Is quantum Hall current density local? (${\bf j}({\bf r}) = \sigma_H {\bf \hat n \times E}({\bf r}) $)

The good old Ohm's law $${\bf j}({\bf r}) = \sigma_O {\bf E}({\bf r})$$ if translated into words would be "the local current density is proportional to a local electric field." In a quantum Hall state (QHS), the linear response to an external electric field is given by the following $${\bf j} = \sigma_H \bf \hat n \times E$$ where $\bf \hat n$ is the surface normal. _**Is it also a local relation between applied field ${\bf E}({\bf r})$ and response current ${\bf j}({\bf r})$?_** If the electric field is non-zero inside the bulk, it means there must be current propagating inside the gapped bulk. I tend to understand it in an intuitive way. As you see in the picture, for a uniform electric field in $x$-direction, the chemical potential is raised up linearly in $x$ and it intersects levels in the bulk. The current in the bulk comes from the chiral edge modes and sink into edge back again. I don't know whether my picture is correct or not. ![As you see, for a uniform electric field in $x$-direction, the chemical potential is raised up linearly in $x$ and it intersects levels in the bulk.](http://i.stack.imgur.com/2edPK.png)


Determination of displacement value in work formula

A net force will always create displacement which approaches infinity if left undisturbed. So how to fix a value of displacement in work formula? Isn't a mention time interval necessary? Or do we measure displacement only while force is acting on the body i.e the moment we stop applying force we also stop measuring displacement.


How do we determine the temperature of a Black Hole? Since we cannot see a Black Hole, which I presume, is because it absorbs light, would it not also prevent radiation from escaping, making temperature determination difficult.


Entropy and Stability

What happens when you increase entropy in a system? Does it increase the instability? Is disorder the same as entropy?


Jordan Wigner Transformation in 1d Majorana chain

So, I was reading the paper by Fidkowski and Kitaev on 1d fermionic phase http://arxiv.org/abs/1008.4138. It explains the classification of 1d fermionic SPT phases with $\mathbb{Z}_2^T$ symmetry for both the interacting and non- interacting cases. I understand why we had a $\mathbb{Z}$ classification in the non-interacting case. Basically, it measures the number of dangling majorana modes when one cuts a TR-invariant majorana chain (see section II of the paper). But then, for the interacting case, apparently, in order to define the interacting invariants, and construct the matrix product states, we need to do a Jordan- Wigner transformation to a bosonic spin chain, so that the original hamiltonian is written in terms of the new bosonic variables, $\sigma_j^x$, $\sigma_j^y$,$\sigma_j^z$ (end of section II). I don't understand why do we need to do this transformation and what good does it do to get the $\mathbb{Z}_8$ classification in the interacting case. I don't see the connection between this transformation and the way he constructed the matrix product states in section III.


Is every QFT non-local in the U.V.?

As much as I understand the renormalization group transformation and the concept of relevant/irrelevant operators, I'd say that if we push the reasoning of only looking at relevant operators when we follow a "renormalization group orbit" (a curve in the space of the parameters) to the infra-red regime the other way, so going to the U.V., we should keep all the operators this time (as they are no more suppressed by huge factors), or there is a infinite number of them, depending on an infinite number of the derivatives, and we end up with a non-local theory. As this reasoning can be applied to every QFT, does this mean that they are all non-local in the U.V.? or the reasoning fooled somewhere (breakdown of the QFT framework somewhere for example...)?


Time dilation at the speed of light

Does a massless particle travelling at the speed on light in a vacuum (c) experience an infinite time dilation effect? i.e. Would the time dilation extend from the perspective of the massless particle past the measurable existence of the particle by a observer outside the dilation.


Imagine the following gedankenexperiment. Observer Alice is right here on Earth. Observer Bob is at say Alpha Centauri. A pair of maximally entangled qubits is formed with one qubit handed over to Observer Alice and the other sealed in a box on Pluto shielded from decoherence. It takes a few years for any signal to reach Observer Bob. Observer Alice measures her qubit and "collapses" it. To her, the pair of qubits is no longer entangled. However, for a few years, will Observer Bob think there is an entanglement? We have to be careful here. Once the qubit on Earth has been measured by Earthlings, the entanglement is "shared". By the well known monogamy of entanglement theorem, any correlation between both qubits by themselves will have become classical. However, Observer Bob, who has a very good working knowledge of quantum mechanics and relativity, knows that for a few hours, until light signals can reach Pluto, the system consisting of an expanding bubble around Earth and the Pluto qubit are in an entangled state. Alice, who lives inside this bubble, will beg to differ. Is quantum entanglement an objective or subjective property? Closely related to Wigner's friend and intersubjectivity in quantum mechanics and Given entanglement, why is it permissible to consider the quantum state of subsystems?.


What is the name of the equation which led to the Schrödinger one?

What is the name of this equation: $$\frac {d^2\psi}{dr^2}+k^2\psi=0?$$ (I want a Wikipedia link for this equation, but I don't know what its name is.) * * * Point: In this equation, the wave function is only function of space $\psi (r)$, for example $\psi=Ae^{\mathrm i kr}$.


Varying the amplitude of a driven wave

I' d like to know whether varying the amplitude of a system at resonance is possible or not and if it is, how? I've calculated the resonance frequency of a material and I'd like to know the displacement this frequency generates and whether I can control it by knowing the relationship between the amplitude (displacement) at resonance and power of driving source. It would make sense that increasing the power would increase the amplitude/displacement of the oscillation, but I need formulas. Thanks!


Why is this classical system seemingly non-deterministic?

So, I was solving this problem1 and I realized that the system given seems to be non-deterministic when analysed with classical laws. ![enter image description here](http://i.stack.imgur.com/Soh81.png) The situation is thus: A rope is wrapped for an angle of $\theta_0$ (symmetrically) around a fixed pole (the circle in the diagram). One end is fixed to a wall, and the other end is pulled with a force/tension $T_0$. The string is massless, but there is friction between the string and the pole of coefficient $\mu$. There is no slipping (the system is static). If necessary, the length of the string and the radius/relative position of the pole can be considered as givens. Now, if you follow my solution2, one gets that the tension in the string along the cylinder, where $\theta=0$ is the part on the right where the string just touches follows the following constraints: $T_0e^{-\mu\theta}\leq T(\theta)\leq T_0e^{\mu\theta}$. Extending this, you get that the tension $T_1$ in the part of the string connected to the wall follows the constraint $T_0e^{-\mu\theta_0}\leq T_1\leq T_0e^{\mu\theta_0}$. However, I don't see any way to _further_ analyse this system with Newtonian mechanics. Note that the inequality appears even if I consider a string with mass, though the calculation becomes more complicated. So basically I'm stuck with an inequality. Note that the system, as given, should be completely defined -- $\theta_0$, $T$,$\mu$, the radius of pole, and the relative positioning of pole are all that should be required to physically create this system. However, the macroscopic, measurable quantity $T_1$ somehow cannot be determined from Newton's laws. How does one resolve this apparent paradox? Classical systems are supposed to be able to fully determine measurable quantities. This doesn't seem to be the case here. 1\. This was a course on Classical (Lagrangian & Hamiltonian) mechanics, and our professor gave us some Newtonian mechanics problems as a first assignment. We're not yet sure why, but most were quite interesting. The assignment due date is over, this is just a residual point that has been bothering me. 2\. This is lifted from the document I submitted. I have blurred out the end because it contains my own resolution of the apparent paradox here. I may post it as an answer later, but for now I'd prefer to give everyone a fresh look at this situation.


The statistical nature of the 2nd Law of Thermodynamics

Ok, so entropy increases... This is supposed to be an absolute statement about entropy. But then someone imagines a box with a 10 particle gas, and finds that every now and then all particles are in the left. Conclusion, the 2nd law holds only in a statistical sense. But then Szilard comes up with a thought experiment with only one particle and a piston that can compress to the left or right. The apparent loss of entropy when finding the particle in the left half is compensated by the very bit of information that indicates where the particle is. So perhaps the 2nd Law actually holds in an absolute sense, except that... > Is there consensus on the absolute vs statistical nature of the 2nd Law, or > is it subject to interpretation? Can the issue be settled within a classical > setting, or does one have to go quantum? **Addendum:** (as per Ben Crowell's request) Here is the paper Szilard, L., 1929, _“On the Decrease of Entropy in a Thermodynamic System by the Intervention of Intelligent Beings”_ , Zeitschrift fur Physik 53: 840–856. English translation in The Collected Works of Leo Szilard: Scientific Papers, B.T. Feld and G. Weiss Szilard (eds.), Cambridge, Massachusetts: MIT Press, 1972, pp. 103–129.


While finding the equivalent resistance why isn't the 8 ohm in series with the 5 ohm?

Is there any way to tell if the circuit below is in series or parallel quickly and accurately? ![refer to image](http://i.stack.imgur.com/uzR8Q.jpg)


Modeling viscous fluid flow in the curved pipe

I want to know if there is any simple (results don't have to be very precise) way of modeling viscous fluid flow throught the curved pipe with elastic walls. I need to gather preassure values and compare them to preassure of the fluid in the straight pipe with the same wall parameters.


Do eigenvectors of quantum operators span the whole Hilbert Space?

I am trying to solve an exercise in Shankar's QM book (concretely 4.2.1), and I am asked the probability of each possible value for the operator $L_x$ when the particle is in a certain eigenstate of another operator $L_z$. I understand that I have to "expand" the state vector in the $L_z$ basis, which in this case is just the appropriate eigenvector, and then change the base to that of $L_x$, i.e to write the $L_z$ eigenstate in the $L_x$ base. The problem is that there is just no way of doing it, since the eigenvectors of $L_x$ don't span the whole Hilbert Space. I thought all quantum operators had a set of eigenvectors that spanned the whole space, I am wrong on that or am I wrong on my algebra?


How to calculate Standing Waves in Electrical Cables?

I have a 20 metre Coaxial Cable. I send digital signals down the cable ranging from 5 KHz to 50 KHz. I have noticed a pattern in the noise ratio, an oscillating wave. I predict this is to do with standing waves in the cable. How can I calculate at what Frequencies these standing waves would occur in the cable?


Is there a chance for Earth to have an additional satellite someday?

I am just an space enthusiast and not a physics professional so pardon me if this is not the right place to ask.   As title says, is there a chance for our Earth to have additional satellite(s) like our Moon? I tried to google it, without luck, however I know here there are very smart people that can perhaps give me some references to read or can give me an answer :)   Thanks


Why is a suspended spinning body of liquid prolate?

Instead of a flat disc one would expect the centrifugal forces to push it in.. The body is not in a container but suspended in air spinning horizontally (i.e. left to right or vice-versa) - how to do this could be challenging - but that is not the issue. **Answer:** It is not prolate: http://www.youtube.com/watch?v=BxyfiBGCwhQ.


Hydrogen atom: potential well and orbit radii

I happened to open up an old solid-state electronics book by Sah, and in it he says: "it is evident that the electron orbit radius is half the well radius at the energy level En" The orbit radius is $r_n=\frac{4\pi\epsilon_0 ℏ^2 n^2}{mq^2}$ and the potential well $V(r_n)=\frac{−q^4m}{(4\pi\epsilon_0)^2ℏ^2n^2}$ Of course the orbit has to be confined in the well, but it's not obvious to me why it should be exactly half the well radius? This isn't something I recall seeing before either in any other text. Thanks


Physical interpretation related to a non-linear partial differential equation

I am doctoral student in pure mathematics working on a particular problem. My question is if this problem has applications to real world phenomena. I will try to explain the direct problem starting from a simple case with known applications, and then generalising it to a nonlinear (actually, quasilinear) case, and then finally explain the inverse problem. I admit to not having studied physics at university level. Hence, the notation and language I use might be exotic, for which I apologise. **Conductivity equation** Suppose we have a conducting body, $\Omega$, with conductivity (or conductance) $\sigma$. I will be using terminology from electricity, but this might also be heat conductance. Conductivity is the inverse of resistance, which I will not write more about. We take the material to be isotropic, which corresponds to conductivity with scalar (real) values. By (differential) Ohm's law the current J is proportional to conductivity and electric field, which we write as the gradient of potential v: $$ J = -\sigma \nabla v$$ By Kirchhoff's law the current is solenoidal, or divergence-free: $\nabla \cdot J = 0$. All in all, we have the conductivity equation $$-\nabla \cdot \sigma \nabla v = 0$$ If the conductivity is constant, say one, this is simply the Laplace equation $-\nabla^2 v = -\Delta v = 0$. **p-Laplace equation** We consider a nonlinear Ohm's law: $$ J = -\sigma |\nabla v|^{p-2} \nabla v $$ with finite parameter p > 1. The case p = 2 is the standard Ohm's law, and also the standard conductivity equation. Combining this with Kirchhoff's law we get the p-Laplace equation (with weight $\sigma$): $$-\nabla \cdot (\sigma |\nabla v|^{p-2} \nabla v) = 0$$ **Calderón's problem** The inverse problem of Calderón asks one to reconstruct the conductivity within the body by making boundary measurements: One prescribes either the current or the voltage on the boundary and measures the other one. These correspond to Neumann and Dirichlet boundary values for the equation - one sets the other and measures the other. This is a steady-state measurement, so one should wait for the current pattern to stabilise before making measurements. For the conductivity equation (p = 2 case) this problem has been studied, there are theoretical and computation results, and several applications, such as electrical impedance tomography which can be used to detect breast cancer, discover oil or image cracks in concrete. **Questions** Are there applications for the Calderón problem in the more general case where we need not have p = 2? That is, is there some medium on which the Ohm's law has the extra term $|\nabla v|^{m}$ for some non-zero m, and where finding out the conductivity by non-invasive boundary measurements would be useful? Is there some other nonlinear Ohm's law that is more realistic in situations where electrical impedance tomography or similar imaging methods would be useful?


How does the construction of a hydrophone differ from a microphone?

Both devices serve the same function, but a hydrophone must do it in a denser medium. What's the major difference in design? (besides being water-proof!) I'm asking about the relevant physics driving design in this case. I think a physicist would be able to explain this so that a good mechanical engineer would be able to design microphones for use alternately in air, underwater, submerged in honey, or at 20,000 feet.


Generalized Unitarity cut of Scalar One-loop Box integral

How does one perform the integrals in four particle cuts in generalized unitarity? It would be helpful how one finds solutions to the simplest case, the fully determined box integral given by: $\int_{\infty}^{\infty} \frac{d^4k}{(2 \pi)^4} \delta(k^2) \delta((k+p1)^2)\delta((k+p1+p2)^2)\delta((k+p1+p2+p3)^2)$ Where the k and $p_i$ are 4-dimensional complex momenta, and the external $p_i$ are massless and on shell i.e., $p_i^2=0$. I know there exist two solutions to the problem, and it would be helpful to be expressed in terms of mandelstam variables s,t,u I'm sure you can take a jacobian of each delta argument expanded explicitly in terms of (k0,k1,...) but I feel there should be an easier solution


What is the Quantum equivalent of chaos on a classical system? (if there's any)

This is a question that bugging me around for some time now. It is not clear to me what is the meaning of a chaos if we consider a quantum system. What is the mathematical formalism (or the quantum analogy) to a orbit that is ergodic in the phase space? Can we still talk of a wave-function describing the particle? I mean, is it still well defined?


Difference between a central potential that is a point and one that is a sphere?

In quantum physics there is a special case known as a particle in a spherically symmetric potential. I have a problem which is similar to the case of a hydrogen atom in that there is one free electron, but the "nucleus" is actually a group of several atoms. So I'm not sure how to find the radial wavefunction for this situation? If I could say that it's a point charge I could use the same radial wavefunction as for the hydrogen atom (I think). But since the group of atoms takes up space, how does that affect the potential/wavefunction?


Moving the plates of a charged capacitor to calculate energy density - where's the flaw in my argument?

For a charged air-dielectric capacitor, let the plates be parallel to the $xy$ plane, with the top carrying a positive charge $+Q$, the bottom a charge $-Q$. The force on an infinitesimal charge contained in a volume $dv$ inside the top plate is $$-\vec E_z\rho(x,y,z)dv$$ Integrating this over the volume of the plate gives the total force on its center of mass as $$-\vec E_z Q$$ which becomes using Gauss's law $$-\hat n_z E^2A\epsilon_0$$ If the plates are moved apart a distance $dz$ by an opposing force $\vec F$, the infinitesimal work done is $E^2\epsilon_0Adz$, and integrating this from zero to their final separation $d$ gives the energy density as $E^2\epsilon_0$. But this is different to the correct value of $1/2E^2\epsilon_0$, so where's the flaw in the above argument?


Compressed Spring Dissolving in Acid

This is just an interesting question a friend's uncle asked me that I was somewhat annoyed I couldn't answer. When a material dissolves in acid there is a chemical process that causes the changes in state/composition of the materials in question. There will be a change in energy due to chemical bonds forming/breaking and potentially a change in temperature. So we take a spring and compress it in a clamp. Obviously now there is a potential energy based on the compression of the spring as well as the chemical energy within the spring itself before compression. This is now placed into a bath of acid that will dissolve the spring but _not_ the clamp. There is a change of energy based on the dissolving of the spring, but **what happens to the potential energy due to it being compressed?** My thoughts were that in a realistic situation one (weak) point on the spring would be attacked, break, it would then probably just ping out of the clamp and lose the energy that way. But in the ideal scenario where is dissolves at a uniform rate, this wouldn't happen, so what happens at the instant that it is completely dissolved? This is obviously a massive assumption, so would be unlikely to be true. I thought maybe some kind of change of temperature would occur, due to the assumption that the compression would cause the atoms to be more tightly compacted causing a repulsion. As you remove layers this would gradually decrease, but there is still an inherent increase in energy for each atom, so this would most likely be expelled as something (like) heat. What are people's thoughts on this? **EDIT:** I'm mainly interested in the detail of the mechanisms for the energy tranfers on an atomic level as opposed to the broad picture. Specifically the transfer from coulomb repulsion to heat in the acid. The more low-level detail the better.


The conservation of energy when a spring is placed in acid

Stretch a spring out and give it a bunch of potential energy. Lower the spring into a vat of acid, disolving the spring into nothingness. Where did the potential energy go?


How is a standard unit divided into equally smaller or fractional units physically/experimentally?

Consider the standard unit of length: meter. How was it divided into decimeter, centimeter, millimeter, etc. when there were no shorter lengths than the standard? What is the physical/experimental process involved?


Circuit with inductor and lightbulb when switch is turned off?

I want to know what happens with a lightbulb immediately after a switch is turned off. We're talking about a circuit with a battery (AC), an inductor, a lightbulb and a switch. I need to know it in case the inductor and lightbulb are in serie and in parallel. I know you can see the lightbulb as a resistor but i'm kinda lost on how to calculate this. Help would be much appreciated! I also need to draw the current in function of the time. Will this be a wave?


What is the difference between the words transparent and translucent?

Merriam Webster defines **_transparent_** as: > Having the property of transmitting light without appreciable scattering so > that bodies lying beyond are seen clearly. And **_translucent_** as: > Transmitting and diffusing light so that objects beyond cannot be seen > clearly. Now if you see any object through a lens or a bottle filled with water then most of the times whatever you see is not at all clear. What does this signify? Are these two transparent or translucent? Or is their behaviour conditional to how we see through them?


Can Hubble red shift be interpreted as time dilation?

Can we interpret the de Sitter universe as a spherical cosmic horizon null surface of finite radius, centered at Earth, and containing the Hubble volume of space where time is dilated and radial dimensions contract closer to the edge in such a way that objects closer to the edge do not recognize that they are radially contracted? Everything is attracted to the edge, but the total radius remains more or less constant and emits de Sitter radiation at finite temperature. ![enter image description here](http://i.stack.imgur.com/xgtGt.png)


What happens when you cut a biconvex lens in half?

Specifically, does the focal length change? How can this be rationalized?


One question about renormalization

The idea of renormalization of "naked" perturbation theory is in principal possibility of addition counterterms which reduce infinity when calculating matrix elements. But I have met such concepts as exact propagator and the exact vertex part. So, the question: are exact propagator and vertex part collected all countreterms of our theory (i.e. all countreterms are convoluted into expressions which are called exact propagators and vertex)? I.e., do they (exact propagators and vertes) contain all information about renormalization of the given theory?


Block on an inclined plane

If you take moments about the centre of mass of a block positioned on an inclined plane so that the gravitational force can be drawn from the centre of mass of the block to one corner of the block, why is there no turning effect due to the reaction?


Why is the universe so organized?

If you think about the Big Bang and the flow of matter in all directions, you would think, how unorganized would this universe be? No matter how long it would take. The idea that matter or most of it will be organized in such a beautiful way is unthinkable. Check this for example. The Moon orbits around our planet in a perfect fashion, not one but millions of system are like this. Our planet circles the Sun, again there are millions of examples like it. Our star circle the galaxy center in a predefined path. Again billions of stars do this! Now if you look at the matter that is organized (has found fixed orbit) vs that that is not, almost 90% of the matter is organized. They have fixed trajectories and have a stable path. In fact, if you want to look at matter that is not unorganized, you will find very little (we can think of our own solar system). Now there is not one galaxy but billions of them, all of them similar shape, like or spiral, which has stars. Outside this galaxy, there are few stars or maybe a supernova somewhere. Why and how is our universe so organized? Is the law incredibly simple, since the law has been repeated billions of times in every galaxy, or is it the work of some phenomenon of which the details are not yet known?


On Bell inequality and bound entangled states

I have recently seen some presentation slides of Michał Horodecki (slide number 77) in which he discussed the following conjecture. > Bound entangled states satisfy all Bell inequalities The conjecture is not true for multi-partite systems, and the references are given in the talk itself. I believe by now, the bipartite case is also settled, one way or the other. Bell theorem has some applications in computer science areas, like cryptography (non-local games). Is there some similar application of the (falsification of) the above conjecture? I mean, what are the interesting applications which can take place by disproving the above conjecture? Advanced thanks for any help/suggestions. **To the Moderators:** I do not know, whether I should have asked this question in Physics or in Cstheory. Considering the fact that there can be physical applications also, I am posted the question here. Please suggest and/or take the required actions.


Organs & Oscillations: An Analysis on the Temperature Dynamics of Solids

Does temperature have an influence on the frequency of an oscillating organ pipe?


How do you calculate vortex shedding frequency?

I am attempting to try to find out if there is any effect of Von Karman vortices on a group of wind speed readings where it is presumed that due to a mountain nearby the data collection spot Von Karman vortices were formed which would register in the data. Based on what I have read the vortex shedding frequency which should be present in the data is simply calculated from the formula $F= \mathbb{St}\cdot \frac{V}{L}$ with $\mathbb{St}$ as the Strouhal number, $V$ as the velocity and $L$ as the characteristic length. From what I understand the Strouhal number is essentially constant for a given flow, but when I looked up how to find the Strouhal number it was defined as $\mathbb{St} = F\cdot \frac{L}V$, which is the exact same relationship as given in the first equation for the computation of the shedding frequency. My question then is in two parts: 1) How do I calculate the Strouhal number for a flow so that I can calculate the shedding frequency? 2) What should I use as my characteristic length for a landscape feature such as a mountain or cliff etc as everything online tells only in the case of uniform cylinders and using their diameter instead of something like a cone which can be used as an approximation of a mountain or other landmass. thank you for any help.


Rotational inertia of a ball

This question refers to the solution of problem 12 here. It involves a spherical shell of mass $M$ filled with frictionless fluid of mass $M$ rolling down an inclined plane. (This is problem 12 of the 2013 F=Ma exam, a high school competitive physics exam). I understand the solution, but I don't understand how the given rotational inertia is obtained. We have a spherical shell of mass $M$ filled with frictionless fluid of mass $M$. The rotational inertia is apparently $$I_{contact} = \frac{2}{3}MR^2 + MR^2 + MR^2$$ Why? In particular, why is the $MR^2$ term added twice?


Why are Navier-Stokes equations needed?

Can't we picture air or water molecules individually? Then, why are Navier- Stokes equations needed, after all? Can't we just aggregate individual ones? Or is it computationally difficult, or inefficient to generate the picture of whole flow?


Do glass beads show any piezoelectric property? Since glass beads is mostly made out of SiO2 which is piezoelectric, will it show any piezoelectric property. Thanks. If anything wrong with my question kindly please tell me. Thanks.


How is the topological $Z_2$ invariant related to the Chern number? (e.g. for a topological insulator)

This question relates to the $Z_2$ invariant defined e.g. for topological insulators: Is it correct to relate $Z_2$ = 1 to an odd Chern number and $Z_2$ = 0 to an even Chern number? If yes, is it also correct to think of an even or odd Chern number in terms of an even or odd number of band crossings across the Fermi energy? (If it's odd, there must be a band connecting the valence to the conduction band and therefore provide a topological protected surface state.) * * * Edit: These lecture notes* (under Point H) state: "The formula (49) was not the first definition of the two-dimensional Z2 invariant, as the original Kane-Mele paper gave a definition based on counting of zeros of the “Pfaffian bundle” of wavefunctions. However, (49) is both easier to connect to the IQHE and easier to implement numerically." and furthermore: "...and the Chern numbers of the two spheres are equal so that the total Chern number is zero. The above argument establishes that the two values of the Z2invariant are related to even or odd Chern number of a band pair on half the Brillouin zone."   * http://socrates.berkeley.edu/~jemoore/Moore_group,_UC_Berkeley/Welcome_to_the_Moore_group,_Department_of_Physics,_UC_Berkeley_files/mitcourse.pdf


Can hair be used as an dielectric?

I was letting my mind freewheel, and it occurred to me that hair, containing keratin, is a polymer. It is also capable of acquiring and maintaining a charge as happens, on occasion, when brushed. So ...   1. Can hair be used as a dielectric in a capacitor?   2. Do any capacitors/manufacturers use hair as an dielectric already ?


The meaning of Goldstone boson equivalence theorem

The Goldstone boson equivalence theorem tells us that the amplitude for emission/absorption of a longitudinally polarized gauge boson is equal to the amplitude for emission/absorption of the corresponding Goldstone boson at high energy. I'm wondering what's the physical meaning of this theorem. Is there any relation between equivalence theorem and Higgs mechanism ?


possible values of quantity in matrix mechanics

Suppose that there is a quantity described by the matrix as the following: $M = \begin{bmatrix} 3 & 0 & -i \\\ 0 & 1 & 0 \\\ i & 0 &3\end{bmatrix}$ How we determine possible values of this quantity? (According to the book, it says that some possible values are 1,2 and 4, but it does not say how it determined this.)


Why did my liquid soda freeze once I pulled it out of the fridge and opened it?

This isn't a duplicate to "Why did my liquid soda freeze once I pulled it out of the fridge?". My question is why soda froze after it was opened. Opening a can or bottle seems to have a larger effect than just jostling it. Is it because of the disturbance noted in the previous question? Is it related to the pressure decrease? Is it because of the release of some CO2 when it was opened?


Exact diagonalization by Bogoliubov transformation

I am developing a model of multiple gaps in a square lattice. I simplified the associated Hamiltonian to make it quadratic. In this approximation it is given by, $$ H = \begin{pmatrix} \xi_\mathbf{k} & -\sigma U_1 & -U_2 & -U_2\\\ -\sigma U_1 & \xi_{\mathbf{k}+(\pi,\pi)} & 0 & 0\\\ \- U_2 & 0 & \xi_{\mathbf{k}+(\pi/2,0)} & 0\\\ \- U_2 & 0 & 0 & \xi_{\mathbf{k}+(0,\pi/2)} \end{pmatrix} $$ And my Nambu operator is given by, $$ ψ_\mathbf{k} = \begin{pmatrix} c_{\mathbf{k},\sigma} \\\ c_{\mathbf{k}+(\pi,\pi),\sigma} \\\ c_{\mathbf{k}+(\pi/2,0),\sigma} \\\ c_{\mathbf{k}+(0,\pi/2),\sigma} \end{pmatrix} $$ I tried to diagonalized by making three Bogoliubov transformations, the first to diagonalize the upper right submatrix of H, and then the two others (a sort of nested transformations). But I get a lengthy result, what I would like to know if there is a smart transformation which allows me to write $$ H = A_1^\dagger A_2^\dagger A_3^\dagger D A_3 A_2 A_1 $$ or simply $$ H = U^\dagger D U $$ Or the only way is to use just brute force? My question may be rephrase as, is there any elegant representation or expression of the diagonalization as for example the one obtained in BCS Superconductivity? Thanks


Equilibrium - uniform circular motion

Maybe this is a bit of a silly question, but let us pretend we have a pendulum in a ideal universe with no friction, drag, or anomalous forces there to affect it. Additionally, our pendulum is subjected to an ideal gravitation force, the force vector of which is unaffected by the position of the ball (hence the responsible gravitating body is so massive that its own spherical geometry does not affect the forces on the pendulum-ball. Now we have our pendulum spin around so that it experiences uniform circular motion and some angle theta is formed between the resultant orientation of the spinning pendulum string and its previous stationary form. My question is, is this system in equilibrium? I know that no work is being done even though there is a force. And that from a relativistic (non-Newtonian) point of view, there is no centrifugal force to cancel the centripetal. So please, could you explain the "answer" from a Newtonian (high-school physics) POV and a relativistic POV.


How viable is reuse of nuclear waste at this point, and what are future prospects?

I keep seeing documentaries describing the pile-up of nuclear waste from nuclear power stations in the US and in other countries. Countries such as Finland are working to deposit waste in old mines in geologically stable regions, and I was under the impression that the US was heading down that track as well, but this has now apparently been put on hold. A case of NIMBY I presume (not in my back yard). The media story is that the waste is piling up, and is being stored in pools at nuclear power stations all over the US - which sounds dangerous considering the prevalence of natural disasters. I wonder what the real truth is? I am not quite sure what the situation is in France and England, but I got the impression they are experimenting with methods of re-use of the waste products as well as storing some material under ground. What is the current state-of-the-art in terms of nuclear waste reuse? Are new reactors capable of burning discarded fuel from older reactors? Are there realistic expectations to be able to eliminate deposition of nuclear waste altogether in the foreseeable future?


Are derivative indices summed in indicial notation?

A paper I'm reading uses indicial notation and the convention that $u_{j,k}$ means the derivative with respect to $x_k$. Which one of these interpretations are correct? $$A_{ijk}u_{j,k} = \sum_{j}A_{ijk} \frac{\delta u_j}{\delta x_k}$$ $$A_{ijk}u_{j,k} = \sum_{j,k}A_{ijk} \frac{\delta u_j}{\delta x_k}$$


When Venus is the brightest when viewed from Earth?

Nowadays Venus is very bright. I can spot it during broad daylight without problem. It's because it's near Earth and appear as a crescent. This made me think: as it's reaching the inferior conjunction the crescent becomes slimmer, so I expect that it will fade a bit. So there should be an optimal phase angle for optimal brightness. So my question is at which phase angle Venus is the brightest when viewed from the Earth?


Gravity and objects at 0K

Let's say I put 2 objects at rest relative to each other with 0 Kelvin. Will they move towards each other because of gravity? Where they will get energy to move? I don't have education in this field, just wondering.


Intuition contradicting answer for calculating pressure

I have a cylindrical container say of some base area $A$. The atmospheric pressure is $P_0$. The cylinder is moved at an acceleration $a$ horizontally right. I need to find which base would have to face more internal pressure. The cylinder is filled with some ideal gas. My intuition, even though I doubt it, says that in the frame of the cylinder the fictitious forces act towards left, so it should be the left face. But on some calculations using the laws of motion, I get equations as $$(P_0-P_1)A=ma$$ $$(P_2-P_0)A=ma$$   ($m$ is the mass of each base of cylinder) On solving I have $$P_1=P_0-{ma\over A},P_2=P_0+{ma\over A}$$ which means the right face faces more internal pressure. What is going wrong? P.S. just give me a hint. ![enter image description here](http://i.stack.imgur.com/3euyA.png)


Difference between primitive unit cell and the associated basis?

As I understand it, the basis is the group of atoms whilst the primitive unit cell is the unit space that fits the total space without any gaps, and only containing one lattice point? How do the two relate to each other? Thanks.


Volume quotient in Carnot-cycle

Problem: > One kilomole of an ideal, monatomic gas undergoes a reversible Carnot- > processes between temperatures 300 °C and 20 °C. The work done during one > cycle is 1500 kJ. > > a) Find the entropy-change in every process and show that total sum of > entropy-changes is zero > > b) What is the ratio between the largest and smallest volume that the gas > takes on during the whole process? I have already done a), it is b) I am struggling with. I realized that the greatest and smallest volume is assumed in the adiabatic processes of the Carnot-cycle, so I applied $T_1V_1^{\gamma -1} = T_1V_1^{\gamma -1}$ but to no avail. How can one do it?


What limits laser pulse duration to single cycle?

1. Why pulses shorter than single cycle cannot propagate in free space ? Is there fundamental reason or it is engineering ?   2. Is it possible for a laser pulse to have bandwidth that can support half cycle pulse and how it will propagate in free space ?


Is the universe a perpetual motion machine?


Centripetal Force Acceleration

In uniform circular motion, acceleration is $\frac{v^2}{r}$ and time which it acts $\rightarrow 0$. So $\Delta v = 0$, but then why/how does direction change, when the acceleration should be producing no effect at all .


Why don't metals bond when touched together?

It is my understanding that metals are a crystal lattice of ions, held together by delocalized electrons, which move freely through the lattice (and conduct electricity, heat, etc.). If two pieces of the same metal are touched together, why don't they bond? It seems to me the delocalized electrons would move from one metal to the other, and extend the bond, holding the two pieces together. If the electrons don't move freely from one piece to the other, why would this not happen when a current is applied (through the two pieces)?


Physics of the point of contact for a spinning top

I understand how spinning tops don't tip over, cf. e.g. this and this Phys.SE questions. What I'm more interested is in identifying the factors that determine the direction the spinning top moves to?


Interpretations of quantum mechanics

It is well known that there are many interpretations of quantum mechanics. I'm wondering if there is a specific reason why the Copenhagen interpretation is the most popular. Why is it that the wavefunction interpreted as a probability distribution is such a useful description. This question came to me when I was trying to write a paper and got a strange wavefunction (it wasn't normalizable, had singularities etc.). I'm a mathematical physics grad student so I was usually more concerned with mathematical elegance than physical interpretations, but this question has been bothering me.


Relativity of simultaneity and cause-effect

Relativity says that two events simultaneous in one reference frame might not be simultaneous in another reference frame. Can we extend this idea a little and say that the order of two events might be be reversed in going from one reference frame to another? And if so, what if they have a cause-effect relation ship in the first frame. Isn't reversing the two in the second frame a contradiction?


Tadpole symmetry factor

Can someone help me with symmetry factor of one-loop tadpole diagram (one loop correction to one point Green function in phi-3 theory)?


Am I talking about the correct units here?

Paolo made a move I knew from dozens of townie soccer games played under lights on the fenced-in macadam pitch next to the elementary school in Badalucco, packed tight between Via Cristoforo Colombo and the bed of the Argentina River. Paolo twisted up his left leg; rotated that leg in an entirely autonomous and anatomically-impossible split-second burst of acetylcholine; power-rammed, with his left foot, the front bumper of our white cheese-tour van. A van, it now became clear, that Paolo had left precariously in neutral. With the parking brake disengaged. Tilted up along the road’s seriously steep grade. Its rear wheels inches from the verge, a sharp drop to the curve below. His cousin’s van, I remembered, as it lurched backward and gathered speed. Here’s the thing, often under-appreciated, about translational kinetic energy:   ET=(1/2)mv2 I’m quite certain that Hugo failed to fully apprehend the ramifications of this formula, given our weighty cheese van and his own position vis-à-vis an initially lumbering, but definitely accelerating, object. Perhaps Hugo had forgotten his boyhood physics class with its repeated exhortation that v = v0 + at.   Gravity, Hugo, gravity. 9.8 m/s² Hugo. But, most important: mass, Hugo. Mass. The van was heavy, no doubt about it, all by itself. Then you add all that luggage. Plus one 100 pound wheel of Parmigiano Reggiano. Plus Jack’s, or Lauren’s, or whoever’s six boxes of Canopic Cookie Jars. Plus whatever illicit items the Arbogasts had inserted into their collection. Plus nearly two weeks of accumulated detritus dumped on the floor by people who’ve spent way too many van-hours for the niceties of proper trash disposal to hold much sway. Plus Lauren. Maybe it was Lauren of whom Hugo was thinking when he jumped not away from but toward the rapidly-accelerating van, its front passenger door still gaping wide; a van that had by then about reached its tipping-point on the steep-shouldered roadside. Maybe it was Lauren. Or maybe Hugo had played linebacker for Texas A & M, back in the day, and still figured he was on top of his game, good to go, stop anything coming at him. But this was no 250 pound running back, Hugo. This was not even a steroidally-enhanced 350 pound running back. No. This was a stout door winged out from a 3000 kilogram white Fiat Ducato made weightier by its 20 gallon auxiliary tank full of used grease. So, let’s say three seconds elapsed between the moment when Paolo’s kick overcame the van’s inertia (and started ‘er rolling) and the instant when Hugo attempted to stop her. For ease of calculation, let’s put the inclined plane on which the Ducato rested at 45 degrees, i.e., pretty steep, and thus the parallel vector of gravity pulling her downhill at about 4.9 meters per second per second. After 3 seconds and change (we’ll ignore the rolling friction of axels and rubber tires) the Ducato would be chugging along at close to 15 meters per second. I think. Over 30 mph. Not so shabby. Which brings us back to ET=(1/2)mv2. 3000 kg at 15 meters per second. At the instant of contact, a direct hit would work on a firmly-planted Hugo with something in the vicinity of 337,500 Joules. Fortunately for Hugo, he was not planted, immovable, like the high-security bollard he believed himself to be, but more along the lines of a wobbly bowling pin. The door dealt Hugo a split-second glancing blow. What we probably need to figure here, as in football, is impact force per unit area. Newtons/m2. Stress. The specifics of this inelastic collision. Consult the law of cosines, maybe? You do the math, Hugo. Hugo was not going to be doing any math for quite a while. More or less simultaneously: I saw the open door plow into a Hugo determined to impede its progress. I heard the loudest noise I’d ever experienced. I saw the windshield explode in frost. I saw Paolo drop from sight. I heard Lauren scream. Then: I saw Hugo go down. I remembered Phil and Massimo. Under the backseat. Gliding toward oblivion.   Which must be when I made my own ill-advised lunge toward the driver’s seat, into the path of the open driver’s side door. A split-second later, as Pippa howled “Sybilla, don’t!” must be when I stumbled over a loose piece of rubble and myself went down. I’m told the bottom of the door just grazed my face. But, as noted, at 15 meters per second and rising, even a graze by a three ton vehicle can hit you with lots of Ns.


The things to consider when choosing a mesh for an electron detector (MCP detector)

I am designing a electron velocity map imaging setup. I need a copper mesh just infront of the detector to accelerate the electrons towards the MCP detector. I am doing the instrument design for the first time. Can anybody please tell me how to choose a mesh and what are the things should I keep in mind while choosing a mesh for the detector (regarding the number of wires per inch, its transmission etc.)


Conservation of Mathematical Constraints when deriving Energy and Momentum from $F=ma$

**Background:**   * Starting from $F = ma$, integrating with respect to time, and using basic calc, one can derive $\int Fdt = m (v_f - v_i)$   * Starting from $F = ma$, integrating with respect to distance, and substituting $a\ ds = v\ dv$ (from calculus), one can derive $\int Fdx = KE_f - KE_i$ (Obviously I care about these results because, when combined with $F_{ab} = -F_{ba}$, they give conservation of momentum and a starting point for conservation of energy, but for this question I'm not going to consider this part of the derivation.) In 1-D, the two bulleted results make sense to me: the starting point is a 2nd-order ODE, and the two results form a coupled system of two 1st-order ODEs. This is precisely what math says should be possible - the number of mathematical constraints has been conserved. **Question:** A similar (slightly more involved) derivation is possible in 3 dimensions, but it's harder for me to classify the resulting mathematical constraints and reassure myself that the transformation doesn't add or remove constraints:   * Starting Point: $\vec{F}=m\ddot{\vec{s}}\ \rightarrow$ 3 coupled second-order ODES    * Result 1: $\int \vec{F}dt = m(\dot{\vec{s}_f} -\dot{\vec{s}_i})\ \rightarrow$ 3 coupled first-order ODEs (viewing initial state as a boundary condition)   * Result 2: $\int \vec{F}\cdot d\vec{s} = \frac{1}{2}m(\dot{\vec{s}_f} \cdot \dot{\vec{s}_f}) - KE_i\ \rightarrow $ Unsure how to classify this      * Rewriting this as $\int \vec{F}(x,y,z)\cdot d\vec{s} = \frac{1}{2}m(\dot{x}_f^2 +\dot{y}_f^2 +\dot{z}_f^2 ) - KE_i $ makes it clearer that this is a single non-linear ODE with the first derivatives of 3 different dependent variables in it.  Math says that a system of 3 coupled second-order ODEs can be rewritten as a system of 6 coupled first-order ODEs, but I obviously only have 4 equations. **What type of ODE is Result 2?** A single first-order ODE? A 'triple' first- order ODE with 3 dependent variables? I'm ultimately looking to reassure myself that commuting the problem 'solve $F=ma$' to the problem 'solve conservation of momentum and conservation of energy' doesn't add or remove mathematical constraints. If anyone can refer me to a textbook which addresses this idea, I'd appreciate that as well. **Edit:** On further consideration, I think I've made a mistake. $F=ma$ is a differential equation with time as the independent variable, so integrating with respect to time (as is necessary to derive conservation of momentum) changes the order, and it's not actually reasonable to expect two coupled first-order differential equations as a result any more. In fact, we could integrate Conservation of Momentum wrt $t$ again, and then we'd no longer have a differential equation at all. Since integrating is a valid technique for solving differential equations, it now kind of appears as though Conservation of Momentum could be viewed as a [/the] solution to $F=ma$, and could thus be expected to describes all the information which is present in the original ODE.   * That would make the $F=ma$-to-Conservation of Momentum derivation logical (from a Conservation of Constraints perspective) in both 1-D and 3-D, and then make the real question 'Where Conservation of Energy come from, from a constraints perspective?'   * On the other hand, the 1-D elastic collision problem can't be solved with Conservation of Momentum alone - Conservation of Energy is also required. Since Conservation of Energy can be derived from $F=ma$ and calculus alone, it must be representing a constraint which is in the original equation. Thus $F=ma$ holds more information than Conservation of Momentum. **Other things I've been thinking about:**   * This question may actually relate to integro-differential equations, which I know nothing about.    * There may be some subtleties related to $F$ which I'm missing, and writing it as $\vec{F}$ instead of $\vec{F}(\vec{s},\dot{\vec{s}},t)$ may be sweeping them under the carpet.


Young's Double Slit experiment question

> _Q-A beam of light consisting of two wavelenghts 600 nm and 450 nm is used > to obtain interference in Young's Double Slit experiment (YDSE). Find the > least distance from the central maximum where the bright fringes due to both > wavelengths coincide. The distance between the two slits is 0.4 mm and > screen is placed at a distance of 1.0 m from the slits._ Now my problem is not with the question but the solution given in my book. It says > _Let at distance $x$ from central bright maxima, the bright fringes due to > both the wavelengths coincide for the first time. It is possible only if > within the distance $x$ there are $n$ fringes for light of 600nm and $(n+1)$ > fringes for light of wavelength 450 nm._ Okay so my problem is that I am unable to figure out how have they made this argument about $n$ and $n+1$ fringes? It's isn't mention anywhere in the question. Can't it be ($n$+any number)? If there's any better solution than this I would be really thankful for that


Does a guitar sound different in zero (or micro) gravity?

Seeing a video of astronaut Chris Hadfield playing a guitar on the International Space Station made me wonder if a guitar or other stringed instrument played in zero-G would sound any different than on earth. It seems that on earth, the downward pull of gravity could cause an asymmetrical oscillation of the guitar string, with a larger amplitude as the string moves downward due to the pull of gravity. (of course, it wouldn't take zero-G to test this since a guitar could be held vertically) With the amount of tension that a guitar string is under, perhaps this effect is so minuscule so as to be unnoticeable?


Design and use of the RLC model for signal processing systems and spatial acoustics

Given a frequency response for a signal processing device, set of devices, or physical space it is possible to construct an equivalent (identical frequency response) series RLC circuit. 1) Does this offer information about the system that the frequency response in and of itself does not? 2) Is the series model the most appropriate model, or is some other configuration more informative? Many thanks in advance Joe


Momentum of a particle?

I really need help to understand what is momentum of a particle (of a photon, proton, an electron...) I see so many definitions! My main questions are: •What exactly is momentum •What are the differences between momentum and spin of a quantum particle. •What is the relation between momentum of a photon and it's wave. I just wanted simple explanations. Because after months hearing about momentum, I still don't know what is it.


Practical physics: where does this current come from?

It's been a while since school and I don't remember the right laws and formulas. So I come here. :) I've got a power cord extender running from the wall outlet to my washing machine. It's about 3 or 4 meters in length. The outlet supplies 220V, but no grounding. The extender however does have grounding support, as does the washing machine. This means that the extender has three wires in it, and the grounding wire terminates at the outlet. When plug in one end of the outlet, and measure the AC voltage between the grounding (disconnected) and zero wires, it gives me about 60V. Which happens to be unpleasant enough when accidentally touching the washing machine (even when it's turned off). **Considering that the cable is so short, can this really be explained by induction?** To rule out a few other theories: the wall outlet does not connect anything to the grounding wire, it really does terminate there. The grounding wire does not touch any of the other wires, the resistance between it and other wires is infinity (or rather, beyond the range of my cheap multimeter).


How is the wattage of a human larynx calculated?

This site says the typical human voice has a wattage of about 1W. Another site indicates it is 80db with about 0.001W of energy. I'm confused. How does one compare 80db of 0.001W against 1W? Most importantly how is the wattage of the human larynx calculated?


Is Biot-Savart law obtained empirically or can it be derived?

Well, there's already a question like this here so that my question could be considered duplicate, but I'll try to make my point clear. Is there a way to derive Biot-Savart law from the Lorentz' Force law or just from Maxwell's Equations? The point is that we usually defined based on experiments that the force felt by a moving charge on the presence of a magnetic field is $F = qv\times B$, now can this be used in some way to obtain Biot-Savart law like we obtain the equation for the electric field directly from Coulomb's Force law? And I wanted to know that because as pointed out in the question I've mentioned, although Maxwell's Equations can be considered more fundamental, those equations are obtained after we know Coulomb's and Biot-Savart's laws, so if we start with Maxwell's Equations to obtain Biot-Savart's having use it to find Maxwell's Equations then I think we'll fall into a circular argument. In that case, without recoursing to Maxwell's Equations the only way to obtain Biot-Savart's law is through observations or can it be derived somehow?


How would I work this continuity problem?

I am trying to design a system for a project and I need some input on if this is right or not. I have $10 \frac{kg}{s}$ of fluid flowing through a pipe. It breaks up into $10$ 1 inch diameter pipes. I am giving a picture below. Does this mean that the feed pipe, the one supplying the $10 \frac{kg}{s}$, should be just 10 inches in diameter instead of 1 inch? In case your wondering, the fluid is ethylene glycol, I dont think that matters ![enter image description here](http://i.stack.imgur.com/90Omj.jpg)


Understanding the energy transffered/used from a power source?

A source of power(from a generator,battery,etc...) is used to power a system of any kind. We can calculate the energy used via this formula $E$ = $IV$ x $t$ ? For example, a battery that is connected to a conductor, that conductor converts electrical energy to heat. $E_B$ = $P$ x $t$ E - energy used from the batter. t - time the battery is connected to the circuit and current flows. Is this the only way to model or analyze the energy transferred?


How fast do electrons travel in an atomic orbital?

I am wondering how fast electrons travel inside of atomic electron orbitals. Surely there is a range of speeds? Is there a minimum speed? I am not asking about electron movement through a conductor.


Neutrinos travels through our bodies: do they contribute for gravitational atractive force?

Are neutrinos traveling through our bodies what really make us attracted to the Earth?


SPEED of LIGHT as photons reach black holes

As photons reach a black hole and begin fall in - is their speed increased as they go down (like a vacuum cleaner does to dust).


Small car colliding with large truck

A small car collides with a large truck. Why do both vehicles experience the same magnitude of force? Wouldn't the large vehicle experience less force than the small one?


Is there a direct relationship between an isotope's neutron count and radioactivity?

In my textbook, it lists isotopes of Carbon: C-12, C-13, and C-14. It noted that C-14 is radioactive (C-12 and C-13 are not). Is there a direct relationship between the number of neutrons and an element's radioactivity? In other words, since we know C-14 is radioactive, does that mean we also know that C-15 (if such a thing exists) would also be radioactive?


Giant leap of the space goat

This problem was inspired by this Math.SE post and similar problems involving tethered animals. A space goat is tethered to the north pole of a spherical asteroid of uniform density and radius $a$ by a rope of length $\pi a$. Becoming bored with grazing the surface, the goat tests the limits of its freedom by making a series of leaps. What is the longest time that it can remain airborne? The answer should be expressed as a multiple of the orbital period of an object circling the asteroid just above its surface. Assume that the goat’s strength is not a limiting factor, the asteroid is very much more massive than the goat, atmospheric resistance and the weight of the rope are negligible, and solutions involving any possibility of the rope dragging across the surface are non-optimal. Note: this is an original problem and I don’t yet have a complete solution. It looks doable but don’t necessarily expect a result in closed form. [Edit: added assumption about relative masses.]


Big Bang snuffed by a black hole?

Wasn't the density of the universe at the moment after the Big Bang so great as to create a black hole? If the answer is that the universe/space-time can expand anyway what does it imply about what our universe looks like from the outside?


Do these equations apply to an object on a zip line or slope?

I was looking at the conversion equations for Potential Energy to Kinetic Energy to Velocity here: http://www.ftexploring.com/energy/PE-to-KE.html Would these apply to an object on a zip-line? What, if any, adjustments need to be made.


About the definition/motivation/properties of the twisted chiral superfield in ${\cal N}=2$ theories in $1+1$ dimensions

The following is in the context of the ${\cal N}=2$ supersymmetry in $1+1$ dimensions - which is probably generically constructed as a reduction from the ${\cal N}=1$ case in $3+1$ dimensions.   * In the $\pm$ notation what is the definition of ${\cal D}_+$ and ${\cal D}_{-}$, which I understand from context to be the gauge covariant superderivatives. (..It would be great if someone can relate them to the usual definition in the notation of say Wess and Bagger..)   * So what is the meaning/motivation of defining a twisted chiral superfield as, $\Sigma = \\{\bar{{\cal D}}_{+}, {\cal D}_{-}\\} $ (..naively this looks like an operator and not a field - I guess there is some way of arguing that the derivative terms which are not evaluated on something actually go to zero..) I am guessing that in the above context it will be helpful if someone can explain as to what is meant by the following decomposition/reduction of the gauge field from $3+1$ dimensions, $\sum _ {\mu = 0}^3 A_\mu dx^\mu = \sum _{\mu =0} ^1 A_\mu dy^\mu + \sigma (dy^2-idy^3) + \bar{\sigma}(dy^2+idy^3)$ ?   * From the above (does it/how does it) follow that one can write $\Sigma$ as,  $\Sigma = \sigma + \theta\lambda + \theta \bar{\theta}(F+iD)$ (..where I am not sure if $F,D,\sigma$ are real or complex scalar fields...and $\lambda$ is a Weyl fermion..)   * What is the R-charge of this twisted chiral super field? (..from some consistency conditions I would think that its 2..but I am not sure..) I guess that the R-symmetry transformations act as,   * The "right" R symmetry keeps $\theta^-$s invariant and maps, $\theta^+ \mapsto e^{i\alpha}\theta^+$, $\bar{\theta}^+ \mapsto e^{-i\alpha}\bar{\theta}^+$   * The "left" R-symmetry keeps $\theta^+$ invariant and maps, $\theta^- \mapsto e^{-i\alpha}\theta^-$, $\bar{\theta}^- \mapsto e^{i\alpha}\bar{\theta}^+$. Though I am not sure and like to understand as to why one wants to think of these two different R-symmetry groups as having two different origins - one coming from the rotation symmetry of the two spatial dimensions of the original $\cal{N}=1$, $1+3$ theory and another coming from R-symmetry of the $\cal{N}=1$, $U(1)$ gauge theory.


Put a sensor at the focal length, behind, or in front?

My intuition tell me that at the focal length a convex lens all the light converges to a point. Following that logic, it would make sense to me that a camera chip would either need to go slightly in front or slightly behind the focal point in order to expose the entire chip to the incoming light. This would produce a normal or inverted image, respectively. Upon investigating this question, I read through these articles:   1) Wikipedia - Focal Length   2) Paragon Press - Focal Length   3) The Photo Forum That all agree that the sensor MUST be placed AT the focal length, not in front or behind. **How does a chip placed at the focal length not just produce a white spot of light at the center?**


Translation Invariance without Momentum Conservation?

Instead of the actual gravitational force, in which the two masses enter symmetrically, consider something like $$\vec F_{ab} = G\frac{m_a m_b^2}{|\vec r_a - \vec r_b|^2}\hat r_{ab}$$ where $\vec F_{ab}$ is the force on particle $a$ due to particle $b$ and the units of $G$ have been adjusted. Whenever the masses are unequal, the forces are not equal and opposite, violating Newton's third law and conservation of momentum in the process. As momentum conservation has been violated, my understanding is that translation invariance should be violated as well by this force. But the force law still depends only on separations rather than absolute coordinates, so the physics seems to be translation invariant. What am I getting wrong?


What are the implications for the AdS/Cft program if AdS is unstable?

To my understanding recent progress in the study of the non linear stability of AdS spacetime suggest that $AdS$ might be unstable. If this is true, what are the physical and mathematical implications for the $AdS/Cft$ approach? The stability of an spacetime warranties that that sufficiently small perturbations remain small. In the case of Minkowski the stability theorems proved by Christodoulou and Klainerman show that sufficiently small perturbations not only remain small but decay to zero with time in any compact region (this stronger type ofstability is called asymptotic stability). In the case of $AdS$ numerical and certain results in nonlinear partial differential equations suggest the posibility that the following conjecture might be true: The $AdS_{d+1}$ space (for d ≥ 3) is unstable against the formation of a black hole for a large class of arbitrarily small perturbations. Now, the $AdS/Cft$ correspondence is a conjectured relationship between two kinds of physical theories. On one side of the correspondence are conformal field theories (CFT) which are quantum field theories, including theories similar to the Yang–Mills theories that describe elementary particles. On the other side are Anti-de Sitter spaces (AdS) which are used in theories of quantum gravity, formulated in terms of string theory or M-theory. In this program there are some examples that one can relate the formation of black holes in the bulk to certain thermodynamic properties of the conformal field theory. In particular,the presence of black holes can be seen as some thermalization of the field theory. How would the results on stability be interpreted in the dual conformal picture? Is there some meaningful thermodynamical process that explains the instability at the classical level?


Cylindrical lens aberration fixed by roll

I have two cylindrical lenses essentially forming a beam expander. The problem that I am having is that I am seeing a very strange aberration. The image becomes "squashed". Cylindrical lenses only focus in one direction unlike spherical lenses so with a circular input beam a crisp vertical line should form at the focal point of the converging lens (If I just use the converging lens...no beam expander). But, at the focal point of just the converging lens instead of seeing a vertical line, the line is slanted. Now looking at the beam expander system the image begins as a nice crisp expanded circular beam but as I continue looking at the progression of the beam the image looks as if it is folding over itself and squashing into a horizontal line. If I add in some relative roll between the converging and diverging lenses The squashing is improved but not perfect. I wanted to know if this kind of aberration is common or if there is a defect in the lenses I am using. In the Zemax simulations this should not be happening. It might be important to note that the converging lens is essentially saturated. The input beam is collimated in such a way that the maximum sized circular beam is sent through the lens without clipping. ![enter image description here](http://i.stack.imgur.com/sEzv8.png)


Normal modes of two wires fastened together

The problem is to find the normal frequencies of the system formed by two fastened wires of length L, and different mass per unit length. I already wrote the boundary conditions, but I need to know if the joint has to be a node. I'm pretty sure it has to, otherwise it wouldn't be in a normal mode, but I can't set it up straight in a rigorous way. The boundary conditions I imposed are: $y_1(0,t)=y_2(2L,t)=0$ $y_1(L,t)=y_2(L,t)$ $\frac{\partial y_1}{\partial x}(L,t)=\frac{\partial y_2}{\partial x}(L,t)$ Where $y_1(x,t)=A_1e^{i(k_1 x-\omega t)}+B_1e^{i(-k_1 x-\omega t)}$ and $y_2$ is written similarly. Thanks!


Earnshaw Theorem for ionic solids

Is a single molecule of sodium chloride (say) or a cluster of molecules of NaCl unstable, although macroscopically NaCl is in fact, stable? How can I reason this based on Earnshaw's theorem?


Could motives aid in the study of the Navier-Stokes equations?

Recently, mathematicians and theoretical physicists have been studying Quantum Field Theory (and renormalization in particular) by means of abstract geometrical objects called motives. Amongst these researchers are Marcolli, Connes, Kreimer and Konsani. You can read about Marcolli's work here*. Now, in the wikipedia article on the Navier-Stokes Equations, there is a short paragraph on Wyld Diagrams. It is stated there that they are similar to the Feynman diagrams studied in QFT. Since motives and other algebraic approaches are currently used to study these Feynman diagrams, I was wondering if these approaches could also aid in studying the Navier-Stokes equations. If so, how? And to what extend could they potentially help solving the (in)famous Millennium Problem? If not, why not? Please note that I'm far from an expert in any of these fields. Thanks in advance. * _(I would like to include more hyperlinks to the books and articles of the respective scientists, but since I am a new user I am incapable of doing so. You can find a lot more literature simply by looking up the names of these researchers.)_


Does it make sense to speak of amplitudes of finite closed boundaries in QFT?

A example of amplitude in Relativistic Quantum Mechanics or specifically in QFT is the amplitude of a field configuration on a space-like hyper-surface of space-time to "lead" to another field configuration on another space-like hyper-surface of space-time. In the path-integral picture one simply integrates over all possible field configurations on the interior, giving each a weight in the normal way. Now if one wants to generalize this to finite closed boundaries, we would get an amplitude for each field configuration on a finite closed boundary of space-time. but how would we interpret this? This question relates to interpretations of quantum mechanics, has anyone investigated this line ?


Determining diameter of effective image circle of large lens in a Camera Obscura

I am in the process of gathering materials and designing a large lensed Camera Obscura. I would like to estimate the approximate diameter of the image circle and the approximate field of view. in degrees, but all the formulae I have encountered require parameters I do not yet know. I am under the impression that a free lens has an intrinsic image circle diameter/FoV. **—The lens** **Diameter:** 304.8mm (12.0"in) **Focal Length:** 1143mm (45.0"in) **Type:** Plano-convex glass **—The Room:** An inflatable inhabitable sphere with a hanging translucent screen about a meter from the lens mounted on the wall. I am using 'clear' Polyethylene sheeting as my ground-glass surface. I have built a lensed Camera Obscura before with a 150mm diameter 2000mm FL glass lens. Documentation of project: https://vimeo.com/94692036 **—The Question:** How does one go about determining the Image circle/field of view of a single lens?


Most particles that have ever been modeled

Two billiard balls collide. Consider the two balls only, how many particles are in that system? Have particle systems of that size been modeled with each individual particle before? What is the largest object ever been modeled with pure particle physics?


Work being done without distance

I have this issue: If you push a 40.0 kg crate at a constant speed of 1.40 m/s across a horizontal floor (µk=0.25 ), at what rate (a) is work being done on the crate by you and (b) is the energy dissipated by the frictional force? For starters, here is what I have so far: RN = mg = 40*9.8 = 392 N Ff = muRN = 0.25*392 = 98 N KE = 0.5mv^2 = 0.5*40*(1.40)^2 = 39.2 J At this point, I don't know how to answer part (a). If W = FD , how do I find the work being done without a distance?


Question about entropy and "smart materials" (that remember shape)

We all know that entropy is the measure of chaos in the system, and it's always increasing in the system. Now comes my question - how does entropy work in smart materials? Here is a youtube video about smart material that I'm talking about, it remembers its shape - you can distort it, but when you heat it, it returns back to its original shape. So effectively it reverses from more chaotic state to a more uniform state: http://www.youtube.com/watch?v=fsBHF_j2FJ4 Therefore my question - how come the system became less chaotic just by adding heat? It looks like entropy decreased, but the 2nd law of thermodynamics says that entropy can only increase.


How can the derivative of this trace be constrained?

I am studying for my exam on relativity and I am going through some problems sets including ones where I was not very successful in so I want to know how to do this problem. > (Convergence of particles): Consider a freely falling reference particle in > vacuum as in Exercise 1. Assume now in addition that the Hessian matrix of > the gravitational potential does not vanish initially in the vicinity of the > reference particle. Recall that the displacement of a nearby particle $y(t)$ > is given in terms of its initial displacement $y(0)$ (we assume that the > particle is intitially at rest relativie to the reference particle) by > $$y(t) = A(t)y(0)$$ NOTE: Just to give some context we say that $y(t)$ is a displacement between two freely falling particles positioned $x_{0}(t)$ and $x(t)$ so $y(t) = x(t)-x_{0}(t)$. We can say that a nearby freely falling particle satisfies to first order in y the tidal equations, mainly we can see that: $$\frac{\mathrm{d}^{2}y}{\mathrm{d}t^2}=-\nabla^2\psi(x_0(t))\cdot{y(t)}.$$ Back to the problem. > We call $A$ the deformation matrix. Show that there is an initial > displacement $y(0)\neq0$ and a time $t^*>0$ such that $y(t^*)=0$. > > Hint. Recall the definition of the strain matrix: > $$\theta=\frac{\mathrm{d}A}{\mathrm{d}t}A^{-1}.$$ Now to solve this the question breaks it down to several small parts. I was able to solve the first part so I will skip it. But here is the part I am having trouble with: > (2) Deduce from the tidal equation the following ordinary differential > inequality: > $$\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{tr}\theta\leq-\frac{1}{3}(\operatorname{tr}\theta)^2$$ > Hint: Decompose $\theta$ into its diagonal and its traceless part. NOTE: We deduced from lecture a long time ago that $\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{tr}\theta=-|\theta|^2-\operatorname{tr}M$ where $M=\nabla^2\psi(x_0(t))$ or in other words $M$ is the hessian of the gravitational potential of the particles that are freely falling. We also know that $\operatorname{tr}M=\Delta\psi(x_0(t))$ and in a vacuum this is equal to 0. So, concerning this problem I figure if we have to break down $\theta$ to a diagonal part and a traceless part that means we can call the diagonal $\sigma$ and the traceless part $\omega$, such that $\theta=\sigma+\omega$. That means that: \begin{align} &\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{tr}(\sigma+\omega) \\\ &\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{tr}(\sigma) \end{align} Since the trace of the omega is 0 but honestly I am not sure what more to do. I think I actually have to move that derivative and maybe I will be able to get that inequality.


Fourier Transform on a Riemannian Manifold

The question is quite simple: What would be the definition of Fourier Transform (and it's inverse) on a Riemannian Manifold? I've found that a similar question has been asked at Mathematics.SE but that question was asking for a FT on a Lorentzian Manifold. The answer given is actually very good but then it gets a little too technical and quite cryptic to me... The first comment on that question actually points at the right direction but could someone give me an example on a specific Riemannian Manifold (aside Euclidean), for example on a 2-cylinder with metric $g=dz^2+r^2d\theta^2$? **PS:** Maybe I should have posted the question on Mathematics.SE but I was hopping for the physicists more direct approach to these kind of problems...


Dark Matter 'Stars'

I'm aware that the Milky Way has a dark matter 'halo' around it, presumably a spherically symmetric distribution. But I'm completely ignorant regarding the theories explaining dark matter... Is there any reason to not expect a star-sized object to also be made of dark matter? I know they'll be extremely difficult to detect, but I'm wondering if it's even physically possible to exist.


What's the corresponding symmetry of enstrophy conservation?

In fluid mechanics, especially 2D turbulence study, people talk about conservation of enstrophy. But I can't really understand enstrophy very well, and what's the corresponding symmetry of enstrophy conservation?


Are there more distinctive names of "null curves" with certain additional properties (absence of "chord curves")?

In this answer (to the question " _In general relativity, are light-like curves light-like geodesics?_ ", PSE/q/76170) a particular example of a curve is discussed whose " _tangent is everywhere null_ " and which is therefore called a " _null curve_ ". I'll restate the example curve explicitly as $$\nu : \mathbb R \rightarrow \cal M,$$ together with a coordinate function $$\mathbf r : \cal M \rightarrow \mathbb R^{1,2}; \qquad \mathbf r := \\{~t,~x,~y~\\}$$ such that $$\mathbf r \circ \nu[~\lambda~] := \\{~t_{\nu}[~\lambda~],~x_{\nu}[~\lambda~],~y_{\nu}[~\lambda~]~\\} = \\{~\lambda,~\text{Cos}[~\lambda~],~\text{Sin}[~\lambda~]~\\}.$$ The calculation of the corresponding " _tangent_ " value proceeds (roughly) via $$\begin{align} \left( \frac{d}{d\lambda}[~t_{\nu}[~\lambda~]~] \right)^2 - \left( \frac{d}{d\lambda}[~x_{\nu}[~\lambda~]~] \right)^2 - \left( \frac{d}{d\lambda}[~y_{\nu}[~\lambda~]~] \right)^2 & = & \\\ 1 - \left( -\text{Sin}[~\lambda~] \right)^2 - \left( \text{Cos}[~\lambda~] \right)^2 & = 0\end{align}$$ for all values $\lambda$ in the domain of " _null curve_ " $\nu$. Now, interestingly, for any two distinct values $\lambda := a$ and $\lambda := b$ from the domain of "null curve" $\nu$, there exist curves (which in the following are suggestively called "chord curves") $$\kappa_{ab} : [~a, ~b~] \subset \mathbb R \rightarrow \cal M,$$ such that $$\kappa_{ab}[~a~] = \nu[~a~], \qquad \kappa_{ab}[~b~] = \nu[~b~]$$ and: the " _tangent_ of $\kappa_{ab}$" is everywhere **positive**. As one concrete case consider $$\mathbf r \circ \kappa[~\lambda~] := \\{~t_{\kappa}[~\lambda~],~x_{\kappa}[~\lambda~],~y_{\kappa}[~\lambda~]~\\} = $$ $$\\{~\small{\lambda,~\text{Cos}[~a~] + \left( \frac{\lambda - a}{b - a} \right) \left( \text{Cos}[~b~] - \text{Cos}[~a~] \right),~\text{Sin}[~a~] + \left( \frac{\lambda - a}{b - a} \right) \left( \text{Sin}[~b~] - \text{Sin}[~a~] \right)~}\\},$$ with the corresponding " _tangent_ " value $$\left( \frac{d}{d\lambda}[~t_{\kappa}[~\lambda~]~] \right)^2 - \left( \frac{d}{d\lambda}[~x_{\kappa}[~\lambda~]~] \right)^2 - \left( \frac{d}{d\lambda}[~y_{\kappa}[~\lambda~]~] \right)^2 = $$ $$\small{1 - \left( \frac{\text{Cos}[~b~] - \text{Cos}[~a~]}{b - a} \right)^2 - \left( \frac{\text{Sin}[~b~] - \text{Sin}[~a~]}{b - a} \right)^2 = 1 - 4 \left( \frac{\text{Sin}[~(b - a)/2~]}{b - a} \right)^2 \gt 0.}$$ (As a sidenote: similarly, for given distinct values $\lambda := a$ and $\lambda := b$ from the domain of "null curve" $\nu$, one may ask for "chord curves" whose " _tangent_ " value should be everywhere **negative** ; but I have not been able to construct a corresponding concrete example. In the following, it is not necessary to distinguish "positive chord curves" from "negative" ones, if cases of the latter exist at all. Relevant is only that the " _tangent_ " value of a "chord curve" exists everywhere and is **not "null" anywhere**.) **My question is:**   Is there some specific name (i.e. more distinctive than "just another _null curve_ ") referring to " _null curves_ " which do **not have any** "chord curves" at all? (Are such special cases of " _null curves_ " perhaps called " _null geodesics_ "?)


How to theoretically determine the angular momentum of an atom?

To determine if an atom is a boson or a fermion I have to count the fermions that constitute the atom (protons, neutrons and electrons). My question is: How to theoretically (as opposed to experimentally) determine the total angular momentum of an atom? Can I just sum the spin of all the electrons and the nucleus spin, and the orbital angular momunta of the electron orbitals? For example: A Hydrogen atom in the ground state of can be considered like a spin 1 particle?


Apparent violation of Newton's 3rd law and the conservation of angular momentum for a pair of charged particles interacting magnetically

Consider a system of the two identical point positive charges situated in the space (isolated from influence of any other external fields) as shown in the figure.Particle1 is at (a,a,0) and Particle2 is at (0,a,0).Velocity of them ( at the instant considered ) are as shown in the figure. Now by applying Biot- Savart law we find that magnetic field due to Particle2 at the position of Particle1 is along +ve Y axis. And thus the force acting on Particle 1 is along +ve Z axis.(By Fleming’s left hand rule) Now similarly analyzing the force on particle 2 we find that no force is there on Particle 2. Now if we check net torque on the system of the two particles about Y axis then it is non zero and along the –ve Y axis…We have no external forces or torques acting on the two particle system considered , still net external torque on the system is non zero…WHY ? also NLM3 seems to be violted ...WHY? I know that the electromagnetic field itself takes away some momentum and moment of momentum about the considered axis but I think that if I consider only two charged particle as my system than the Abraham Lorentz force can be assumed to be acting upon the system and that is sufficient to make sure that we have considered the momentum being carried away by the electromagnetic field itself. Even after considering the action of the AbrahamLorentz force for the two particle system, both the Newton's third law, the angular momentum conservation rule and also the linear momentum conservation rule seem to be violated..... Because the Abraham Lorentz forces do not exactly counter balances the force and the torques on the considered two particle system due to simple magnetic field..... ![enter image description here](http://i.stack.imgur.com/0weBR.jpg)


How can things be chaotic on a quantum level, yet tangible on a classical level?

This may seem basic, but I am wondering if anyone has any input on this topic. It doesn't make any sense to me (I mean I don't need to use the Schrödinger equation to find my cell phone...). I just do not understand how things can appear to be highly statistical when observing the rudiments of matter, yet when observing the large scale result, very consistent (for all practical reasons). How can things with no position make up something with a position? Is it possible that our detection methods are flawed beyond what Heisenberg suggested? Is there some intuition here that I could be missing?


Wall stress of a hexagonal pressure vessel

**Problem:** I want to calculate the stress in the walls of a hexagonal pressure vessel but I can't manage to get coherent results. For long vessels, cylinders are supposed to have the lowest hoop stress but I obtain out-of-wall stresses with close or lower magnitudes with my method, which is wrong... **Wrong method:** 1) I cut the cross section where I assume the max stress will be 2) drew a free body diagram where $\sigma$ is the unknown and $F_p$ is the outward force that the pressure generates on each wall. I considered 2 options, one where the pressure is always normal to the walls (not represented), and one where it's omnidirectional coming from the center (represented). I am not sure about which one applies. 3) Respectively, $F_p=p*h*a$ and $F_p = 2\int^{30°}_{0°}p*h\frac{a}{2}cos(\theta)d\theta$ ; $F_{\sigma}=\sigma *h*t$ 4) When projecting on X and Y, $\sigma$ appears to be out-of-wall (no Y component), and considering only $2*\sigma_x*cos(30°)*h*t=2F_p$, I end up with 87MPa and 38MPa at the corners (lower elsewhere) compared to 75MPa for a cylinder (t=1mm,a=375mm)) - which does not sound right at all. a is the length of a side, h the height of the vessel (flat ends), p the internal pressure, t the thickness of the walls and $\theta$ the angle between the pressure vector and the normal to the considered face. ![enter image description here](http://i.stack.imgur.com/fKQUL.png) **Question** What is the right method?


Will a sound composed of the frequencies 450Hz, 650Hz 850Hz have a clearly defined musical pitch? Why?

According to my lecturer, the perceived pitch of a sound composed of the following harmonics: 750Hz, 1000Hz, 1250Hz is equal to the fundamental frequency which is the highest common factor of the harmonic frequencies; so 250Hz. He also said that the harmonic frequencies 450Hz, 650Hz 850Hz do not have a clearly defined musical pitch. How can this be true when the harmonic frequencies have a highest common factor and therefore a fundamental frequency of 50Hz?


Notation for Propability Amplitudes

I've recently stumbled upon a certain piece of notation that doesn't quite seem clear to me. When discussing the amplituhedron, my teacher mentioned the relation between the volume and the amplitude scattering, and then he wrote the following on the board: $$[1-,\,1+,\,3-,\,3+]$$ I assume these are probability amplitudes, but I don't quite what the integers and the signs mean.


Effective Mass and Fermi Velocity of Electrons in Graphene:

In graphene, we have (in the low energy limit) the linear energy-momentum dispersion relation: $E=\hbar v_{\rm{F}}|k|$. This expression arises from a tight-binding model, in fact $E =\frac{3\hbar ta}{2}|k|$ where $t$ is the nearest-neighbor hopping energy and $a$ the interatomic distance. But how does one know that the Fermi velocity is given by $v_{\rm{F}} = \frac{3ta}{2}$? Normally one would use $m_{\rm{eff}}^{-1} =\frac{1}{\hbar^{2}} \frac{\partial^{2}E}{\partial k^{2}}$ and $v_{\rm{F}} = \frac{\hbar k_{\rm{F}}}{m_{\rm{eff}}}$, but in this case $m_{\rm{eff}} = \infty$.


Cavity dynamics from traveling waves to standing waves

This is a question on mathematical physics. The conventional reasoning about the characteristic wave modes in a cavity is to apply the boundary conditions, namely, wave amplitude is equal to 0, and obtain discrete cavity-mode frequencies. My question here is, supposing we start by generating some arbitrarily shaped traveling wave inside the cavity. If this traveling wave contains both cavity-mode and non-cavity-mode frequencies, after bouncing around inside the cavity for while, how are those cavity-mode "standing" waves kept and other non-cavity-mode waves disappeared? In a math-physics setting, the dynamics obey a usual wave eq, with proper boundary (decaying) conditions. This is solved and the full dynamics in the solution should give us the answer. Has this problem already been fully treated and discussed in some math-physics books? if anyone knows such references, please kindly point me to them. Thanks! [Modification] \--- Below is the revised question --- Let's consider a Fabry-Perot type wave filter, in which the input waves enter from one side and leave on the other end. Due to the constructing and destructing superposition at the output end, the output waves only take certain resonant frequencies. For example, $\omega_f = n\pi c/d$, where d is the distance between the plates. In this set-up, input waves would be expanded over various non-resonant modes, which are in the functional space larger than the cavity-mode space. My question is: as the decay rate $\gamma \rightarrow 0$, how do non-resonant modes "disappear" at the output end? I have assumed the decay is so weak that it wouldn't be responsible for the "disappearance" of non-resonant modes. So my original intention of the question, was to look into a detailed mathematical solution to this process. By doing that, I might better understand the dynamics of "disappearance" of non-resonant modes. I believe the actual mechanism must be the conversion from non-resonant modes to resonant modes. Another interesting thing is, this dynamic process should be independent of the specific material that the reflecting plates are made of.


What is the significance of Planck force?

I have been curious to find what could be the significance of Planck force? It is calculated by the formula $c^4/G$, where $c$ is the speed of light; $G$ is the gravitational constant. Thus (the speed of light^4) / gravitational constant = 1.21031359 $\times 10^{44}$ newtons.


Integration of Dirac propagator

I've got confused with Dirac (or Spinor) Propagator. Everywhere in books I have examples of integration Klein-Gordon propagators, which are quite easy. But I don't understand how to integrate 4-component function. My second question - how to deal with a mix of Dirac and KG propagators in one equation? I know that if we have many scalar vertices we have to regularize our integral. And examples are shown in lot of books - Peskin, Maggiore, many more. But I couldn't find case with Dirac propagators. My equations were with: $$S_a(x_1-x)D(x-y)S_b(x-y)S_a(y-z)S_b(z-y_1)D(z-y_2)$$ Where $x_1$ were position of initial particle (it's 1->2 decay process) and $y_i$ are positions of final particles. $x,y,z$ coordinates are for internal vertices. We started from a-fermion particle (that's why there is $S_a$ propagator) and we ended with b-fermion and $\phi$ - scalar particles. Interaction Hamiltonian contained only these 3 particles. If there is required, I can even post Feynman diagram here for process. EDIT: Here are the two Feynman Diagrams for decay process, in third order of perturbation. Described problem was for the first one: ![enter image description here](https://fbcdn-sphotos-h-a.akamaihd.net/hphotos-ak- prn2/v/t34.0-12/10363750_801409816536818_1693252894_n.jpg?oh=ef70b424d50f193856bc928c48909a19&oe=537F435A&__gda__=1400883615_ce6af1cbc461f014e8453ea50db92632) Ah: our theory is $\mathcal{H}_I=k\varphi\bar{\psi_b}\psi_a$


Why does strong interaction increase with distance?

I read numerous times that strong interaction increases with distance. But how can one actually derive the force-distance relation from the lagrangian (quark field + gluon field + gauge coupling)? Since I can't use perturbation theory, I don't know what to do with it.


How does one refute a Machian mechanism for inertial emergence?

**Introduction:** Consider the diagrams representing the duality between the weak and strong principles of equivalence. Now based on how these diagrams were taught to us (at least how they were taught to me); each of the two graphics are displayed for the sake of demonstrating how two separate situations are actually physically indistinguishable for the inside observer (Einstein himself). ![enter image description here](http://i.stack.imgur.com/QwjKc.png) After careful consideration and analyzation, I claim that these diagrams in no way admits any information which can be used to suggest that in each diagram the two scenarios are in any way physically or _situation-ally nonequivalent_. That is; free fall may not be described as any sort of new class of inertial motion as the Wikipedia article on "General Relativity" suggests. Equivalently gravitational forces may not be thought of as a new class of accelerated motion. In my opinion this idea that free fall represents a new class of inertial motion is simply an artifact of the assumption that gravitational fields are not present at a distance far removed from a gravitational source. Essentially I believe all frames in which we don't feel inertial forces _must be free falling frames_ (such as orbit or flat space drift) Even though it initially doesn't seem that way (Since the gravitational source cannot be seen from a far distance). Perhaps this is already considered obvious to some but I know it is not to everyone (Since so much work is being done to understand inertia by other means...such as say radiation reaction forces(1)) I don't want anyone to misinterpret my meaning; I COMPLETELY agree with the principle of equivalence, in fact I believe it is far more powerful than most physicists give it credit for. I only claim that the diagrams are _misleading_ because each diagram uses two seemingly different yet _already fully equivalent_ scenarios as a means to describe how they are actually (surprise!) physically equivalent from the point of view of the inside of an elevator _but makes no claims about the equivalence on the outside as well_ ; In each diagram there is an indirect suggestion being made; that the two situations are _actually distinguishable_. It is as if Einstein pops his head out the door and says "Oh I thought I was free falling, turns out my elevator was just drifting in locally Minkowskian space-time, silly me". I say he was always free falling, just not in the traditional way. In my opinion all Minkowskian frames are free falling ones. The geodesic equation says that mass-energy follows its natural geodesic path through space-time but mass can only be measured via force probe if there is a resistance to that natural geodesic path. When you accelerate an object in a way which admits an inertial property of matter(One that is resisting its natural geodesic path through space-time) such as... the pushing force on our backsides as we accelerate in car, it MUST be the result of an objects coupling to the gravitational sources throughout the universe (and Earth too!) being resisted. This is essentially the meaning of a Machian principle of equivalence. $\frac{d^{2}x^{\lambda }}{d\tau^{2} }+\Gamma ^{\lambda }_{\mu \nu }\frac{dx^{\mu }}{d\tau}\frac{dx^{\nu }}{d\tau }=0$ For the sake of clarity I will augment the geodesic equation to demonstrate how inertia may be calculated as an Machian effect of changing the natural geodesic path of an object. (The point here is to show that the connection coefficient is BUILT from all mass energy in the universe.) $m\frac{d^{2}x^{\lambda }}{d\tau^{2} }+m\Gamma ^{\lambda }_{\mu \nu }\left ( \left ( T_{\mu \nu } \right )_{Universe} \right )\frac{dx^{\mu }}{d\tau}\frac{dx^{\nu }}{d\tau }=F^{\lambda }_{inertial}$ Also when most of us speak of the geodesic equation, we refer to it's description of an object in an inertial frame since it is always equated to zero but the equation is equivalent to the four force equation which is not necessarily zero. This is precisely why inertia manifests. When an object is free falling. $F^{\lambda }_{inertial}=0$ When an object is feeling inertial forces. $F^{\lambda }_{inertial}\neq 0$ (1) I intend no disrespect to the legitimacy of radiation reaction forces or to those undertaking research in radiation reaction forces, only that if there are radiation reaction forces then they are available here, raining upon us on the surface of the Earth all the time since we are always standing in an accelerated frame. I would even go as far to say that radiation reaction research may serve as a means to understand inertia and gravitation far better than we currently do. **Conclusions:** 1a) Weight and inertia are one-in-the-same as told by the strong principle of equivalence. 1b) Applied forces and gravitational normal forces are one-in-the-same as told by the strong principle of equivalence. 2) Free fall, free drift and orbit are one in-the-same as told by the weak equivalence principle. 3) An objects mass-energy content can only be defined by effective measurement of an obstruction (by a force probe) to the object's natural geodesic path through space-time. For this reason mass-energy MUST be defined this way. 4) The principle of equivalence is Machian by virtue of the augmented geodesic equation. (NOT IN THE SENSE THAT MACH ORIGINALLY PROPOSED ~ SE USERS: PLEASE DO NOT REFERENCE THIS VERSION OF MACH'S PRINCIPLE WHEN COMMENTING OR ANSWERING BECAUSE IT IS NOT THE CONCERN HERE) Yet most physicist claim any sort of Machian principle is erroneous. Additional note: When I approached my professor with this information; I originally suggested that if all objects with mass-energy in the universe (which share time-like separations) contributes to the natural geodesic paths of every other object in the universe. Then inertia can be described in terms of a Machian framework due to the principle of equivalence itself. He claimed that my argument was useless since that no one has been able to explain WHY mass energy contributes to space-time curvature. While I agree that the mechanism which somehow curves space-time due to the presence of mass energy is not well understood, I disagree with him that the information is pointless since these ideas must remain valid once we finally do figure out the mystery behind gravitational mechanism. The certainty of a Machian process for inertial emergence should not change once we understand gravitational mechanism. In my opinion, the understanding of gravitation in terms of a Machian principle of equivalence seems beneficial for solving new problems or setting new boundary conditions. **Questions:** 1) Where is my logic/thought process flawed? 2) How can a Machian principle of equivalence of this type be refuted? (Since it is explicitly described by the geodesic equation and the principle of equivalence.) 3) If an object's mass energy content can only be defined by a measure of an obstruction to its natural geodesic path through space-time, then how can two separate interactions such as electromagnetism and gravitation both generate by experiment the exact same values for an objects mass-energy if the two interactions are not fundamentally one in the same? (Let me say that the lack of measurable gravitational forces and the availability of measurable electromagnetic forces at microscopic scales compared to macroscopic scales where the opposite is true seems all too artificial.)


Interpretation Klein-Gordon equation

What is the problem if we try to interpret KG equation as a single particle equation? Also I wish to know whether the born interpretation of wavefunction is applicable in relativistic quantum mechanics.


Why search for theoretical particle?

Just out of curiosity, is there any special purpose in human life, the search for some theoretical particle for its uncertain benefit, to prove theory of everything which doesn't serve any advancement towards humanity (yet), using huge amount of resources which is better of allocated to fund research for alternative renewable energy? To enable warp drive? Anti-matter as alternate fuel? To know the ultimate answer of universe, life and everything?