theory.</div> |
1188e91c187ef8d9 | Physics, Rogue Science? / Evaluating twentieth century physics
Physics, Rogue Science?
Evaluating twentieth century physics
Modern theoretical physics, physics post 1900, is badly flawed.
It started out with a set of bold ideas:
There is no ether, and hence
Everything is relative
Light is a particle
Light travels at the same constant speed for all observers.
These appeared at the time to have the potential to solve the serious problems that existed in the wake of the theoretical analysis of Maxwell and the experimental observations of Michelson.
It is now clear that each of these ideas is incorrect:
They conflict with each other, as detailed here
They conflict with observational evidence, as detailed across this site and summarised below
They are as a consequence illogical.
Several dozen physicists have examined the analysis in these pages, in whole or in part. Some have dismissed it as not agreeing with what they believe, a few errors were identified and removed, but the core has survived. And this analysis condemns modern physics as illogical, inconsistent, wilfully unscientific, and defensive to the point of repeatedly lying to outsiders.
As a result, those further theories that are built on this foundation are questionable at best, and it is unsurprising that they often appear somewhat bizarre.
Status of the three theories
Special relativity is based on a clearly stated principle that has been abandoned for all practical purposes, except in the classroom and fringe usage. It is derived in an imagined spacetime of simplicity and purity that doesn’t exist, and its supposedly validating observations occur in situations and ways that run directly counter to those assumptions. Clocks in motion, for example, do not exhibit relativism in any recognisable form, and clearly indicate a preferred frame of reference denied by relativity.
Quantum theory is an excellent set of mathematical descriptions of particle and sub-atomic behaviour, but the verbal depictions, explanations and discussions that accompany them are flawed in more ways than will fit in a paragraph, ranging through contradictory, deliberately vague, metaphysical and unphysical. The descriptions (but not the mathematics) are based on conclusions about the nature of light that have not been properly substantiated.
General relativity is mathematically flawed in a variety of ways. Some of the components of the Schwarzschild metric have been validated, while others have not, and two core assumptions are very easily demonstrated to be false, namely the equivalence principle and the use of a single metric for the motion of both light and matter.
There is a further problem afflicting modern theory that goes beyond arguments about the flaws just described. This is that the discrepancies between quantum theory and general relativity are a certain indication that there are more basic assumptions than can be accommodated in one theory. Regardless of whether you accept the individual criticisms of the theories, there are more assumptions in modern physics than can in principle be true, and yet modern theorists carelessly use any and all when constructing further theory.
This is a certain recipe for theoretical chaos and scientific disaster, and means that none of the more recent theories based on this core can be considered reliable or even scientific, and that is the reason why we have not examined them here in more detail.
Science is at its best when it tells us something we didn’t expect
Most established ideas in modern physics are nonsensical on first hearing, but then so were evolution and continental drift (plate tectonics). My sense of the world is that I am a special creature, different in nature to other animals, stationary, on a flat world, on ground that will last forever, but science has revealed that none of this is true.
At the start of the twentieth century, physics took this a step further, concluding from determinist analysis that determinism itself had failed, and you can see on this site the disgraceful theoretical and pedagogic consequences of that conclusion. One of the central unspoken beliefs of science is that it is a superior method of enquiry that will overcome assumption and prejudice, and it will – eventually. But at present theoretical physics is an unscientific mess.
Some of the wild ideas that have been shown for what they are:
Relativity: shifting definition, but the core idea repeatedly fails
Particle light: unable to explain key properties
Constant light speed: contradicted by key observations
Duality: waffle, not model; and light cannot be both localised and non-localised at the same time
Entanglement: poor reasoning from invalid assumptions
Black holes: both the physics and the mathematics have been abused
The big bang: holds the record for the most irrational, a-causal, ad hoc add-ons in science
Inflation, expanding space, dark energy, dark matter …
The physical is no more than mathematics
This idea keeps coming back whenever a theorist with sufficient profile has an idea they can only express in mathematical form.
The thinking, such as it is, is as follows. The mathematics of special relativity, general relativity and quantum mechanics works, in the important sense that it correctly models reality and that it correctly predicts events, but the associated ideas do not. We should therefore learn from this that the mathematics, and not the physical, is the reality.
This is never stated this clearly, in deference to colleagues who are still selling the failed ideas, but this is the underlying reasoning whenever this claim is made.
One of the unstated and perhaps unintended effects is to deflect attention from the obvious failings in the original ideas. Furthermore, while the ideas are undoubtedly flawed, and mostly incorrect, there are also problems with some of the core mathematics, detailed here.
Bell and von Neumann
John von Neumanni and John Bellii both produced famous ‘proofs’ of the non-existence of a physically describable reality that they called ‘hidden variables’. One is mathematically simple and the other hugely complex. The mathematical ‘proof’ is only as good as its initial assumptions, and both are flawed. Von Neumann’s ‘proof’ is long and highly mathematical, but his error was exposed by Bell.
Bell’s error was a simple one. He assumed that light is particulate, and his ‘proof’ only relates to light corpuscles that have key properties of simple macroscopic particles. The possibility for wave light does not feature in his key work and neither does the ‘duality model’.
Milonni correctly observes that modern theory of the photoelectric effect ‘allows Einstein’s relation to be deduced without photons: Once electrons are described by the Schrödinger equation, it follows that a classical light wave of frequency ν can induce an electron to change its state…’iii
The physicist mind
The focus of theoretical physics in the twenty-first century, for professional practitioners and critics alike, is on a cacophony of new and novel theories each based on the assumption that some core part of the originating theories of the early twentieth century is true. As stated above, it is abundantly clear that this cannot be assumed.
The original theories are still taught as if valid, and such is the confusion in theoretical physics that even abandoned or comprehensively modified parts of these theories are taught uncritically. Examples are the original formulations of the relativity principle, the particle photon and the expanding universe.
What we have in physics is a huge community of individuals who have studied science, are recognised and qualified as scientists, who for most of their working day do real, valuable science and yet abandon – feel forced to abandon – their scientific rationality when dealing with fundamental theory.
Reasoning illogically is something we all fall prey to at times, but in theoretical physics this is endemic and entrenched. When it extends to a deliberate lack of clarity, known falsehoods as required and obfuscation as a way of life – as it too often does – it amounts to a crime against science and against education.
We need to understand that most physicists really have no alternative. To express one’s doubts and criticisms publicly is career suicide, and you could lose a lot of friends. There are no longer any experts, no one to approach with one’s doubts, to re-evaluate established belief. So individuals survive and prosper, while science and education are knowingly corrupted.
I like physicists. Many are my friends. I understand their dilemma and their often-subconscious decisions. From this tolerance I exempt those who conspire with broadcast and print media to propagate nonsensical ideas. Some of what they proclaim they know is false, and others they should know, if only they asked some simple and obvious questions, such as those posed on this site; in other words, if they behaved as a scientist should, checking their facts and reasoning before telling untruths to others.
Some of this is sociological, pressure to conform, the threat of excommunication, a fate that has befallen a number. Some of it is psychological, the reasonable belief that those who taught them understood that which they do not. Some is philosophical, a preference for anti-scientific metaphysics over scientific training, for a chaotic, incomprehensible universe over a deterministic one.
But mostly it is the abrogation of proper scientific enquiry and debate, the shameful inaction of an entire profession.
In spite of considerable efforts to have the ideas and reasoning checked, it is of course possible that some of the ideas, arguments and criticisms expressed may still contain flaws, but the problems exposed are so numerous, so fundamental, and the sloppy thinking in physics so blatant, that it is not possible that any current theory is fundamentally correct, or indeed that theoretical physics overall can properly be considered a functioning science.
Return to top of page
i. John von Neumann, 1903 – 1957, Mathematische Grundlagen der Quanten-Mechanik (Springer-Verlag, Berlin 1932). The English translation is: Princeton UP, NJ, 1955
ii. John S. Bell, 1928 – 1990, On the Problem of Hidden Variables in Quantum Mechanics, Reviews of Modern Physics, Volume 38:3 (July 1966) 447-452; quote on page 449
iii. P.W. Milonni, Los Alamos National Laboratory, in Am. J. Phys. (January 1997) |
1f34f6c6496710db | Wednesday, June 30, 2010
Déjà vu
Last week, on the workshop in Bonn, I was in for a nasty surprise. Sitting there, listening to one talk after the other about black holes, I saw pictures reappear that I had made. Four different pictures of mine, in four different talks. All without picture credits. When I told the speakers later that they've been using a picture that took me in some cases hours to make without even putting my name below it, they apologized. One shrugged shoulders and said "It came up in Google." I checked that, it did come up when doing a Google image source for "Black Hole Evaporation," the source being my home page. I'm not surprised by this, my homepage has always been well indexed by Google. Apparently I was expecting too much when thinking people could at least look at the front page and find my name.
I will admit that I am very dismayed by this. Yes, I too sometimes do use other people's figures and plots in my talks, but I usually add a source, if possible to find. It's more complicated with photos, who will typically appear in so many copies on some dozen websites that it's next to impossible to find out who originally took the photo. In any case, some of the pictures I saw reappearing in those talks I don't even hold the copyright on. They were published in one of my papers, and with that the copyright went to the publisher.
I don't mind at all if people use my pictures, otherwise I wouldn't upload them to my website. I receive the occasional email from somebody asking if they can use one or the other for a talk or a paper and I always say yes. (I once was asked for a picture to be reprinted in a popular science book, but when the publisher of my picture was asked for the reprint permission they said no for reasons I still don't understand.) But of course I do expect that people add at least my name below it. It has previously happened that I saw pictures of mine reappear, this one showing an evaporating black hole seems to be the favorite
but that workshop convinced me to add my name in a corner of all these pictures. Sure, one can cut it out, but it takes a deliberate effort.
This also reminds me that I once received a paper for peer review. It was written in dramatically bad English, then all of a sudden there were two paragraphs that weren't only readable but sounded eerily familiar. A quick check confirmed my suspicion that it was an introduction from one of my own papers. They had cited my paper somewhere, but it was by no means clear they had copied half a page from it. Again, my paper was published, the copyright was with the publisher. The paper I reviewed wasn't only badly written but also wrong, so it didn't get published. However, I later wrote to the authors making it very clear that this is not an appropriate way to cite. They either mark it as a quotation, or they rewrite it. They apologized and then rearranged a few words here and there. I know other people who have made exactly the same experience with one of their papers.
I find it very worrisome that more and more people make so unashamedly use of other's work without even thinking about it. My mother is a high school teacher and as a standard procedure she'll have to check every essay for whether it's been copied elsewhere. Evidently, there's still kids stupid enough to try nevertheless. I know these checks are being done in many other places too, there's even software for it so you don't have to Google every sentence manually. An extreme case that I know of was a PhD candidate who had copied together half of his thesis from other people's review articles, including equations, references and footnotes. He did cite the papers he used, but certainly didn't mark the "borrowed" pieces as quotations.
It is clear that when thousands of people write introductions to the same topic, then many of them will sound quite similar. I also understand that when you find a nice picture for your talk online it seems superfluous to spend time yourself on what Google gives it to you on a silver plate. Certainly you have better things to do than making a pictures for your talk, right? But what you're doing is simply using someone else's effort and selling them as your own. So next time, spend the three seconds and check whose homepage you've been downloading your pictures from.
And here's a recent copyright story that I found hilarious "Greek man sues Swedish firm over Turkish yoghurt pic"
"A Greek man has sued a dairy firm in southern Sweden after his picture ended up on a Turkish yoghurt product. The man whose picture adorns the Turkish yoghurt product, manufactured by Lindahls dairy in Jönköping, argues that the company does not have permission to use his image [...]
The man, who lives in Greece, was made aware of the use of his picture on the popular Swedish product when an acquaintance living in Stockholm recognized his bearded friend [...]
In his writ the man has underlined that he is not Turkish, he is Greek, and lives in Greece, and the use of his picture is thus misleading both for those who know him and for buyers of the product.
Lindahls dairy has expressed surprise at the writ and argues that the image was bought from a picture agency [...]"
Monday, June 28, 2010
The left-handed Piano
As a left-hander, I have an early hands-on experience with the concept of chirality, or handedness: It can be quite difficult to cut a piece of paper with the left hand using standard scissors; the blades usually do not close precisely, resulting in a frayed cut. And of course, scissors with modern, "ergonomically-formed" handles cannot be used with the left hand in the first place.
There is a small niche market for all kinds of chiral partners of standard right-handed everyday products and tools: left-handed scissors, left-handed can-openers, left-handed pencil sharpeners. However, I do not utilize any of them, and use standard instruments with the right hand instead.
Today, I heard on the radio about something really amazing on the market on left-handed products: There are left-handed pianos!
Invented by Geza Loso, musician, piano teacher, left-hander and father of three left-handed kids, they are exact mirror images of usual pianos, with the pitch rising from the right to the left. As Geza Loso explains on his website: For the first time left-handed people receive a real chance to learn how to play the piano on an adequate instrument. Left-handed people would basically use their right hand to accompany and the skilled hand to handle the main functions of a piano-play, to play the melody. This is very decisive for every artistic interpretation.
The left-handed piano will be distributed by the Leipzig piano-manufacturing company Blüthner. Chief executive Christian Blüthner doesn't expect a big commercial success, but thinks that the left-handed piano demonstrates his company's inventiveness. And I am wondering if my career with the piano may have have been longer than a couple of lessons if the instrument would have been left-handed.
Saturday, June 26, 2010
Hello from Bonn
Stefan and I, we are currently in Bonn for a workshop on "Black Holes in a Violent Universe." Bonn is the former German capital and a quite charming city, though not what you'd expect from a capital. So probably a good thing Berlin has taken over the burden. Germany is collectively in a good mood these days since the Germans won Friday's soccer game, and everybody is looking forward to Sunday's game.
We're staying in a small hotel near the river Rhine. Needless to say, our room is on the 4th floor without elevator. On the other hand, we have a small roof patio. And here's what we found looking out of the window on the side opposite the patio: A small staircase leading to a platform (the top of the downstairs windows) with railing. That little walkway ends then, leaving you with the only option of a 4 floors' jump down on the paved street. I was thinking it might be the emergency exit, but the evacuation plan on our door points another direction. So not sure what this is. An invitation for suicide? A diving platform in case the river floods?
My talk about the black hole information loss problem went very well (slides here). I wish you all a great weekend.
Thursday, June 24, 2010
Guestpost: Marcelo Gleiser
[A month ago, I was at a workshop at Perimeter Institute and I reported on a talk by Marcelo Gleiser. Marcelo's talk was very interesting and thought-stimulating. It touched upon very many different topics, from the process of knowledge discovery to the question of whether we should be searching for a fundamental theory of everything. In my post I expressed my opinion that of course believing in a theory of everything, if you take the name literally, is religion not science because if we had one we would never know if not one day we'd discover something that the theory would not explain. But the whole question of whether it exists is somewhat besides the point, the actual question (for me, the pragmatist) is what is a promising approach to take that will lead to progress.
Marcelo has now written a reply to some of the points that came up in my post and the comments, and to some other reactions that he got. This reply can also be found at his blog 13.7.]
To Unify Or Not To Unify: That Is (Not) The Question
My latest book, A Tear at the Edge of Creation, came out in the US early April. In it, I present a critique of some deeply ingrained ideas in physics. In particular, I examine the question of unification and the search for a theory of everything, arriving at conclusions that—judging from some of the reactions I’ve been getting in lectures and in various blogs around the world—are shocking to many people.
Of course, I welcome criticism and skepticism. We are used to this in scientific debates. What’s surprising to me, and perhaps alarming, is the speed with which superficial commentary in the blogosphere quickly escalates into complete misunderstanding of what it is that I am saying and why. So, I think the time is ripe for sketching a reply, even though the space here won’t do justice to the details of the argument. I do hope, however, that this will at least inspire critics and skeptics to actually read the book and judge for themselves and not through a few lines on a blog post.
Among other things, in the book I suggest that the notion of a final theory, that is, a theory that encompasses complete knowledge of how matter particles interact with one another, is impossible. First, note that “final theory” here deals only with fundamental particle physics. Any claim that physical theories could be complete in the sense of describing (and predicting) all natural phenomena, including why you’re reading this, shouldn’t be taken seriously.
First, we must consider if a complete theory of matter does exist. Second, assuming it does, if we can ever get to it. The first question is quite nebulous. We have no way of knowing if such a complete theory exists. We don’t even know what a “complete” theory is. You may believe it does and spend your life searching for it. That’s a personal choice. Or, like most physicists, you may believe this is nonsense, more metaphysics than physics. The second question, though, is tangible. Can humans achieve complete knowledge of the subatomic world?
To answer this question, we must look at how science actually works. In a post at her blog Back Reaction, physicist Sabine Hossenfelder expressed her surprise at my statement that it took me 15 years to figure out that the notion of a final theory is faulty. Sorry Sabine, I guess old habits are hard to break. At least, I did see the light in the end. Happily, she agreed with my basic argument, that since what we know of the world depends on our measurements of the world, we can never be sure that we arrived at a final explanation: as tools advance, there is always room for new discoveries. Knowledge is limited by discovery.
I go on to describe how the unifications that we have achieved so far, beautiful and enlightening as they are, are approximations and not “perfect” in any sense. The electroweak theory, a unification of the electromagnetic and the weak nuclear forces, is not a true unification but a mixing of the two interactions. Even electromagnetism, the paradigm of unification, only works flawlessly in the absence of sources. To be a truly perfect unification, objects called magnetic monopoles would have to exist. And even though they could still be found, their properties are clearly very different from the ubiquitous electric monopoles, e.g. point-like particles like electrons. We have partial unifications and we should keep on looking for more of them. This is the job of theoretical physicists. The mistake is made when symmetry, a very useful tool in physics, is taken as dogma.
I don’t agree with Sabine when she says that it doesn’t matter what you believe in as long as the search “helps you in your research.” I think beliefs are very important, and to a large extent drive what it is that we are searching and the cultural context in which research is undertaken. Wrong beliefs can have very negative consequences. And can keep us blind for a long time.
So, one of the points I make is that science is a construction that evolves in time to expand our body of knowledge through a combination of intuition and experimental consensus. There is no end point to it, no final truth to arrive at.
Now, here are some of the things that have been said about my arguments:
“Marcelo is disillusioned with unification; he has closed up his mind to string theory; he couldn’t find a Theory of Everything and now thinks no one can find one as well; he’s just frustrated; he doesn’t understand the role of symmetry in physics (!); he’s timing is bad because the LHC will be revealing new physics.” George Musser, at a Scientific American blog post wrote “My own reaction was that although it’s useful to caution against clinging to preconceived ideas about a final theory, Gleiser was too insistent on seeing the glass of physics as half-empty.” Musser goes on to say how much we do know about Nature and how much of that is due to the fact that simple laws govern natural phenomena.
It’s true that Musser (and Sabine) were basing their comments on a lecture I gave recently at the Perimeter Institute and not on my book (you can watch the video here). Even so, as I tried to make clear in my text, I would never put down the remarkable achievements of science and much less be foolish to say that there are no patterns and symmetries in Nature! After all, that is how science works, by searching for simplifying explanation of natural phenomena. Having the LHC turned on and able to probe physics at energies higher than ever before is a very exciting prospect.
The same general defensive zeitgeist was echoed by Neil Turok, the current director of the Perimeter Institute. We recently participated in a televised debate hosted by TV Ontario on Stephen Hawking’s ideas. We were a group of six physicists, hosted by Steve Paikin and had a great time. But at the end, when I made my arguments about final unification and the limits of knowledge, Turok accused me of pessimism!
If anything, my book is a celebration of the human mind and all that we have achieved in such a short time. The fact that I point out that science has limitations doesn’t detract from all of its achievements. Or from all that lies ahead.
I’m not disillusioned for not having found a TOE or for believing it doesn’t exist. I’m actually relieved!
The reactions that I have encountered only reinforce my point, that there is great confusion these days about the cultural role of science and scientists. Science is not a new form of religion, scientists are not holy men and women, and we don’t have or can have all the answers.
As I wrote in Tear at the Edge of Creation, “Human understanding of the world is forever a work in progress. That we have learned so much, speaks well of our creativity. That we want to know more, speaks well of our drive. That we think we can know all, speaks only of our folly.”
Hopefully, this acceptance of our perennial ignorance won’t be interpreted as an opening to religion and supernatural explanations. Let me make my position clear: behind our ignorance there is only the science we still don’t know.
Monday, June 21, 2010
Friday, June 18, 2010
The summer solstice is near and days here in Stockholm are getting longer and longer. The other day I woke up early and, looking out of the window, saw that it was dawning already. Or so I thought. The clock revealed that it wasn't the dawn I was seeing, but that the sun hadn't even set. My biorhythm seems to be a little confused these days.
Along with midsummer also the long awaited wedding of Sweden's Crown Princess Victoria is coming closer. Tomorrow Victoria will exchange I-do's in Stockholm Cathedral with her former personal trainer Daniel Westling. It's a giant marketing event: The Swedes have declared Stockholm's airport Arlanda the "Official Love Airport 2010" and the two weeks before the wedding we had to endure the "LOVE Stockholm 2010," a "two-week festival of love, right in the centre of Stockholm." You can buy postcards and posters of the happy couple in every supermarket here, together with loads of blue-yellow decorations. Busy cityworkers have planted yellow and blue flowers all over the place. Just the weather isn't really playing along, today it's rainy at 17° C.
My Swedish isn't good enough to actually understand the traffic report on the radio, but I understand as much as a long list of streets separated by stängt stängt stängt stängt (closed). I for certain will stay as far away as possible from the city center tomorrow. If your national TV station doesn't broadcast the event, you can follow the wedding ceremonies live tomorrow via SVT. I think it's great the two get married tomorrow because that way I was able to grab a slot for the laundry room on Saturday morning.
Next week, I'll be on a short trip to Bonn for a workshop on quantum black holes, where I'll give a talk about my paper with Lee on the black hole information loss. I wish you all a lovely weekend :-)
Thursday, June 17, 2010
Science Metrics
Tuesday, June 15, 2010
Why do people get tattooed?
Last night, I had a weird dream. A white haired man with a long beard insisted on tattooing my shoulder. I couldn't get him to drop his plans, so he started punching. I asked him what the image will be. “I'm doing a circle,” he said. He continued his circle but when he finished it didn't close. “Now I have to walk around with a stupid non-closing circle!” I complained and he poured his ink over me. Then I woke up.
You're welcome to analyze this dream, but not allowed to use the words “string” and “loop.”
If you read science blogs frequently you'll probably have come across one or the other posting of a science related tattoo. (See eg here for a nice compilation.) It always leaves me wondering what drives people to do that. It's one of these emerging social and cultural trends that are so complex even the people doing it don't know why they're doing it. It is, from an evolutionary perspective, very interesting what weird behaviors intelligent creatures can develop in large groups. My attempt to understand humans recently brought me across the paper “Modifying the body: Motivations for getting tattooed and pierced” (Wohlrab, Stahl and Kappeler, Body Image 4 (1007) 87). They start with an interesting historical summary (please see paper for references):
“[Tattooing and body piercing] have a long history and are well known from various cultures in Asia, Africa, America, and Oceania. There is also evidence for the prevalence of tattoos in Europe, dating back over 5000 years. Although the appearance of tattoos and body piercings varied geographically, they always possessed a very specific meaning for the particular culture. Piercings were often used in initiation rites, assigning their bearer to a certain social or age group, whereas tattoos were utilized to signal religious affiliations, strength or social status. In Europe, the practice of tattooing was predominant among sailors and other working class members from the beginning of the 20th century onwards. Later on, tattoos assigned affiliations to certain groups, such as bikers or inmates. In the 1980s the punk and the gay movement picked up invasive body modification, mainly as a protest against the conservative middle class norms of society.
Until the 1990s, body modifications remained a provocative part of various subcultures. In the last decade tattoos and piercings have increased tremendously in popularity, rising not only in numbers but also involving a broader range of social classes.”
Thus, historically tattoos seem to predominantly have been used to signal affiliation to or sympathy with a group. The paper is basically a literature survey, and the authors then identify ten motivations for getting tattooed that have been studied. These are: 1) Beauty, art and fashion 2) Individuality 3) Personal narrative 4) Physical endurance 5) Group affiliation and commitment 6) Resistance 7) Spirituality and cultural tradition 8) Addiction (to obtaining the tattoo) 9) Sexual motivation (in the case of tattoos: expressing affectation or emphasizing the own sexuality) 10) No specific reason (eg under the influence of drugs).
For what science tattoos are concerned, I think we can forget about the last category. It seems quite unlikely to me the average guy on the street will get drunk and wake up the next morning with the Wheeler-DeWitt equation on his shoulder. For what point 4) is concerned, I think we can leave this aside as well. I don't think the physical endurance is higher for scientific motives. Unless maybe there's a mistake in the equation.
For what sexual motivations are concerned, it is in this context interesting to draw upon a recent survey, conducted in Germany (sample size approximately 2500, as reported in “Machen Tattoos sexy?” forschung SPEZIAL. Das Magazin der Deutschen Forschungsgemeinschaft, 2/07, 22-25). More than 10% of men and more than 8% of women were tattooed. The age range that currently dominates the wedding market (18-36 years) has the largest fraction of tattooed people. Men are more likely to be tattooed on arms and legs, whereas women prefer places that can easily be covered by cloths: back, belly, bottom. Not so surprisingly, men prefer designs with skulls, weapons and such, whereas women prefer flowers and animals. Maybe the most interesting fact though is that while only 8% of women had a tattoo, 56% of the participants with a tattoo had a partner who was also tattooed. So there's clearly some matching going on there. Another study in which participants were shown images of tattooed people revealed that both women and men judged people with tattoos to be more “aggressive” and “dominant.” Maybe for some, that is a desired effect?
Needless to say, all that reading didn't really explain why people want to have an equation on their arm. I can relate to the beauty/fashion motivation to some extend, but I suspect that if your fashion statement are Maxwell's equations you'll get more confused than admiring looks. I suppose the most likely motives are thus personal narrative and showing group affiliation and commitment. Or maybe we're seeing an attempt of resistance to anti-intellectualism? Not to mention that you can upload the photo to your blog and collect cheers. As for myself, I've fleetingly considered getting tattooed once or twice, but my tastes are at the best metastable and whatever the design, I'd probably get fed up with it after a few months, so tattoos are not for me.
Anyway, it is sometimes very refreshing to read an article in a journal I had never heard of before like Body Image. The most amusing part was this sentence from the abstract, right out of the ivory tower:
“[A] profound understanding of the underlying motivations behind obtaining tattoos and body piercings nowadays is required.”
Sure, I mean, unstable financial systems are ruining the lives of millions of people, climate change is about to erode the basis of many economies posing a threat for global political and social stability, each year about 5 million people still die because they don't have enough to eat, but what's really required is a profound understanding of why people punch needles through their nipples. If you replace “motivations behind” with “structure of” and “obtaining tattoos and body piercings” with your favourite physics term, I'm sure you'll find the same sentence in a significant fraction of arxiv papers...
Saturday, June 12, 2010
Book review: From Eternity to Here by Sean Carroll
By Sean Carroll
Dutton Adult (January 7, 2010)
Most of you will know Sean Carroll, who blogs at Cosmic Variance. Sean is a Senior Research Associate at CalTech and his research focuses on cosmology, general relativity and the standard model, as well as extensions thereof. He has written a textbook on General Relativity, and the lecture notes that gave rise to the book are available online. I've met Sean a few times, he's an interesting person and gives great talks. Sean has a special interest in the arrow of time, and that is also the topic of book “From Eternity to Here.” The arrow of time is, in a nutshell, the question why the past is different from the present.
I bought the book for three reasons. One is that for many years I've been using the PDF version of his lecture notes as a handy quick reference when on travel and had a bad consciousness for never buying the book. The second one is that from reading Sean's blog I know he writes well. The third reason is that adding a second book to the order rendered delivery free.
“From Eternity to Here” is a very well written book that communicates a lot of science, both textbook science and contemporary science, while at the same time being amazingly accurate. The biggest part of the book - all but the last chapter - is dedicated to accurately framing the question. Why is it interesting to ask why the past was what it was? What exactly is it that we don't understand? How do we get a grip on the problem? For this, Sean covers first of all the second law of thermodynamics, then special relativity, general relativity, cosmology, quantum mechanics, black hole physics, and finally inflation and the multiverse. In the last chapter, he then discusses possible solutions to the question he has posed and puts forward his own solution as the most plausible one. Along the way he scratches on topics like the vacuum energy, structure formation, the AdS/CFT duality and magnetic monopoles.
Sean is very careful with distinguishing between established science and unconfirmed speculations. The only glitch is the section on the holographic principle where he fails to point out that there is no experimental evidence for such a feature of Nature to be true in all generality. I am somewhat sick of being misinterpreted on this point so let me be very clear here. All I am saying is that, absent experimental evidence, scientists should be very careful with what they put forward as a true description of Nature. Theoretical evidence can very easily be biased simply because a topic that attracts attention may mount one-sided “evidence.” This can never replace actual tests of a hypothesis. The holographic principle certainly does not rest on the same basis as ΛCDM or the Schrödinger equation and I wish its status had been framed more clearly. Anyway, Sean needs the holographic counting of degrees of freedom for the rest of his argument.
I was very pleased that Sean's explanations of physical concepts are not as superficial and vague as one frequently finds in popular science books. He does not shy away from the phase space, using logarithms, and discusses the amplitude of the wave function. The chapter on quantum mechanics however somewhat suffers from the overuse of cats and dogs. The book has plenty of footnotes with additional explanations, and offers many references so that the interested reader will easily be able to find the relevant keywords and dig deeper, should they wish so. On several occasions I took a note that Sean had forgotten to point out a specific assumption that entered his argument or left out some exceptions. In every single case, these points were later addressed, so I am left with nothing to complain about.
I personally don't have a large interest in the topic and don't care very much about the whole discussion. I think the question is ill-posed and when we have a better understanding of quantum gravity we'll see why. Sean's book didn't succeed in increasing my interest. Nevertheless, it was a pleasure to read. Sean has a good sense of humor, but doesn't overdo it. The story he tells is also well embedded into its scientific history and I learned a thing or two here that I hadn't known before. Both the historical and the philosophical aspects however play a secondary role and don't take over the scientific discussion. All together, the book is very well balanced and a recommendable read. It has something to offer for anybody who has an interest in modern cosmology and/or the arrow of time. I'd give this book 5 out of 5 stars.
From January through April, Sean offered a book club at his blog, each weak discussing another chapter. You might find this a useful addition to the book itself.
Wednesday, June 09, 2010
Perimeter Institute is looking for a Scientific IT specialist
Two years ago, I organized a conference on Science in the 21st Century, focusing on topics at the intersection on science, society and information technology. (I wrote about the conference here, a summary is here and a brief write-up of my own talk is here.) There are three aspects to the changes that the use of information technologies are bringing to science. One is the improving communication with the public - this blog is an example for such a change. The second one is that advances in hard- and software allow us to better understand the process of knowledge discovery and the dynamics of the scientific communities itself - the Maps of Science are an example for this. The third aspect, and probably the one most interesting for the scientist at work, is the development of new tools that support research and researchers in their every day work.
As I learned the other day, Perimeter Institute is now looking for a person who works at exactly this intersection. The job description reads as follows:
The Perimeter Institute for Theoretical Physics (PI) is looking for a Scientific IT specialist -- a creative individual with experience in both scientific research and information technology (IT). This is a new, hybrid, research/IT position within the Institute, dedicated to helping PI’s scientific staff make effective use of IT resources. It has two clear missions. First, to directly assist researchers in using known, available IT tools to do their research. Second, to uncover or develop cutting-edge IT resources, introduce and test them with PI researchers, and then share the things we create and discover with the worldwide scientific community.
By "tools", we mean almost anything. Coding techniques are an obvious example. Collaboration and communication technologies are another: tools for peer-to-peer interactions (such as skype), virtual whiteboards, video conferencing tools, platforms for running virtual conferences (that can do justice to talks in the mathematical sciences), and novel ways of presenting research results such as archives for recorded seminars, blogs, and wikis. Further examples include tools for helping researchers organize information (e.g., specialized search engines and filtering schemes), and end-user software that facilitates bread-and-butter scientific activities like writing papers collaboratively, preparing presentations, and organizing references.
We are seeking a person who brings an independent and ambitious vision that will help define this vision. The job is as yet quite malleable in its scope and duties! We're looking for someone who is inspired by the possibility that new IT tools can improve or perhaps even revolutionize the way that physics research is done, and someone who can take full advantage of a mandate to create and implement that vision.
Some Duties and Responsibilities:
- Act as a knowledge broker among Researchers. That is, find and test new programs and practices, advertise them, and be prepared to train others in their use.
- Participate in the creation of a high quality “standard" Researcher IT environment (desktop hardware, software set-up), built from a mix of open source software and popular commercial packages.
- Help with High Performance Computing demands.
- Maintain expert level knowledge in the use of the main packages used by Researchers, including Mathematica, Maple, LaTex, etc.
For the official job ad, go here.
[Via Rob Spekkens]. The deadline for applications is Friday, July 2, 2010. The Albert Einstein Institute in Potsdam meanwhile offers an almost identically sounding position. I've been told PI was first, but their posting is not dated.
I very much like this development. My requirements on IT staff these days are however very modest. I am happy when the printer spits out my paper without chewing up some pages or leaving them blank. My biggest wish would be not a virtual whiteboard but an actual whiteboard with a plugin to my computer so I could use the board for equations and figures during a skype call. The equations are usually cumbersome but still doable, in the worst case by typing them in LaTex into the chat interface. But diagrams are a disaster. Drawing with a mouse yields no sensible results and the drawing pads that I've tried weren't too convincing either, even neglecting the problem on how to incorporate them into the call. On occasion I've thus drawn on a paper and held it into the camera. This however only works for figures with few details and necessitates plenty of additional explanations.
What is the software or hardware you dream of for your research life?
Saturday, June 05, 2010
Diamonds in Earth Science
To clarify the situation, experiments would need to push above 120 Gigapascal and 2500 Kelvin. I [...] started laboratory experiments using diamond-anvil cell, in which samples of mantle-like materials are squeezed to high pressure between a couple of gem-quality natural diamonds (about two tenths of a carat in size) and then heated with a laser. Above 80 Gigapascal, even diamond—the hardest known material—starts to deform dramatically. To push pressure even higher, one needs to optimize the shape of the diamond anvils's tip so that the diamond will not break. My colleagues and I suffered numerous diamond failures, which cost not only research funds but sometimes our enthusiasm as well.
(From The Earth's Missing Ingredient)
But in the end, Kei Hirose and his group succeeded in subjecting a small sample of magnesium silicate to the pressure and temperature that prevails in the lower Earth's mantle, about 2700 kilometer below our feet.
Planet Earth has an onion-like structure, as has been revealed by the analysis of seismological data: There is a central core consisting mostly of iron, solid in the inner part, molten and liquid in the upper part. On top of this follows the mantle, which is made up of silicates, compounds of silicon oxides with magnesium and other metals. The solid crust on which we live is just a thin outer skin.
The lower part of the mantle down to the iron core was long thought to consist of MgSiO3 in a crystal structure called perovskite. However, seismological data also revealed that the part of the mantle just above the CMB (in earth science, that's the core-mantle boundary, not the cosmic microwave background... ) somehow is different from the rest of the mantle. This lower-mantle layer was dubbed D″ (D-double-prime, shown in the light shade in the figure), and it was unclear if the difference was by chemical composition or by crystal structure.
As Kei Hirose describes in the June 2010 issue of the Scientific American, his group started a series of experiments to study the properties of magnesium silicate at a pressure up to 130 Gigapascal (water pressure at an ocean depth of 1 kilometer is 0.01 GPa) and a temperature exceeding 2500 Kelvin ‒ the conditions expected for the D″ layer of the lower mantle.
To achieve such extreme conditions, one squeezes a tiny piece of magnesium silicate between the tips of two diamonds, and heats up the probe by a laser. The press used in such experiments is called "laser-heated diamond anvil cell".
The figure shows the core of a diamond anvil cell: The sample to be probed is fixed by a gasket between the tips of two diamonds. The diameter of the tips is about 0.1 millimeter, so applying a moderate force results in huge pressure.
Diamonds are used because of their hardness, but they have the additional bonus of being transparent. Hence, the probe can be observed, or irradiated by a laser for heating, or x-rayed for structure determination.
The diamonds are fixed in cylindrical steel mounts, but creating huge pressure does not require huge equipment: The whole device fits on a hand! (Photo from a SPring-8 press release about Kei Hirose's research.)
Actually, the force on the diamond tips is applied in such a device by tightening screws by hand.
In the experiment, the cell was mounted in a brilliant, thin beam of x-rays created by the SPring-8 synchrotron facility in Japan. This allows to monitor the crystal structure of the probe by observing the pattern of diffraction rings.
It was found that under the conditions of the D″ layer of the lower mantle, magnesium silicate forms a crystal structure unknown before for silicates, which was called "Post-Perovskite". The formation of post-perovskite in the lower mantle is a structural phase transition of the magnesium silicate, and this transition can explain the existence of a separate the D″ layer, and many of its peculiar features. It also facilitates heat exchange between core and mantle, which seems to have quite important implications for earth science.
And here is the heart of the experiment (from the "High pressure and high temperature experiments" site of the Maruyama & Hirose Laboratory at the Department of Earth and Planetary Sciences, Tokyo Institute of Technology) ‒ a diamond used in a diamond anvil pressure cell:
High-quality diamonds of this size cost about US $500 each.
Thursday, June 03, 2010
Impressions from the PI workshop on the Laws of Nature
As you know, 2 weeks ago I was at Perimeter Institute for the workshop on the Laws of Nature: Their Nature and Knowability. It was a very interesting event, bringing together physicists with philosophers, a mix that isn't always easy to deal with.
People (them)
On the list of participants, you'll find some well known names. Besides the usual suspects Julian Barbour and Lee Smolin, Paul Davies was there (though only for the first day), Anthony Aquirre (the event was sponsored by FQXi) and of course several people from PI and the University of Waterloo. In my previous post, I already wrote about Marcelo Gleiser's talk. Marcelo is from Brazil, and he is apparently well known there for his popular science books (which was confirmed by Christine in an earlier post.) I had frankly never heard of him before. I talked to him later over dinner, and he told me he writes for a group blog called 13.7 together with, among others, Stuart Kauffman who is also well known for his popular science books. (13.7 is the estimated age of the universe in billion years. What will they do if that number gets updated?)
Another interesting name on the list of participants is Roberto Unger, who is a well-known Brazilian politician and besides that a professor for law at Harvard Law School, and author of multiple books on social and political theory. He apparently has an interest not only in the laws of societies, but also in the laws of Nature*. And finally let me mention George Musser was also at the workshop. George writes for Scientific American and is author of The Complete Idiot’s Guide to String Theory. He turned out to be a very nice guy with the journalist's theme "I want to know more about that."
Talks (their)
Now let me say a word about the talks. First, and most important, all the talks were recorded and are available on PIRSA here. The talks on the first day were heavily philosophical. I will admit that I often have problems making sense of that. Not because I don't have an interest in philosophy, but because one frequently ends up arguing about the meaning of words which is, at the bottom of things, a consequence of lacking definitions and thus a waste of time. Yes, my apologies, I'm, duh, a theoretical physicist with some semesters maths on my CV. If I don't see a definition and an equation, I get lost easily. In some cases it seems the philosophers imply some specific meaning that they just never bother to explain. But in other cases they'll start arguing about it themselves, and that's when I usually zoom out wondering what's the point in arguing if they don't know what they're arguing about anyway.
The most interesting event on the first day was arguably Lee Smolin's and Roberto Unger's shared talk "Laws and Time in Cosmology". Let me add that I've heard Smolin talking about the "reality of time" several times and I still can't make sense of it. The problem I have is simply that I don't know what he's talking about. This recent talk didn't change anything about my confusion, but if you haven't heard it before, you might find it inspiring. Unger's talk is very impressive on the rhetorical side. Unfortunately, it made even less sense to me than Lee's talk. For all I can see, there's no tension neither between a block-universe and a notion of simultaneity nor between a block-universe and causality, as I think I heard Unger saying (thus my question in the end). Point is, I don't understand the problem they're attempting to address to begin with. I see no problem. As Barbara Streisand already told us "Life is a moment in space" and "In love there is no measure of time." Consequently, a universe where time is real must be loveless. I don't like that idea.
On that note, let me recommend Julian Barbour's talk "A case for geometry". Julian is a charming British guy and he has his own theory of a lovely, timeless universe. I don't buy a word of what he says, but his talk is very accessible and fun to listen to. It makes your head spin what he's saying, just try it out, it's very intriguing. I am curious to see how these ideas will develop, it seems to me they might be on the brink of actually making predictions. (A somewhat more detailed explanation of his ideas is here, audio becomes audible at 3:30 min.)
On the second day, we had several talks discussing concrete proposals for how one could think of the laws of Nature off the trodden path. You probably won't be surprised to hear that one of the suggestions is that of "Law without Law: Entropic Dynamics" by Ariel Caticha. It is not directly related to Erik Verlinde's entropic gravity, but certainly plays in the same corner of the room: exploiting the possibility that fundamentally all our dynamics is simply a consequence of the increase of entropy. Ariel's talk however isn't really recommendable, it sits on a funny edge between too many and too few details.
Another approach is Kevin Knuth's who put forward in his talk "The Role of Order in Natural Law" the idea that on the basis of all, there's order - in a well-defined mathematical sense. I can't avoid the impression though that even if this worked out to reproduce the standard model, it would merely be a reformulation. Kevin's talk was basically a summary of this recent paper. And Philip Goyal gave a very nice talk on "The common symmetries underlying quantum theory, probability theory, and number systems." I have a lot of sympathy for the attempt to reconstruct quantum theory, it's just that I don't understand why literally all the quantum foundations guys hang themselves up on the measurement process in quantum mechanics. For what I'm concerned, quantum field theory is the thing, and I'm still waiting for somebody to reconstruct the non-commutativity of annihilation and creation operators.
Finally, let me mention Kevin Kelley's talk "How does simplicity help science find true laws?" Kelley is a philosopher from Carnegie Mellon, and in his talk he explored whether it is possible to put Ockham's Razor on a rational basis. Unfortunately, while the theme could in principle have been very interesting, his talk is not particularly accessible. He assumed way too much knowledge from the audience. At least I get very easily frustrated when technical terms are dropped and procedures are mentioned without being explained, since it's not a field I work in. In any case, I'll spare you the time watching the full thing and just mention an interesting remark that came up in the discussion. Apparently there have been efforts to create a computer software that could simulate a "scientist," in this case for the example of trying to extract a theory from data of the motion of the planets. At least so far, such attempts failed (if anybody knows a reference, it would be highly appreciated.) So it seems, for the time being, scientists will not be replaced by computers.
At the end of the last day we had a discussion session, moderated by Steven Weinstein, wrapping up some of the topics that came up the previous days and some others. One of them is the question about the power of mathematics and if there are limits to what humans can grasp (a theme we have previously discussed here). For a fun anecdote making the point well, watch Steven at 1:13:50 min ("I remember distinctively being in a graduate quantum mechanics class by Bob Wald...") Of course Tegmark's mathematical universe made an appearance as well, another topic we have previously discussed on this blog. For what I am concerned, declaring that all is mathematics may be some sort of unification of the laws of Nature, alright, but it's eventually a completely useless unification. And that brings me to...
Thoughts (mine)
On several occasions at the workshop, I felt like the stereotypical physicist among philosophers, and it took me a while to figure out what I found lacking at this workshop. You could say I'm a very pragmatic person. There's even an ism that belongs to that! If you talk about reality and truth, I don't know what you mean, and I actually don't care. This is just words. I'll start caring if you tell me what it's good for. If you want to reformulate the laws of physics, fine, go ahead. But if you want me to spend time on it, you'll have to tell me what the advantage is. If there's two theories and they make the same predictions, that doesn't cause me headaches. For what I'm concerned, if they make the same predictions, they're the same theory.
What matters in the end about a law or a theory or a model is not whether it's philosophically appealing and not even if there's a rational process by which it's been selected (and btw, what means "rational" anyway), but simply whether it's useful. And usefulness is eventually a notion deeply connected to human societies and values. For that reason I think to understand the scientific method and its success one inevitably needs to take into account the dynamics of the communities and the embedding of scientific knowledge into our societies. (It should be clear that with usefulness I don't necessarily mean technical applications as I have recently expressed in this post.)
Leaving aside that I found this aspect entirely missing to the discussions about the process of science itself and its possible limitations, the workshop has given me a lot to think about. Having said that the pragmatist in me searches for the use in all that enters my ears, I nevertheless have enough fantasy to imagine that some of the themes discussed at the workshop will become central to shaping our thinking about the laws of Nature in the future and thus eventually prove their usefulness. It was a very stimulating meeting and the approaches that were presented are all as bold as courageous. It will be interesting to follow the progress of these thoughts.
*I once made an attempt to read one of Unger's books, What should the left propose? I had to look up every second word in a dictionary, and even that didn't always help. When I had, after an hour or so, roughly deciphered the meaning of a page it seemed to me one could have said the same in one simple sentence, avoiding 3 or more syllable words. I gave up on page 20. Sorry for being so incredibly unintellectual, but to me language is first and foremost a means of communication. If you want to be heard, you better use a code that the receiver can decipher. Friedrich Engels, for example, was an excellent writer...
Tuesday, June 01, 2010
Update on the ESQG 2010
• What to sacrifice?
• The Future of Particle Physics.
• Experiments and Thought Experiments
|
66a175de8013a6d2 | Physical, Environmental and Mathematical Sciences
The interaction of the surfaces of spacecraft with the space environment has significant and poorly understood effects on the spacecraft's performance. In conjunction with UNSW Canberra Space, several projects are available investigating typical spacecraft surface interactions. Two particular targets for investigation are energy transfer and accretion processes in collisions with rarefied gasses, and understanding the factors controlling surface coating though photo-initiated polymerisation of adsorbates.
Predicting the structure of molecular solids is vital for many aspects of materials science. Packing densities and patterns have a critical effect on many important mechanical and materials function properties. This project shall develop new methods to predict molecular packing in crystals, based on multiple hierarchies of approximations leading to accurate, quantum chemistry calculation of solid phase atomic structure.
Quantum mechanics is the most accurate framework known for investigating and simulating chemical reactions, incorporating all significant physical effects in most cases. However, the practicality of using quantum mechanical descriptions of the dynamics of reactions is limited in practice by the computational cost of performing accurate quantum simulations.
Many applications exist for interrogating large, scattered, high-dimensional data sets to find a group of nearby points for an arbitrary test point. Examples range from advanced methods to simulate chemical reactions to facial recognition software. This project will develop fast neighbour searching algorithms and implementations with a focus on applications in chemistry and physics.
Suitable candidates will have a mathematical background suitable for applied mathematics research, and engage in scientific programming.
Many advances have been made recently in applying Gaussian basis functions in the modelling of the quantum nature of chemical reactions. A typical approach uses overlapping Gaussians that follow likely molecular trajectories as a basis set within which to solve the time dependent Schrödinger equation that describes the quantum behaviour of a molecular system that is undergoing a reaction.
Photosynthesis is the source of all biological solar energy capture, and the source of most atmospheric oxygen. All photosynthetic oxygen production occurs in the oxygen evolving centre of a structure known as photosystem II. However, much remains to be understood about the detailed mechanism of using captured solar energy to generate free charges which go on to oxidise water into oxygen. |
6d14098b6920a3dd |
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Earth Science
Scientific Cruise Meets Perfect Storm, Inspires Extreme Wave Research 107
Posted by Unknown Lamer
from the creative-punishment-for-copyright-infringers-discovered dept.
An anonymous reader writes "The oceanographers aboard RRS Discovery were expecting the winter weather on their North Atlantic research cruise to be bad, but they didn't expect to have to negotiate the highest waves ever recorded in the open ocean. Wave heights were measured by the vessel's Shipborne Wave Recorder, which allowed scientists from the National Oceanography Centre to produce a paper titled 'Were extreme waves in the Rockall Trough the largest ever recorded?' It's that paper, in combination with the first confirmed measurement of a rogue wave (at the Draupner platform in the North Sea), that led to 'a surge of interest in extreme and rogue waves, and a renewed emphasis on protecting ships and offshore structures from their destructive power.'"
Scientific Cruise Meets Perfect Storm, Inspires Extreme Wave Research
Comments Filter:
• by Anonymous Coward on Monday April 16, 2012 @11:06PM (#39707379)
This scientific cruise also proved that the only kind of cruise where nobody gets laid is a "scientific cruise"
• by cplusplus (782679) on Monday April 16, 2012 @11:15PM (#39707423) Journal
I only RTFAs to find out how high the waves were - it turns out they were up to 29.1 meters (95.5 feet).
• Rogue waves (Score:3, Funny)
by gstrickler (920733) on Monday April 16, 2012 @11:23PM (#39707453)
Outlaw them and put out a bounty (or a Bounty?)
• 2006 (Score:5, Informative)
by Anonymous Coward on Monday April 16, 2012 @11:32PM (#39707491)
The article was published in 2006. How is this 'new?'
• The article was published in 2006. How is this 'new?'
I guess it's some sort of tie in with the 100th anniversary of the Titanic making it almost all the way across the Atlantic.
• The wave was so high that the ship did a loopty-loop, causing a rift in time where they just ended up here. The same phenomenon can be seen if you can swing high enough on a swingset to go around once
• by jlehtira (655619)
Well, I agree with your point. But six years is a good time to let scientific papers simmer. Less than that is not enough time for other scientists to evaluate the correctness and value of some paper.
• by Anonymous Coward
Many researchers were lost during the peer-review of this paper.
• by dreemernj (859414)
2006? Wasn't that around the time a rogue wave was recorded on The Deadliest Catch?
• Data collected in 2000. Paper published in 2006. Reported in /. in 2012. The pace of good science is slow and deliberate.
• by Anonymous Coward on Monday April 16, 2012 @11:43PM (#39707553)
look up Schrodinger wave equations and apply them to ocean waves. You will get 30+ meter tall waves with a trough next to the "wall" of water, (the wave is tall and narrow - like a wall). This trough adds to the great difficulty in surviving one of these waves. Ships that are designed to withstand forces of 10 tons/m2 have to content with 10 times that force. I believe there was a study in which someone, (don't remember her name :( ) mapped the entire earth over a two week period and found something on the order of 20 of these waves. Fascinating stuff.
• by phantomfive (622387) on Tuesday April 17, 2012 @02:21AM (#39708089) Journal
Oh yeah, just found it []. They found about 10 giant waves.
• by Anonymous Coward
FYI the Schrodinger wave equation does not describe ocean waves. Water waves are described by the Navier-Stokes (N-S) equations. Turbulence models fall out of N-S, however only electrons sometimes fall out from Schrodinger :)
• There is a non-relativistic version of the Schrödinger equation. Some theories attempt to explain rogue waves in the open sea using these non-linear equations as a model, because the distribution of wave heights that would result from the linear model substantially underpredicts the occurrence and size of rogue waves.
• by Anonymous Coward
The nonlinear Schordinger equation is one of the many various equations that can be used to describe the behaviour of water waves in various regimes, with a tiny bit about it on Wikipedia here []. Although the NLS is mostly used for behaviour of the envelope of deep water waves, which means you can show soliton based rouge wave like behaviour, but not say much about trough to peak steepening as in the grandparent post.
The set of equations and theories used to model nonlinear water waves is quite diverse, wit
• by WaffleMonster (969671) on Monday April 16, 2012 @11:52PM (#39707605)
For those looking for more details about this voyage []
• Specifically in 1998, a 120ft wave off the east coast of tasmania []
• Since extreme waves were not the subject of their expedition, they had not read all the prior literature.
• by TapeCutter (624760) on Tuesday April 17, 2012 @01:48AM (#39708017) Journal
The Tasman sea is notorious for rouge waves. Many moons ago I worked a fishing trawler in Bass Straight, I never saw anything like 120ft but the regular waves were tall enough that the radar was blocked by the peaks when the boat was in a trough, I'm guessing the radar mast was about 30ft above the water line. A lot like riding in a giant roller coaster carriage really, slowly climb up one wave, crest, then race down the other side and watch the bow dig under the next one, throw the water over the wheel house as the bow pops up to the surface, and starts the next climb. From what I've heard, the problem with rouge waves is not so much their height but the fact that they are too steep to climb.
• Wow, that is incredibly exciting.
• I detect a hint of sarcasm but to be honest it was downright fucking scary the first trip but after a few trips it became as exciting to me as an old fashioned roller coaster is to the guy who stands up on it all day operating the brake. Although a stingray the size of a family dinner table flapping about on an 8X12 deck was never boring.
• No sarcasm at all. If the human lifespan weren't so short I would definitely consider going down and trying it out for a few years. I don't know about that stingray thing, though. I know people who go ocean kayaking but that's nothing in comparison.
• by tlhIngan (30335)
Waves are never boring, especially big ones. The key is to cut through them - if you let them hit the side, you risk capsizing. The only way to do this is engine power (run
• by serbanp (139486)
Does this mean that the "the Perfect Storm" depiction of how the Andea Gail sunk was technically inaccurate? In that film, the ship went with its bow straight into the freak wave but could not reach the top and fell over.
• Yep, it's a lot like a plane, if engine is fucked, gravity takes over and you basically fall of the wave..
• That article claims 42.5m is 120 feet - it's actually 140 feet. The wave was probably recorded as 120 feet and someone mangled the conversion rather than the other way round.
• by Sarten-X (1102295) on Monday April 16, 2012 @11:53PM (#39707611) Homepage
Rogue waves: Demonstrating yet again that reality is a fascinatingly weird place.
• by iamhassi (659463)
And we don't understand our planet as much as we think. We are always focused on exploring strange new worlds, to seek out new life and new civilizations, to boldly... um, you get the idea, but look, there's new things happening on our own planet. How can we understand new planets when we don't understand the one we are on? Not saying never explore space, just saying maybe we should focus on what we have.
• by Anonymous Coward
How can we understand this planet when we have nothing to compare it to?
Rethorical questions only caters to peoples emotional response but they don't make much of an argument.
• by Sarten-X (1102295)
Reminds me of the TV show seaQuest... for almost a whole season, they had interesting episodes based around real weirdness in the oceans.
What fascinates me even more is the emergent behavior observable in simple systems, such as growing crystals, diffusing liquids, convection currents... all of those delightfully complex results from simple principles. There's beauty in the result, and simplicity in the process.
• by Anonymous Coward
Although the paper might have spurred interest in rogue waves, the wave in the paper linked in the summary wouldn't really be considered a rogue wave. Usually a cut-off is arbitrarily picked at 2 times the significant wave height (the average of the highest third of waves). In this case, the wave was about 1.5 times the significant wave height. Statistically speaking, you would expect about 1 in a 100 waves to be 1.5 times the wave height, just from the mixing and constructive interference of waves, whil
• Big waves (Score:4, Interesting)
by MarkRose (820682) on Tuesday April 17, 2012 @12:06AM (#39707665) Homepage
Waves over 20 m (60 ft) tall are actually pretty common in some places. My dad is senior keeper at Triple Island Lightstation [], located just off the BC coast. In severe winter storms, the waves will often crest over the square part of the building, which is about 20 m above sea level. This January, one such wave blew in a storm window on the top floor -- several tons of water will sometimes do that. The building stays up because it's constructed with 2 ft thick rebar concrete walls.
• Re:Big waves (Score:5, Informative)
by tirerim (1108567) on Tuesday April 17, 2012 @12:39AM (#39707811)
TFA is talking about waves in the open ocean, though. Waves get higher when they reach shallower water, so the 20 m waves you're talking about would have been significantly smaller in the open ocean -- which makes 29 m open ocean waves that much more impressive.
• Nice traditional exterior, but sad to see the drop ceiling [] on the interior. At least the wood floor is original.
• Interesting link but some of the text is reminiscent of Julian and Sandy ( from "Round the Horne", I mean, "The Triple Island light was built to guide mariners through the rocky waters of Brown Passage, on their way to the port of Prince Rupert.", I ask ya!
• It's interesting how often myth and legend end up being scientific fact. There has been talk since sailors took to the sea of rogue waves that reached a 100' or more. Science has been confirming these myths in recent years. Most myths have an element of truth in them. On the practical side it's a serious concern since surviving a 100' rogue wave is not something all sea worthy ships can do yet they can face them without warning. I read years ago the theoretical limit was twice what has been recorded so the
• The paper is from 2006, and describes a wave observed in 2000.
Satellite-based radar altimeters produce a lot of data about wave height world wide, but they don't, apparently, have quite enough resolution yet to see this kind of thing. A view of such waves from above, over a few minutes, would tell us a lot. Is it an intersection of two or more waves? How far does it travel? How long does it persist?
The U.S. Navy has put considerable effort into answering questions like that.
• bad statistics (Score:4, Interesting)
by Tom (822) on Tuesday April 17, 2012 @03:24AM (#39708283) Homepage Journal
What has fascinated me about freak/rogue waves is that sailors have known about them for decades if not centuries, but scientists were telling them it can't be.
And the reason is badly understood statistics. I've recently read Black Swan, and that gave me a few new concepts to work with, but the basic idea is exactly that: We don't really have a good understanding of statistics and probabilities, especially about extremely low probabilities in big numbers.
Or, as Tim Minchin put it: One-in-a-million things happen all the time.
And it's not just in the oceans. The entire financial crisis was caused by the people in charge taking huge (but low probability) risks, ignoring that once enough people have taken enough of those "low probability" risk, they become very likely to actually happen.
Freak waves are cool because they are in the gray area between the normal distribution and the really freaky - thus they happen often enough that they are rare, but not bigfoot-rare. We can actually study them.
• Re: (Score:3, Interesting)
by edxwelch (600979)
There's an interesting article about that, here: []
Apparently, there are two scientific models, linear, which says freak waves are impossible and Quantum physics which says they are possible.
• by Tom (822)
The problem is that a gaussian approach to the numbers assumes that random fluctuations will even out. But the equations used in quantum physics allow for waves to combine, and that's what is happening - interference, just not between 2 waves as in the double-slit experiment, but between dozens or maybe hundreds of waves.
This article here: [] shows towards the bottom how massive peaks you can get with mult
• by Anonymous Coward
Linear wave theory allows for interference and combining of waves (that is kind of actually one of the major properties of linear theories in a lot of situations). The statistics on linear theory waves (which ends up being a Rayleigh distribution, not a Gaussian) is what says that waves much larger than those around it are very unlikely. What nonlinear theories add is not just overlapping like interference, but soliton like solutions, where a single wave or small wave train much larger than neighboring wa
• by Tom (822)
Thanks, AC. In 12+ years of /. this was one of the most informative AC comments I've come across.
• We have bigger waves in Texas!
• I've never understood that particular idiocy. Texans know they don't live in the biggest US state, right? Texas is less than half the size of Alaska.
• by dtmos (447842) * on Tuesday April 17, 2012 @06:09AM (#39708623)
My uncle retired as a US Navy Captain. For many years he had two photographs displayed in his house, which he ascribed to Admiral "Bull" Halsey's "second" typhoon [], in June 1945. At that time my uncle was an ensign, assigned to a destroyer, and on his first sea voyage.
The two photographs were of a sister destroyer. In the first photograph, all one sees is a giant wave, with the bow of the destroyer sticking out of one side, and the stern sticking out of the other. The middle of the ship, including the masts and superstructure, is submerged and not visible.
In the second photo, taken a few seconds later, the middle of the ship is now visible, but both the bow and stern are now submerged in the wave train. And as a kid, the part that fascinated me the most: You could see an air gap below the middle of the ship, between the ship's keel and the wave trough below.
• I'm surprised I can't get for my boat (or raft) a platform with accelerometers that operates a hydraulic piston to compensate for wave action. It might need some lateral actuator too, as wave motion is circular. But it might not, if the light floats slide along the surface as the piston pushes down on them keeping the heavy inertial payload in place.
Just accelerometers, hydraulic pistons, and DSP. Big bonus points for a device that harvests that energy moving through the site to power the hydraulics.
Brain damage is all in your head. -- Karl Lehenbauer |
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As we know, the solution space of Schrödinger equation is a Hilbert space, however, what about it of Nonlinear Schrödinger equations such as $$i\partial_t\psi=-{1\over 2}\partial^2_x\psi+\kappa|\psi|^2 \psi$$?
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2 Answers 2
up vote 3 down vote accepted
Although,the set of solutions of the nonlinear Schrodinger equation (NLS) is not a Hilbert space and the field $\psi$ cannot be interpreted as a wave function, this does not mean that the NLS cannot be quantized. It can if we interpret $\psi$ as a classical field.
In this case the space of solutions or equivalently, the space of initial conditions (configurations) (by considering a solution of a this PDE as an evolution of an intial condition or configuration) can be interpreted as a classical phase space (It turns out to be an infinite dimensional symplectic manifold).
It is a quite general property that the space of solutions or equivalently,the space of the initial data of a wide class of partial differential equations is a symplectic manifold. This happens in ordinary mechanics. Also in the case of the linear Schrodinger equation or in field theories having linear equations of motion, this symplectic manifold is the projective Hilbert space of (the Hilbert space) of solutions. This point constitutes the main answer to the question and it is mutualto the linear and nonlinear Schrodinger equations.
Not only that,in the case of the NLE, the evolution of the classical configurations is Hamiltonian (i.e., half of the parameters can be interpreted as positions and theother half as momenta). There are choices of the initial parameters which satisfy almost canonical commutation relations such as the inverse scattering parameters. In this case,the quantization can be performed quite straightforwardly.
The only difference between this procedure and the familiar second quantization of the linear Schrodinger field, is that the solutions of the NLE depend nonlinearly on the initial parameters. Of course, it required a great deal of ingenuity to derive these solutions.
This principle has been applied in other cases of quantization of nonlinear field theories such as the Chern Simons theory.
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Okay, it is really helpful, thank you very much. – Popopo Sep 22 '12 at 6:27
Do you have a reference for the symplecticity of spaces of solutions? for which kinds of differential equations does this hold? – Arnold Neumaier Nov 18 '12 at 14:44
@Arnold, Please see for example fiz.uni.opole.pl/pgar/documents/IJMPA87.pdf by Piotr Garbaczewski – David Bar Moshe Nov 18 '12 at 15:12
Thanks, David. But this seem to be about particular integrable PDEs, whereas your answer seemed to promise ''the space of the initial data of a wide class of partial differential equations is a symplectic manifold''. – Arnold Neumaier Nov 18 '12 at 15:22
@Arnold, please see the Crncovic-Witten and Zuckerman's articles given in Urs Schreiber's answer physics.stackexchange.com/questions/26883/…. The Crncovic-Witten's link is not working, but you can find their article in the book: books.google.co.il/… – David Bar Moshe Nov 18 '12 at 15:45
This is not a question about physics. As has been stressed numerous times here, solutions of the NLS cannot be interpreted as quantum mechanical wave-functions. Their evolution is not unitary. As a consequence, the solution space has much less physical relevance.
The cubic NLS you wrote down appears in various approximations to nonlinear dispersive waves (including KdV, nonlinear perturbations of Klein-Gordon waves, and water waves); it describes the modulation profile of slowly varying wave packets with small amplitude.
The equation is Hamiltonian, with the ``energy'':
$\frac{1}{2}\int |\nabla \psi|^2\,\mathrm{d}x+\frac{\kappa}{4}\int|\psi|^4\,\mathrm{d}x$
and the mass
$\int |\psi|^2\,\mathrm{d}x$
as conserved quantities. These conserved quantities allow you to solve the equation for all time given initial data in the Sobolev space $H^1$ (this just means that the integrals of $|\nabla \psi|^2$ and $|\psi|^2$ are convergent) in case $\kappa > 0$ or, if $\kappa < 0$, whenever the nonlinearity is weak enough to be controlled by the gradient term for all times. This last condition can be expressed in terms of the Sobolev embedding. In dimension one, it is satisfied for power nonlinearities that are less than quintic. If the nonlinearity is too severe (for example, in dimension greater than 2 for the cubic case you asked about) perfectly nice solutions can blow up in finite time. In that case speaking of ``solution space'' does not make much sense since we cannot associate uniformly a time evolution to every vector.
Mathematicians have expended a lot of effort to solve this equation in various spaces of rough functions. Like $H^1$, most of them happen to be Hilbert spaces, but this has little physical (and no quantum mechanical) relevance. It is just a matter of convenience.
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Okay. You say that solutions of the NLS cannot be interpreted as quantum mechanical wave-functions, then does $\psi$ have the same physical meaning as it in LS? You know, by the orthodox interpretation in linear quantum mechanics $\psi$ denotes the probability amplitude, and linear Hermitian operators denote physical quantities. So does orthodox interpretation also works in nonlinear quantum mechanics? – Popopo Sep 20 '12 at 16:11
@Popopo:No--- the field $\psi$ is the density of a self-interacting superfluid, with repulsions when two particles are touching. The equation is exactly solvable in 1d. – Ron Maimon Sep 21 '12 at 6:37
It is a question about physics, as the NLS equation arises as a semiclassical approximation of nonrelativistic quantum field theories. – Arnold Neumaier Nov 18 '12 at 16:06
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2f715ae1bf23bbb5 | Take the 2-minute tour ×
Is the Born rule a fundamental postulate of quantum mechanics, or can it be inferred from unitary evolution?
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As the page about postulates you linked to correctly says, the Born-like rules to calculate probabilities from state vectors and operators are among the general postulates of quantum mechanics. It doesn't mean that they can't be derived from some other assumptions. However, the other assumptions clearly have to be connected with the notion of "probability" in one way or another, so they will be either a special or generalized formulation of the Born rule, anyway. Saying that the evolution is unitary doesn't say anything about probabilities - it can't "replace" the Born rule. – Luboš Motl Nov 23 '12 at 17:43
@LubošMotl I felt that if experimental apparatus must obey the same laws as the system under observation, then Born rule must follow from unitary evolution in all situations. Please can you elaborate on this comment "However, the other assumptions clearly have to be connected with the notion of "probability" in one way or another, so they will be either a special or generalized formulation of the Born rule" – Prathyush Nov 24 '12 at 4:28
I gave a derivation of the Born rule in the last answer to the question physics.stackexchange.com/q/19500 – Stephen Blake Aug 6 '13 at 20:37
related or duplicate: physics.stackexchange.com/q/73329 – Ben Crowell Aug 6 '13 at 22:27
5 Answers 5
up vote 2 down vote accepted
The Born rule is a fundamental postulate of quantum mechanics and therefore it cannot be derived from other postulates --precisely your first link emphasizes this--.
In particular the Born rule cannot be derived from unitary evolution because the rule is not unitary
$$A \rightarrow B_1$$ $$A \rightarrow B_2$$ $$A \rightarrow B_3$$ $$A \rightarrow \cdots$$
The Born rule can be obtained from non-unitary evolutions.
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This argument is actually not valid because it does not count in unknown states from the environment which could differ for different outcomes. – A.O.Tell Nov 24 '12 at 13:35
That is not true. Adding the environment and its equation of evolution gives an isolated system whose exact evolution is non-unitary. – juanrga Nov 26 '12 at 11:15
You are arguing that the same input state gives different output states, which is not unitary. That argument is false because you don't know that the input state is different for different outcomes, simply because you don't know the state of the unknown environment, by definition, that leads to the different outcomes. I'm not saying that your conclusion is wrong, but your argument certainly is. – A.O.Tell Nov 26 '12 at 12:36
Either if you assume that the same initial environment state $A\otimes E$ or not $A\otimes E_1,A\otimes E_2,A\otimes E_3\dots$ the evolution of the composite isolated system continues being non-unitary. von Neuman understood this and introduced his non-unitary evolution postulate in orthodox QM. – juanrga Nov 26 '12 at 20:46
That's not what you wrote in your answer however – A.O.Tell Nov 26 '12 at 21:34
Strictly speaking, the Born rule cannot be derived from unitary evolution, furthermore, in some sense the Born rule and unitary evolution are mutually contradictory, as, in general, a definite outcome of measurement is impossible under unitary evolution - no measurement is ever final, as unitary evolution cannot produce irreversibility or turn a pure state into a mixture. However, in some cases, the Born rule can be derived from unitary evolution as an approximate result - see, e.g., the following outstanding work: http://arxiv.org/abs/1107.2138 (accepted for publication in Physics Reports). The authors show (based on a rigorously solvable model of measurements) that irreversibility of measurement process can emerge in the same way as irreversibility in statistical physics - the recurrence times become very long, infinite for all practical purposes, when the apparatus contains a very large number of particles. However, for a finite number of particles there are some violations of the Born rule (see, e.g., the above-mentioned work, p. 115).
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Unfortunately the article is completely wrong. I know two of the authors and their works on perpetual machines and supposed violations of the second law of thermo. – juanrga Nov 24 '12 at 11:28
Thank you, I will take a look at the article referred to see if there is any weight in their arguments. Probably they are wrong as juanrga says, as most papers in this field are. – Prathyush Nov 24 '12 at 12:58
@juanrga: Maybe you're right, and the article is indeed completely wrong, but until you offer some specific arguments, why should I believe you, rather than the authors and the referees of their published articles? You mentioned their articles on other topics, but I am not sure this is relevant. – akhmeteli Nov 24 '12 at 13:34
@Prathyush: You may wish to start with their article arxiv.org/abs/quant-ph/0702135 , which is much shorter (see references to their journal articles there). – akhmeteli Nov 24 '12 at 13:53
@akhmeteli Thank you I will look into it, Indeed since I haven't gone deeply into the article, I must not comment on its factual accuracy. May I ask what you thought about the article? – Prathyush Nov 24 '12 at 17:32
The use of the word "postulate" in the question may indicate an unexamined assumption that we must or should discuss this sort of thing using an imitation of the axiomatic approach to mathematics -- a style of physics that can be done well or badly and that dates back to the faux-Euclidean presentation of the Principia. If we make that choice, then in my opinion Luboš Motl's comment says all that needs to be said. (Gleason's theorem and quantum Bayesianism (Caves 2001) might also be worth looking at.) However, the pseudo-axiomatic approach has limitations. For one thing, it's almost always too unwieldy to be usable for more than toy theories. (One of the only exceptions I know of is Fleuriot 2001.) Also, although mathematicians are happy to work with undefined primitive terms (as in Hilbert's saying about tables, chairs, and beer mugs), in physics, terms like "force" or "measurement" can have preexisting informal or operational definitions, so treating them as primitive notions can in fact be a kind of intellectual sloppiness that's masked by the superficial appearance of mathematical rigor.
So what can physical arguments say about the Born rule?
The Born rule refers to measurements and probability, both of which may be impossible to define rigorously. But our notion of probability always involves normalization. This suggests that we should only expect the Born rule to apply in the context of nonrelativistic quantum mechanics, where there is no particle annihilation or creation. Sure enough, the Schrödinger equation, which is nonrelativistic, conserves probability as defined by the Born rule, but the Klein-Gordon equation, which is relativistic, doesn't.
This also gives one justification for why the Born rule can't involve some other even power of the wavefunction -- probability wouldn't be conserved by the Schrödinger equation. Aaronson 2004 gives some other examples of things that go wrong if you try to change the Born rule by using an exponent other than 2.
The OP asks whether the Born rule follows from unitarity. It doesn't, since unitarity holds for both the Schrödinger equation and the Klein-Gordon equation, but the Born rule is valid only for the former.
Although photons are inherently relativistic, there are many situations, such as two-source interference, in which there is no photon creation or annihilation, and in such a situation we also expect to have normalized probabilities and to be able to use "particle talk" (Halvorson 2001). This is nice because for photons, unlike electrons, we have a classical field theory to compare with, so we can invoke the correspondence principle. For two-source interference, clearly the only way to recover the classical limit at large particle numbers is if the square of the "wavefunction" ($\mathbf{E}$ and $\mathbf{B}$ fields) is proportional to probability. (There is a huge literature on this topic of the photon "wavefunction". See Birula 2005 for a review. My only point here is to give a physical plausibility argument. Basically, the most naive version of this approach works fine if the wave is monochromatic and if your detector intercepts a part of the wave that's small enough to look like a plane wave.) Since the Born rule has to hold for the electromagnetic "wavefunction," and electromagnetic waves can interact with matter, it clearly has to hold for material particles as well, or else we wouldn't have a consistent notion of the probability that a photon "is" in a certain place and the probability that the photon would be detected in that place by a material detector.
The Born rule says that probability doesn't depend on the phase of an electron's complex wavefunction $\Psi$. We could ask why the Born rule couldn't depend on some real-valued function such as $\operatorname{\arg} \Psi$ or $\mathfrak{Re} \Psi$. There is a good physical reason for this. There is an uncertainty relation between phase $\phi$ and particle number $n$ (Carruthers 1968). For fermions, the uncertainty in $n$ in a given state is always small, so the uncertainty in phase is very large. This means that the phase of the electron wavefunction can't be observable (Peierls 1979).
I've seen the view expressed that the many-worlds interpretation (MWI) is unable to explain the Born rule, and that this is a problem for MWI. I disagree, since none of the arguments above depended in any way on the choice of an interpretation of quantum mechanics. In the Copenhagen interpretation (CI), the Born rule typically appears as a postulate, which refers to the undefined primitive notion of "measurement;" I don't consider this an explanation. We often visualize the MWI in terms of a bifurcation of the universe at the moment when a "measurement" takes place, but this discontinuity is really just a cartoon picture of the smooth process by which quantum-mechanical correlations spread out into the universe. In general, interpretations of quantum mechanics are explanations of the psychological experience of doing quantum-mechanical experiments. Since they're psychological explanations, not physical ones, we shouldn't expect them to explain a physical fact like the Born rule.
Aaronson, "Is Quantum Mechanics An Island In Theoryspace?," http://arxiv.org/abs/quant-ph/0401062
Bialynicki-Birula, "Photon wave function", 2005, http://arxiv.org/abs/quant-ph/0508202
Carruthers and Nieto, "Phase and Angle Variables in Quantum Mechanics", Rev Mod Phys 40 (1968) 411; copy available at http://www.scribd.com/doc/147614679/Phase-and-Angle-Variables-in-Quantum-Mechanics (may be illegal, or may fall under fair use, depending on your interpretation of your country's laws)
Caves, Fuchs, and Schack, "Quantum probabilities as Bayesian probabilities", 2001, http://arxiv.org/abs/quant-ph/0106133; see also Scientific American, June 2013
Fleuriot, A Combination of Geometry Theorem Proving and Nonstandard Analysis with Application to Newton's Principia, Springer, 2001
Halvorson and Clifton, "No place for particles in relativistic quantum theories?", 2001, http://philsci-archive.pitt.edu/195/
Peierls, Surprises in Theoretical Physics, section 1.3
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Since the Born rule has to hold for the electromagnetic "wavefunction," and electromagnetic waves can interact with matter, it clearly has to hold for material particles as well, or else we wouldn't have a consistent notion of the probability that a photon "is" in a certain place and the probability that the photon would be detected in that place by a material detector. Could you explain this in more detail? – Sebastian Henckel Aug 8 '13 at 20:52
@SebastianHenckel: This is not completely thought out and may be wrong. But suppose that the rule for electrons is not the Born rule but a rule saying that probability is $\propto|\Psi|^p$, where $p\ne 2$. If you scatter an EM wave off of an electron, they interact through some wave equation such that the scattered part of $\Psi$ is proportional to the amplitude of the EM wave: amplitude is proportional to amplitude. But then the electron is acting like a detector, and $p\ne 2$ means that the probability of detection isn't proportional to the probability that the photon was there. – Ben Crowell Aug 8 '13 at 21:08
I like this argument. The interaction between the photon and the electron however is quantum electrodynamics all the way through, and that's something I don't know much about. However, thanks for making a connection between electrons and waves I never thought about. The pure de Broglie argument always seemed very ad hoc, and this makes it somewhat more plausible. – Sebastian Henckel Aug 8 '13 at 21:30
It took me a while to read this answer. you said "The OP asks whether the Born rule follows from unitarity. It doesn't, since unitarity holds for both the Schrödinger equation and the Klein-Gordon equation, but the Born rule is valid only for the former." Isn't Born rule applicable even in Relativistic Quantum mechanics(Any field theory in general), not in the sense of KG equation but the KG field. Also Would you comment on my recent answer on a related topic, physics.stackexchange.com/questions/76132/… – Prathyush Sep 4 '13 at 8:54
@Prathyush: My relativistic field theory is pretty weak, so if you want a really coherent explanation of why the Born rule doesn't apply to the KG equation, you're probably better off posting that as a question and letting someone more competent answer. But basically I think the concept is that in relativistic QM, we have to give up on the idea of having eigenstates of position, so the whole Copenhagen-ish interpretation of a position measurement as projecting the wavefunction down to a delta function doesn't really work. – Ben Crowell Sep 4 '13 at 15:45
It is independent, but it is not fundamental, as it applies only to highly idealized kinds of measurements. (Realistic measurements are governed by POVMs instead.)
In fact, the role of Born's rule in quantum mechanics is marginal (after the standard introduction and the derivation of the notion of expectation). It is hardly ever used for the analysis of real problems, except to shed light on problems in the foundations of quantum mechanics.
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One day I will learn about POVM's its been on my list of To Do's for a long time. – Prathyush Nov 24 '12 at 19:23
POVMs can be regarded as Born type measurements in a larger space, so you're back where you started. – A.O.Tell Nov 24 '12 at 22:45
@A.O.Tell: On the formal level, yes. But in this larger space, one never does any measurements that would deserve that name. – Arnold Neumaier Nov 26 '12 at 9:37
That statement would require an exact definition of what a measurement is and how it is applied to a subsystem. Also, it makes no practical difference. If you know how a Born style measurement works you understand how a POVM works. – A.O.Tell Nov 26 '12 at 12:39
@A.O.Tell: It is enough to know what is really measured. Measure the mass of the sun, the halflife of Technetium, or the width of a spectral line in the Balmer series, and try to express it in terms of the Born rule! – Arnold Neumaier Nov 26 '12 at 12:55
The idea of deriving the Born rule (and in fact the whole measurement postulate) from the usual unitary evolution of quantum systems is at the very heart of a realist interpretation of quantum theory. If the quantum state really describes a the true internal state of a system and measurement is just a certain kind of interaction, then there should be only one single law for the time evolution.
Quantum theory however is fundamentally non-local and separating systems is conceptually hard, which makes observer and experiment impossible to describe separately. There should be a system containing both parts however and which follows a simple law of time evolution. Of course, the obvious candidate for such a law is unitary evolution, simply because that is what we observe for systems that we isolate as good as possible.
It is usually argued that this route leads to the Everett interpretation of quantum theory, where observations are relative to the observer and realized by entangled states. There have been several attempts to derive the Born rule in this context, but all that seem valid require additional assumptions that are questionable (and may in fact be inconsistent with the realist approach or other fundamental assumptions).
The reason why there cannot be a derivation that just uses ordinary unitary evolution and results in the Born rule is not even unitarity but the linearity of the theory. Say there is an evolution that takes out input to the measurement output, and we decide to measure a|A>+b|B> in the basis {|A>,|B>}. Then independently from the environment the Born rule predicts that |A> and |B> are invariant under measurement. A superposition (|A>+B>)/sqrt(2) should end up in either |A> or |B> depending on a possible environment state if the Born rule applies. The linearity of the theory requires that the outcome is a superposition of |A> and |B> however (the phase may change though).
Everett's answer to this problem is that the superposition comes out, but with the outcomes entangled with the observer seeing either outcome. But this creates two observers that are unaware of their own amplitude. Because of the linearity their future evolution is independent from the branch amplitude, and it's therefore hard to argue that any aspects of their perceived reality would depend on the branch amplitude.
Interestingly approaches to fix this issue, like the use of decision theory, advanced branch counting, etc, in some form introduce a nonlinear element to the theory. Be it a measure of branch amplitude, a cutoff amplitude or amplitude discretization, a stability rule (envariance or quantum darwinism). There are also approaches that don't hide the nonlinearity in additional assumptions that may collide with the linear evolution. Those are explicit nonlinear variations of the Schroedinger equation that can in fact produce an evolution that allows the Born rule to emerge. Of course, this is not something that most theorists embrace, simply because the linearity of quantum theory is such an attractive feature.
But there's one more approach that I personally favor. The nonlinearity could be only subjective to an observer, caused by incomplete knowledge about the universe. An observer, i.e. a local mechanism realized within quantum theory, can only gather information by interacting with his environment. Certain information however is inaccessible dynamically, hidden outside the observer's light cone or just not available for direct interaction. Considering this, it can be shown that reconstructing the best possible state description an observer can come up with must follow a dynamic law that is not unitary all the time, but also contains sudden state jumps with random outcomes driven by incoming priorly unknown information from the environment. It can be shown that a photon from the environment with entirely unknown polarization can cause a subjective state jump that corresponds exactly to the Born rule. This is of course a bold claim. But please see http://arxiv.org/abs/1205.0293 for a proper derivation and discussion of the details. If you you would like to look at a more gently introduction to the idea you can also read the (less complete but more intuitive) blog I've set up for this: http://aquantumoftheory.wordpress.com
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I don't know if environment is a necessary concept in the measurement problem, For example, will a photographic Plate work in perfect vacuum. Thought I don't have the opportunity of experiment with such a situation, I believe a photographic must work normally in vacuum where there is no environment or extraneous photons. – Prathyush Nov 25 '12 at 17:02
Even in your perfect vacuum you always have an interacting environment. And of course the environment may not be needed for the resolution of the measurement problem, but it might possibly be necessary, and so you cannot simply exclude it. It is at least a plausible source for randomness due to our lack of information about its state. – A.O.Tell Nov 25 '12 at 17:11
In some situations where you cannot remove it from the experimental setup you will have to include the environment in the theory. What do you mean even in perfect vacuum you have the interaction environment? The basic process in a photographic plate is a light sensitive chemical reaction right? So an environment wont play a role – Prathyush Nov 25 '12 at 17:16
The environment always plays a role in quantum theory. You cannot remove the quantum fields from space, no matter how perfect your vacuum is. There will always be interaction on some level, and ignoring that is surely not helpful for understanding the properties of quantum systems. You seem to be thinking is more or less classical terms with your photographic plate example. – A.O.Tell Nov 25 '12 at 17:21
Also, in order to see if your plate has been affected by light you have to look at it. So at the very latest then you will subject it to massive interaction with an unknown environment – A.O.Tell Nov 25 '12 at 17:22
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1814c1400eefa428 | Introduction to Computational Chemistry
David Young
Cytoclonal Pharmaceutics Inc.
Table of Contents
Recent years have seen an increase in the number of people doing theoretical chemistry. Many of these newcomers are part time theoreticians, who work on other aspects of chemistry as well. This increase has been facilitated by the development of computer software which is increasingly easy to use. It is now easy enough to do computational chemistry that you do not have to know what you are doing to do a computation. As a result, many people don't understand even the most basic description of how the calculation is done and are therefore sucessufully doing a lot of work which is, frankly, garbage.
Many universities are now offering classes, which are an overview of various aspects of computational chemistry. Since we have had many people wanting to start doing computations before they have had even an introductory course, this document has been written as step one in understanding what computational chemistry is about. Note that this is not intended to teach the fundamentals of chemistry, quantum mechanics or mathematics, only most basic description of how chemical computations are done.
The term theoretical chemistry may be defined as the mathematical description of chemistry. The term computational chemistry is usually used when a mathematical method is sufficiently well developed that it can be automated for implementation on a computer. Note that the words exact and perfect do not appear in these definitions. Very few aspects of chemistry can be computed exactly, but almost every aspect of chemistry has been described in a qualitative or approximate quantitative computational scheme. The biggest mistake that a computational chemists can make is to assume that any computed number is exact. However, just as not all spectra are perfectly resolved, often a qualitative or approximate computation can give useful insight into chemistry if you understand what it tells you and what it doesn't.
Although most chemists avoid the true paper & pencil type of theoretical chemistry, keep in mind that this is what many Nobel prizes have been awarded for.
Ab Initio
The term "Ab Initio" is latin for "from the beginning". This name is given to computations which are derived directly from theoretical principles, with no inclusion of experimental data. Most of the time this is referring to an approximate quantum mechanical calculation. The approximations made are usually mathematical approximations, such as using a simpler functional form for a function or getting an approximate solution to a differential equation.
The most common type of ab initio calculation is called a Hartree Fock calculation (abbreviated HF), in which the primary approximation is called the central field approximation. This means that the Coulombic electron-electron repulsion is not specifically taken into account. However, it's net effect is included in the calculation. This is a variational calculation, meaning that the approximate energies calculated are all equal to or greater than the exact energy. The energies calculated are usually in units called Hartrees (1 H = 27.2114 eV). Because of the central field approximation, the energies from HF calculations are always greater than the exact energy and tend to a limiting value called the Hartree Fock limit.
The second approximation in HF calculations is that the wave function must be described by some functional form, which is only known exactly for a few one electron systems. The functions used most often are linear combinations of Slater type orbitals exp(-ax) or Gaussian type orbitals exp(-ax^2), abbreviated STO and GTO. The wave function is formed from linear combinations of atomic orbitals or more often from linear combinations of basis functions. Because of this approximation, most HF calculations give a computed energy greater than the Hartree Fock limit. The exact set of basis functions used is often specified by an abbreviation, such as STO-3G or 6-311++g**.
A number of types of calculations begin with a HF calculation then correct for the explicit electron-electron repulsion, referred to as correlation. Some of these methods are Mohlar-Plesset perturbation theory (MPn, where n is the order of correction), the Generalized Valence Bond (GVB) method, Multi-Configurations Self Consistent Field (MCSCF), Configuration Interaction (CI) and Coupled Cluster theory (CC). As a group, these methods are referred to as correlated calculations.
A method, which avoids making the HF mistakes in the first place is called Quantum Monte Carlo (QMC). There are several flavors of QMC .. variational, diffusion and Green's functions. These methods work with an explicitly correlated wave function and evaluate integrals numerically using a Monte Carlo integration. These calculations can be very time consuming, but they are probably the most accurate methods known today.
An alternative ab initio method is Density Functional Theory (DFT), in which the total energy is expressed in terms of the total electron density, rather than the wavefunction. In this type of calculation, there is an approximate Hamiltonian and an approximate expression for the total electron density.
The good side of ab initio methods is that they eventually converge to the exact solution, once all of the approximations are made sufficiently small in magnitude. However, this convergence is not montonic. Sometimes, the smallest calculation gives the best result for a given property.
The bad side of ab initio methods is that they are expensive. These methods often take enormous amounts of computer cpu time, memory and disk space. The HF method scales as N4, where N is the number of basis functions, so a calculation twice as big takes 16 times as long to complete. Correlated calculations often scale much worse than this. In practice, extremely accurate solutions are only obtainable when the molecule contains half a dozen electrons or less.
In general, ab initio calculations give very good qualitative results and can give increasingly accurate quantitative results as the molecules in question become smaller.
Semiempirical calculations are set up with the same general structure as a HF calculation. Within this framework, certain pieces of information, such as two electron integrals, are approximated or completely omitted. In order to correct for the errors introduced by omitting part of the calculation, the method is parameterized, by curve fitting in a few parameters or numbers, in order to give the best possible agreement with experimental data.
The good side of semiempirical calculations is that they are much faster than the ab initio calculations.
The bad side of semiempirical calculations is that the results can be eratic. If the molecule being computed is similar to molecules in the data base used to parameterize the method, then the results may be very good. If the molecule being computed is significantly different from anything in the parameterization set, the answers may be very poor.
Semiempirical calculations have been very successful in the description of organic chemistry, where there are only a few elements used extensively and the molecules are of moderate size. However, semiempirical methods have been devised specifically for the description of inorganic chemistry as well.
Modeling the solid state
The electronic structure of an infinite crystal is defined by a band structure plot, which gives energies of electron orbitals for each point in k-space, called the Brillouin zone. Since ab initio and semiempirical calculations yield orbital energies, they can be applied to band structure calculations. However, if it is time consuming to calculate the energy for a molecule, it is even more time consuming to calculate energies for a list of points in the Brillouin zone.
Band structure calculations have been done for very complicated systems, however the software is not yet automated enough or sufficiently fast that anyone does band structures casually. If you want to do band structure calculations, you had better expect to put a lot of time into your efforts.
Molecular Mechanics
If a molecule is too big to effectively use a semiempirical treatment, it is still possible to model it's behavior by avoiding quantum mechanics totally. The methods referred to as Molecular Mechanics set up a simple algebraic expression for the total energy of a compound, with no necessity to compute a wave function or total electron density. The energy expression consists of simple classical equations, such as the harmonic oscillator equation in order to describe the energy associated with bond stretching, bending, rotation and intermolecular forces, such as van der waals interactions and hydrogen bonding. All of the constants in these equations must be obtained from experimental data or an ab initio calculation.
In a molecular mechanics method, the data base of compounds used to parameterize the method (a set of parameters and functions is called a force field) is crucial to it's success. Where as a semiempirical method may be parameterized against a set of organic molecules, a molecular mechanics method may be parameterized against a specific class of molecules, such as proteins. Such a force field would only be expected to have any relevance to describing other proteins.
The good side of molecular mechanics is that it allows the modeling of enormous molecules, such as proteins and segments of DNA, making it the primary tool of computational biochemists.
The bad side of molecular mechanics is that there are many chemical properties that are not even defined within the method, such as electronic excited states. In order to work with extremely large and complicated systems, often molecular mechanics software packages have the most powerful and easiest to use graphical interfaces. Because of this, mechanics is sometimes used because it is easy, but not necessarily a good way to describe a system.
Molecular Dynamics
Molecular dynamics consists of examining the time dependent behavior of a molecule, such as vibrational motion or Brownian motion. This is most often done within a classical mechanical description similar to a molecular mechanics calculation.
The application of molecular dynamics to solvent/solute systems allows the computation of properties such as diffusion coeficients or radial distribution functions for use in statistical mechanical treatments. Usually the scheme of a solvent/solute calculation is that a number of molecules (perhaps 1000) are given some initial position and velocity. New positions are calculated a small time later based on this movement and this process is itterated for thousands of steps in order to bring the system to equilibrium and give a good statistical description of the radial distribution function.
In order to analyze the vibrations of a single molecule, many dynamics steps are done, then the data is Fourier transformed into the frequency domain. A given peak can be chosen and transformed back to the time domain, in order to see what the motion at that frequency looks like.
Statistical Mechanics
Statistical mechanics is the mathematical means to extrapolate thermodynamic properties of bulk materials from a molecular description of the material. Much of statistical mechanics is still at the paper and pencil stage of theory, since the quantum mechanicians can't solve the Schrödinger equation exactly yet, the statistical mechanicians don't really have even a good starting point for a truly rigorous treatment. Statistical mechanics computations are often tacked onto the end of ab inito calculations for gas phase properties. For condensed phase properties, often molecular dynamics calculations are necessary in order to do a computational experiment.
Thermodynamics is one of the most well developed mathematical chemical descriptions. Very often any thermodynamic treatment is left for trivial pen and paper work since many aspects of chemistry are so accurately described with very simple mathematical expressions.
Structure-Property Relationships
Structure-property relationships are qualitative or quantitative empirically defined relationships between molecular structure and observed properties. In some cases this may seem to duplicate statistical mechanical results, however structure-property relationships need not be based on any rigorous theoretical principles.
The simplest case of structure-property relationships are qualitative thumb rules. For example, an experienced polymer chemist may be able to predict whether a polymer will be soft or brittle based on the geometry and bonding of the monomers.
When structure-property relationships are mentioned in current literature, it usually implies a quantitative mathematical relationship. These relationships are most often derived by using curve fitting software to find the linear combination of molecular properties, which best reproduces the desired property. The molecular properties are usually obtained from molecular modeling computations. Other molecular descriptors such as molecular weight or topological descriptions are also used.
When the property being described is a physical property, such as the boiling point, this is refered to as a Quantitative Structure-Property Relationship (QSPR). When the property being described is a type of biological activity (such as drug activity), this is refered to as a Quantitative Structure-Activity Relationship (QSAR).
Symbolic Calculations
Symbolic calculations are performed when the system is just too large for an atom-by-atom description to be viable at any level of approximation. An example might be the description of a membrane by describing the individual lipids as some representative polygon with some expression for the energy of interaction. This sort of treatment is used for computational biochemistry and even microbiology.
Artifical Intelligence
Techniques invented by computer scientists interested in artificial intelligence have been applied mostly to drug design in recent years. These methods also go by the names De Novo or rational drug design. The general scenario is that some functional site has been identified and it is desired to come up with a structure for a molecule that will interact with that site in order to hinder it's functionality. Rather than have a chemist try hundreds or thousands of possibilities with a molecular mechanics program, the molecular mechanics is built into an artificial intelligence program, which tries enormous numbers of "reasonable" possibilities in an automated fasion. The number of techniques for describing the "intelligent" part of this operation are so diverse that it is impossible to make any generalization about how this is implemented in the program.
How to do a computational research project
When using computational chemistry to answer a chemical question, the obvious problem is that you need to know how to use the software. The problem that is missed is that you need to know how good the answer is going to be. Here is a check list to follow.
What do you want to know? How accurately? Why? If you can't answer these questions, then you don't even have a research project yet.
How accurate do you predict the answer will be? In analytical chemistry, you do a number of identical measurements then work out the error from a standard deviation. With computational experiments, doing the same thing should always give exactly the same result. The way that you estimate your error is to compare a number of similar computations to the experimental answers. There are articles and compilations of these studies. If none exist, you will have to guess which method should be reasonable, based on it's assumptions then do a study yourself, before you can apply it to you unknown and have any idea how good the calculation is. When someone just tells you off the top of their head what method to use, they either have a fair amount of this type of information memorized, or they don't know what they are talking about. Beware of someone who tells you a given program is good just because it is the only one they know how to use, rather than the basing their answer on the quality of the results.
How long do you expect it to take? If the world were perfect, you would tell your PC (voice input of course) to give you the exact solution to the Schrödinger equation and go on with your life. However, often ab initio calculations would be so time consuming that it would take a decade to do a single calculation, if you even had a machine with enough memory and disk space. However, a number of methods exist because each is best for some situation. The trick is to determine which one is best for your project. Again, the answer is to look into the literature and see how long each takes. If the only thing you know is how a calculation scales, do the simplest possible calculation then use the scaling equation to estimate how long it will take to do the sort of calculation that you have predicted will give the desired accuracy.
What approximations are being made? Which are significant? This is how you avoid looking like a complete fool, when you successfully perform a calculation that is complete garbage. An example would be trying to find out about vibrational motions that are very anharmonic, when the calculation uses a harmonic oscillator approximation.
Once you have finally answered all of these questions, you are ready to actually do a calculation. Now you must determine what software is available, what it costs and how to use it. Note that two programs of the same type (i.e. ab initio) may calculate different properties, so you have to make sure the program does exactly what you want.
When you are learning how to use a program, you may try to do dozens of calculations that will fail because you constructed the input incorrectly. Do not use your project molecule to do this. Make all your mistakes with something really easy, like a water molecule. That way you don't waste enormous amounts of time.
Data visualization is the process of displaying information in any sort of pictorial or graphical representation. A number of computer programs are now available to apply a colorization scheme to data or work with three dimensional representations.
Further information
For an introductory level overview of computational chemistry see
G. H. Grant, W. G. Richards "Computational Chemistry" Oxford (1995)
A more detailed description of common computational chemistry techniques is contained in
A. R. Leach "Molecular Modelling Principles and Applications" Addison Wesley Longman (1996)
F. Jensen "Introduction to Computational Chemistry" John Wiley & Sons (1999)
There are many books on the principles of quantum mechanics and every physical chemistry text has an introductory treatment. The work which I am listing here is a two volume set with each chapter broken into a basic and advanced sections making it excellent for both intermediate and advanced users.
C. Cohen-Tannoudji, B. Diu, F. Laloe "Quantum Mechanics Volumes I & II" Wiley-Interscience (1977)
For an introduction to quantum chemistry see
D. A. McQuarrie "Quantum Chemistry" University Science Books (1983)
A graduate level text on quantum chemistry is
I. N. Levine "Quantum Chemistry" Prentice Hall (1991)
An advanced undergraduate or graduate text on quantum chemistry is
P. W. Atkins, R. S. Friedman "Molecular Quantum Mechanics" Oxford (1997)
For quantum Monte Carlo methods, order the following book using ISBN 981-02-0322-5 because the title is listed incorrectly in 'Books in Print'.
B. L. Hammond, W. A. Lester, Jr., P. J. Reynolds "Monte Carlo Methods in Ab Initio Quantum Chemistry" World Scientific (1994)
A good review article on density functional theory is
T. Ziegler Chem. Rev. 91, 651-667 (1991)
For density functional theory see
R. G. Parr, W. Yang "Density-Functional Theory of Atoms and Molecules" Oxford (1989)
For a basic understanding of solid state modeling see
R. Hoffmann "Solids and Surfaces : A Chemist's View of Bonding in Extended Structures", VCH (1988)
For a graduate level description of statistical mechanics see
D. A. McQuarrie "Statistical Mechanics" Harper Collins (1976)
Any physical chemistry text will have a description of thermodynamics but I will recommend
I. N. Levine "Physical Chemistry" McGraw Hill (1995)
Another nice introduction to computational chemistry is
S. Profeta, Jr. "Kirk-Othmer Encyclopedia of Chemical Technology Supplement" 315, John Wiley & Sons (1998).
There is a comprehensive listing of all available molecular modeling software and structural databanks, free or not, in appendix 2 of
"Reviews in Computational Chemistry Volume 6" Ed. K. B. Lipkowitz and D. B. Boyd, VCH (1995)
There is a write up on computer aided drug design at
Mathematical challenges from theoretical/computational chemistry
An online text on molecular modeling using molecular mechanics
A Computational Chemistry Primer
An online text on computational chemistry
Another online text on quantum chemistry
An online introduction to quantum mechanics is at
Citation: This article was originally published on the web. It has now appeared in print in D. Young, Chem. Aust. 11, 5 (1998).
An expanded version of this article will be published in "Computational Chemistry: A Practical Guide for Applying Techniques to Real World Problems" by David Young, which will be available from John Wiley & Sons in the spring of 2001.
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The main application of Feynman path integrals (and the primary motivation behind them) is in Quantum Field Theory - currently this is something standard for physicists, if even the mathematical theory of functional integration is not (yet) rigorous.
My question is: what are the applications of path integrals outside QFT? By "outside QFT" I mean non-QFT physics as well as various branches of mathematics.
(a similar question is Doing geometry using Feynman Path Integral?, but it concerns only one possible application)
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The path integral has many applications:
Mathematical Finance:
In mathematical finance one is faced with the problem of finding the price for an "option."
An option is a contract between a buyer and a seller that gives the buyer the right but not the obligation to buy or sell a specified asset, the underlying, on or before a specified future date, the option's expiration date, at a given price, the strike price. For example, an option may give the buyer the right but not the obligation to buy a stock at some future date at a price set when the contract is settled.
One method of finding the price of such an option involves path integrals. The price of the underlying asset varies with time between when the contract is settled and the expiration date. The set of all possible paths of the underlying in this time interval is the space over which the path integral is evaluated. The integral over all such paths is taken to determine the average pay off the seller will make to the buyer for the settled strike price. This average price is then discounted, adjusted for for interest, to arrive at the current value of the option.
Statistical Mechanics:
In statistical mechanics the path integral is used in more-or-less the same manner as it is used in quantum field theory. The main difference being a factor of $i$.
One has a given physical system at a given temperature $T$ with an internal energy $U(\phi)$ dependent upon the configuration $\phi$ of the system. The probability that the system is in a given configuration $\phi$ is proportional to
$e^{-U(\phi)/k_B T}$,
where $k_B$ is a constant called the Boltzmann constant. The path integral is then used to determine the average value of any quantity $A(\phi)$ of physical interest
$\left< A \right> := Z^{-1} \int D \phi A(\phi) e^{-U(\phi)/k_B T}$,
where the integral is taken over all configurations and $Z$, the partition function, is used to properly normalize the answer.
Physically Correct Rendering:
Rendering is a process of generating an image from a model through execution of a computer program.
The model contains various lights and surfaces. The properties of a given surface are described by a material. A material describes how light interacts with the surface. The surface may be mirrored, matte, diffuse or any other number of things. To determine the color of a given pixel in the produced image one must trace all possible paths form the lights of the model to the surface point in question. The path integral is used to implement this process through various techniques such as path tracing, photon mapping, and Metropolis light transport.
Topological Quantum Field Theory:
In topological quantum field theory the path integral is used in the exact same manner as it is used in quantum field theory.
Basically, anywhere one uses Monte Carlo methods one is using the path integral.
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Some of your examples are about Wiener integrals (minus sign in the exponent rather than imaginary phase), which are matematically well-defined rather than Feynman path integrals which are successfully defined only in some special cases. – Zoran Skoda Apr 6 '10 at 21:03
I agree. All examples I am aware of outside of QFT exchange the i for a -1. Are you aware of any non-QFT examples that have an i in the exponent? – Kelly Davis Apr 6 '10 at 22:26
I usually think of a path integral as just a very glorified and specific version of a simple and general construction from probability. Namely, a path integral is basically an element of an ordered product of matrices belonging to some semigroup. So under this interpretation, "path integrals" are ubiquitous when this sort of object is being considered--particularly in Markov processes. Every time you're computing a multi-step transition probability, you're doing a path integral, and vice versa.
In discrete-time Markov processes you take a power of the transition matrix. Each element of it encodes all the ways in which you can get from the initial to the final state in the appropriate number of steps, along with their proper weights. In continuous time it's the same basic idea, but a bit more involved. The idea is covered here for inhomogeneous continuous-time processes in the course of demonstrating a fairly general form of the Dynkin formula.
Here's the gist in physics:
We can arrive at a formal solution to the Schrödinger equation via a time evolution operator, i.e. $\vert \psi(t) \rangle = U(t) \lvert \psi(0) \rangle$, $U(t) = e^{-itH}$. But equivalently, the quantum initial-value problem is solved once we have the propagator/transition amplitude/Green function $U(x,t,x_0,t_0) = \langle x \lvert U(t-t_0) \rvert x_0 \rangle$, since $\psi(x,t) = \int dx_0 U(x,t,x_0,t_0) \psi(x_0,t_0)$. The transition amplitudes enable us to obtain transition probabilities by the simple expedient of taking squared norms.
The transfer matrix is an infinitesimal time evolution operator: i.e., $T = U(\Delta t) = \exp(-i \Delta t \cdot H) = I - i\Delta t \cdot H$, where these equalities are up to $o(\Delta t)$. Since time evolution operators belong to a semigroup, we have after a simple manipulation that
$U(x_N, t_0 + N \Delta t, x_0, t_0) = \langle x_N \lvert T^N \rvert x_0 \rangle$.
Following Feynman, we can also obtain the propagator from the Lagrangian point of view. But the idea is still basically the same.
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Witten, I think, deserves much of the credit for getting mathematicians interested in the path integral, with his paper Quantum field theory and the Jones polynomial. In particular, path integrals are closely related to questions about (quantum) groups.
For one direction, namely the perturbative Feynman path integral, you should check out Dror Bar-Natan's thesis and later work.
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Also, wasn't it Atiyah who first asked whether there was a physics explanation of the Jones polynomial? – Kevin H. Lin Apr 6 '10 at 0:25
For a pretty narrow definition of mathematicians: analysts, operator algebraists, and integrable systems people had been thinking about path integrals in various contexts long before Witten got involved. I'm not saying Witten hasn't been influential, especially in geometry and topology, but path integrals and mathematics didn't meet for the first time in the early 80s. – userN Apr 6 '10 at 1:28
Kevin and AJ both make good points, and I apologize for misrepresenting the history. My only excuse is that the Witten paper is a nice place to start a history of the topics I've been most interested in. (Incidentally, I originally posted only Dror's thesis, and then decided that perhaps I should mention Witten's motivation for it.) – Theo Johnson-Freyd Apr 6 '10 at 2:08
Feynman Path Integral is connected with saddle point method and stationary phase method . In fact is used as generating function for certain factors in perturbation series. So it can be used wherever this technique may be used, if problem requires certain normalizations. If You are looking for variational solution and You cannot find exact solution, path integral is always an option, specially if You know zeroth configuration, and You want to amount certain perturbations which are polynomial potentials ( because then You may account it as functional derivatives see
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Path integral is NOT in general "related" to stationary phase; rather the stationary phase is an asymptotic method for integrals with rapidly oscillating phase, whose infinite dimensional version (that version is to large extent non-rigorous and underdeveloped mathematically) can be sometimes meaningfully APPLIED to the path integral. This is a path integral version of the WKB approximation of the usual approach to QM (nlab). Approximating variational extrema by path integral is equally OK in certain asymptotic regime. – Zoran Skoda Apr 7 '10 at 0:34
Yes You are right - my mistake and inconsistency - in general ( from point of view of some kind of the definition, for example by means of general propagator composed in time ordered points). You are right. But please, could You give me an example of this approach without such method? Possibly the only one is Gaussian path integral in quantum oscillator. Other ones usually are treated in perturbation given by saddle point method. – kakaz Apr 7 '10 at 12:20
One application is to computer graphics. When simulating the effect of lighting a translucent material (see my avatar!) you often need to integrate over all possible paths from the light source to the camera via the material. This is similar to the Feynman integral in quantum mechanics, but note that this is an integral in the domain of classical geometric optics, not quantum field theory.
I believe it was Jerry Tessendorf who pioneered this approach in the graphics world. You may have watched movies with effects rendered using techniques derived from Tessendorf's!
I should add that this is a particular case of what Steve Huntsman describes in his answer.
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Some expansions in deformation theory, Lie theory, study of graph cohomology etc. are Feynman integral like expansions and one can formally define "theories leading to them". See for example articles by Dror Bar-Natan for some of such combinatorial and Lie-theoretic aspects. Kontsevich's own deformation quantization formula is usual quantum mechanics is governed by a theory called Poisson sigma model (this was the intuition behind Kontsevich's formula, though he did not explicitly write it that way, but it was later rediscovered by Cattaneo and Felder).
On the other hand, I find very fascinating Sasha Goncharov's "theory" giving a Feynman diagram expansion giving "correlators" formally like in physics, but in fact consisting of Hodge theoretic information on Kähler manifolds:
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If we understand QFT as the framework that unites quantum mechanics and special relativity, then I'd refer to
Hagen Kleinert: "Path integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets"
for non-QFT non-pure-mathematical applications.
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5a3a586b976db5fc | From Wikipedia, the free encyclopedia
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For a mathematical treatment of spin-1/2, see Spinor.
In quantum mechanics, spin is an intrinsic property of all elementary particles. Fermions, the particles that constitute ordinary matter, have half-integer spin. All known elementary fermions have a spin of 1/2.[1][2][3]
Heuristic depiction of spin angular momentum cones for a spin-1/2 particle.
Particles having net spin 1/2 include the proton, neutron, electron, neutrino, and quarks. The dynamics of spin-1/2 objects cannot be accurately described using classical physics; they are among the simplest systems which require quantum mechanics to describe them. As such, the study of the behavior of spin-1/2 systems forms a central part of quantum mechanics.
A spin-1/2 particle is characterized by an angular momentum quantum number for spin s of 1/2. In solutions of the Schrödinger equation, angular momentum is quantized according to this number, so that total spin angular momentum
However, the observed fine structure when the electron is observed along one axis, such as the z-axis, is quantized in terms of a magnetic quantum number, which can be viewed as a quantization of a vector component of this total angular momentum, which can have only the values of ±1/2ħ.
Note that these values for angular momentum are functions only of the reduced Planck constant (the angular momentum of any photon), with no dependence on mass or charge.[4]
Stern–Gerlach experiment[edit]
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 heterogeneous 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 3 parts, corresponding to atoms with Lz = −1, 0, and +1. The conclusion was that silver atoms had net intrinsic angular momentum of 1/2.[1]
General properties[edit]
Spin-1/2 objects are all fermions (a fact explained by the spin statistics theorem) and satisfy the Pauli exclusion principle. Spin-1/2 particles can have a permanent magnetic moment along the direction of their spin, and this magnetic moment gives rise to electromagnetic interactions that depend on the spin. One such effect that was important in the discovery of spin is the Zeeman effect, the splitting of a spectral line into several components in the presence of a static magnetic field.
Unlike in more complicated quantum mechanical systems, the spin of a spin-1/2 particle can be expressed as a linear combination of just two eigenstates, or eigenspinors. These are traditionally labeled spin up and spin down. Because of this, the quantum-mechanical spin operators can be represented as simple 2 × 2 matrices. These matrices are called the Pauli matrices.
Creation and annihilation operators can be constructed for spin-1/2 objects; these obey the same commutation relations as other angular momentum operators.
Connection to the uncertainty principle[edit]
One consequence of the generalized uncertainty principle is that the spin projection operators (which measure the spin along a given direction like x, y, or z) cannot be measured simultaneously. Physically, this means that it is ill-defined what axis a particle is spinning about. A measurement of the z-component of spin destroys any information about the x- and y-components that might previously have been obtained.
Complex phase[edit]
A single point in space can spin continuously without becoming tangled. Notice that after a 360° rotation, the spiral flips between clockwise and counterclockwise orientations. It returns to its original configuration after spinning a full 720°.
When a spinor is rotated by 360° (one full turn), it transforms to its negative, and then after a further rotation of 360° it transforms back to its initial value again. This is because in quantum theory the state of a particle or system is represented by a complex probability amplitude (wavefunction) ψ, and when the system is measured, the probability of finding the system in the state ψ equals | ψ |2 = ψ*ψ, the square of the absolute value of the amplitude.
Suppose a detector that can be rotated measures a particle in which the probabilities of detecting some state are affected by the rotation of the detector. When the system is rotated through 360°, the observed output and physics are the same as initially but the amplitudes are changed for a spin-1/2 particle by a factor of −1 or a phase shift of half of 360°. When the probabilities are calculated, the −1 is squared, (−1)2 = 1, so the predicted physics is the same as in the starting position. Also, in a spin-1/2 particle there are only two spin states and the amplitudes for both change by the same −1 factor, so the interference effects are identical, unlike the case for higher spins. The complex probability amplitudes are something of a theoretical construct which cannot be directly observed.
If the probability amplitudes rotated by the same amount as the detector, then they would have changed by a factor of −1 when the equipment was rotated by 180° which when squared would predict the same output as at the start, but experiments show this to be wrong. If the detector is rotated by 180°, the result with spin-1/2 particles can be different to what it would be if not rotated, hence the factor of a half is necessary to make the predictions of the theory match the experiments.
In terms of more direct evidence, physical effects of the difference between the rotation of a spin-1/2 particle by 360° as compared with 720° have been experimentally observed in classic experiments [5] in neutron interferometry. In particular, if a beam of spin-oriented spin-1/2 particles is split, and just one of the beams is rotated about the axis of its direction of motion and then recombined with the original beam, different interference effects are observed depending on the angle of rotation. In the case of rotation by 360°, cancellation effects are observed, whereas in the case of rotation by 720°, the beams are mutually reinforcing.[5]
Mathematical description[edit]
NRQM (Non-relativistic quantum mechanics)[edit]
The quantum state of a spin-1/2 particle can be described by a two-component complex-valued vector called a spinor. Observable states of the particle are then found by the spin operators Sx, Sy, and Sz, and the total spin operator S.
When spinors are used to describe the quantum states, the three spin operators (Sx, Sy, Sz,) can be described by 2 × 2 matrices called the Pauli matrices whose eigenvalues are ±ħ/2.
For example, the spin projection operator Sz affects a measurement of the spin in the z direction.
The two eigenvalues of Sz, ±ħ/2, then correspond to the following eigenspinors:
These vectors form a complete basis for the Hilbert space describing the spin-1/2 particle. Thus, linear combinations of these two states can represent all possible states of the spin, including in the x- and y-directions.
The ladder operators are:
Since S± =Sx ± i Sy, Sx = 1/2(S+ + S) and Sy =1/2i(S+S). Thus:
Their normalized eigenspinors can be found in the usual way. For Sx, they are:
For Sy, they are:
RQM (relativistic quantum mechanics)[edit]
While NRQM defines spin 1/2 with 2 dimensions in Hilbert space with dynamics that are described in 3-dimensional space and time, RQM define the spin with 4 dimensions in Hilbert space and dynamics described by 4-dimensional space-time.
As a consequence of the four-dimensional nature of space-time in relativity, relativistic quantum mechanics uses 4×4 matrices to describe spin operators and observables.
Spin as a consequence of combining quantum theory and special relativity[edit]
When physicist Paul Dirac tried to modify the Schrödinger equation so that it was consistent with Einstein's theory of relativity, he found it was only possible by including matrices in the resulting Dirac Equation, implying the wave must have multiple components leading to spin.[6]
See also[edit]
2. ^ Atkins, P. W. (1974). Quanta: A Handbook of Concepts. Oxford University Press. ISBN 0-19-855493-1.
3. ^ Peleg, Y.; Pnini, R.; Zaarur, E.; Hecht, E. (2010). Quantum Mechanics (2nd ed.). McGraw Hill. ISBN 978-0-071-62358-2.
4. ^ Nave, C. R. (2005). "Electron Spin". Georgia State University.
5. ^ a b Rauch, Helmut; Werner, Samuel A. (2015). Neutron Interferometry: Lessons in Experimental Quantum Mechanics, Wave-Particle Duality, and Entanglement. USA: Oxford University Press.
6. ^ McMahon, D. (2008). Quantum Field Theory. USA: McGraw Hill. ISBN 978-0-07-154382-8.
Further reading[edit] |
74a2941cd6ed27c1 | TRLan software package This software package implements the thick-restart Lanczos method. It can be used on either a single address space machine or a distributed parallel machine. The user can choose to implement or use a matrix-vector multiplication routine in any form convenient. Most of the arithmetic computations in the software are done through calls to BLAS and LAPACK. The software is written in Fortran 90. Because Fortran 90 offers many utility functions such functions such as dynamic memory management, timing functions, random number generator and so on, the program is easily portable to different machines without modifying the source code. It can also be easily accessed from other language such as C or C++. Since the software is highly modularized, it relatively easy to adopt it for different type of situation. For example if the eigenvalue problem may have some symmetry and only a portion of the physical domain is discretized, then the dot-product routine needs to be modified. In this software, this modification is limited to one subroutine. It also can be instructed to write checkpoint files so that it can be restarted is a later time.
References in zbMATH (referenced in 44 articles )
Showing results 1 to 20 of 44.
Sorted by year (citations)
1 2 3 next
1. Campos, Carmen; Roman, Jose E.: Restarted Q-Arnoldi-type methods exploiting symmetry in quadratic eigenvalue problems (2016)
2. Kestyn, James; Polizzi, Eric; Tang, Ping Tak Peter: Feast eigensolver for non-Hermitian problems (2016)
3. Li, Ruipeng; Xi, Yuanzhe; Vecharynski, Eugene; Yang, Chao; Saad, Yousef: A thick-restart Lanczos algorithm with polynomial filtering for Hermitian eigenvalue problems (2016)
4. Teng, Zhongming; Zhou, Yunkai; Li, Ren-Cang: A block Chebyshev-Davidson method for linear response eigenvalue problems (2016)
5. Vecharynski, Eugene; Yang, Chao; Xue, Fei: Generalized preconditioned locally harmonic residual method for non-Hermitian eigenproblems (2016)
7. Aishima, Kensuke: Global convergence of the restarted Lanczos and Jacobi-Davidson methods for symmetric eigenvalue problems (2015)
8. Potts, Daniel; Tasche, Manfred: Fast ESPRIT algorithms based on partial singular value decompositions (2015)
9. Wu, Lingfei; Stathopoulos, Andreas: A preconditioned hybrid SVD method for accurately computing singular triplets of large matrices (2015)
10. Zhou, Yunkai; Chelikowsky, James R.; Saad, Yousef: Chebyshev-filtered subspace iteration method free of sparse diagonalization for solving the Kohn-Sham equation (2014)
11. Baglama, James; Reichel, Lothar: An implicitly restarted block Lanczos bidiagonalization method using Leja shifts (2013)
12. Huang, Tsung-Ming; Kuo, Yueh-Cheng; Wang, Weichung: Computing extremal eigenvalues for three-dimensional photonic crystals with wave vectors near the Brillouin zone center (2013)
13. Wu, Gang; Zhang, Ying; Wei, Yimin: Accelerating the Arnoldi-type algorithm for the PageRank problem and the ProteinRank problem (2013)
14. Campos, Carmen; Roman, Jose E.: Strategies for spectrum slicing based on restarted Lanczos methods (2012)
15. Stoll, Martin: A Krylov-Schur approach to the truncated SVD (2012)
16. Du, Kui: GMRES with adaptively deflated restarting and its performance on an electromagnetic cavity problem (2011)
17. Specogna, Ruben; Trevisan, Francesco: A discrete geometric approach to solving time independent Schrödinger equation (2011)
18. Abdel-Rehim, Abdou M.; Morgan, Ronald B.; Nicely, Dywayne A.; Wilcox, Walter: Deflated and restarted symmetric Lanczos methods for eigenvalues and linear equations with multiple right-hand sides (2010)
19. Anderson, Christopher R.: A Rayleigh-Chebyshev procedure for finding the smallest eigenvalues and associated eigenvectors of large sparse Hermitian matrices (2010)
20. Huang, Tsung-Ming; Chang, Wei-Jen; Huang, Yin-Liang; Lin, Wen-Wei; Wang, Wei-Cheng; Wang, Weichung: Preconditioning bandgap eigenvalue problems in three-dimensional photonic crystals simulations (2010)
1 2 3 next |
6c1d9f70d98268bf | Saturday, July 28, 2007
I am guilty of frequently using physics speech in daily life, an annoying habit I also noticed among many of my colleagues [1]. You'll find me stating "My brain feels very Boltzmannian today", or "The customer density in this store is too high for my metastable mental balance". I have a friend who calls Chinese take out "the canonical choice" and another friend who, when asked whether he had made a decision, famously explained "I don't yet want my wave-function to collapse". My ex-boyfriend once called it "the physicist's Tourette-syndrome" [2].
One of my favourite physics-speech words is self-consistent. Self-consistency is tightly related to nothing. You know, that "nothing" that causes your wife to conclude her whole life is a disaster, we're all going to die in a nuclear accident, her glasses vanished (again!), and btw that's all your fault (obviously). But if you ask her what's the matter. Well, nothing.
"There's nothing I hate more than nothing
Nothing keeps me up at night
I toss and turn over nothing
Nothing could cause a great big fight
Hey -- what's the matter?
Don't tell me nothing."
~Edie Brickell, Nothing
1. Self-consistent
Science is our attempt to understand the world we live in. We observe and try to find reliable rules upon which to build our expectations. We search for explanations that are useful to make predictions, a framework to understand our environment and shape our future according to our needs. If our observations disagree with our rules, or observations seemingly disagree with each other (I swear I left my glasses in the kitchen), we are irritated and try to find a mistake. Something being in contradiction with itself [3] is what I mean with not self-consistent (What's the matter? - Nothing!).
On a mathematical basis this is very straight forward. E.g. If you assume my mood is given by a real valued continuous function f on the compact interval [now, then] with f(now)f(then) smaller than 0, this isn't self-consistent with the expectation it can do so without having a zero [4]. For more details on my mood, see sidebar.
Self-consistency is a very powerful concept in theoretical physics: if one talks about a probability, that probability better should not be larger than one. If one starts with the axioms of quantum mechanics, it's not self-consistent to talk about a particle's definite position and momentum. The speed of light being observer independent is not compatible with Galileo invariance and the standard addition law for velocities. Instead, self-consistency requires the addition law to be modified. This lead Einstein to develop Special Relativity.
A particularly nice example comes from multi-particle quantum mechanics, where an iterative approach can be used to find a 'self-consistent' solution for the electron distribution e.g. in a crystal or for an atom with many electrons (see self-consistent field method or Hartree-Fock method). A state of several charged particles will not be just a tensor product of the single particles, since the particles interact and influence each other. One starts with the tensor product as a 'guess' and applies the 'rules' of the theory. That is, by solving the Schrödinger equation with the mean- field potential which effectively describes the interaction, a new set of single particle wave functions can be computed. This result will however in general not agree with the initial guess: it is not self-consistent. In this case, one repeats the procedure with using the result as an improved guess. Given that the differential equations behave nicely, this iterative procedure leads one to find a fixed point with the properties that the initial distribution agrees with the resulting one: it is self-consistent.
A similar requirement holds for quantum corrections. A theory that is subject to quantum corrections but whose initial formulation does not take into account the existence of such extra terms is strictly speaking not self-consistent (see also the interesting discussion to our recent post on Phenomenological Quantum Gravity).
There are some subtleties one needs to consider, most importantly that our knowledge is limited in various regards. Self-consistency might only hold under certain assumptions or in certain limiting regimes, like small velocities (relative to the speed of light), large distances (relative to the Planck length) or at energies below a certain threshold. Likewise, not being self-consistent might be the result of having applied a theory outside these limits (typically, using an expansion outside a radius of convergence). In some cases (gravitational backreaction), violations of self-consistency can be negligible.
However, one might argue if it is possible at all to arrive at such a disagreement then at least one of the assumptions was unnecessary to begin with, and could have been replaced by requiring self-consistency. Unfortunately, this is often more easily said than done -- physics is not mathematics. We rarely start with writing down a set of axioms which one could check for self-consistency. Instead, in many cases one starts with little more than a patchwork of hints, and an idea how to connect them. Self-consistency in this case is somewhat more subtle to check. My friends and I often kill each others ideas by working out nonsensical consequences. Here, at least as important as self-consistency is that a theory in physics also has to be consistent with observation.
2. Consistent with Observation
The classical Maxwell-Lorentz theory is self-consistent. However, it is in disagreement with the stability of the atom. According to the classical theory, an electron circling around the nucleus should radiate off energy. The solution to this problem was the development of quantum mechanics. The inconsistency in this case was one with observation. Without quantizing the orbits of the electron, atoms would not be stable, and we would not exist.
This requirement is specific to sciences that describe the real world out there. Such a theory can be 'wrong' (not consistent with observation) even though it is mathematically sound. Sometimes however, these two issues get confused. E.g. in a recent Discover issue, Seth Lloyd wrote:
"The vast majority of scientific ideas are (a) wrong and (b) useless. The briefest acquaintance with the real world shows that there are some forms of knowledge that will never be made scientific [...] I would bet that 99.8 percent of ideas put forth by scientists are wrong and will never be included in the body of scientific fact. Over the years, I have refereed many papers claiming to invalidate the laws of quantum mechanics. I’ve even written one or two of them myself. All of these papers are wrong. That is actually how it should be: What makes scientific ideas scientific is not that they are right but that they are capable of being proved wrong."
~Seth Lloyd, You know too much
The current issue now had a letter in reply to this article:
"I was taken aback by Seth Lloyd's assertion that "99.8 percent of ideas put forth by scientists are [probably] wrong" and even more so by his statement that "of the 0.2 percent of ideas that turn out to be correct ... [t]he great majority of them are relatively useless." His thesis omits a basic trait of what we call science -- that it is a continuous fabric, weaving all provable knowledge together [...] we do science for a science sake, because a fundamental principle of science is that we never know when a discovery will be useful"
~Eric Fisher, Springfield, IL.
Well, the majority of my scientific ideas are definitely (a) wrong and (b) useless, but these usually don't end up in a peer review process. However, the reply letter apparently referred to the word 'correct' as 'provable knowledge', and to science as the 'weave' of all that knowledge. It might indeed be that the mathematical framework of a theory that is not consistent with observation turns out to be useful later but that doesn't change the fact that this idea is 'wrong' in the meaning that it does not describe nature. Peer review today seems to be mostly concerned with checking self-consistency, whereas being non-consistent with observation is ironically increasingly tolerated as a 'known problem'. Like, the CC being 120 orders of magnitude too large is a known problem. Oohm, actually the result is just infinity. But, hey, you've turned your integration contour the wrong way, the result is not infinity, but infinity + 2 Pi.
The requirement of consistency with observation was for me the main reason to chose theoretical physics over maths. The world of mathematics, so I found, is too large for me and I got lost in following runaway thoughts, or generalizing concepts just because it was possible. It is the connection to the real world, provided by our observations, that can guide physicists through these possibilities and lead the way. (And, speaking of observations and getting lost, I'd really like to know where my glasses are.)
3. Self-contained
Unlike maths, theoretical physics aims to describes the real world out there. This advantageous guiding principle can also be a weakness when it comes to the quantities we deal with. Mathematics deals with well defined quantities whose properties are examined. In physics one wants to describe nature, and the exact definitions of the quantities are in many cases subject of discussion as well. Consider how our understanding of space and time has changed over the last centuries!
In physics it has often happened that concepts of a theory's constituents only developed with the theory itself (e.g. the notion of a tensor or the Fock-space). As such it happens in physics that one can deal with quantities even though the framework does not itself define them. One might say in such a case the theory is incomplete, or not self-contained.
Due to this complication, I've known more than one mathematician who frowned upon approaches in theoretical physics as too vague, whereas physicists often find mathematical rigour too constraining, and instead prefer to rely on their intuition. Joe Polchinski expressed this as follows:
"[A] chain of reasoning is only as strong as its weakest step. Rigor generally makes the strongest steps stronger still - to prove something it is necessary to understand the physics very well first - and so it is often not the critical point where the most effort should be applied. [A]nother problem with rigor [is]: it is hard to get it right. If one makes one error the whole thing breaks, whereas a good physical argument is more robust."
~Joe Polchinski, Guest Post at CV
When it comes to formulating an idea, physicists often set different priorities than mathematicians. In some cases it might just not be necessary to define a quantity because one can sit down and measure it (e.g. the PDFs). Or, one can just leave a question open (will be studied in a forthcoming publication) and get a useful theory nevertheless. All of our present theories leave questions open. Despite this being possible, it is unsatisfactory, and the attempt to make a theory self-contained has lead to many insights throughout the history of science.
Newton's dynamics deals with forces, yet there is nothing in this framework that explains the origin of a force. It contains masses, yet does not explain the origin of masses. Maxwell's theory provides an origin of a force (electromagnetic). It has a source term (J), yet it does not explain the dynamics of the source term. This system has to be closed, e.g. with minimal coupling to another field whose dynamics is known. The classical Maxwell-Lorentz theory does this, it is self-contained and self-consistent. However, as mentioned above, this theory is not consistent with observation. Today we know the sources for the electromagnetic field are fermions, they obey the Dirac equation and Fermi statistic. However, if you look at an atom close enough you'll notice that quantum electrodynamics alone also isn't able to describe it satisfactory...
Besides the existence of space and time per se, the number of space-time dimensions is one of these open questions that I find very interesting. It has most often been an additional assumption. An exception is string theory where self-consistency requires space-time to have a certain number of dimensions. However - if it also contains an explanation why we observe only three of them, nobody has yet found it. So again, we are left with open questions.
4. Simple and Natural [5]
The last guiding principle that I want to mention is simplicity, or the question whether one can reduce a messy system of axioms and principles to something more simple. Is there a way to derive the parameters of the standard model from a single unified approach? Is there a way to derive the axioms of quantization? Is there a way to derive that our spacetime has dimension three, or Lorentzian signature?
In my opinion, simplicity is often overrated compared to the first three points I listed. We tend to perceive simplicity as elegance or beauty, concepts we strive to achieve, but these guidelines can turn out to be false friends. If you can find your glasses, look around and you'll notice that the world has many facettes that are neither elegant nor simple (like my husband impatiently waiting for me to finish). Even if you'd expect the underlying laws of nature to be simple, you'll still have to make the case that a certain observable reflects the elementary theory rather than being a potentially very involved consequence of a complex dynamical system, or an emergent feature. A typical example are the average distances of planets from the sun, a Sacred Mystery of the Cosmos that today nobody would try to derive from a theory of first principles (restrictions apply).
Also, we tend to find things simpler the more familiar we are with them, up to the level of completely forgetting about them (did you say something?). E.g. we are so used to starting with a Lagrangian that we tend to forget that its usefulness rests on the validity of the action principle. It is also quite interesting to note that researchers who are familiar with a field often find it 'simple' and 'natural'... I therefore support Tommaso's suggestions to renormalize simplicity to the generalized grandmother.
In this regard I also want to highlight the argument that one can allegedly derive all the parameters in the standard model 'simply' from today's existence of intelligent life. Notwithstanding the additional complication of 'intelligent', could somebody please simply explain 'existence' and 'life'?
Much like classical electrodynamics, Einstein's field equations too have a source term whose dynamics one needs to know. The system can be closed with an equation of state for each component. This theory is self-consistent [6], and it is consistent with all available observations. It reaches its limits if one asks for the microscopic description of the constituents. The transition from the macro- to the microscopic regime can be made for the sources of the gravitational field, but not also for the coupled gravitational field (oh, and then there's the CC, but this is a known problem).
Two theories that yield the same predictions for all observables I'd call equivalent (if you don't like that, accept it as my definition of equivalence.) But our observations are limited, and unlike the case of classical electrodynamics not being consistent with the stability of the atom, there is presently no observational evidence in disagreement with classical gravity.
For me this then raises the question:
Is there more than one theory that is self-consistent, self-contained and consistent with all present observations?
In a recent comment, Moshe remarked:"To paraphrase Ted Jacobson, you don't quantize the metric for the same reason you don't go about quantizing ocean waves." That sounds certainly reasonable, but if I look at water close enough I will find the spectral lines of the hydrogen atom and evidence for its constituents. And their quantization. To me, this just doesn't satisfactory solve the question what the microscopic structure of the 'medium', here space-time, is.
And what have we learned from all this...?
Let me go back to the start: If you ask a question and the answer is 'Nothing', you most likely asked the wrong question, or misunderstood the answer.
Ah... Stefan found my glasses (don't ask).
See also: Self-Consistency at The Reference Frame
[1]This habit is especially dominant -- and not entirely voluntarily -- among the not native English speakers, whose vocabulary naturally is most developed in the job related area.
[2] Unintentional cursing and uttering of obscenities, called Coprolalia, is actually only a specific feature of the Tourette syndrom.
[3] However, some years ago I was taught the word 'self-consistency' in psychology has a different meaning, it refers to a person accumulating knowledge from his/her own behaviour. A person whose thoughts and actions are in agreement and not in contradiction is called 'clear'. (At least in German. I couldn't find any reference to this online, and I'm not a psychologist, so better don't trust me on that.).
[4] See:
Bolzano's theorem.
[5] "Woman on Window", by F.L. Campello.
For more, see here.
[6] Note that this theory is self-consistent at arbitrary scales as long as you don't ask for the microscopic origin of the sources.
TAGS: , ,
The most spherical object ever made... used for the gyroscopes in NASA's Gravity Probe B. Launched in April 2004, Gravity Probe B tests two effects predicted by Einstein's theory: the geodetic effect and the frame-dragging (see here for a brief intro).
In order for Gravity Probe B to measure these tiny effects, it must use a gyroscope that is nearly perfect—one that will not wobble or drift more than 10-12 degrees per hour while it is spinning.
"A nearly-perfect gyroscope must be nearly perfect in two ways: sphericity and homogeneity. Every point on its surface must be exactly the same distance from the center (a perfect sphere), and its structure must be identical from one side to the other [...]
After years of research and development, Gravity Probe B produced just such a gyroscope. It is a 1.5-inch sphere of fused quartz, polished and “lapped” to within a few atomic layers of perfect sphericity. A scan of its surface shows that only .01 microns separate the highest point from the lowest point. Transform the gyroscope into the size of the Earth and its highest mountains and deepest ocean trenches would be a mere eight feet from sea level!"
Thursday, July 26, 2007
FIAS, the Frankfurt Institute for Advanced Studies
This week, I was again at the new campus of my old university. The science departments of the Johann Wolfgang Goethe University are all moving out of downtown Frankfurt into the fields of Niederursel, where new buildings keep springing up at an extraordinary rate. One of these new buildings is especially eye-catching with its bright-red finish.
This is the new building of FIAS, the Frankfurt Institute for Advanced Studies, and it's interesting not only because of its colour - it's one of the first public research institutes in Germany financed to a large extent by the money of private sponsors.
Universities in Germany have traditionally been financed by public money of the state and federal governments, and they usually don't have large funds at their own. Frankfurt University is a bit special in this respect, since it has been founded in 1914 by wealthy Frankfurt citizens. While today it is a publicly funded university as it is common in Germany, there is a strong tradition of private sponsoring of research and higher education.
So, a few years ago, theoretical physicist Walter Greiner and neuroscientist Wolf Singer started using their connections to raise private funds to establish a new kind of institute, which was supposed to be legally independent, but closely connected to the university and its science departments. It should bring together theorists from such diverse areas as biology, chemistry, neuroscience, physics, and computer science in order to address problems all revolving around a common theme: The study of structure formation and self-organization in complex systems.
This was the beginning of FIAS.
Today, there are more than 50 scientists, guests and students working together on cooperative phenomena on length scales ranging from quarks in colour superconductivity and heavy ion collisions over atoms in atomic clusters and macromolecules to cells in the immune system and the brain. Details and more links can be found on the pages of the FIAS scientists.
The training of graduate students is organized in a Graduate School. Last summer, I was involved in the compilation of a brochure presenting the FIAS, and I was fascinated by the really inspiring atmosphere among the students, who come from all over the world and form very diverse scientific backgrounds, but were always involved in interesting discussions.
In September, the FIAS is supposed to move into the new, red building, which was built for the institute by a private sponsor, the Giersch Foundation. There, FIAS scientist will have a place to work and think - it will be interesting to follow the outcome of this kind of "experiment".
Tuesday, July 24, 2007
Don't fart
Okay, it's unlikely you visit this blog to hear my opinion about farting, but I just read this article in New Scientist
How the obesity epidemic is aggravating global warming
(Issue June 30th - July6th, p. 21)
which is the most ridiculous fart line up of weak links designed to support a specific opinion that I've come across lately. The argumentation of the author, Ian Roberts (a professor of public health in London), is roughly: if you're fat you are wasting energy. Either by storing fat such that it can't even be used as bio fuel, or by moving it around with the help of gasoline powered transportation devices.
To begin with, despite of what the title says, the author does not actually talk about global warming, but about wasting energy. The connection between both is just assumed in the first sentence with 'we know humans are causing [global warming]', and not even once addressed after this. On the other hand, also the connection between wasting energy and obesity is constructed to make the point that you should loose weight to save the earth:
"[...] it is becoming clear that obese people are having a direct impact on the climate. This is happening through their lifestyles and the amount and type of food they eat, and the worse the obesity epidemic gets the greater its impact on global warming."
Well, if one wants to criticize a lifestyle, then one should criticise a lifestyle, but not add several associative leaps after that. Let us start with asking what exactly is a 'waste' of energy? Using energy for purposes that do not necessarily improve our well-being could generally be considered a waste. That goes for breaking a cellphone (consider all the energy needed to produce it), browsing the web the whole day (your home wireless doesn't run on vacuum energy) as well as for unnecessary consumption of food for whose production energy was needed.
However, whether that food is actually eaten or thrown away is completely irrelevant in this context. Also, on an equal footing one can argue that the mere presence of diet products damages the climate: it takes energy to produce and transport them, but the energy gain after consumption is lowered. Is there any reason to waste energy on producing diet coke when one can as well drink water? And while we're at it, is there any reason to go jogging every morning - isn't that just a waste of energy? Come to think about it, civilization itself seems to be a waste of energy.
The article goes on arguing
"[...] his greater bulk and higher metabolic rate will cause him to feel the heat more in the globally warmed summers, and he will be the first to turn on the energy intensive air conditioning."
If one argues that overweight people turn on the AC more often because they sweat more easily, one might want to take into account that underweight (or generally sickly) people tend to turn on the heating more often. People who suffer from back pain, arthritis and shortness of breath might use their car more often (as the article states), but this must not necessarily be a cause of obesity. The only thing one can state is that being healthy and well adapted to the part of the world you live in minimizes the additional energy needed to survive and feel comfortable (how 'needed' relates to 'actually used' is a completely different question).
I am definitely in favor of more sidewalks, of increased awareness for health risks caused by obesity, and I totally agree that we should save energy. But I would appreciate a scientific discussion of these issues, and not a mixed up mesh of several issues all drowned in politcal correctness.
In a similar spirit I read last week several articles claiming "Meat is murder on the environment" or likewise, a 'conclusion' based on a paper "Evaluating environmental impacts of the Japanese beef cow–calf system by the life cycle assessment method" (published in Animal Science Journal 78 (4), 424–432)
"a kilogram of beef is responsible for the equivalent of the amount of CO2 emitted by the average European car every 250 kilometres"
Being a vegetarian myself, I could give you a good number of reasons to drop the meat, but nothing you wouldn't find online in some thousand other places, so let me just focus on the issue at hand. If you want to save energy with the food you buy and eat, the most important factor to consider is origin and transportation.
• Your apple from New-Zealand, labeled 'bio' or not, doesn't tunnel to you. In fact you could say since, unlike beef, vegetables and friuts consist mostly of water, the amount of gasoline needed per energy content (joule) of transported food is higher for greens. So, preferably buy stuff that was not transported all around the globe whenever you can.
• If you buy products from countries where slash and burn is still practiced, you're damaging the environment more than if you support your local farmer - even if he's somewhat more expensive than Safeway.
• And, needless to say, don't buy stuff you don't need. Each time you have to throw something away, you are throwing away all the energy that was necessary to produce it. That doesn't only go for food, but for everything else including wrappings.
I want to add that much like cows, human flatulence as well release methane, which is said to contribute to global warming. So maybe we should consider a national anti-fart campaign? Regarding the vegetarian factor, also please note that "The cellulose in vegetables cannot be digested, therefore vegetarians produce more gas than people with a mixed diet." [source]
The bottomline of this writing is: don't construct or publish ridiculous cross-relations that are scientifically doubtful for a catchy headline.
See also: Global Warming
Monday, July 23, 2007
This and That
• I am very proud to report that I eventually managed to install a recent-comments-box in the sidebar!! Thanks go via several detours back to Clifford.
• Flip has an excellent post on The Braneworld and the Hierarchy in the Randall Sundrum (I) model
• Hey America, Germany is catching up.
• Idea of the day: I suggest that journals which reject more than 70% of submitted manuscripts should offer a consolidation gift. What I have in mind is a shirt saying "My manuscript went to PRD and all I got was this lousy T-shirt".
• Ever felt like your brain is too small? Think twice (if you have capacity left): Man with tiny brain shocks doctors
• Coincidentally, I came across the German version of Lee Smolin's book Warum gibt es die Welt? (Life of the Cosmos), which I found somewhat disturbing (I mean, even more than the English version). Among other things (that concern Japanese surfer) I learned that New York is the largest city on the planet (such the re-translation). Apologies to the translator*, but should you consider buying that book, I strongly recommend the English version (to read the original sentence go to amazon, and search inside for "irrelevant content" - amazingly the result is only one hit).
• Quotation of the day:
"The days come and go like muffled and veiled figures sent from a distant friendly party, but they say nothing, and if we do not use the gifts they bring, they carry them as silently away."
Ralph W. Emerson, in Society and Solitude [Vol 7], Chapter VII: Works and Days
* It turned out my husband knows him personally. It's a small world...
Sunday, July 22, 2007
GZK cutoff confirmed
In an earlier post, Bee explained the physics behind the GZK (Greisen, Zatsepin and Kuzmin) cutoff: protons traveling through outer space will - when their energy crosses a certain threshold - no longer experience the universe as transparent. If their energy is high enough, the protons can scatter with the omnipresent photons of the Cosmic Microwave Background, and create pions. As a result, their mean free paths drops considerably and only very little of them are expected to reach earth. This threshold for photopion production for ultra high energetic protons is known as the GZK cutoff.
The presence of this cutoff had been observed by the HiRes cosmic ray array (Observation of the GZK Cutoff by the HiRes Experiment, arXiv:astro-ph/0703099), but had been disputed by the results from the Japanese detector AGASA (Akeno Giant Air Shower Array) which caused excitement when it failed to see the cut-off in data obtained up to 2004. A third experiment, the Pierre Auger Observatory on the plains of the Pampa Amarilla in western Argentina, which started taking data last year, now settled the question:
"If the AGASA had been correct, then we should have seen 30 events [at or above 1020 eV], and we see two," says Alan Watson, a physicist from the University of Leeds, U.K., and spokesperson for the Auger collaboration [source]. According to Watson, the data also suggests that these highest energy rays comprise protons and heavier nuclei, the latter of which don't feel the GZK drag.
The results were announced on the 30th International Cosmic Ray Conference in Merida, Yucatan, Mexico, and had a brief mentioning in Nature. The Nature article also points out that there is prospect of identifying the regions of the sources of the highest energetic particles, but these data are preliminary. "Unless I talk in my sleep, even my wife doesn't know what these regions are", as Watson was quoted in Nature.
And of course, now that there is new data, somebody is around to claim one needs an even larger experiment to understand it: "Now we understand that above the GZK cutoff there are ten times less cosmic rays than we thought 10 years ago, so we may need a detector ten times as big as Auger," says Masahiro Teshima of the Max Planck Institute for Physics in Munich, Germany, who worked on AGASA and is working on the Telescope Array [source].
The recent paper by the Pierre Auger collaboration with more details was on the arxiv last week:
The UHECR spectrum measured at the Pierre Auger Observatory and its astrophysical implications
T.Yamamoto, for the Pierre Auger Collaboration, arXiv:0707.2638
Abstract: The Southern part of the Pierre Auger Observatory is nearing completion, and has been in stable operation since January 2004 while it has grown in size. The large sample of data collected so far has led to a significant improvement in the measurement of the energy spectrum of UHE cosmic rays over that previously reported by the Pierre Auger Observatory, both in statistics and in systematic uncertainties. We summarize two measurements of the energy spectrum, one based on the high-statistics surface detize. The large sample of data collected so far has led to a significant improvement in the measurement of the energy spectrum of UHE cosmic rays over that previously reported by the Pierre Auger Observatory, both in statistics and in systematic uncertainties. We summarize two measurements of the energy spectrum, one based on the high-statistics surface detector data, and the other of the hybrid data, where the precision of the fluorescence measurements is enhanced by additional information from the surface array. The complementarity of the two approaches is emphasized and results are compared. Possible astrophysical implications of our measurements, and in particular the presence of spectral features, are discussed.
The upper end of the cosmic ray energy spectrum as measured by the Pierre Auger Observatory: The black dots represent data points, the blue and red curves are expectations derived from different models for the composition and energy distribution of the cosmic ray particles, all based on well-established physics including the GZK cutoff mechanism. Two events cannot be understood as stemming from protons, but may well be explained by heavier nuclei. (Figure from T. Yamamoto, The UHECR spectrum measured at the Pierre Auger Observatory and its astrophysical implications, ICRC'07; Credits: Auger Collaboration, technical information)
More plots and data can be found on the websites of the Pierre Auger Observatory.
Saturday, July 21, 2007
The LHC at Nature Insight
With less than a year to go before the start of the Large Hadron Collider at CERN, there has been a lot of media coverage about this huge collider lately - see e.g. at NYT, The New Yorker, and of course Bee's post The World's Largest Microscope.
Much more in-depth information on the physics, the history, and the engineering aspects of the LHC can be found in this week's Nature Insight: The Large Hadron Collider. Unfortunately, a subscription is required for the full content, but two interesting articles are freely available:
How the LHC came to be, by former CERN Director-General Chris Llewellyn Smith, on the political and organisational struggles involved with the building such an international, multi-billion euro machine, and Beyond the standard model with the LHC, by CERN theorist John Ellis (the guy with the penguins - see page 5), on the different options on possible new physics that might be discovered at the LHC.
Have a nice weekend!
Wednesday, July 18, 2007
Phenomenological Quantum Gravity
[This is the promised brief write-up of my talk at the Loops '07 in Morelia, slides can be found here, some more info about the conference here and here.
When I submitted the title for this talk, I actually expected a reply saying "Look. This is THE international conference on Quantum Gravity. We already have ten people speaking about phenomonelogy - could you be a bit more precise here?". But instead, I found myself joking I am the phenomenology of the conference. Therefore, I added a somewhat extended motivation to my talk which I found blog-suitable, so here it is.]
The standard model (SM) of particle physics [1] is an extremely precise theory and has demonstrated its predictive power over the last decades. But it has also left us with several unsolved problems, question that can not be answered - that can not even be addressed within the SM. There are the mysterious whys: why three families, three generations, three interactions, three spatial dimensions? Why these interactions, why these masses, and these couplings? There are the cosmological puzzles, there is dark matter and dark energy. And then there is the holy grail of quantum gravity (see also: my top ten unsolved physics problems).
There are two ways to attack these problems. The one is a top-down approach. Stating with a promising fundamental theory one tries to reach common ground and to connect to the standard model from a reductionist approach. The difficulty with this approach is that not only one needs that 'promising candidate for the fundamental theory', but most often one also has to come up with a whole new mathematical framework to deal with it. Most of the talks on the conference [2] were top down approaches. The other way is to start from what we know and extend the SM in a constructivist approach. Examples for that might be to take the SM Lagrangian and just add all kinds of higher order operators, thereby potentially giving up symmetries we know and like. The difficulty with this approach is to figure out what to do with all these potential extensions, and how to extract sensible knowledge about the fundamental theory from it.
I like it simple. Indeed, the most difficult thing about my work is how to pronounce 'phenomenology' (and I've practiced several years to manage that). So I picture myself somewhere in the middle. People have called that 'effective models' or 'test theories'. Others have called it 'cute' or 'nonsense'. I like to call it 'top-down inspired bottom-up approaches'. That is to say, I take some specific features that promising candidates for fundamental theories have, add them to the standard model and examine the phenomenology. Typical examples are e.g. just asking what the presence of extra dimensions lead to. Or the presence of a minimal length. Or a preferred reference frame. You might also examine what consequences it would have if the holographic principle or entropy bounds would hold. Or whether stochastic fluctuations of the background geometry would have observable consequences.
These approaches do not claim to be a fundamental theory of their own. Instead, they are simplified scenarios, suitable to examine certain features as to whether their realization would be compatible with reality. These models have their limitations, they are only approximations to a full theory. But to me, in a certain sense physics is the art of approximation. It is the art of figuring out what can be neglected, it is the art of building models, and the art of simplification.
"Science may be described as the art of systematic over-simplification."
~Karl Popper
One can imagine more beyond the standard model than just QG! So, if we are talking about phenomenology of quantum gravity we'll have to ask what we actually mean with that. To me, quantum gravity is the question how we can reconcile the apparent disagreements between classical General Relativity (GR) and QFT. And I say 'apparent' because nature knows how quantum objects fall, so there has to be a solution to that problem [3]. To be honest though, we don't even know that gravity is quantized at all.
I carefully state we don't 'know' because we've no observational evidence for gravity to be quantized whatsoever. (The fact that we don't understand how a quantized field can be coupled to an unquantized gravitational field doesn't mean it's impossible.) Indeed one can be sceptical about whether it's observable at all. This is reflected very aptly in the below quotation from Freeman Dyson, which I think is deliberately provocative and basically says my whole field of work doesn't exist:
"According to my hypothesis, the gravitational field described by Einstein's theory of general relativity is a purely classical field without any quantum behavior [...] If this hypothesis is true, we have two separate worlds, the classical world of gravitation and the quantum world of atoms, described by separate theories. The two theories are mathematically different and cannot be applied simultaneously. But no inconsistency can arise from using both theories, because any differences between their predictions are physically undetectable."
~Freeman Dyson [Source]
Well. Needless to say, I do think there there is phenomenology of QG that is in principle observable, even though we might not yet be able to observe it. And I do think that observing it will lead us a way to QG.
However, there are various scenarios that could be realized at Planckian energies. Gravity could be quantized within one or the other approach. Also, higher order terms in classical gravity could become important. Or, there could be semi-classical effects coming into the game. Now one tries to take some insights from these approaches, leading to the above mentioned phenomenological models. Already here one most often has a redundancy. That is, various scenarios can lead to the same effect. E.g. modified dispersion relations, or the Planck scale being a fundamental limit to our resolution are effects that show up in more than one approach. In addition, there's a second step in which these models are then used to make predictions. Again, various models, even though different, could yield the same predictions. That's what I like to call the 'inverse problem': how can we learn something about the underlying theory of quantum gravity from potential signatures?
In the figure below I stress 'new and old' phenomenology because a sensible model shouldn't only be useful to make new predictions, it should also reproduce all that stuff we know and like. I have a really hard time to take seriously a model that doesn't reproduce the standard model and GR in suitable limits.
Now here are some approaches in this category of 'top down inspired buttom up approaches' that I find very interesting (for some literature, see e.g. this list):
(And possibly we can maybe soon add macroscopic non-locality to that list, an interesting scenario that Fotini, Lee and Chanda are presently looking into.)
However, whenever one works within such a model one has to be aware of its limitations. E.g. the models with large extra dimensions are in my opinion such a case in which has been done what sensibly could be done. And now we'll have to turn on the LHC and see. After the original ideas had been outlined, many people began to build more and more specific models with a lot of extra features. It's not that I don't find that interesting, but it's somewhat besides the point. To me it's like building a house and worrying about the color of the curtains before the first brick has been laid.
Now, all of the approaches I've mentioned above are attempts to get definitive signatures of QG, but so far none of these predictions on its own would be really conclusive. Take e.g. a possible modification of the GZK cutoff - could have been 'new' physics, but not clear which, or maybe just some ununderstood 'old' physics, like the showers not being created by protons from outside our galaxy as generally assumed?
So, my suggestion to make progress in this regard is to construct models that are suitable to investigate observables in varios different areas. In such a way, we could be able to combine predictions and make them more conclusive. Think about the situation with GR at the beginning of the last century: It predicted a perihelion precession of Mercury, but there were other explanations like an additional planet, a quadrupole moment of the sun, or maybe a modification of Newtonian gravity. It took another observable - in this case light deflection by the sun - that was predicted within the same framework, and confirmed GR was the correct description of nature [4]. And please note, a factor 2 mattered here [5].
I personally am very optimistic about the future progress in quantum gravity - and that not only because it's hard to beat Dyson's pessimism. I think it doesn't matter where we start from, may it be a top-down, a buttom-up approach or somewhere in the middle. I also think it doesn't matter which direction each of us starts into. The history of science tells us that there often are various different ways to arrive at the same conclusion. A particularly nice example is how Schrödinger's wave formulation and Heisenberg's matrix approach turned eventually out to be part of the same theory.
I think as long as we listen to what our theories tell us, if we take into account what nature has to say, are willing to redirect our research according to this - and if we don't get lost in distractions along the way, then I think we have good chances to find a way to quantum gravity. And this finally solves the mystery of the quotation on the last slide of my talk:
'The problem is all inside your head' she said to me
The answer is easy if you take it logically
I’d like to help you in your struggle to be free
There must be fifty ways to [quantum gravity]
[1] In my notation the SM includes General Relativity.
[2] The exception being the very recommendable talk on
Effective Quantum Gravity by John F. Donoghue.
[3] Though 3 years living in the US have tought me there's actually no such thing as a 'problem' - it's called a challenge. One just has to like them, eh?
[4] Admittedly, what the measurement actually said was not as straight forward as one would have wished. I leave it to my husband to elaborate on this interesting part of the history of science.
[5] The resulting deviation can be reproduced in the Newtonian approach up to a factor 1/2.
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PS on Zeitgeist...
More at
Tuesday, July 17, 2007
AvH's 10 point plan
The Alexander von Humboldt Foundation is the master of science networking among the German non-profit foundations. If you've managed to get one of their scholarships you become part of their brotherhood for a lifetime, including a membership card - Unfortunately I don't know about the secret handshake, since I've never even applied. The largest drawback of their scholarships is that one can only apply to a host who is also a member (Humboldtianer!), which was the reason for me to choose the German Academic Exchange Service (DAAD) instead.
However, I've just found that AvH came up with a ten point plan of recommendations "for making Germany more attractive for international cutting-edge researchers". Their suggestions make a lot of sense to me and I find the press release worth mentioning. Even though some of it (2./7.) addresses specifically German problems, especially the points 9. and 10. apply to many other countries as well, so does 4., and 3. is generally a good idea (that I too have mentioned repeatedly, and in my opinion an issue that will become more important the more complex and global the scientific community becomes). Let us hope that all these pretty word-ideas will have concrete consequences in the not to far future.
For the full text, see here. In brief the points are:
1. More jobs for scientists and scholars
On average, German professors supervise 63 students. This is more than twice as many as the average at top-rank international universities.
2. Academic careers need planning certainty: establishing tenure track as an option for junior researchers
German universities must take measures to plan the career stage between a doctorate and a secure professorship and make it compatible internationally. On the pattern of the Anglo-Saxon tenure track, clear, qualifying steps should be defined at which decisions are made about remaining at an institution.
3. Career support as an advisory and supervisory task of academic managers
Senior academics as well as university and/or institute directors must play an active role in human resources development for their junior researchers. Young scientists and scholars need careers advice.
4. Promoting early independence by taking risks in financing research
By international comparison, young academics in Germany have less scope for decision-making and action. Funding programmes for early, independent research must be strengthened. Especially for researchers at an early stage in their careers, procedures should be profiled for research work involving an unknown risk factor.
5. Making recruitment and appointments more professional
Appointment procedures must have an open outcome and be transparent. To this end, commissions charged with appointments must include external or independent expert reviewers. Good academics should be appointed quickly. Internationally respected universities can no longer afford to take years over appointments, particularly as universities and research establishments now actively have to recruit junior researchers internationally to a much greater extent than they did in the past.
6. Dissolve staff appointment schemes and adapt management structures
Rigid staff appointment schemes must make way for flexible appointment options, or be dissolved. Independent junior research group leaders must be put on a par with junior professors within the universities and in collaborations between universities and non-university research establishments.
7. Creating special regulations for collective wage agreements in the academic sector
According to many of those involved, the new wage agreement for the public service sector is not commensurate with appropriate remuneration for academic and non-academic staff at non-university and university research establishments. By comparison with other pay-scales, it is not competitive, either nationally or internationally, it restricts mobility, and its rigid conditions do not take account of the special features of academic life.
8. Internationally competitive remuneration
It must be ensured that cutting-edge researchers can be offered internationally competitive remuneration. The framework for allocating remuneration to professors currently valid at universities leaves too little scope for this.
9. Internationalising social security benefits
Internationally mobile researchers often have to accept major disadvantages or financial losses with regard to pension rights.
10. Increasing transparency and creating an attractive working environment
• Academic employers in Germany must be put in a position to offer organisational and financial support for removal and relocation which is already the norm in other countries, especially when top-rank academic personnel are appointed.
• Child-care facilities for internationally mobile researchers at universities and non-university research establishments must be expanded quickly and extensively. International appointments in Germany still often fail because there is a lack of child-care facilities.
• Careers advice and support for (marital) partners seeking employment as well as so-called dual career advice or support for academic couples are required to attract internationally mobile researchers. Examples from abroad indicate that this does not necessarily mean concrete job offers ( which are often difficult to find), rather, intelligent counselling can satisfy many people's needs.
Related: See also The LHC Theory Initiative, The Terrascale Alliance, Temporary Display, and Temporary Display - Contd.
... is not only a German word that I've never heard a German actually using [1], but also the title of the new Smashing Pumpkins album. By coincidence, I've been wearing my ancient ZERO shirt last week, so I felt like it was my duty to pick up the CD.
It is an interesting album, but overall very disappointing. To begin with, I never liked Billy Corgan's voice, but if there's no way around it, it definitly goes better with melancholy and infinite sadness than with revolution. I mean, come on, he's composing a song in 2006 titled United States with lyrics saying "fight! I wanna fight! I wanna fight! revolution tonight!" and manages to sing such that it could as well have been about, say, compactification on Calabi Yau manifolds [2].
There are more politically flavored tracks on the album: For God and Country ("it's too late for some, it's too late for everyone") and Doomsday Clock ("it takes an unknown truth to get out, I'm guessing I'm born free, silly me") but the only thing worth mentioning about them is the fact there presently is a market for this. This tells a lot more about the 'Zeitgeist' than the music itself [3].
Most of the tracks on the CD sound extremely similar, drowned in an ever present electric guitar soup and exchangeable melodies. Billy Corgan is at his best with the slower and more thoughtful titles like e.g. Neverlost ("If you think just right, if you'll love you'll find, certain truths left behind").
Favourite tracks from previous albums: Disarm, To Sheila, Bullet with Butterfly Wings, 1979
[1] My husband proudly reports he can testify at least one incident in which one of his uncles, a Prof. for theology and philosophy, successfully used the word.
[2] That's why I call it a science blog.
[3] And while I am at it: the German 'ei' is pronounced like the English 'I' (or the beginning of the word 'aisle') in both places (whereas the German 'i' is pronounced like the English 'ee'). The German 'Z' is pronounced close to 'ts'. That is with 'Tsaitgaist', you'll make yourself understood better than with 'seetgeest'.
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Monday, July 16, 2007
What's new?
Nothing. Well, almost nothing.
• I dyed my hair. The color is galled 'ginger'. I'd have called it pumpkin. It actually looks like foul apricots. Say of the day so far 'What happened to your hair?' - 'It's an allergic reaction.' - 'To what?' - 'Stupid questions.' (As one can easily deduce, my conversation partner in this case obviously was not Canadian.)
• Though the plan was this year it would not be necessary to pack my household into boxes and drag them around, I will actually be moving twice before the end of the year. Don't ask. At least I am staying in town.
• My last plant, which suffered significantly during my previous trip, has surprisingly recovered (well, at least half of it), and is so not looking forward to my upcoming trip. This is to warn you that I'll be flying to Europe on Thursday, and be off and away for a while.
• I've found six degrees of freedom.
• I just saw this paper on the arxiv:
Search for Future Influence from L.H.C
By Holger B. Nielsen, Masao Ninomiya
Abstract: We propose an experiment which consists of pulling a card and use it to decide restrictions on the running of L.H.C. at CERN, such as luminosity, beam energy, or total shut down. The purpose of such an experiment is to look for influence from the future, backward causation. Since L.H.C. shall produce particles of a mathematically new type of fundamental scalars, i.e. the Higgs particles, there is potentially a chance to find hitherto unseen effects such as influence going from future to past, which we suggest in the present paper.
which features the idea that the nature of the Higgs field is such that it attempts to avoid its own production: "When the Higgs particle shall be produced, we shall retest if there could be influence from the future so that, for instance, the potential production of a large number of Higgs particles in a certain time development would cause a pre-arrangement so that the large number of Higgs productions, should be avoided."
Therefore - if this hypothesis is true - the LHC is likely to suffer an accident and has to be shut down. The argument is supported by the cancellation of the Superconducting Supercollider: "Thus it is really not unrealistic that precisely at the first a large number of Higgs production also our model-expectations that is influence from the future would show up. Very interestingly in this connection is that the S.S.C. in Texas accidentally would have been the first machine to produce Higgs on a large scale. However it were actually stopped after a quarter of the tunnel were built, almost a remarkable piece of bad luck."
The authors therefore propose to give backwards causation an economically less damaging possibility to avoid Higgs production by means of a card game that settles runs for the LHC, and permits for the possibility to shut down completely in a quiet and undesastrous way.
One should take this very seriously: "It must be warned that if our model were true and no such game about restricting strongly L.H.C. were played [...] then a “normal” (seemingly accidental) closure should occur. This could be potentially more damaging than just the loss of L.H.C. itself. Therefore not performing [...] our card game proposal could - if our model were correct - cause considerable danger."
I find this interesting as it gives a completely new spin to postdiction. See, we now can have a theory that disables its own observability by backward causation. So, one can actually post-dict something before it has happened, and then go back into the future. Makes me wonder though why the universe hasn't disabled itself even before nucleosynthesis. Maybe God doesn't playing dice with the universe, but instead card games?
• Have a good start into the week!
Saturday, July 14, 2007
First Light for the Gran Telescopio Canarias
Last night, the Gran Telescopio Canarias (GTC) at the Observatorio del Roque de los Muchachos of the European Northern Observatory (ENO) in La Palma, Canary Islands, Spain, saw its "First Light". The first star observed was Tycho 1205081, close to Polaris - a bit more photogenic is this shot of the pair of interacting galaxies UGC 10923 with extended star formation regions, taken with an exposure time of 50 seconds:
Interacting galaxies UGC 10923 seen with the eyes of the World's largest telescope (Credits: Gran Telescopio Canarias, Instituto de Astrofisica de Canarias)
The primary mirror of the new telescope consists is made up of 36 separate, hexagonal segments, fabricated at the Glaswerke Schott in Mainz, just around the corner from Frankfurt. Taken together, the segments have a light-collecting surface of 75.7 m2, which corresponds the a circular mirror with a diameter of 10.4 metres. At this size, it is the currently largest telescope for optical and near-infrared light!
The Gran Telescopio Canarias in La Palma, Canary Isles, in September 2006 (Credits: GTC project webcam)
This was in the news these days here (see e.g.,, or Le Monde), but the European Northern Observatory somehow has managed to issue a press release only in Spanish, so I am a bit at loss to find more details. Actually, the report in the FAZ is very good, and recalls the developments that lead to the construction of these huge telescopes:
I remember from the popular astronomy book I read as a kid that at that time the 5-metre mirror of the Mount Palomar telescope was thought to be the endpoint of the growth of telescope mirror size: Larger solid mirrors are to heavy and deform when the telescope is moved, and moreover, the image gets blurred anyway by the distortions caused to the light as it passes through the atmosphere. As a case in point, a 6-metre telescope in the Soviet Union was mentioned, which produced pictures of not as high a quality as expected from its size. I was quite disappointed when I read that.
Fortunately, both obstacles could be overcome with new technologies first realised in the 1990s: Active Optics, which means that the mirror is always kept in perfect shape by an array of motors and can therefore be lightweight, and large, and Adaptive Optics, which manages to compensate for the fluctuations of the density of air and allows for a seeing nearly as good as in space.
Among the big optical telescopes using these techniques - the Keck, Subaru and Gemini-North telescopes in Hawaii, the four mirrors of the Very Large Telescope and the Gemini-South telescope in Chile, the Large Binocular Telescope in Arizona, the Hobby-Eberly-telescope in Texas, and the South African Large Telescope in the South African Karoo - the Gran Telescopio Canarias is currently the largest one.
The good news is that all these telescopes will continue to take great shots of the Universe for the professionals and for armchair astronomers like me, even when the Hubble Space Telescope will once have stopped working.
Potentially Insane
If you have a look at the sidebar, you'll see that even the internet is presently bored! Here is what PI residents do when they go bonkers.
PI stands for... Probably Improbable, Politically Incorrect, Potentially Insane, Preon Infected, Problems Included, Proudly Ignorant, Promising Insults, Positively Irrational, Presently Insignificant, Philosophical Illusions, Physics Inside
Contributed submissions:
Promoting Ideas, Prain Included, Pump It, Plotting Infinity, Position Independent, Pissing Ion, Perfectly Intolerant, Protecting Insanity, Post Inflation, Plutonium Injection, Pain Intensifier, Premature Interruption, Positive Impact, Private Intrusion
And here is what Wikipedia had to add, see PI (disambiguation):
Primitive Instinct (sometimes), Public Intoxication (definitly), People's Initiative (more than useful), Principal Investigator (haven't seen one), Primary Immunodeficiency (not yet), Predictive Index (none), Provider Independent (that's what I dream of), Pass Interference (my job), Programmed Instruction (absent)
My apologies to the whole public outreach department. I expect a sentence of 4 months snow.
See also: 3.141592653589793238462...
Thursday, July 12, 2007
I once read a science fiction about the not-too far future. Our planet's flora became fed up with mankind, and decided to strike back. It began with plumbing problems - tree's roots destroying pipes, went on to grass breaking through the pavement and ivy growing over houses. I have to think about this each time when I see a tree causing cracks in a walkway, or grass growing in every possible and impossible place.
Tuesday, July 10, 2007
Shrinking Earth
No, this is not about a resuscitation of old ideas about the history of planet Earth, but these days I could learn that the Earth Is Smaller Than Assumed, according to geodesist from the University of Bonn who have discovered that the blue planet is really smaller than originally thought. Well - not really, I would say: these guys are talking about 5 millimetre, or 0.2 inch.
Anyway, this accurate result is really impressive! It results from the combined analysis of radio signals from distant quasars, observed by a worldwide net of more than 70 radio telescopes. Characteristic features in the radio signals from quasars are received at slightly different times at different places on Earth, and the combination of these measurements using the technique of Very Long Baseline Interferometry allows a very precise determination of the relative distance of the radio telescopes: These relative distances can be deduced up to 2 millimetre on 1000 km, or up to 2 parts per billion (ppb). From the network of radio telescopes distributed all around the globe, it is possible to calculate its dimension very precisely. This analysis, accomplished with improved precision over previous similar work by the Bonn geodesist, yields a diameter of the Earth 5 millimetre smaller than supposed so far. According to a report in the New Scientist about this result, the total diameter of the Earth at the equator is around 12,756.274 kilometres (7,926.3812 miles).
Axel Nothnagel of the University of Bonn, who heads the team that provided new and more accurate data about the diameter of the Earth. (Credits: University of Bonn Press Release, July 5, 2007, Frank Luerweg)
A propos shrinking Earth: Earth was shrinking by a huge step, in a metaphorical way, 45 years ago today, as I heard this morning on the radio: On July 10, 1962, TELSTAR was launched from Cape Canaveral, the first communications satellite which allowed live TV broadcast between Europe and North America, bridging by the speed of light a distance that is steadily growing by 18 millimetre per year...
The TELSTAR communications satellite, launched 45 years ago today (Source: Wikipedia on Telstar)
PS: The paper by the Axel Nothnagel team is: The contribution of Very Long Baseline Interferometry to ITRF2005, by Markus Vennebusch, Sarah Böckmann and Axel Nothnagel, Journal of Geodesy 81 (2007) 553-564, DOI: 10.1007/s00190-006-0117-x. If someone can tell me where I can find the 5 millimetre in that paper, I am very grateful ;-)
Today on the Arxiv
Today I came across this very entertaining paper
Hollywood Blockbusters: Unlimited Fun but Limited Science Literacy
By C.J. Efthimiou, R.A. Llewellyn
Abstract: In this article, we examine specific scenes from popular action and sci-fi movies and show how they blatantly break the laws of physics, all in the name of entertainment, but coincidentally contributing to science illiteracy.
I didn't even know there is an arxiv for Physics and Society. The authors conclude with
"Hollywood is reinforcing (or even creating) incorrect scientific attitudes that can have negative results for the society. This is a good reason to recommend that all citizens be taught critical thinking and be required to develop basic science and quantitative literacy."
It's hard to disagree with that recommendation, even without reading the paper. Though I have to say if somebody has the scientific attitude he might survive a jump from the 15th floor, I guess natural selection will take care of that. For most cases I think we've all been taught from earliest childhood on not to mix up fiction with reality... That is, except for those of us who end up in theoretical physics, involuntarily or on purpose bending and breaking the laws of nature on our notebooks.
Update: See also The Physics of Nonphysical Systems.
Monday, July 09, 2007
Monday Links
In case you're just sitting at breakfast looking for a good read:
Sunday, July 08, 2007
The LHC Theory Initiative
Want proof that the grass is always greener on the other side? I just read this article
Refilling the Physicist Pool
about the LHC theory initiative:
"We are behind the Europeans, and we believe very strongly that we shouldn't just leave this work to the Europeans," Baur said in a UB statement. [...]
Funding in the US for particle physics as a whole and theoretical particle physics in particular has declined significantly over the past 15 years, Baur said. In addition, physics departments in US universities tend to hire faculty members who develop innovative ideas, whereas in Europe, the physics culture puts equal emphasis on novel research and solid calculations that help advance the field as a whole. But with the Large Hadron Collider -- the world's largest particle accelerator -- coming online in the next year or sooner, Baur said, the US cannot afford to fall behind."
It's interesting that in the US ideas are 'innovative' whereas in Europe they are 'novel' (especially since both refers to a field that is several decades old, and hasn't seen very much novelty lately). Admittedly, I find the perspective of a 'physics culture' that produces 'solid' Next-to-next-to-next-to-next-to leading order calculations somewhat depressing.
For German counterpart, see also the Terrascale Alliance.
Saturday, July 07, 2007
I spent half of the day trying to sort through all that stuff which has accumulated on my desk while I was away. My efforts where impressively unsuccessful. The only thing that came out of this was the poem below. I think I'll go for a walk, buy a lighter and then give it a second try.
Cardboard boxes, paper piles,
Unread books, and many files,
Coffee cups and empty cans,
Post-its, trash and broken pens.
Unpaid bills, forgotten friends,
Pieces, broken in my hands,
Wedding photos in between
Notebooks and a magazine.
Plastic plants, a moving box,
And a pair of unmatched socks,
Unfinished, and missing pieces,
Leave me wondering where peace is.
[For more, check my website]
... I actually think I have a lighter... if only I could find it... what a mess!
Friday, July 06, 2007
It's all about sex...
... yes, we already knew that. Men are intelligent to impress women, and women are intelligent to find the best men. That's why you're sitting on your desk, chewing a pen, trying to quantize gravity.
Here's what Psychology tells us today (Source: Ten Politically Incorrect Truths About Human Nature, by Alan S. Miller and Satoshi Kanazawa):
"Women often say no to men. Men have had to conquer foreign lands, win battles and wars, compose symphonies, author books, write sonnets, paint cathedral ceilings, make scientific discoveries, play in rock bands, and write new computer software in order to impress women so that they will agree to have sex with them. Men have built (and destroyed) civilization in order to impress women, so that they might say yes."
Well, and once you've destroyed a civilization and sufficiently impressed every women that was 'fit' enough to survive, keep in mind that by your human nature you are actually polygamous because it's an evolutionary advantage:
"Relative to monogamy, polygyny creates greater fitness variance (the distance between the "winners" and the "losers" in the reproductive game) among males than among females because it allows a few males to monopolize all the females in the group. The greater fitness variance among males creates greater pressure for men to compete with each other for mates. Only big and tall males can win mating opportunities. Among pair-bonding species like humans, in which males and females stay together to raise their children, females also prefer to mate with big and tall males because they can provide better physical protection against predators and other males."
And I'm sure 6 feet 4 also come in handy for changing light-bulbs. On the other hand, there are certain natural selection mechanism in societies which tolerate polygamy. As you'll also learn from the above article, suicide terrorists are dominantly Muslim because a) polygamy increases competition among men and b) because they are promised 72 virgins in heaven. (If only things were that simple. I still think airline passengers should stroke pigs before boarding, definitly preferable to throwing away my Coke each time I go through security.)
Also, sorry to report, but having children is statistically seen a bad idea for men when it comes to the peak of the crime-and-creativity curve:
"These calculations have been performed by natural and sexual selection, so to speak, which then equips male brains with a psychological mechanism to incline them to be increasingly competitive immediately after puberty and make them less competitive right after the birth of their first child. Men simply do not feel like acting violently, stealing, or conducting additional scientific experiments, or they just want to settle down after the birth of their child but they do not know exactly why."
I especially like the part with 'they don't know why'. And finally, a Harvard professor solved the puzzle why men prefer D-cups:
"Until very recently, it was a mystery to evolutionary psychology why men prefer women with large breasts, since the size of a woman's breasts has no relationship to her ability to lactate. But Harvard anthropologist Frank Marlowe contends that larger, and hence heavier, breasts sag more conspicuously with age than do smaller breasts. Thus they make it easier for men to judge a woman's age (and her reproductive value) by sight—suggesting why men find women with large breasts more attractive."
Well, I think there's truth in it, as my age seems to be incredibly hard to judge. Related, you'll be interested to hear that a recent study shows Women Don't Talk More Than Guys:
"The researchers placed microphones on 396 college students for periods ranging from two to 10 days, sampled their conversations and calculated how many words they used in the course of a day. The score: Women, 16,215. Men, 15,669.The difference: 546 words: "Not statistically significant," say the researchers."
Have a nice weekend. Have fun. Reproduce. Go, discover a new country or write a sonnet.
Thursday, July 05, 2007
The Planck Scale
The Planck scales - a length and a mass* - indicate the limits in which we expect quantum gravitational effects to become important
Gravity coupled to matter requires a coupling constant G that has units of length over mass. One finds the Planck scale if one lets quantum mechanics come into the game. For this, let us consider a quantum particle of a (so far unknown) mass mp with a Compton wavelength lp, the relation between both given by the Planck constant
This is the quantum input. Now consider that particle to be as localized as it is possible taking into account its quantum properties. That is, the mass mp is localized within a space-time region with extensions given by the particle's own Compton wavelength. The higher the mass of that particle, the smaller the wavelength. However, we know that General Relativity says if we push a fixed amount of mass together in a smaller and smaller region, it will eventually form a black hole. More general, one can ask when the perturbation of the metric that this particle causes will be of order one:
which then can be solved for the mass, and subsequently for the length scale we were looking for. If one puts in some numbers one finds
These Planck scales thus indicate the limit in which the quantum properties of our particle will cause a non-negligible perturbation of the space-time metric, and we really have to worry about how to reconcile the classical with the quantum regime. Compared to energies that can be reached at the collider (the LHC will have a center of mass energy of the order 10 TeV), the Planck mass is huge. This reflects the fact that the gravitational force between elementary particles is very weak compared to the the other forces that we know, and this is what makes it so hard to experimentally observe quantum gravitational effect.
Max Planck introduced these quantities in 1899, the paper (it's in German) is available online
(Credits to Stefan for finding it). You'll find the natural mass scales introduced on page 479ff. He didn't call them 'Planck' scales then, and it is also interesting why he found them useful to introduce, namely because the aliens would also use them
"It is interesting to note that with the help of the [above constants] it is possible to introduce units [...] which [...] remain meaningful for all times and also for extraterrestrial and non-human cultures, and therefore can be understood as 'natural units'."
Coincidentally, yesterday I saw a paper on the arxiv
What is Special About the Planck Mass?
By C. Sivaram
Abstract: Planck introduced his famous units of mass, length and time a hundred years ago. The many interesting facets of the Planck mass and length are explored. The Planck mass ubiquitously occurs in astrophysics, cosmology, quantum gravity, string theory, etc. Current aspects of its implications for unification of fundamental interactions, energy dependence of coupling constants, dark energy, etc. are discussed.
which gives a nice introduction into the appearances of various mass scales in physics, with some historical notes.
* With the speed of light set to be equal 1, in which case a length is the same as a time. It you find that confusing, just define a Planck time by dividing the length through the speed of light. |
c2f23f416398a25d | Tuesday, August 31, 2010 ... Français/Deutsch/Español/Česky/Japanese/Related posts from blogosphere
Why complex numbers are fundamental in physics
I have written about similar issues in articles such as Wick rotation, The unbreakable postulates of quantum mechanics, and Zeta-function regularization, among others.
But now I would like to promote the complex numbers themselves to the central players of the story.
History of complex numbers in mathematics
Around 1545, Girolamo Cardano (see the picture) was able to find his solution to the cubic equation. He already noticed the quadratic equation "x^2+1=0" as well. But even negative numbers were demonized at that time ;-) so it was impossible to seriously investigate complex numbers.
Cardano was able to additively shift "x" by "a/3" ("a" is the quadratic coefficient of the original equation) to get rid of the quadratic coefficient. Without a loss of generality, he was therefore solving the equations of the type
x3 + bx + c = 0
that only depends on two numbers, "b, c". Cardano was aware of one of the three solutions to the equation; it was co-co-communicated to him by Tartaglia (The Stammerer), also known as Niccolo Fontana. It is equal to
x1 = cbrt[-c/2 + sqrt(c2/4+b3/27)] +
+ cbrt[-c/2 - sqrt(c2/4+b3/27)]
Here, cbrt is the cubic root. You can check it is a solution if you substitute it to the original equation. Now, using the modern technologies, it is possible to divide the cubic polynomial by "(x - x_1)" to obtain a quadratic polynomial which produces the remaining two solutions once it is solved. Let's assume that the cubic polynomial has 3 real solutions.
The shocking revelation came in 1572 when Rafael Bambelli was able to find real solutions using the complex numbers as tools in the intermediate calculations. This is an event that shows that the new tool was bringing you something useful: it wasn't just a piece of unnecessary garbage for which the costs are equal the expenses and that should be cut away by Occam's razor: it actually helps you to solve your old problems.
Consider the equation
x3 - 15x - 4 = 0.
Just to be sure where we're going, compute the three roots by Mathematica or anything else. They're equal to
x1,2,3 = {4, -2-sqrt(3), -2+sqrt(3)}
The coefficient "b=-15" is too big and negative, so the square root in Cardano's formula is the square root of "(-15)^3/27 + 4^2/4" which is a square root of "-125+4" or "-121". You can't do anything about that: it is negative. The argument could have been positive for other cubic polynomials if the coefficient "b" were positive or closer to zero, instead of "-15", but with "-15", it's just negative.
Bombelli realized the bombshell that one can simply work with the "sqrt(-121)" as if it were an actual number; we don't have to give up once we encounter the first unusual expression. Note that it is being added to a real number and a cube root is computed out of it. Using the modern language, "sqrt(-121)" is "11i" or "-11i". The cube roots are general complex numbers but if you add two of them, the imaginary parts cancel. Only the real parts survive.
Bombelli was able to indirectly do this calculation and show that
x1 = cbrt(2+11i) + cbrt(2-11i) = (2+i) + (2-i) = 4
which matches the simplest root. That was fascinating! Please feel free to verify that (2+i)^3 is equal to "8+12i-6-i = 2+11i" and imagine that the historical characters would write "sqrt(-1)" instead of "i". By the way, it is trivial to calculate the other two roots "x_2, x_3" if you simply multiply the two cubic roots, cbrt, which were equal to "(2+-i)", by the two opposite non-real cubic roots of unity, "exp(+-2.pi.i/3) = -1/2+-i.sqrt(3)/2".
When additions to these insights were made by John Wallis in 1673 and later by Euler, Cauchy, Gauss, and others, complex numbers took pretty much their modern form and mathematicians have already known more about them than the average TRF readers - sorry. ;-)
Fundamental theorem of algebra
Complex numbers have many cool properties. For example, every N-th order algebraic (polynomial) equation with real (or complex) coefficients has exactly "n" complex solutions (some of them may coincide, producing multiple roots).
How do you prove this statement? Using powerful modern TRF techniques, it's trivial. At a sufficiently big circle in the complex plane, the N-th order polynomial qualitatively behaves like a multiple of "x^N". In particular, the complex phase of the value of this polynomial "winds" around the zero in the complex plane N times. Or the logarithm of the polynomial jumps by 2.pi.i.N, if you wish.
You may divide the big circle into an arbitrarily fine grid and the N units of winding have to come from some particular "little squares" in the grid: the jump of the logarithm over the circle is the sum of jumps of the logarithm over the round trips around the little squares that constitute the big circle. The little squares around which the winding is nonzero have to have the polynomial equal to zero inside (otherwise the polynomial would be pretty much constant and nonzero inside, which would mean no winding) - so the roots are located in these grids. If the winding around a small square is greater than one, there is a multiple root over there. In this way, you can easily find the roots and their number is equal to the degree of the polynomial.
Fine. People have learned lots of things about the calculus - and functions of complex variables. They were mathematically interesting, to say the least. Complex numbers are really "new" because they can't be reduced to real diagonal matrices. That wouldn't be true e.g. for "U-complex" numbers "a+bU" where "U^2=+1": you could represent "U" by "sigma_3", the Pauli matrix, which is both real and diagonal.
Complex numbers have unified geometry and algebra. The exponential of an imaginary number produces sines and cosines - and knows everything about the angles and rotations (multiplication by a complex constant is a rotation together with magnification). The behavior of many functions in the complex plane - e.g. the Riemann zeta function - has been linked to number theory (distribution of primes) and other previously separate mathematical disciplines. There's no doubt that complex numbers are essential in mathematics.
Going to physics
In classical physics, complex numbers would be used as bookkeeping devices to remember the two coordinates of a two-dimensional vector; the complex numbers also knew something about the length of two-dimensional vectors. But this usage of the complex numbers was not really fundamental. In particular, the multiplication of two complex numbers never directly entered physics.
This totally changed when quantum mechanics was born. The waves in quantum mechanics had to be complex, "exp(ikx)", for the waves to remember the momentum as well as the direction of motion. And when you multiply operators or state vectors, you actually have to multiply complex numbers (the matrix elements) according to the rules of complex multiplication.
Now, we need to emphasize that it doesn't matter whether you write the number as "exp(ikx)", "cos(kx)+i.sin(kx)", "cos(kx)+j.sin(kx)", or "(cos kx, sin kx)" with an extra structure defining the product of two 2-component vectors. It doesn't matter whether you call the complex numbers "complex numbers", "Bambelli's spaghetti", "Euler's toilets", or "Feynman's silly arrows". All these things are mathematically equivalent. What matters is that they have two inseparable components and a specific rule how to multiply them.
The commutator of "x" and "p" equals "xp-px" which is, for two Hermitean (real-eigenvalue-boasting) operators, an anti-Hermitean operator i.e. "i" times a Hermitean operator (because its Hermitean conjugate is "px-xp", the opposite thing). You can't do anything about it: if it is a c-number, it has to be a pure imaginary c-number that we call "i.hbar". The uncertainty principle forces the complex numbers upon us.
So the imaginary unit is not a "trick" that randomly appeared in one application of some bizarre quantum mechanics problem - and something that you may humiliate. The imaginary unit is guaranteed to occur in any system that reduces to classical physics in a limit but is not a case of classical physics exactly.
Completely universally, the commutator of Hermitean operators - that are "deduced" from real classical observables - have commutators that involve an "i". That means that their definitions in any representation that you may find have to include some "i" factors as well. Once "i" enters some fundamental formulae of physics, including Schrödinger's (or Heisenberg's) equation, it's clear that it penetrates to pretty much all of physics. In particular:
In quantum mechanics, probabilities are the only thing we can compute about the outcomes of any experiments or phenomena. And the last steps of such calculations always include the squaring of absolute values of complex probability amplitudes. Complex numbers are fundamental for all predictions in modern science.
Thermal quantum mechanics
One of the places where imaginary quantities occur is the calcuation of thermal physics. In classical (or quantum) physics, you may calculate the probability that a particle occupies an energy-E state at a thermal equilibrium. Because the physical system can probe all the states with the same energy (and other conserved quantities), the probability can only depend on the energy (and other conserved quantities).
By maximizing the total number of microstates (and entropy) and by using Stirling's approximation etc., you may derive that the probabilities go like "exp(-E/kT)" for the energy-E states. Here, "T" is called the temperature and Boltzmann's constant "k" is only inserted because people began to use stupidly different units for temperature than they used for energy. This exponential gives rise to the Maxwell-Boltzmann and other distributions in thermodynamics.
The exponential had to occur here because it converts addition to multiplication. If you consider two independent subsystems of a physical system (see Locality and additivity of energy), their total energy "E" is just the sum "E1+E2". And the value of "exp(-E/kT)" is simply the product of "exp(-E1/kT)" and "exp(-E2/kT)".
This product is exactly what you want because the probability of two independent conditions is the product of the two separate probabilities. The exponential has to be everywhere in thermodynamics.
Fine. When you do the analogous reasoning in quantum thermodynamics, you will still find that the exponential matters. But the classical energy "E" in the exponent will be replaced by the Hamiltonian "H", of course: it's the quantum counterpart of the classical energy. The operator "exp(-H/kT)" will be the right density matrix (after you normalize it) that contains all the information about the temperature-T equilibrium.
There is one more place where the Hamiltonian occurs in the exponent: the evolution operator "exp(H.t/i.hbar)". The evolution operator is also an exponential because you may get it as a composition of the evolution by infinitesimal intervals of time. Each of these infinitesimal evolutions may be calculated from Schrödinger's equation and
[1 + H.t/(i.hbar.N)]N = exp(H.t/i.hbar)
in the large "N" limit: we divided the interval "t" to "N" equal parts. If you don't want to use any infinitesimal numbers, note that the derivative of the exponential is an exponential again, so it is the right operator that solves the Schrödinger-like equation. So fine, the exponentials of multiples of the Hamiltonian appear both in the thermal density matrix as well as in the evolution operator. The main "qualitative" difference is that there is an "i" in the evolution operator. In the evolution operator, the coefficient in front of "H" is imaginary while it is real in the thermal density matrix.
But you may erase this difference if you consider an imaginary temperature or, on the contrary, you consider the evolution operator by an imaginary time "t = i.hbar/k.T". Because the evolution may be calculated in many other ways and additional tools are available, it's the latter perspective that is more useful. The evolution by an imaginary time calculates thermal properties of the system.
Now, is it a trick that you should dismiss as an irrelevant curiosity? Again, it's not. This map between thermal properties and imaginary evolution applies to the thermodynamics of all quantum systems. And because everything in our world is quantum at the fundamental level, this evolution by imaginary time is directly relevant for the thermodynamics of anything and everything in this world. Any trash talk about this map is a sign of ignorance.
Can we actually wait for an imaginary time? As Gordon asked, can such imaginary waiting be helpful to explain why we're late for a date with a woman (or a man, to be really politically correct if a bit disgusting)?
Well, when people were just animals, Nature told us to behave and to live our lives in the real time only. However, theoretical physicists have no problem to live their lives in the imaginary or complex time, too. At least they can calculate what will happen in their lives. The results satisfy most of the physical consistency conditions you expect except for the reality conditions and the preservation of the total probabilities. ;-)
Frankly speaking, you don't want to live in the imaginary time but you should certainly be keen on calculating with the imaginary time!
Analytic continuation
The thermal-evolution map was an example showing that it is damn useful to extrapolate real arguments into complex values if you want to learn important things. However, thermodynamics is not the only application where this powerful weapon shows its muscles. More precisely, you surely don't have to be at equilibrium to see that the continuations of quantities to complex values will bring you important insights that can't be obtained by inequivalent yet equally general methods.
The continuation into imaginary values of time is linked to thermodynamics, the Wick rotation, or the Hartle-Hawking wave function. Each of these three applications - and a few others - would deserve a similar discussion to the case of the "thermodynamics as imaginary evolution in time". I don't want to describe all of conceptual physics in this text, so let me keep the thermodynamic comments as the only representative.
Continuation in energy and momentum
However, it's equally if not more important to analytically continue in quantities such as the energy. Let us immediately say that special relativity downgrades energy to the time component of a more comprehensive vector in spacetime, the energy-momentum vector. So once we will realize that it's important to analytically continue various objects to complex energies, relativity makes it equally important to continue analogous objects to complex values of the momentum - and various functions of momenta such as "k^2".
Fine. So we are left with the question: Why should we ever analytically continue things into the complex values of the energy?
A typical laymen who doesn't like maths too much thinks that this is a contrived, unnatural operation. Why would he do it? A person who likes to compute things with the complex numbers asks whether we can calculate it. The answer is Yes, we can. ;-) And when we do it, we inevitably obtain some crucial information about the physical system.
A way to see why such things are useful is to imagine that the Fourier transform of a step function, "theta(t)" (zero for negative "t", one for positive "t"), is something like "1/(E-i.epsilon)". If you add some decreasing "exp(-ct)" factor to the step function, you may replace the infinitesimal "epsilon" by a finite constant.
Anyway, if you perturb the system at "t=0", various responses will only exist for positive values of "t". Many of them may exponentially decrease - like in oscillators with friction. All the information about the response at a finite time can be obtained by continuing the Fourier transform of various functions into complex values of the energy.
Because many physical processes will depend "nicely" or "analytically" on the energy, the continuation will nicely work. You will find out that in the complex plane, there can be non-analyticities - such as poles - and one can show that these singular points or cuts always have a physical meaning. For example, they are identified with possible bound states, their continua, or resonances (metastable states).
The information about all possible resonances etc. is encoded in the continuation of various "spectral functions" - calculable from the evolution - to complex values of the energy. Unitarity (preservation of the total probabilities) can be shown to restrict the character of discontinuities at the poles and branch cuts. Some properties of these non-analyticities are also related to the locality and other things.
There are many links over here for many chapters of a book.
However, I want to emphasize the universal, "philosophical" message. These are not just "tricks" that happen to work as a solution to one particular, contrived problem. These are absolutely universal - and therefore fundamental - roles that the complex values of time or energy play in quantum physics.
Regardless of the physical system you consider (and its Hamiltonian), its thermal behavior will be encoded in its evolution over an imaginary time. If Hartle and Hawking are right, then regardless of the physical system, as long as it includes quantum gravity, the initial conditions of its cosmological evolution are encoded in the dynamics of the Euclidean spacetime (which contains an imaginary time instead of the real time from the Minkowski spacetime). Regardless of the physical system, the poles of various scattering amplitudes etc. (as functions of complexified energy-momentum vectors) tell you about the spectrum of states - including bound states and resonances.
Before one studies physics, we don't have any intuition for such things. That's why it's so important to develop an intuition for them. These things are very real and very important. Everyone who thinks it should be taboo - and it should be humiliated - if someone extrapolates quantities into complex values of the (originally real) physical arguments is mistaken and is automatically avoiding a proper understanding of a big portion of the wisdom about the real world.
Most complex numbers are not "real" numbers in the technical sense. ;-) But their importance for the functioning of the "real" world and for the unified explanation of various features of the reality is damn "real".
And that's the memo.
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snail feedback (24) :
reader gezinorgiva said...
Learn Geometric Algebra and then you won't need complex numbers anymore (for physics)
Complex numbers are nothing more than a subalgebra of GA/Clifford algebra.
Nothing special about them at all.
reader Lumo said...
Holy cow, gezinorgiva.
There is everything fundamental and special about the complex numbers as you would know if you have read at least my modest essay about them.
The complex numbers may be a subset of many other sets but the complex numbers are much more fundamental than any of these sets.
The nearest college or high school is recommended.
reader publius said...
There is an interesting article related to the topic of this post by C.N. Yang in the book "Schrodinger, Centenary celebration of a polymath" E. C.W. Kilmister, entitled "Square root of minus one, complex phases and Erwin Schrodinger". There Yang quotes Dirac as saying that as a young man he thought that non commutativity was the most revolutionary and essentially new feature of quantum mechanics, but as he got older he got to think that that was the entrance of complex numbers in physics in a fundamental way (as opposite to as auxiliary tools as in circuit theory). He describes Schrodinger struggles to come to terms with that, after unsuccessfully trying to get rid of "i". Also included is the role of previous work by Schrodinger in Weyl's seminal gauge theory ideas in his discovering of quantum mechanics.
reader publius said...
This comment has been removed by the author.
reader CarlBrannen said...
Noncommutativity implies complex numbers; the Pauli spin matrices sigma_x sigma_y sigma_z multiply to give i.
reader Lumo said...
Carl, please... Since you're gonna be a student again, you will have to learn how to think properly again.
Your statement is illogical at every conceivable level.
First, the "product" of the three Pauli matrices has nothing directly to do with noncommutativity. The latter is a property of two matrices, not three matrices. The product is not a commutator (although it's related to it).
Second, the fact that the product includes an "i" is clearly a consequence of the fact that in the conventional basis, one of the Pauli matrices - namely sigma_{y} - is pure imaginary. This imaginary value of sigma_{y} is the reason, not a consequence, of the product's being imaginary.
Third, it's easy to see that noncommutativity doesn't imply any complex numbers in general. The generic real - non-complex - matrices (e.g. the non-diagonal ones) are noncommutative but their commutator is always a real matrix.
Noncommutativity by itself is completely independent of complexity of the numbers. And indeed, complex numbers themselves are commutative, not non-commutative. The only way to link noncommutativity and complex numbers is to compute the eigenvalues of the commutator of two Hermitean operators. Because their commutator is anti-Hermitean, its eigenvalues are pure imaginary. For example, the commutator can be an imaginary c-number, e.g. in xp-px.
For more general operators, the eigenvalues are typically computed from a characteristic equation that will contain (x^2+r^2) factors, producing ir and -ir as eigenvalues.
reader ObsessiveMathsFreak said...
It's actually impossible to avoid the existence of complex numbers even in real analysis—or at least to avoid their effects.
Consider the Taylor series of the function f(x)=1/(1-x^2) centered around x=0. The series is given by f(x)=1+x^2+x^4+x^6+... . It can be seen that this Taylor series is divergent for |x|>1 and so the Taylor series will fail for large x. This isn't very surprising as it can be seen that f(x) has obvious singularities at x=-1,+1 and so the Taylor series could not possibly extend beyond these points.
However, more interesting is the same approach to the function g(x)=1/(1+x^2). This function is perfectly well behaved, having no singularities of any order in the real number. Yet its Taylor series g(x)=1-x^2+x^4-x^6+... is divergent for |x|>1, despite there seemingly being no corresponding singularity as in the previous case.
Analysis in the reals leads to the idea of a radius of convergence, but gives no clear idea where this comes from. In fact using complex numbers the reason becomes clear. g(x) has singularities at x=-i,+i. Despite these existing only in the complex plane, their effects can be felt for the real function. In fact the radius of converge of a Taylor series is the distance from the central point to the nearest singularity—be it in the real or complex plane(See the book "Visual Complex Analysis" for more).
Complex numbers become fundamental and indeed in some sense unavoidable the moment we introduce multiplication and division into our algebra. This is because these operations—and most (all?) elementary operations—hold for complex numbers in general and not just for the real numbers.
Once we write expressions like (x^2+7)/(x^4-3), while we may mean for x to be a purely real number, the complex numbers will work in this equation just as well, and indeed more importantly, will continue to work as we perform all elementary algebraic operations on the expression; "BOMDAS" operations, radixes, even taking exponents and logs.
This should not come as too much of a surprise, and we could have started—like the Pythagoreans—by meaning for the expression to be restricted to rational numbers and even disregarding irrational numbers entirely. But irrational numbers will work in the original expression and in through all our rational manipulations. What we claim holds for a subset of numbers holds for the large set too. And the existence of this larger set has concrete implications for expressions on the subset.
So, when we write down any equation at all, we must be careful. We may mean for it to hold for some restricted class of numbers, but there may be much wider implications. While I am not a physicist, I suspect a similar situation arise. We may want or expect the quantities we measure to expressible in purely real numbers; but the universe may have other ideas.
reader CarlBrannen said...
Sorry, getting old. I meant "Clifford or geometric algebra" rather than "noncommutative". I was continuing the comment by gezinoriva.
And the "i" is not "clearly a consequence" of a basis choice. The product sigma_x sigma_y sigma_z is an element of the Clifford algebra that commutes with everything in the algebra and squares to -1. That's what makes it's interpretation "i" and this does not depend on basis choice.
reader Lumo said...
Dear Carl, it's completely unclear to me why you think that you have "explained" complex numbers.
A number that squares to minus one is the *defining property* of the imaginary unit "i". You just "found" one (complicated) application - among thousands - where the imaginary unit and/or complex numbers emerge.
The condition that a quantity squares to a negative number appears at thousands of other places, too. For example, it's the coefficient in the exponent of oscillating functions - that are eigenvectors under differentiation. Why do you think that Clifford algebras are special?
reader CarlBrannen said...
Dear Lumos; The Clifford algebras are special as they are related to the geometry of space-time. For example, (ignoring the choice of basis and only looking at algebraic relations) Dirac's gamma matrices are a Clifford algebra. Generalizing to higher dimension people expect that the generalization of the gamma matrices will also be a Clifford algebra.
On this subject I sort of follow David Hestenes; his work geometrizes the quantum wave functions, but I prefer to geometrize the pure density matrices. But other than that, his work explains some of the justification. See his papers at geocalc.clas.asu.edu
My concentration on this subject is due to my belief that geometry is more fundamental than symmetry. (This belief is a "working belief" only, that is, what I really believe is that it's more useful for me to assume this belief than to assume the default which almost everyone else assumes.)
A beautiful example of putting geometry ahead of symmetry are Hestenes' description of point groups in geometric / Clifford algebra. I'm sure you'll enjoy these: Point Groups and Space Groups in GA and Crystallographic Space Groups
reader Lumo said...
Apologies, Carl, but what you write is a crackpottery that makes no sense. Clifford algebras are related to the geometry of spacetime?
So is the Hartle-Hawking wave function, black holes, wormholes quintic hypersurface, conifold, flop transition, and thousands of other things I can enumerate. In one half of them, complex numbers play an important role.
Also, what the hell do you misunderstand about the generalization of gamma matrices to higher dimensions - which are still just ordinary gamma matrices - that you describe them in this mysterious way?
You just don't know what you're talking about.
reader CarlBrannen said...
Mathematics is an infinite subject and uses complex numbers in an infinite number of ways. Who cares. What's important is when they appear in the definition of space itself, before QM or SR or GR.
In the traditional physics approach, the Pauli spin matrices are just useful matrices for describing spin-1/2. But they can be given a completely geometric meaning and i falls out as the product. See equation (1.8) of Vectors, Spinors, and Complex Numbers in Classical and Quantum Physics
Regarding the relationship between higher dimensions and gamma matrices see the wikipedia article Higher dimensional gamma matrices It defines the higher dimensional gamma matrices as matrices that satisfy the Clifford algebra relations. But this is well known to string theorists, why are you asking? I must be misunderstanding you.
reader Lumo said...
Dear Carl,
your comment is a constant stream of nonsense.
First, in physics, one can't define space without relativity or whatever replaces it. You either have a space of relativistic physics, or space of non-relativistic physics, but you need *some* space and its detailed physical properties always matter because they define mathematically inequivalent structures.
So it's not possible to define "space before anything else" such as relativity: space is inseparably linked to its physical properties. In particular, space of Newtonian physics is simply incorrect for physics when looked at with some precision - e.g. in the presence of gravity or high speeds.
Second, the examples I wrote were also linked to space - and they were arguably linked to space much more tightly than your Clifford algebra example. So it is nonsensical for you to return to the thesis that your example is more "space-related" or more fundamental than mine. I have irrevocably shown you that it's not.
Third, it's just one problem with your statements that the Clifford algebra is not "the most essential thing" for space. Another problem is the fact that space itself is not more fundamental than many other notions in physics. Space itself is just one important concept in physics - and there are many others, equally important ones, and they're also linked to complex numbers. All of them can be fundamental in some descriptions, all of them - including space - may be emergent. It's just irrational to worship the concept of space as something special.
So even your broader assumption that what is more tightly linked to space has to be more fundamental is a symptom of your naivite - or a quasi-religious bias.
Fourth, it was you, not me, who claimed that he has some problems with totally elementary things such as Dirac matrices in higher dimensions. So why the fuck are you now reverting your statement? Previously, you wrote "Generalizing to higher dimension people expect that the generalization of the gamma matrices will also be a Clifford algebra."
I have personally learned Dirac matrices for all possible dimensions at the very first moments when I encountered the Dirac matrices, I have always taught them in this way as well, and that's how it should be because the basic properties and algorithms of Dirac matrices naturally work in any dimension - and only by doing the algorithm in an arbitrary dimension, one really understands what the algorithm and the properties of the matrices are. So why are you creating a non-existent controversy about the Dirac matrices in higher dimensions? It's a rudimentary piece of maths. Moreover, in your newest comment, you directly contradicted your previous comment when you claimed that it was me, and not you, who claimed that there was a mystery with higher-dimensional matrices.
There are about 5 completely fundamental gaps in your logic. One of them would be enough for me to think that the author of a comment isn't able to go beyond a sloppy thinking. But your reasoning is just defective at every conceivable level. I just don't know how to interact with this garbage. You're just flooding this blog with complete junk.
reader Chuck said...
Some of your readers should look at Gauss on biquadratic residues. The simple fact is that Professor Hawking should return to the black hole that god made for him since he advances no argument beyond those offered many years ago by the fakers Laplace and Lagrange. For the uninformed mathematical physicists, those who don't know up from down (and these are the vast majority), "god" is the nickname among mathematicians for one Kurt Gödel .
(See discussion on "Is it possible that black holes do not exist? " on Physics Forums
http://www.physicsforums.com/showthread.php?t=421491 for relevant citations.)
In any case all rational scientific discourse has been effectively banned since the illegal shutdown of the first international scientific association and journal in 1837 by the Duke of Clarence, Ernest Augustus. See Percy Byssh Shelley's Mask of Anarchy for a pertinent depiction of the Duke of Clarence, the face behind Castlereagh. A simple google search for "("magnetic union" OR "Magnetischer Verein") AND ("Göttingen Seven" OR "Göttinger Sieben") gauss weber" shows that there has been no serious discussion of that action on the subsequent development of scientific practice.
We must assume therefore that the concurrent and congruent Augustin-Louis Cauchy scientific method of theft, assassination, plagiarize at leisure remains hegemonic. Chuck Stevens 571-252-0451 stevens_c@yahoo.com
reader Señor Karra said...
Dear Lubos,
i don't agree that i has to be represented as a c-number. In fact i think many of the posters have been trying to say (poorly) the following:
i can be definitely defined algebraically as a c-number
can be wrote in the representation of a conmutative subalgebra of SO(2) defined by the isomorphism:
a + ib <=> ( ( a , b) , ( -b , a) )
(sorry, i had to write a matrix as a list of rows, i hope its clear)
reader Luboš Motl said...
Dear seňor karra,
of course, I realize this isomorphism with the matrices. But it's just a convention whether you express the "number that squares to minus one" as a matrix or as a new letter. It's mathematically the same thing.
The important thing is that you introduce a new object with new rules. In particular, in "your" case, you must guarantee that the matrices you call "complex numbers" are not general matrices but just combinations of the "1" and "i" matrices. In the case of a letter "i", you must introduce its multiplication rules.
reader Hugo said...
Clifford algebra is the generalization of complex numbers and quaternions to arbitrary dimensions. Just google it. Therefore it should be no controversy here. Clifford algebra (or geometric algebra) has been very successful in reformulating every theory of physics into the same mathematical language. That has, among other tings, emphasized the similarities and differences between the theories of physics in a totally new way. One elegant feature of this reformulation is to reduce Maxwell's equations into one single equation.
The reason why Clifford algebra has lately been renamed "geometric algebra" is that quantities of the algebra are given geometric interpretations, and the Clifford product are effective in manipulating these geometric quantities directly. Together with the extension of the algebra to a calculus this formalism has the power to effectively model many geometries such as projective, conformal, and differential geometry.
In the geometric algebra over three dimensions most quantities are interpreted as lines, planes and volumes. Plane and volume segments of unit size are represented with algebraic objects that square to minus one.
In the reformulation of quantum mechanics with geometric algebra (describes geometry of the three dimensions of physical space), the unit imaginary from the standard treatment is identified with several different quantities in the algebra. In some situations, as in the Schrödinger equation, the unit imaginary times h bar is identified with the spin of the particle by the geometric algebra reformulation. This, together with other results of the reformulation suggests that spin is an intrinsic part of every aspect of quantum mechanics, and that spin may be the only cause of quantum effects.
If you have the time and interest I strongly suggest reading a little about geometric algebra. Geometric algebra is not on a collision course with complex numbers. In fact, geometric algebra embrace, generalize and deploy them to a much larger extent than before.
reader Luboš Motl said...
Dear Hugo, the very assertion that "the Clifford algebra is the generalization of complex numbers to any dimension" is largely vacuous. Complex numbers play lots of roles and they're unique in the most important roles.
One may hide his head into the sand and forget about some important properties of the complex numbers - e.g. the fact that every algebraic equation of N-th degree has N solutions, not necessarily different, in the complex realm (something that makes C really unique) - but if he does forget them, he's really throwing the baby out with the bath water.
Of course that if you forget about some conditions, you may take the remaining conditions (the subset) and find new solutions besides C, "generalizations". But it's surely morally invalid to say that the Clifford algebra is "the" generalization. It's at most "a" generalization in some particular direction - one that isn't extremely important.
reader Hugo said...
Ok, that's a semi-important point for the physicist; Clifford algebra is _a_ generalization of complex numbers and quaternions. It is puzzling that all you managed to extract from my comment was that I should have written "a" in stead of "the". My comment was about the role of Clifford algebra in physics. When you state that Clifford algebra is not important you should consider explaining why, if you don't want to be regarded as ignorant and "not important" yourself.
reader Luboš Motl said...
Dear Huge, your "the" instead of "a" was a very important mistake, one that summarizes your whole misunderstanding of the importance of complex numbers.
This article was about the importance of complex numbers in physics and the branches of mathematics that are used in physics. This importance can't be understated. Clifford algebras simply came nowhere close to it. They're many orders of magnitude less important than complex numbers.
There may exist mathematical fundamentalists and masturbators who would *like* if physics were all about Clifford algebras but the fact still is that physics is not about them. They're a generalization of complex numbers that isn't too natural from a physics viewpoint. After all, even quaternions themselves have an extremely limited role in physics, too.
The relative unimportance of Clifford algebras in physics may be interpreted in many different ways. For example, it is pretty much guaranteed that a big portion of top physicists don't even know what a Clifford algebra actually is. Mostly those who were trained as mathematicians do know it.
Others who "vaguely know" will tell you that it's an algebra of gamma matrices for spinors, or something like that, but they won't tell you why you would talk about them with such a religious fervor because the relevant maths behind gamma matrices is about representations of Lie groups and Lie algebras, not new kinds of algebras.
Moreover, many of them will rightfully tell you that the overemphasis of Clifford algebras means an irrational preference for spinor representations (and pseudo/orthogonal groups) over other reps and other groups (including exceptional ones). It's just a wrong way of thinking to consider the concept of Clifford algebras fundamental. Physicists don't do it because it's just not terribly useful to talk in this way but even sensible mathematicians shouldn't be thinking in this way.
reader Luboš Motl said...
"Huge" should have been "Hugo".
One more comment. People who believe that Clifford algebras are important and start to study physics are often distracted by superficial similarities that hide big physics differences.
For example, Lie superalgebras are very important in physics (although less than complex numbers, of course), generalizing ordinary Lie algebras in a way that must be allowed in physics and is used in Nature.
However, people with the idea that Clifford algebras are fundamental often try to imagine that superalgebras are just a special case etc. See e.g. this question on Physics Stack Exchange.
The answer is, of course, that superalgebras don't have to be Clifford algebras. They may be more complicated etc. Moreover, the analogy between the algebra of Dirac matrices on one hand and Grassmann numbers on the other hand is just superficial. In physics, it's pretty important we distinguish them. The gamma matrices may anticommute but they're still matrices of Grassmann-even numbers which are different objects than Grassmann-odd numbers.
When we associate fields to points in spacetime, the difference between Grassmann-odd and Grassmann-even objects is just huge, despite the same "anticommutator".
When talking about objects such as spinors, the fundamental math terms are groups, Lie groups, Lie algebras, and their representations. For fields, one also adds bundles, fibers, and so on, perhaps, although the language is only used by "mathematical" physicists. But Clifford algebras are at most a name given by one particular anticommutator that appears once when we learn about spinors etc. and it never appears again. It doesn't bring a big branch of maths that should be studied for a long time. It's just a name for one equation among thousands of equations. It's not being manipulated with in numerous ways like we manipulate complex numbers or Lie algebras.
The Clifford algebras are the kind of objects invented by mathematicians who predetermined that a particular generalization should be ever more important except that the subsequent research showed the assumption invalid and some people are unwilling to see this fact.
reader anky don said...
A number whose square is less than or equal to zero is termed as an imaginary number. Let's take an example, √-5 is an imaginary number and its square is -5. An imaginary number can be written as a real number but multiplied by the imaginary unit.in a+bi complex number i is called the imaginary unit,in given expression "a" is the real part and b is the imaginary part of the complex number. The complex number can be identified with the point (a, b).
one-to-one correspondence
reader Rosy Mota said...
is the fundamental reason to explain the absolute asymmetry between left-right handed rotation frames
in the non-euclidean geometry generated by the double torsion given by complex numbers and its comjugate complexes,or best the quaternions,through of anticommutativity to 4-dimensions that connect space and time into spacetime continuos.the biquaternions calcule the motion to curve manifolds to 4-dimensions.
reader M J said...
I stumbled across this point while Googling Dirac's famous comment that it took people many years to become comfortable with complex numbers, so it was likely it would take another 100 years before they are comfortable with spinors.
It is not quite what I was looking for, but it is certainly a good article. But I must admit that having more of a mathematician's inclination than a physicist's, I don't see what the fuss is all about. Mathematicians accept imaginary numbers as 'real' for a number of reasons, and our insight into their reality deepens over time, so that now I would say the reason we accept them as real and interesting can be summarized in a way that appears at first glance very different from the valid historical reasons Lubos gives: that the complex numbers are a perfectly valid algebraic extension of the reals, an extension with unique properties (among all other algebraic extensions) that explain many of the historical reasons for our interest in complex numbers.
But if the reader finds that too obscure, there is always the matrix representation of complex numbers, one of the discoveries that put to rest many of the historical doubts about the 'reality' of complex numbers: represent a complex number a+bi as a matrix a00=a, a01=-1, a10=b a11=a. Such matrices are certainly real; their simplicity and symmetry suggest they should be both significant and easy to study. Lubos's post lists many of the reasons that suggestion has been amply justified over the years. |
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Optical computers Optical computers Document Transcript
• University of Ljubljana Faculty for Mathematics and Physics Department of Physics Seminar Optical Computers Author: Žiga Lokar Mentor: prof. Igor Poberaj Ljubljana, 31.3.10Summary:This seminar will first explain benefits and problems with optical computers, then explain all vitalparts of a computer and how can they be replaced with optical equivalents. First, the seminar willdiscuss feasibility to completely replace copper wires with optical fibres in motherboards, thenexplain how is computation with photons possible. In the end, all optical storage and memory will bediscussed.
• 2. IntroductionFor computing, light is currently used for signal transmission, mostly for long distancecommunication. It offers several benefits compared to electric signal transmission, most importantly,much higher bandwidth. Furthermore, energy requirements are lower and photons also interactweakly with electromagnetic field, meaning signal transmission is less prone to errors. The limitingfactor for long distance communication are currently electronic parts of the system. Signals need tobe cleaned of noise after some distance. This is usually realised with OEO (optic-electronic-optic)devices; signal is converted to electronic one, cleaned and retransmitted as an optical signal. Alloptical signal regeneration is also possible, experimentally demonstrated by various research groups[1, 2]. Another desire that would also greatly speed up long distance communication is fast all opticalsignal routing, also demonstrated [3].SOASemiconductor optical amplifiers are currently used mostly for signal amplification before detection,where only amplitude is increased. Using nonlinear effects exhibited by these devices, signal can bealso reshaped and retimed. This is called 3R regeneration.Image 1: Signal regeneration. 1R means amplification, 2R adds re-shaping while 3R adds re-timing as well. On x axis there is time, with amplitude on y axis.Such regenerators work at much higher bitrates than electronic signal regeneration and there is noneed to convert signals. But their problem is small amplification and regeneration for amount ofenergy needed. Only amplifiers are used in newer fibers, so there is no need to re-transmit the signalevery couple of kilometers, as it was needed in older fibers.All demonstrated technologies show promise for fast all optical systems for long distancecommunication, although there are still many serious problems how to scale the device and decreasepower consumption.For computers, on the other hand, small system size is also required. Seminar will explain currentstatus of all-optical computers.
• 3. Backplane (motherboard)Backplanes are vital parts of computers, however, they are not known under this name in personalcomputing. For personal computers, as well as quite many servers nowadays, term motherboard isused. These terms are not identical; motherboards offer some integrated processing, as well as asocket for the central processing unit, CPU. Backplanes are only used to route signals betweenvarious add-on cards, and a CPU is just one of the inserted cards if there is a central processing uniton the backplane at all. However, these differences are mostly semantic as both parts have identicalimportant role – interconnection between different components.I will use term backplane during this seminar, as work in this area is based on providing fastcommunication.Optical backplanes consist of similar parts than long distance communication. Signal source is aVCSEL, vertical-cavity surface-emitting laser. Benefit of these lasers is a low power requirement and avery high efficiency, but their maximum power is very low as well. Signal is modulated using Kerreffect, which will be discussed under processing unit. The signal propagates through optical fiber,similar to the one used in long distance communication. No regeneration is needed, as distance isshort. On the receiving point, a photodiode is used to convert optical signal back to electric one.Current demonstrations of optical backplanes do not plan to have optical signal routing, optics serveonly as a point-to-point link.Major benefit of the optical backplane is its very high transfer speed. Current demonstrations exceed10 Gbit/s/channel/fiber. Their major difficulty is its high cost of equipment, while there are someother, physical limitations.One of the problems is various sources of noise – noise on the laser, the detector and a fundamentalphysics limit, shot noise. This effect is known to anyone investing some time in photography. Whenthe number of photons collected is small, this causes a large variation of the signal. Underassumption there will be a Poisson distribution of photons and the detector has a constantpercentage to collect a photon, signal to noise ratio equals √ . This noise presents lowest possiblepower consumption, as system is characterized by minimum signal to noise ratio. Electric systemscurrently operate with less noise for short distance. Therefore, the advantage of decreasedinteraction with environment is reduced due to power requirement of the connection. This onlyholds true if the wire is very short, otherwise resistivity of wires greatly decreases performance –signal attenuation due to distance travelled is much higher for copper wires than it is for opticalfibers.
• Image 2: Shot noise on a photograph. http://en.wikipedia.org/wiki/Shot_noise; 21.3.10Furthermore, frequency of light used in long distance communication fibers is 1550 nm. Thistherefore limits minimum diameter of the wire to about 750, as light cannot travel through awaveguide that is less than thick. Further consequence is the imposed limit on the laserdimensions, the very same 750 nm. This limit is much more important than the noise limit; itpresents a great difficulty to scale the system to on-chip communication, especially if routing isneeded.Despite all difficulties, end result is in favor of photonics with higher transfer speed and lower powerneed per GB/s, while cost problem remains. In HPC (high performance computing) connectionsbetween 2 backplanes have been optical for some time now, due to mentioned benefits. Migrationtowards optical backplanes has not started yet, although technology is ready and offers higher speedthan electronics. Shorter transfer length (even on chip optical communication) has beendemonstrated in laboratory, but practical use is not as close as it is for backplane opticalcommunication. [4, 5]
• Image 3: IBMs idea of merging photonics, memory and processing on the same chip in different layers. Estimated by IBM to appear around 2018. [4]For high performance computing, communication is therefore migrating towards photonics. Mainreasons are or need to become cost per bit for backplanes and communication between racks. Foron-chip communication, power per bit and maximum bandwidth is said to bring advantage tophotonic communication over copper wires.For personal use, optical communication is further away. The first announced optical connectionbetween devices is Intel’s Light Peak, optical connection between a PC and other devices, such asmonitors, TVs, hard disk drives and so on. All-optical motherboards or smaller connections are notexpected yet. 4. Processing unitOut of all components of an optical computer, processing unit is the most desired.As there are many nonlinear optical effects, many different possibilities have been tried in order tocreate an optical logic gate. They usually rely on Kerr effect. Kerr effect is a quadratic electro-opticaleffect, where material alters its refraction coefficient when electric field is applied. (1)Where λ is the wavelength of the light, K is is the Kerr constant and E is electric field.This principle is used to modulate signals for communication, as response of the material is veryswift. Electric field could also be a consequence of another beam, if the beam intensity is highenough, making the process a third order nonlinear effect. Third order optical nonlinearity meanspolarization depends on the cube of electric field, (2)
• Such transistors rely on cross phase modulation, where phase of one beam is influenced by another. { } (3) is the wavelength of light in free space, l length of the material where beams interacts, issecond field and is third order nonlinear tensor, where only part responsible for cross phasemodulation is taken.Response to the light is swift; therefore their possible operating frequency is high, much higher thanof current electronic ones. However, most materials have a very small nonlinear index. For areasonably small transistor (to compete with electronic ones in terms of frequency/size), a very highelectric field is needed, meaning powerful lasers are required. Optical transistor major problem isalso material inability to sustain laser power without damage. When frequency of pulses gets highenough to compete with silicon transistors, optical ones incinerate, limiting their maximum operatingfrequency to less than 1 Mhz. Either a material with very high nonlinearity needs to be found or amaterial needs to sustain very high power of lasers in order to increase this limit. First is preferred, asa very powerful laser would be also impractical for computing. Wall Street Journal (Jan. 30, 1990)labeled such a material "unobtainium,” as throughout all the years and funding, nothing was found.Situation has not improved much since then.Another material optical transistors could be based on are photonic crystals, or photonic bandgapcrystals they are also called. These crystals are made of a material with periodically varying refractionindex. While 1D photonic crystals have been known for over a century and frequently appear innature, first 2D crystal for optical wavelength has been fabricated in 1996. Their major problem is stillfabrication, as there is no known efficient method for creating such an array in 3D without defects.Currently, majority of research on this field is focused on producing the crystals, mainly through self-assembly. [7]For computation using photonic crystals, nonlinear defects in crystals are required. Even a simplescheme, where only 3 nonlinear rods are inserted in a 2D photonic crystal, exhibits bistabletransmission. This can be used to create a simple switching device, where one beam sets the path forthe second.
• Image 4: Effects of nonlinear rods on transmission in a configuration shown in inset. Nonlinear rods are marked with black circles. [7]Another phenomenon that showed promise and is being researched for optical computers is asurface plasmon, a standing wave of free electrons on a metallic surface. When a photon hits metal,electrons ripple with a specific frequency. These standing waves propagate through a metal, butlosses are very high. If such waves are confined only to the metallic surface and most energy travelsthrough a dielectric layer that is in contact with metal, losses are dramatically reduced and the signalcan travel much further even through a very narrow waveguide, below conventional limit. [8]Although most research in this field is not useful for an all-optical computer at this stage, plasmonsare very promising for creating small lasers and waveguides. Both were demonstrated, but majoradvances are still needed in order to be of practical use.Optical logic gates also have the problem they need a lot of signal regeneration after gate. As opticalsignal regeneration is difficult, some predict that optical computing and possibly even routing isdead, or at least is not progressing the way scientists hoped for. [6]On the other hand, some are more optimistic: ““For the last five years or so it has been possible tobuild an optical computer chip, but with all-optical components it would have to measure somethinglike half a meter by half a meter and would consume enormous power. With plasmonics, we canmake the circuitry small enough to fit in a normal PC while maintaining optical speeds,” explainsAnatoly Zayats , a researcher at The Queens University of Belfast in the United Kingdom.” - Quotefrom [9].Current status: Such a computer is possible, but not feasible. Many significant improvements arerequired in order to compete with electronic processors for computers.
• 5. StorageWrite once, read many (WORM) optical storage has been around for quite a long time, since the firstCDs in 80s, which even enabled more storage density than magnetic disks at the time. CD wasfollowed by a DVD, this one by Blu-ray. After Blu-ray, holographic storage is predicted as the nextstop. First stop for holographic storage is to create a write once disk, same as with other types ofoptical storage. But, in order to make a proper computer storage using this technology, it is requiredthat a disk can also be rewritten multiple times. Rewritable holographic storage could be based onphotorefractive effect, an effect where a material alters its refractive index when exposed to light. Ifa material exhibits Pockels effect, change of refraction coefficient is (4)Where n is a refraction coefficient without applied electric field, r is an electro-optic coefficient and Eis the electric field.For the effect to be applied to holograms, process works in several steps. First, interference betweenreference and signal beams creates a pattern of light and dark fringes throughout the crystal. Inregions with bright fringes, electrons absorb photons and promote into conductive band. Electronsdiffuse around the crystal (or drift due to photovoltaic effect), while holes must remain stationary.Electrons may, with some probability, recombine with holes or fall back to the traps made byimpurities, where they cannot move. With more electrons in dark areas and holes in bright, there is anet electric field, called space charge field, which causes, via electro-optic effect, refractive index tochange, creating a grating. Grating reflects light, recreating the original signal beam pattern, whenreference beam strikes the grating at the same angle.The following setup does enable multiple writes, but every read would excite some of the trappedelectrons and decrease strength of the hologram. After several reads, image would not berecognizable anymore.The idea would need to be realized using two-photonic write in order to be practical. Two-photonicsetup uses 2 different dopants in order to create shallow and deep traps. When writing an image, apulse of short wavelength excites electrons to conduction band. They quickly fall back to a shallowtrap. From shallow traps, signal and reference beam can excite them back in conduction band, wherethey diffuse like before. Afterwards, electrons fall back, first to a shallow trap, then to a deep. Whenin deep traps, reference beam does not have enough energy to excite them back to conduction band,greatly increasing durability of the hologram.
• Image 5: Two-photonic write, described above. [10]Additionally, storage could be written on in the Fourier space, meaning a single point of data doesnot map to a single part of crystal, but rather to the whole, thus decreasing defects due toimperfections.Using all these techniques, holographic rewritable storage could at least compete with magneticdisks with respect of reliability. Polymeric holographic storage, on the other hand, is among the mostdurable WORM storages and could compete even with magnetic tape in terms of reliability.There is also a possibility to create several holograms on the same space using different angle ofreference beam, or some other way of multiplexing signal (discussed in [10]). This increases storagedensity, but decreases speed as reflection is weaker. Practical upper limit is around one hundredholograms in a material 1 mm thick.As the capacity of magnetic disks increase, holographic storage seems unable to bring any significantadvantages. First announced holographic disk would offer only 300 GB of capacity with up to 1.6 TBto follow. Magnetic disks offer more capacity now than proposed storage could in the future. Butthere is one major possible benefit of holographic storage – speed. Holographic drive enables readingand writing whole set of data in parallel, enabling very high read and write speed. On the other hand,all current data is written sequentially.Due to lack of any moving parts, access time would be very low as well compared to magnetic drives,where needle needs to move. Announced products, on the other hand, would be first severelylimited by write speed of only 20MB/s, which was said to increase later on, to be comparable withcurrent disks. [11]
• The other possible benefit compared to current drives is longevity and reliability, both mostly valuedfor permanent storage and not as useful for home computers as speed and capacity are. Magnetictape is currently dominant in this space with no real alternative.Major difficulty of holographic storage and the reason it has not appeared yet is the material. Thematerial needs to have traps of correct energy, enable high write density and speed, would bewithout defects that decrease performance and cheap enough to make, in order to be in commercialproducts. Most researched material was Lithium Niobate, while many others have been tried inlaboratories, including proteins. Polymers are also currently used, but for permanent holograms.One of the most noticeable companies, working on holographic storage, InPhase technologies, spunoff Bell Labs in 2000. The company made several claims when will they release their storage, first as300GB disks with up to 1.6TB later, but nothing was released to the market. Recently, the companyhad their assets seized, as they were unable to pay taxes. Similarly, Optware made several claimsabout their upcoming products, while none actually appeared on the market in the end, and thecompany does not exist anymore either.General Electric and IBM continue research on holographic storage, but neither announcedupcoming products yet. 6. MemoryMemory for systems seeks to have as high bandwidth as possible with lowest latency possible, whilehaving the largest size possible. These requests are mutually excluding, so compromises are made,usually in favor of speed and latency. Optical competition to traditional memory seem very promisingas replacement, theoretically enabling high bandwidth and low latency, while being able to scale sizeas well. Such optical system could be based on several effects. First possibility was discussed withhard disks and is holographic data storage. Further option is a temporary storage of light pulses.This storage relies on light to excite electrons in conduction band, generating pairs of electron-hole.Electric field is applied to modulate potential and trap electrons and holes in spatially separatepotential minimums, limiting recombination. This energy can be later re-emitted in a short flash oflight, when potential is released. Due to the mechanism, only energy can be stored, coherence is notpreserved. The idea was shown to work with temperatures around 100K, light was stored for several , while energy was released on demand. Further improvements could enable such storage for alonger period of time and at a room temperature. As coherence cannot be preserved, this is not asinteresting storage possibility as the holographic one. [12]In addition to these possibilities, light can be also slowed down or trapped in a resonator. Resonatorbased storage would require mirrors with total reflectivity as there would be 300 reflections ofmirrors if we wanted to store light in 1 m long resonator for only 1 . Similarly, fiber based delayrequires low signal attenuation and ability to release stored pulses on demand at once. Bothsolutions are possible, but more difficult to realize, compared to holographic storage. Anotherpossibility is to stop light and release it on demand.Light stopping is a process in which light is slowed from vacuum speed at least two decades down.However, for a practical memory cell, speed needs to be reduced to 0 on demand, otherwise
• maximum storage time would be limited by memory length. Using a material with very highrefraction coefficient would also result in very high losses; therefore this is not a practical solution.Possibility is to use Rabi oscillations, a quantum phenomenon where two states are coupled throughelectric field.Rabi oscillationsSuppose we have a system with two eigenstate energies, ⟩ , ⟩ , and we shine alight with frequency of We start perturbatively, where perturbative term ofhamiltonian is [ ⟩⟨ ⟩⟨ ] (5)New wave function is a linear superposition of our ground states. | ⟩ ⟩ ⟩ (6)Inserting wave function in Schrödinger equation, we get relations between coefficients A and B: ̈ ̈ ( ) (7)Probability of excited state therefore varies with a typical frequency, called Rabi frequency. (8)Frequency can also be written in generalized form, when field frequency is not equal to energydifference, √ | | (9)Intuitively, light behaves as if the matter was periodically absorbing and then re-emitting photonsdue to stimulated emission. Each such cycle is called a Rabi cycle. [14]If we calculate eigenstates for the new Hamiltonian, we see that states split due to polarization,energy difference between new states equals . These new states are called dressed states.Therefore, matter is more transparent to the light it would be without state splitting.We can use the effect with 3 states in configuration, where 1 wave, called coupling, couples states2 and 3, while second (called probe) couples 1 and 3. As states 2 and 3 require much higher energy,they are not occupied at start. First, coupling wave is activated, but as states 2 and 3 are empty,there is no oscillation. Probe wave is then activated adiabatically. Due to probe wave, state 3 splits.Oscillations interfere, resulting in impossibility to have any particles in state 3. Therefore, absorptionis not only a superposition of 2 absorption lines, but its imaginary part does fall to 0 in the middle,while real part of the refraction index has a very large derivative.
• Image 6: Real and imaginary part of refraction index without coupling field (dashed line) and with coupling field (solid line) [13]If we deactivate coupling wave with light in material, energy is stored in state 2, meaningpropagation of light stops, till coupling wave is reactivated, when the light resumes propagation. Asthere is spontaneous state transition through emission, strength of the signal decays exponentially.Currently, lowest speed in a viable material was 80 , achieved in ruby vapour at 360K. Lower speedsand even total stop were achieved, but vapour needed to be cooled to several hundred nK, waybelow practical usage for storage. Light was also stopped only for some . Using current memoryproduction techniques and materials, light was stopped to 1/100 c. Therefore, it is possible to slow oreven stop light using this process, but it is not currently useful for memory.Out of all mentioned technologies, only holographic seems achievable in not too distant future, whilenone are actually close to commercial production. 7. ConclusionOptical computing is, at the moment, possible, but not practical. For servers, process of migrationtowards optical connections even on a smaller scale has already started, while other systemcomponents are currently not being focused on. For personal computers, situation is even lesspromising, as technology is driven by its cost, not absolute performance for any price. Estimationswhen all-optical computers could appear are currently only guesses, as technology is not ready evenfor prototype computers, let alone massive commercial production.
• 8. References[1]: www.news.cornell.edu/stories/Feb08/4waveregen.ws.html; 21.3.2010[2]: http://www.fujitsu.com/global/news/pr/archives/month/2005/20050304-01.html; 21.3.2010[3]: M. Takenaka, K. Takeda, Y. Kanema, M. Raburn, T. Miyahara, H. Uetsuka, Y. Nakano, 320Gb/soptical packet switching using all-optical signal processing by an MMI-BLD optical flip-flop, Proc. 32ndEur. Conf. Opt. Commun. (ECOC 2006), TH4.5.2, Cannes, France, 2006.[4]: http://www.research.ibm.com/photonics/publications/ecoc_tutorial_2008.pdf 21.3.2010[5]: Board-to-Board Optical Interconnection System Using Optical Slots; In-Kui Cho et al., IEEEPhotonics Technology Letters, Vol. 16, no. 7, July 2004[6]: All-Optical Computing and All-Optical Networks are Dead; Charles Beeler, El Dorado Ventures andCraig Partridge, BBN[7]: http://braungroup.beckman.illinois.edu/photonic.html, 21.3.2010[8]: Lasers go nano, Francisco J. Garcia-Vidal and Esteban Moreno, NATURE|Vol 461|1 October 2009[9]: http://www.alphagalileo.org/ViewItem.aspx?ItemId=61986&CultureCode=en 21.3.2010[10]: Holografsko shranjevanje podatkov, Blaz Kavcic, Ljubljana, November 2007[11]: http://www.inphase-technologies.com/, 21.3.2010[12]: Semiconductor based photonic memory cell, S. Zimmermann, et al., Science 283, 1292 (1999)[13]: Pocasna svetloba, Martin Strojnik, May 2008[14]: “Laser Electronics”, J. Verdeyen, 3rd ed., Chapt. 14. |
925d9e9ff53c141f | Sunday, April 2, 2017
Biphoton Inspiral
The matter-energy equivalence principle shows that the energy of a photon of light is equivalent to mass and the mass of an atom therefore increases when it absorbs light. In fact, the sun's gravity bends the path of a photon just like a the sun's gravity bends the path of a passing asteroid and so sufficiently energetic photons will attract each other and merge into matter. The Higgs boson at 125 GeV collision of two protons is consistent with the inspiral merger of two photons, a biphoton, at 125 GeV to make two hydrogen atoms along with a lot of other particles.
Just like the inspiral merger of two black holes, a photon pair inspiral merger is what makes up each particle of matter with complementary photons trapped in each others gravity wells. Thus all matter is equivalent to a bound photon pair resonance that we interpret as electrons, protons, and neutrons of matter.
Photons travel at the speed of light, c, and the photon pair emits a gravity wave as they inspiral and eventually merge into matter at an event horizon. But matter is not stable until certain photon thresholds and so the electron is the simplest photon superposition. Spinning black holes are large matter accretions that likewise involve the inspiral of photons.
The biphoton nature of matter is completely consistent with the electrons, protons, and neutrons that science observes along with the particle zoo of higher energy matter. The biphoton hydrogen exists because of the emission of a Rydberg photon at the CMB creation, where all matter condensed from the primordial cold photon vapor. The Rydberg photons of all matter exist today as the CMB and their entanglement with matter today is what we call gravity, the basic force that holds biphotons together as matter.
Charge force is then a particular resonance between the electron and proton biphoton that satisfies the quantum action of the Schrödinger equation and h/c2. The Rydberg biphoton is the archetype of the universe and forms the inner and outer forces that science calls charge and gravity. While the Rydberg photon emitted at the CMB creation is responsible for gravity, the Rydberg photon exchange is the bond between an electron and proton in hydrogen. |
7eb296506b0a2186 | Tsunami: Progress in Prediction, Disaster Prevention and
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This page was last modified on 23 July 2016, at 16:52. He had received an offer of a permanent position at the Institute of Advanced Studies at Princeton during his visit there in the spring of 1934 for giving an invited lecture. So, if you have z, a typical name people use for a complex number, having two components. When a wave passes through a gap the diffraction effect is greatest when the width of the gap is about the same size as the wavelength of the wave. Taylor GEO 442/PHY 442GeodynamicsAn advanced introduction to setting up and solving boundary value problems relevant to the solid earth sciences.
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ISBN: 0792334833
Waves of matter: Matter can also behave as a wave http://shop.goldmooreassociates.co.uk/books/supergravity-and-superstrings-a-geometric-perspective-a-geometric-perspective. Suppose you know the wavelength of light passing through a Michelson interferometer with high accuracy. Describe how you could use the interferometer to measure the length of a small piece of material. 11. A Fabry-Perot interferometer (see figure 1.20) consists of two parallel half-silvered mirrors placed a distance d from each other as shown , source: download epub. There is a connection, described in scientific quantum physics, between the quantum level and everyday level, but the connection is not what is claimed by advocates of unscientific mystical physics download epub. Point B is 3 m distant from A at 30◦ counterclockwise from the x axis. Point C is 2 m from point A at 100◦ counterclockwise from the x axis. (a) Obtain the Cartesian components of the vector D1 which goes from A to B and the vector D2 which goes from A to C. (b) Find the Cartesian components of the vector D3 which goes from B to C. (c) Find the direction and magnitude of D3. 2 ref.: portraitofacreative.com. In constructive interference, areas of high probability add to give areas of very high probability: In destructive interference, areas of high probability cancel out to give low (or zero) probability: Then, at some point, you actually measure the position of the photon portraitofacreative.com. Observable properties, such as the position of an atom or the momentum of an electron, arise from projecting the wave function onto an eigenstate. However, each projection only reveals a portion of the underlying wave function and often destroys uniquely quantum features, like superposition and entanglement. The full quantum state is only realized by statistically averaging over many measurements http://portraitofacreative.com/books/string-theory-volume-1-an-introduction-to-the-bosonic-string.
Never the less, you are commanded to compute the following quantity. This quantity is also by definition what we call the expectation value of the Hamiltonian in the state psi. I love the fact, the remarkable fact that we're going to show now, is that this thing provides an upper bound for the ground state energy for all psi. So let me try to make sure we understand what's happening here , e.g. http://portraitofacreative.com/books/fundamentals-of-electronic-circuit-design-1-st-first-edition. He believes quantum theory is incomplete but dislikes pilot-wave theory. Many working quantum physicists question the value of rebuilding their highly successful Standard Model from scratch. “I think the experiments are very clever and mind-expanding,” said Frank Wilczek, a professor of physics at MIT and a Nobel laureate, “but they take you only a few steps along what would have to be a very long road, going from a hypothetical classical underlying theory to the successful use of quantum mechanics as we know it.” “This really is a very striking and visible manifestation of the pilot-wave phenomenon,” Lloyd said. “It’s mind-blowing — but it’s not going to replace actual quantum mechanics anytime soon.” In its current, immature state, the pilot-wave formulation of quantum mechanics only describes simple interactions between matter and electromagnetic fields, according to David Wallace, a philosopher of physics at the University of Oxford in England, and cannot even capture the physics of an ordinary light bulb. “It is not by itself capable of representing very much physics,” Wallace said. “In my own view, this is the most severe problem for the theory, though, to be fair, it remains an active research area.” Pilot-wave theory has the reputation of being more cumbersome than standard quantum mechanics lucassnell.com.
Whenever we make a measurement on a Quantum system, the results are dictated by the wavefunction at the time at which the measurement is made. It turns out that for each possible quantity we might want to measure (an observable) there is a set of special wavefunctions (known as eigenfunctions) which will always return the same value (an eigenvalue) for the observable. e.g.... pdf. Take, for instance, the infamous “collapse of the wave function,” wherein the quantum system inexplicably transitions from multiple simultaneous states to a single actuality pdf. However, his innovative ideas were often misunderstood and he was frequently ridiculed for his vocal involvement in politics and social issues. The birth of the Manhattan Project yielded an inexorable connection between Einstein's name and the atomic age. However, Einstein did not take part in any of the atomic research, instead preferring to concentrate on ways that the use of bombs might be avoided in the future, such as the formation of a world government ref.: read for free. How broad is it when it reaches the moon, which is 4 × 105 km away? Assume the wavelength of the light to be 5 × 10−7 m. Figure 2.21: Graphical representation of the dispersion relation for shallow water waves in a river flowing in the x direction ref.: http://makeavatars.net/?library/pluralism-and-the-idea-of-the-republic-in-france. We also should avoid the reverse mistake, of simplistically extrapolating from small-scale to large-scale by assuming, as in mystical physics, that quantum descriptions of small-scale events (involving electrons,...) can be applied to large-scale events. You must avoid both mistakes, especially the second mistake — because what happens on a small-scale is not the same as what happens on a large-scale — if you want to understand why "things are strange, but not as strange as some people say they are." Here’s the square well at its most basic: This is a one-dimensional well, so you’re concerned only with the x direction; therefore, the Schrödinger equation looks like this: The wave function looks like this: where A and B are constants. Submicroscopic harmonic oscillators are popular quantum physics problems because harmonic oscillators are relatively simple systems — the force that keeps a particle bound here is proportional to the distance that the particle is from the equilibrium point , e.g. http://portraitofacreative.com/books/the-bethe-wavefunction.
How does the dispersion relation for relativistic waves simplify if the rest frequency (and hence the particle mass) is zero http://portraitofacreative.com/books/fields-and-waves-in-communication-electronics-3-rd-ed? Presents classical thermodynamics, which derives relations between various quantities, and statistical methods used to derive classical thermodynamics from the atomic point of view. Presents Brownian motion, random walks, and fluctuation. Gives applications of the second law to the production and uses of energy read epub. Furthermore, since the quantum jump is random, no signal or other causal effect is superluminally transmitted. On the other hand, a deterministic theory based on subquantum forces or hidden variables is necessarily superluminal. Thus quantum mechanics, as conventionally practiced, describes quantum leaps without too drastic a quantum leap beyond common sense. Certainly no mystical assertions are justified by any observations concerning quantum processes http://weatherfor.net/library/guide-to-international-recommendations-on-names-and-symbols-for-quantities-and-on-units-of. In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of a quantum system changes with time. It was formulated in late 1925, and published in 1926, by the Austrian physicist Erwin Schrödinger. [1] In classical mechanics, Newton's second law (F = ma) is used to make a mathematical prediction as to what path a given system will take following a set of known initial conditions http://lucassnell.com/ebooks/the-scattering-and-diffraction-of-waves-harvard-monographs-in-applied-science. Meets the general education “writing intensive” requirement. In a transverse wave, the motion of the particles is _____ the wave's direction of propagation. In a longitudinal wave, the motion of the particles is _____ the wave's direction of propagation. A sound wave is an example of a _____ wave. Wave speed is _____ the period of a wave. A wave has a speed of 10 m/s and a frequency of 100 Hz amazonie-decouverte.com. Links to html pages are allowed, and the viewing in classroom http://portraitofacreative.com/books/non-linear-waves-in-dispersive-media-international-series-of-monographs-in-natural-philosophy. But here are some reasons to reject a claim that we are powerful: The human action is limited to arranging a situation in which a physical interaction causes the attribute to manifest, and this occurs due to physical interaction interaction-during-observationthat is described by the Uncertainty Principle) or human consciousness , cited: http://portraitofacreative.com/books/singularities-and-oscillations-the-ima-volumes-in-mathematics-and-its-applications. Quark model: model in which all particles that interact via the strong interaction are composed of two or three quarks. Radiation: electromagnetic waves that carry energy. Radioactive decay: spontaneous change of unstable nuclei into other nuclei. Radioactive materials: materials that undergo radioactive decay , source: portraitofacreative.com. This is true for any orbital in any sublevel of any main energy level www.louis-adams.com. Here it is: Quantum states are represented by wave functions, which are vectors in a mathematical space called Hilbert space. Wave functions evolve in time according to the Schrödinger equation. Quite a bit simpler — and the two postulates are exactly the same as the first two of the textbook approach. Everett, in other words, is claiming that all the weird stuff about “measurement” and “wave function collapse” in the conventional way of thinking about quantum mechanics isn’t something we need to add on; it comes out automatically from the formalism http://conversion-attribution.de/books/oscillations-in-finite-quantum-systems-cambridge-monographs-on-mathematical-physics.
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8c80578c0d8ed25f | Wiener process
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A single realization of a one-dimensional Wiener process
A single realization of a three-dimensional Wiener process
In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion process or Brownian motion due to its historical connection with the physical process known as Brownian movement or Brownian motion originally observed by Robert Brown. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics, quantitative finance, and physics.
The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time martingales. It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in stochastic calculus, diffusion processes and even potential theory. It is the driving process of Schramm–Loewner evolution. In applied mathematics, the Wiener process is used to represent the integral of a white noise Gaussian process, and so is useful as a model of noise in electronics engineering (see Brownian noise), instrument errors in filtering theory and unknown forces in control theory.
The Wiener process has applications throughout the mathematical sciences. In physics it is used to study Brownian motion, the diffusion of minute particles suspended in fluid, and other types of diffusion via the Fokker–Planck and Langevin equations. It also forms the basis for the rigorous path integral formulation of quantum mechanics (by the Feynman–Kac formula, a solution to the Schrödinger equation can be represented in terms of the Wiener process) and the study of eternal inflation in physical cosmology. It is also prominent in the mathematical theory of finance, in particular the Black–Scholes option pricing model.
Characterisations of the Wiener process[edit]
The Wiener process is characterised by the following properties:[1]
1. a.s.
2. has independent increments: for every the future increments , are independent of the past values ,
3. has Gaussian increments: is normally distributed with mean and variance ,
4. has continuous paths: With probability , is continuous in .
The independent increments means that if 0 ≤ s1 < t1s2 < t2 then Wt1Ws1 and Wt2Ws2 are independent random variables, and the similar condition holds for n increments.
An alternative characterisation of the Wiener process is the so-called Lévy characterisation that says that the Wiener process is an almost surely continuous martingale with W0 = 0 and quadratic variation [Wt, Wt] = t (which means that Wt2t is also a martingale).
A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent N(0, 1) random variables. This representation can be obtained using the Karhunen–Loève theorem.
Another characterisation of a Wiener process is the Definite integral (from zero to time t) of a zero mean, unit variance, delta correlated ("white") Gaussian process.[citation needed]
The Wiener process can be constructed as the scaling limit of a random walk, or other discrete-time stochastic processes with stationary independent increments. This is known as Donsker's theorem. Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed neighborhood of the origin infinitely often) whereas it is not recurrent in dimensions three and higher[citation needed]. Unlike the random walk, it is scale invariant, meaning that
is a Wiener process for any nonzero constant α. The Wiener measure is the probability law on the space of continuous functions g, with g(0) = 0, induced by the Wiener process. An integral based on Wiener measure may be called a Wiener integral.
Wiener process as a limit of random walk[edit]
Let be i.i.d. random variables with mean 0 and variance 1. For each n, define a continuous time stochastic process
This is a random step function. Increments of are independent because the are independent. For large n, is close to by the central limit theorem. Donsker's theorem proved that as , approaches a Wiener process, which explains the ubiquity of Brownian.[2]
Properties of a one-dimensional Wiener process[edit]
Basic properties[edit]
The unconditional probability density function, which follows normal distribution with mean = 0 and variance = t, at a fixed time t:
The expectation is zero:
The variance, using the computational formula, is t:
Covariance and correlation[edit]
The covariance and correlation:
The results for the expectation and variance follow immediately from the definition that increments have a normal distribution, centered at zero. Thus
The results for the covariance and correlation follow from the definition that non-overlapping increments are independent, of which only the property that they are uncorrelated is used. Suppose that t1 < t2.
we arrive at:
Since W(t1) = W(t1) − W(t0) and W(t2) − W(t1), are independent,
Wiener representation[edit]
Wiener (1923) also gave a representation of a Brownian path in terms of a random Fourier series. If are independent Gaussian variables with mean zero and variance one, then
represent a Brownian motion on . The scaled process
is a Brownian motion on (cf. Karhunen–Loève theorem).
Running maximum[edit]
The joint distribution of the running maximum
and Wt is
To get the unconditional distribution of , integrate over −∞ < wm :
And the expectation[3]
If in the Wiener process has a known value , it is possible to calculate the conditional probability distribution of the maximum in interval (cf. Probability distribution of extreme points of a Wiener stochastic process).
A demonstration of Brownian scaling, showing for decreasing c. Note that the average features of the function do not change while zooming in, and note that it zooms in quadratically faster horizontally than vertically.
Brownian scaling[edit]
For every c > 0 the process is another Wiener process.
Time reversal[edit]
The process for 0 ≤ t ≤ 1 is distributed like Wt for 0 ≤ t ≤ 1.
Time inversion[edit]
The process is another Wiener process.
A class of Brownian martingales[edit]
If a polynomial p(x, t) satisfies the PDE
then the stochastic process
is a martingale.
Example: is a martingale, which shows that the quadratic variation of W on [0, t] is equal to t. It follows that the expected time of first exit of W from (−c, c) is equal to c2.
More generally, for every polynomial p(x, t) the following stochastic process is a martingale:
where a is the polynomial
Example: the process
is a martingale, which shows that the quadratic variation of the martingale on [0, t] is equal to
About functions p(xa, t) more general than polynomials, see local martingales.
Some properties of sample paths[edit]
The set of all functions w with these properties is of full Wiener measure. That is, a path (sample function) of the Wiener process has all these properties almost surely.
Qualitative properties[edit]
• For every ε > 0, the function w takes both (strictly) positive and (strictly) negative values on (0, ε).
• The function w is continuous everywhere but differentiable nowhere (like the Weierstrass function).
• Points of local maximum of the function w are a dense countable set; the maximum values are pairwise different; each local maximum is sharp in the following sense: if w has a local maximum at t then
The same holds for local minima.
• The function w has no points of local increase, that is, no t > 0 satisfies the following for some ε in (0, t): first, w(s) ≤ w(t) for all s in (t − ε, t), and second, w(s) ≥ w(t) for all s in (t, t + ε). (Local increase is a weaker condition than that w is increasing on (t − ε, t + ε).) The same holds for local decrease.
• The function w is of unbounded variation on every interval.
• The quadratic variation of w over [0,t] is t.
• Zeros of the function w are a nowhere dense perfect set of Lebesgue measure 0 and Hausdorff dimension 1/2 (therefore, uncountable).
Quantitative properties[edit]
Law of the iterated logarithm[edit]
Modulus of continuity[edit]
Local modulus of continuity:
Global modulus of continuity (Lévy):
Local time[edit]
The image of the Lebesgue measure on [0, t] under the map w (the pushforward measure) has a density Lt(·). Thus,
for a wide class of functions f (namely: all continuous functions; all locally integrable functions; all non-negative measurable functions). The density Lt is (more exactly, can and will be chosen to be) continuous. The number Lt(x) is called the local time at x of w on [0, t]. It is strictly positive for all x of the interval (a, b) where a and b are the least and the greatest value of w on [0, t], respectively. (For x outside this interval the local time evidently vanishes.) Treated as a function of two variables x and t, the local time is still continuous. Treated as a function of t (while x is fixed), the local time is a singular function corresponding to a nonatomic measure on the set of zeros of w.
These continuity properties are fairly non-trivial. Consider that the local time can also be defined (as the density of the pushforward measure) for a smooth function. Then, however, the density is discontinuous, unless the given function is monotone. In other words, there is a conflict between good behavior of a function and good behavior of its local time. In this sense, the continuity of the local time of the Wiener process is another manifestation of non-smoothness of the trajectory.
Related processes[edit]
Wiener processes with drift (blue) and without drift (red).
2D Wiener processes with drift (blue) and without drift (red).
The generator of a Brownian motion is ½ times the Laplace–Beltrami operator. The image above is of the Brownian motion on a special manifold: the surface of a sphere.
The stochastic process defined by
is called a Wiener process with drift μ and infinitesimal variance σ2. These processes exhaust continuous Lévy processes.
Two random processes on the time interval [0, 1] appear, roughly speaking, when conditioning the Wiener process to vanish on both ends of [0,1]. With no further conditioning, the process takes both positive and negative values on [0, 1] and is called Brownian bridge. Conditioned also to stay positive on (0, 1), the process is called Brownian excursion.[4] In both cases a rigorous treatment involves a limiting procedure, since the formula P(A|B) = P(AB)/P(B) does not apply when P(B) = 0.
A geometric Brownian motion can be written
It is a stochastic process which is used to model processes that can never take on negative values, such as the value of stocks.
The stochastic process
is distributed like the Ornstein–Uhlenbeck process.
The time of hitting a single point x > 0 by the Wiener process is a random variable with the Lévy distribution. The family of these random variables (indexed by all positive numbers x) is a left-continuous modification of a Lévy process. The right-continuous modification of this process is given by times of first exit from closed intervals [0, x].
The local time L = (Lxt)xR, t ≥ 0 of a Brownian motion describes the time that the process spends at the point x. Formally
where δ is the Dirac delta function. The behaviour of the local time is characterised by Ray–Knight theorems.
Brownian martingales[edit]
Let A be an event related to the Wiener process (more formally: a set, measurable with respect to the Wiener measure, in the space of functions), and Xt the conditional probability of A given the Wiener process on the time interval [0, t] (more formally: the Wiener measure of the set of trajectories whose concatenation with the given partial trajectory on [0, t] belongs to A). Then the process Xt is a continuous martingale. Its martingale property follows immediately from the definitions, but its continuity is a very special fact – a special case of a general theorem stating that all Brownian martingales are continuous. A Brownian martingale is, by definition, a martingale adapted to the Brownian filtration; and the Brownian filtration is, by definition, the filtration generated by the Wiener process.
Integrated Brownian motion[edit]
The time-integral of the Wiener process
is called integrated Brownian motion or integrated Wiener process. It arises in many applications and can be shown to have the distribution N(0, t3/3)[5], calculus lead using the fact that the covariance of the Wiener process is .[6]
Time change[edit]
Every continuous martingale (starting at the origin) is a time changed Wiener process.
Example: 2Wt = V(4t) where V is another Wiener process (different from W but distributed like W).
Example. where and V is another Wiener process.
In general, if M is a continuous martingale then where A(t) is the quadratic variation of M on [0, t], and V is a Wiener process.
Corollary. (See also Doob's martingale convergence theorems) Let Mt be a continuous martingale, and
Then only the following two cases are possible:
other cases (such as etc.) are of probability 0.
Especially, a nonnegative continuous martingale has a finite limit (as t → ∞) almost surely.
All stated (in this subsection) for martingales holds also for local martingales.
Change of measure[edit]
A wide class of continuous semimartingales (especially, of diffusion processes) is related to the Wiener process via a combination of time change and change of measure.
Using this fact, the qualitative properties stated above for the Wiener process can be generalized to a wide class of continuous semimartingales.[7][8]
Complex-valued Wiener process[edit]
The complex-valued Wiener process may be defined as a complex-valued random process of the form Zt = Xt + iYt where Xt, Yt are independent Wiener processes (real-valued).[9]
Brownian scaling, time reversal, time inversion: the same as in the real-valued case.
Rotation invariance: for every complex number c such that |c| = 1 the process cZt is another complex-valued Wiener process.
Time change[edit]
If f is an entire function then the process is a time-changed complex-valued Wiener process.
Example: where
and U is another complex-valued Wiener process.
In contrast to the real-valued case, a complex-valued martingale is generally not a time-changed complex-valued Wiener process. For example, the martingale 2Xt + iYt is not (here Xt, Yt are independent Wiener processes, as before).
See also[edit]
1. ^ Durrett 1996, Sect. 7.1
2. ^ Steven Lalley, Mathematical Finance 345 Lecture 5: Brownian Motion (2001)
3. ^ Shreve, Steven E (2008). Stochastic Calculus for Finance II: Continuous Time Models. Springer. p. 114. ISBN 978-0-387-40101-0.
4. ^ Vervaat, W. (1979). "A relation between Brownian bridge and Brownian excursion". Annals of Probability. 7 (1): 143–149. JSTOR 2242845. doi:10.1214/aop/1176995155.
5. ^ "Interview Questions VII: Integrated Brownian Motion – Quantopia". Retrieved 2017-05-14.
6. ^ Forum, "Variance of integrated Wiener process", 2009.
7. ^ Revuz, D., & Yor, M. (1999). Continuous martingales and Brownian motion (Vol. 293). Springer.
8. ^ Doob, J. L. (1953). Stochastic processes (Vol. 101). Wiley: New York.
9. ^ Navarro-moreno, J.; Estudillo-martinez, M.D; Fernandez-alcala, R.M.; Ruiz-molina, J.C. (2009), "Estimation of Improper Complex-Valued Random Signals in Colored Noise by Using the Hilbert Space Theory" (PDF), IEEE Transactions on Information Theory, 55 (6): 2859–2867, doi:10.1109/TIT.2009.2018329, retrieved 2010-03-30
• Kleinert, Hagen (2004). Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets (4th ed.). Singapore: World Scientific. ISBN 981-238-107-4. (also available online: PDF-files)
• Stark, Henry; Woods, John (2002). Probability and Random Processes with Applications to Signal Processing (3rd ed.). New Jersey: Prentice Hall. ISBN 0-13-020071-9.
• Durrett, R. (2000). Probability: theory and examples (4th ed.). Cambridge University Press. ISBN 0-521-76539-0.
• Revuz, Daniel; Yor, Marc (1994). Continuous martingales and Brownian motion (Second ed.). Springer-Verlag.
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