title: Reaction force in electron spin measurements

text: Consider the following (thought) experiment, where an electron is emitted, then deflected by a magnetic field, and then detected: ![enter image description here](http://i.stack.imgur.com/SKqat.png) Because the momentum of the electron changes when it gets deflected, it seems intuitively clear that there should be a reaction force on the magnet, which could in principle be detected. But now consider the following modified setup: ![enter image description here](http://i.stack.imgur.com/motIz.png) Here electrons are emitted, with their spin aligned in some arbitrary direction. The first Stern-Gerlach magnet, A, can deflect electrons either up or down. B and C deflect the paths of the two beams, and the final magnet, D, combines them back into a single beam, effectively reversing the action of A. This system of magnets is followed by a detector, which can measure the spin of the electron in the $z$ direction, perpendicular to the plane of the rest of the diagram. According to my understanding of quantum mechanics, the measurements from this detector should be the same as if the magnets were not there. That is, if the electrons are emitted with their spins always pointing in the $z$ direction, the detector should also find that their spins are always 'up' in the $z$ direction. The problem is that if we can measure the reaction force on any of the four deflecting magnets then we can work out which of the two paths the electron took, and thus we would have measured its spin in two directions simultaneously. Clearly this isn't possible, so where have I gone wrong? Is there a reaction force on the magnets? If there isn't, what happened to the conservation of momentum? Is it just that the reaction force does exist but we can't measure it? (And if so, what prevents us from doing so?) Or have I just made a mistake in reasoning about how this setup would behave?


title: Optical Raytracing by using Adiabatic Hamiltonian Method

text: I'm looking into raytracing a Lüneburg Lens which is a gradient index (GRIN) optical element with a radially varying refractive index: $$ n(\rho)=n_0\sqrt{2-\left(\frac{\rho}{R}\right)^2}, 0<\rho<R $$ The good paper by Hanhong Gao et.al, "Design of thin-film photonic metamaterial Lüneburg lens using analytical approach" describes the theoretical background and applies it to the Lüneburg lens, but does not indicate any code which can be used to carry out the raytracing. I'm considering to try out an implementation in Mathematica, however if there is any comparable example of the application of Hamiltonian optics raytracing this would give me a head start. In the book "Geometric Optics" by Antonio Romano has provided an online site of Mathematica notebooks which were offering some implementation of the Hamiltonian formulations but the Link to these notebooks seems to be broken. Any info about already existing implementations on Optical Raytracing with Hamiltonian Method would thus be most welcome!


title: How to make the bubbles in my soda drink last longer?

text: I would like to make the bubbles in my soda drink last longer. For example, one good thing to do is to keep the bottle shut when you don't drink. But what else I can do? Should I try to minimize the surface between the drink and the air from the bottle? How about the surface of contact with the bottle? The bottle should be made of glass, or plastic? Does the temperature matter? Should I keep the bottle in a vertical position, to ensure more pressure at least for a part of it? Other, less realistic methods, are to keep it under pressure all the time, use a carbonating device, or drink it on the ocean floor. Or never open the bottle :), but assume that I want to open and drink from time to time.


title: Bug in linear thermal expansion, $L_0$ must be $0$

text: Assume we change the temperature of an object with negligible size in $2^{nd}$ and $3^{rd}$ dimensions from $T_0$ to $T_1$ to $T_2$, with all of them pairwise different. We choose a substance with coefficient $\alpha \ne 0$. $\alpha L_0(T_2 - T_0) = L_2 - L_0 = (L_2 - L_1) + (L_1 - L_0) = \alpha L_1(T_2 - T_1) + \alpha L_0(T_1 - T_0)$   $\Leftrightarrow L_1T_2 - L_1T_1 + L_0T_1 - L_0T_2 = 0$   $\Leftrightarrow (T_2 - T_1)(L_1 - L_0) = 0$   $\Leftrightarrow \alpha L_0(T_2 - T_1)(T_1 - T_0) = 0$   $\Rightarrow L_0 = 0$ Where is my mistake?


title: Franck and Hertz graph function

text: I need a function whom graph is the following (the Frank and Hertz experiment graph): > ![](http://i.stack.imgur.com/WifSY.png) Currently I wrote the following function:               function comptuteValue (value) {              var period = 4.9           , low = 6           ;              function addOrSubstract (x) {                  if (x < low) {                 if (x <= period) {                     return Math.pow(x, 3/2);                 } else {                     return Math.pow(period + low - x, 3/2) - x * 1.1;                 }             }                  return addOrSubstract (x - period);         }              return (2 * (Math.floor(value / period + 0.1)) + addOrSubstract(value) * 1.1989).toFixed(2);     }      And the graph looks like: > ![](http://i.stack.imgur.com/gPPca.png) As you can see there are differences between the two graphs. What's the function of the Franck and Hertz experiment graph? $f(x) = ???$ For example: $f (0) = 0$ $f (4.9) = 12$ $f (5) = 6$ And so on.


title: Why the wave-particle duality cannot be explained as a traveling-standing wave duality?

text: This would explain why speed and position cannot be measured at the same time, since either the wave would be traveling (speed) or enclosed and standing (position). The act of enclosing it (to be observed) could explain the effects of observation as well. I asked a similar question some time ago and apparently it was deleted. In the previous question I asked for the name of the theory that holds this. Apparently there was something I misunderstood, there is no such a theory and this is impossible for some reasons that were not specified (because that was not the question). So that is the question now. I understand the reasons may be long to explain. If that is the case then a reference should be enough and the pointer would be greatly appreciated. If it is possible to provide a short explanation then the explanation will be even more appreciated. In any case thank you very much for your answers and sorry for my ignorance.


title: Double Compound Pendulum: why use inertia about the center of mass for bottom pendulum?

text: I'm trying to wrap my head around the kinetic energy of a double compound pendulum, like the one shown in the Wikipedia article on double pendulums. I know for computing the kinetic energy of the first (top) pendulum you can either use the angular velocity with the moment of inertia calculated from the end of the rod (i.e., $\frac{1}{2}I_{1,end}\dot\theta_1^2$), OR the linear plus angular velocity about the center of mass (i.e., $\frac{1}{2}M_1 v_1^2+\frac{1}{2}I_{1,cm}\dot\theta_1^2$). This makes sense to me because this is basically equivalent to applying the parallel axis theorem to change the moment of inertia from the center of mass to the end of the top pendulum. However, it's not so clear to me why this works for the second (bottom) pendulum. The talk section of the article says it's a lot simpler to model the kinetic energy of the bottom rod by summing the linear plus angular velocities about the rod's center of mass (i.e., $\frac{1}{2}M_2 v_2^2+\frac{1}{2}I_{2,cm}\dot\theta_2^2$) where the position of the center of mass is a function of the two generalized coordinates, $\theta_1$ and $\theta_2$. Can somebody please explain to me why this works? I can understand computing the linear kinetic energy about the center of mass, but it's unclear why the rotational energy is also chosen to be computed at the center of mass. Is this also just an application of the parallel axis theorem, but about some rotational axis which is more difficult to imagine? All help appreciated! Thanks!


title: Why do nucleons feel a repulsive force when less than 1 fm?

text: My Modern Physics textbook by Taylor states that when nucleons are less than 1 fm apart, there is a strong repulsive force between them. I am fairly certain that it is not the Pauli Exclusion Principle that the book is talking about. So if this is true, what is the mechanism for such a force?


title: What determines the angle of the cushion on a pool table?

text: If you look at the cushions (bumpers) on a pool table, you'll see that they're not vertical. They're tilted inwards. About 10 years ago, I came across a physics exam in which one of the problems stated a set of physical assumptions and then asked the examinee to determine the optimal angle so that if a ball was rolling without slipping when it hit the cushion, it would rebound such that it was again rolling without slipping. The implication seemed to be that this was the angle used on actual pool tables. I didn't have access to the examiner's own solution, and when I analyzed the problem myself, I found that I couldn't reproduce the actual angles found on pool tables. There seems to have been some speculation that somewhere back in the mists of time, somebody set the angle empirically by maximizing the rebound distance for balls incident along the normal, and that this angle was well explained by this type of calculation. In fact, high-speed video (Mathaven 2009) shows that in snooker, a ball that hits a cushion along the normal _doesn't_ rebound by rolling without slipping; it slips for about 0.1 s before the torque due to kinetic friction brings it into the non-slipping state. (I don't think the physical situation is vastly different between pool and snooker.) Some data. Let $r$ be the radius of a pool ball, and let the angle of the cushion be such that the point of contact between the ball and the cushion is above the center of the ball by a height $b$. Real pool tables have $b/r\approx 0.26$. In snooker, the coefficient of kinetic friction, as measured by Mathaven, is $\mu_k \approx .18-.24$, and the deceleration due to rolling resistance is about $a/g\approx 0.0127-0.0129$. My question: Is there any physical explanation for the angle of the cushions on a pool table? It doesn't seem to be optimal for normal incidence, but is it possible that it's optimized in the sense of an average over all possible angles of incidence? Are there models and possibly computer simulations that are sufficiently sophisticated to address this kind of question? Mathaven, S., et al., "Application of high-speed imaging to determine the dynamics of billiards," American Journal of Physics Vol. 77, No. 9, pp. 788-794, 2009. http://billiards.colostate.edu/physics/ajp_09_hsv_article.pdf


title: Why can I tell a flute from a trumpet?

text: The usual story I've heard describing the difference between a 440 Hz note played by a flute and a trumpet is that the overtones are different. That is, if you play a note at 440 Hz, there will also be Fourier components at 880 Hz, 1320 Hz, etc. The difference in the relative amplitudes of those higher harmonics constitutes the difference between instruments. However, if you mess around with a Fourier applet (like this one, the first from a Google search), I think you'll quickly become convinced that you can't make a sound like a trumpet, flute, or pretty much anything other than an electronic box with it. What's the essential difference between the idealized sound source and the real sound source?


title: Explaining Arrow of Time with Entropy

text: WARNING: Possibly a naive post. Hi guys. I just watched on BBC, [some show] yesterday (sunday) where the host talked about Arrow of Time, where by 2nd law of thermodynamics states that "nature" is always low entropy to high entropy OR "ordered/structured" to "unordered/unstructured" state. The "intiutive" examples used, also found elsewhere, are events like iceberg "spontaneously" melting, but never freeze up again; perfume diffuse out of bottle but not entering back in it on its own; etc. Then what about the other things that appear to naturally create order like formation of cells and life, crystalization, snowflakes etc? Am I missing something?


title: What gonna happen to beer on the moon?

text: Let's assume that we have bottle of beer on the Moon. What would happen to it in sunlight? Some cases:   1. Beer can be dark or lager.   2. It can be contained in glass bottle (transparent or dark) or aluminium can. Will it freeze or vaporize or what?


title: Spin 1/2 finite-difference field simulator?

text: Is there a finite-difference field simulator for spin 1/2 fields, something like meep for electromagnetism (spin 1)? Looking for something free (GNU, MIT or other open/free style license) and easy to set up and run simulations for educational purposes, not a heavy research tool.


title: Criteria for stable equilibrium

text: A homogeneous wooden bar of length 10 cm, thickness 4 cm and weight 1 Kg is balanced on the top of a semicircular cylinder of radius R as shown below. What is the minimum radius of the semicircular cylinder if the wooden bar is at stable equilibrium? ![enter image description here](http://i.stack.imgur.com/0C9r6.jpg) What should be the general approach for solving these type of problems?


title: Which device amplifies the optical signal in fiber optics?

text: My teacher ask me one question. How the optical signal or information was transfered in fiber optics communication? Then i explained it and use the word optical amplifier then teacher ask me what is optical amplifier i said which amplifies optical signal then again which device amplifies the optical signal in fibre optics communication? I was dump at that moment i dont have the answer for it and i google it then i read about Doped fiber amplifier(DFA) but i didn't get it so can you give me any suggestion to understand it like any website or explain it.


title: Should the term Watt's Law be used?

text: I'm revising some electrical curriculum for a technical training program. In the curriculum students have to calculate values using Ohm's law and the equation Power = Current * Voltage (or P = IV). Some of my coworkers, who do not have science backgrounds, have started calling the equation P = IV "Watt's Law." When I told my co-worker it was appropriate to call P = IV the power equation she told me I was crazy and "everyone is calling it Watt's Law" according to her internet research. Am I going crazy? I've only every heard P= IV referred to as the power equation (as it applies to circuits). I've never used the term "Watts Law" in the 10+ years I've been studying and teaching physics. An if I were to call something Watts law it would be in reference to content in an energy unit not an electricity unit.


title: Orbits of maximally entangled mixed states

text: It is well known (Please, see for example Geometry of quantum states by Bengtsson and Życzkowski ) that the set of $N-$dimensional density matrices is stratified by the adjoint action of $U(N)$, where each stratum corresponds to orbits with a fixed type of degeneracy structure of the density matrix spectrum. These orbits are flag manifolds, for example the orbit of density matrices of spectrum $(1, 0, 0, 0)$ (pure states) is $\mathbb{C}P^3$ and the orbit of density matrices of spectrum $(0.5, 0.5, 0, 0)$ (and also $(0.4, 0.4, 0.1, 0.1)$) is the complex Grassmannian $Gr(4,2, \mathbb{C})$. These spaces - being coadjoint orbits - are known to be Kahler-Einstein. An observation by Ingemar Bengtsson which was stated and proved in the following article, asserts that in the composite system of two $N-$ state quantum systems $\mathcal{H}^N \otimes \mathcal{H}^N$, whose orbit of pure states is $\mathbb{C}P^{N^2-1}$, the orbit of maximally entangled pure states is $\mathbb{R}P^{N^2-1}$, which is a minimal Lagrangian submanifold.This observation was also mentioned in the first reference. My questions:   1. Does this result generalize to non-pure states, for example is the orbit of maximally entangled states in $Gr(N^2,2, \mathbb{C})$ (the orbit of density matrices of the type $(0.5, 0.5, 0, 0, . . .)$ in $\mathcal{H}^N \otimes \mathcal{H}^N$) isomorphic to $Gr(N^2,2, \mathbb{R})$.   2. Is there a physical interpretation of this result (a question left open by Bengtsson). Update: This is a clarification following Peter Shor's remark. Among the biparticle pure states, the maximally entangled states, have the property that their partial trace with respect to one system is totally mixed (please, see the discussion following equation 22 in the second reference). As a generalization I wish to know the local orbits of the states within a fixed biparticle density matrix orbit whose partial trace has the maximal Von Neumann entropy relative to all other states in the same biparticle orbit. My motivation is that if the local orbits will happen to be the real flag Lagrangian submanifolds of the complex flag manifolds defining the biparticle state orbits (which is the case for the pure states), these manifolds have well known geometries and this can contribute to our understanding of mixed state entanglement.


title: Complete set and Klein-Gordon equation

text: In http://www.physics.ucdavis.edu/~cheng/teaching/230A-s07/rqm2_rev.pdf, it says that when there is some external potential, the Klein-Gordon equation is altered, and it says the following: > The solution $\Psi$ can always be expressed as a superposition of free > particle solutions, provided that the latter form a complete set. They from > a complete set only if the negative energy components are retained, so they > cannot be simply discarded. Can anyone explain what this is talking about? First of all, what is a complete set? And why does forming a complete set require negative energy components being retained? Why is the equation (solutions, I guess) need to be a complete set?


title: Graph Invariants and Statistical Mechanics

text: Many intuitive knot invariants including Jones' polynomial are inspired by statistical mechanics. Further profound connections have been explored between knot theory and statistical mechanics. I was looking if similar connection have been found for graph invariants. I am interested to know if any attempts have been made form statistical physics community to solve graph isomorphism problem. Since both knot problem and graph isomorphism are problem about testing structural equivalence there might be connection between them.


title: Quantum Eraser under Lorentz Boost

text: Suppose I am conducting the Quantum Eraser experiment. The results of this experiment are easy to understand with the traditional quantum mechanical interpretation of a pair of entangled photons. Specifically, suppose that I am performing the "eraser" part of the experiment in which one photon is diagonally polarized so that when the entangled partner passes through the circular polarizers of the double-slit apparatus it cannot be determined through which slit it passed (thus reproducing an interference pattern). Now suppose (It's like supposition _Inception_! Supposings inside supposings inside supposings.) that I Lorentz boost into a reference frame in which the particle passing through the double-slit apparatus is received at the detector (and consequently has its polarization measured) **before** its entangled partner gets diagonally polarized. It appears that there are two possible outcomes: either (a) I disagree with the results recorded by an observer in the commoving frame of the experiment (i.e. I see no interference pattern.) or (b) I must conclude that the **future** polarization of the entangled partner somehow reached back in time and changed the outcome of my measurement. So which is it? Do I get different results or do I need to embrace a further layer of quantum non-locality? This is a tough one for me to understand because of its complementary experiment: Imagine putting the polarizer much farther away than the double slit apparatus so that there is no interference pattern created (because the entangled partner does not get diagonally polarized until **after** the measurement has already occurred). In that case, I could boost into a frame where an interference pattern _should_ occur because the partner gets polarized first. (Or am I missing something? Will there be - or not be - an interference pattern regardless?) If the answer is that I get the same results as the commoving measurement, then why is the commoving frame preferred? Why is it _its_ results that are maintained in all reference frames and not the (expected) results from my boosted frame (which should be just as valid)? Also, I think that this version of the experiment is fundamentally different from the Delayed-Choice Quantum Eraser experiment and Wheeler's thought experiment since both of those experiments are easily explained by representing the particle in a different eigenbasis (namely, as being in a superposition of the interfering and the non-interfering states - instead of being in a "collapsed" state of another observable). However, I am also interested to know whether I am wrong in this estimation. Are these experiments basically the same? Can conclusions about or explanations of one of them be generalized to the others? **Edit** : Because of the question in the comments, I've added a little additional explanation for why I think the experiments are basically different. Primarily, the difference comes from the fact that both Wheeler's thought experiment and the Delayed-Choice Quantum Eraser can be explained in such a way that apparent "cause" always precedes "effect." Causality is still violated, but at least its not giving us information about the future. For instance, if we didn't know whether or not the entangled partner beam would get polarized a year from now (in the quantum eraser experiment), we could predict whether it does or not **today** by determining whether or not there was an interference pattern at the double-slits. If there exists any boosted frame in which the beam gets polarized first, then that result must carry over to the frames where it gets polarized after. So "future-telling" can happen in the relativistic quantum eraser, but not in the other two experiments. For example, in the Wheeler experiment, we polarize the photon and then _after the fact_ decide at random whether to measure its polarization or not. If we measure the polarization, we get no interference; if we don't measure the polarization, we do get interference. Often this is ascribed to a sort of "predictive" non-locality, but just as easily, we could say that the polarized state of the photon is actually a superposition of states (namely, interfering and non-interfering) and when we measure its polarization we collapse it into the non-interfering state (and if we don't, we collapse it into interference). Likewise for the delayed-choice quantum eraser: instead of thinking that the beam-splitter in the future decided the interference in the past, you could just as easily conclude that the interference (or not) in the past decided the outcome of the beam-split in the future.


title: Will a magnet loose too much strength it cut with a hacksaw?

text: Can a magnet be cut with a hacksaw, by hand, without losing its essential characteristic of being magnetic? I know i would have to be very careful with heat but I don't know how careful


title: Nicholas Kollerstrom article on the history of Calculus

text: Today, Newton´s birthday, I read an article posted in the arXiv by Nicholas Kollerstrom http://www.arxiv.org/abs/1212.2666 That basically claims that Newton did not invent Calculus. The article does not cite many works. It would be a major break from my understanding of the history of science if it was to true. Question: Does any one know of other historians having the same historical take as this article of Kollerstrom? Anything that actually got published in a peer reviewed journal would be preferable.


title: Is it possible to enlarge the output angle of a projector with outside lens?

text: I have a very small projector, with a small output angle (30 degree). In my current demand I want a bigger output angle(90 degree), so that I can get a bigger image within a given distance. Is it possible to get it by outside lens? The projector: ![enter image description here](http://i.stack.imgur.com/fp4XE.jpg) What I expect: ![enter image description here](http://i.stack.imgur.com/LOb0F.jpg) If it's possible, how to design the lens? If not, why? What would I lose? (resolution,brightness,distortion etc)


title: Path integral with zero energy modes

text: Consider the field integral for the partition function of a free non- relativistic electron in a condensed matter setting, i.e. $$ Z = ∫D\bar\psi D\psi \exp\left(-\sum_{k,ω} \bar\psi_{k,ω} (-iω + \frac{k^2}{2m} - \mu) \psi_{k,ω}\right) $$ where the action is written in Fourier representation and $\mu$ denotes the chemical potential. Now, this integral is well known to be the determinant $$ Z = \det\left(β\left(-iω+\frac{k^2}{2m} - \mu\right)\right) $$ which is equal to the product of all eigenvalues of the quadratic form in brackets. But here is my problem: > How to calculate a quadratic path integral if the quadratic form has some > _eigenvalues_ that are equal to _zero_? If the chemical potential $\mu$ is positive, then all all momenta with $\frac{k^2}{2m} = \mu$ (the Fermi surface) will represent an eigenvalue equal to zero and would force the determinant to become zero. Of course, I could just drop the problematic eigenvalues from the determinant and call it a day, but unfortunately, I would like to understand quantum anomalies a bit better and > zero energy eigenmodes are important for understanding the axial quantum > anomaly for example of the $1+1D$ Schwinger model. Fujikawa's book on quantum anomalies argues that the axial anomaly comes from an asymmetry of zero modes of the Dirac operator, but I am very confused because a zero mode would make the determinant of the Dirac operator and hence the path integral vanish. How to make sense of this?


title: Is there a nice way to write Navier-Stokes equations in exterior calculus

text: I'm considering to study some high-dimensional Navier-Stokes equations. One problem is to do write the viscous equation for vorticity, helicity and other conserved quantities. I think it might be better if it is possible to work with differential form and exterior calculus? Is there any reference that I may find somewhere?


title: Meaning of a canonical transformation "preserving" a differential form?

text: In Chapter 9 of Arnold's _Mathematical Methods of Classical Mechanics_ , we find the following definition: > Let $g$ be a differentiable mapping of the phase space $\mathbb R^{2n}$ to > $\mathbb R^{2n}$. The mapping $g$ is called _canonical_ if $g$ preserves the > $2$-form $\omega^2 = \sum dp_i \wedge dq_i$. My question is, what does it mean to preserve a differential form? If we write the mapping as $g : (p, q) \mapsto (P, Q)$, does it mean that $\sum dP_i \wedge dQ_i = \sum dp_i \wedge dq_i$?


title: Resonance of a tube of air in case of more complex shapes

text: I've been thinking about posting this question on Music Performance stack, but finally I think it fits more here since I'm interrested in technical details. The subject of resonance of a tube of air is pretty well documented for simple shapes, like cylinders (open of closed), cones, closed rectangular boxes, spheres et cetera. I even found formulas in my high school books. But I am interested in much more complex case. Let's imagine we have a solid tube with a certain shape with holes in it filled with vibrating air of frequency $f_i$. How can we predict the frequency $f_o$ of air that come outside of the tube? For now I can imagine only two ways.   * Decompose the problem into sub-problems, in this case complex shape into much simpler shapes (cylinders, cones, rectangular boxes, spheres etc). But still I have no idea how a whole in the tube would affect the frequency, I suppose we would need to take in count the placement of this whole and its size.   * Run a computer simulation of such system. This would require knowledge how does the frequency propagade through air in such solid tube. To simplify a little bit this complex problem let's forget conditions like air temperature, pressure and humidity. I'm more interrested in second approach, to create a tool that works a little bit like fuild mechanics simulation software, that help for example design heating installations etc cetera, except that this tool would be usefull for acoustic installations, architecture or instrument design. If anyone could share some knowledge usefull to understand this phenomena I would be greatfull. What I have found in school and university books or wikipedia is not enough, or I'm not searching for right words.


title: Are operators in quantum mechanics linear transformations?

text: Observables in quantum mechanics correspond to self-adjoint linear operators. If $\psi$ is an eigenvector of $\hat A$, then $\hat A\psi=\alpha\psi$ where $\alpha$ is the eigenvalue of $\psi$. Otherwise, if the initial state $\psi$ is not one of the eigenvectors of $\hat A$, then the terminal state will be one of $\hat A$'s orthonormal eigenbasis with probability determined by the projection.   1. Then what is the physical meaning of $\hat A\psi$ in this situation?    2. Shouldn't observables be more appropriately described by maps from the Hilbert space to the set of probability distributions on that space?


title: Can an elliptical orbit take the same time as a circular obit?

text: In the picture below you can see two orbits of potential pbjects. The main aspect of the orbits is that they have a collision point at the maximum of the elliptical orbit. My question is, could the object travelling elliptical orbit take the same amount of time as the object taking the circular orbit? The reason behind this question is that I am doing an analysis on the movie called Gravity. The only information provided in the movie is that the debris cloud will collide every 90 minutes (with the Hubble telescope) and that it is going at 50000 miles per hour (which is much faster that the Hubble telescope). Obviously, since they are going at different speeds, they have to be in different orbits. So this is the only possibility I have come up with where they could possibly collide every 90 minutes, or that the orbit time of the elliptical orbit would take 45 minutes, or 22.5 minutes (any factor of 90). ![enter image description here](http://i.stack.imgur.com/g99H4.png)


title: Conservation of phase space volume in Rindler space-time

text: Let us consider Rindler space-time, i.e. Minkowski space-time as seen by a constantly accelerating observer. My question is, does Liouville's theorem, i.e. the conservation of phase space volume in classical Hamiltonian mechanics, hold in Rindler space-time? In other words, for an ensemble defined on any arbitrary system of interacting classical particles and fields with a Lorentz-invariant Lagrangian, is the phase space volume at Rindler time $t=t_1$ necessarily equal to the phase space volume at $t=t_2$? More or less equivalently, if we transform a Hamiltonian system from Minkowski space-time coordinates to Rindler ones, is the result still a Hamiltonian system? I'm interested in the answer to this for two reasons. The first is that it would immediately provide an answer to another question of mine, Phase space volume and relativity. The other is that if the answer is "yes" then there is no analogue of the black hole information paradox in Rindler space-time (at least classically), whereas if it's "no" then there is, and it might be easier to think about the solution in terms of Rindler space-time than in terms of black holes. I guess the quantum version of this question would be something like "does unitarity hold for a quantum field theory defined on the Rindler wedge?" I am also interested in this question, although I haven't studied QFT so I might not understand an overly technical answer. I guess the answer to this quantum version is "no", with the lack of unitarity at the horizon being the origin of Unruh radiation -- but I'd like to know if this intuition is correct.


title: What will happen if the charge do not exist?

text: It is just a naive idea, and I want some discussions to help me think deeper. I don't know the reason of indispensability of charge, maybe some physical laws guarantee that the charge must be exist. If it is ture, could you tell me the detail about it? My interest is what will happen if the charge do not exist in our world or universe. Without charge, the electromagnetic field still exist, such as electromagnetic wave. But we cannot observe or measure it directly, because it will not interact with neutral matter. But, in fact, the electromagnetic field still has some properties such that we can know it exist, for example, by gravity. The special relativity and the general relativity still correct without charge, even though the electromagnetic field will alway be a free field without electromagnetic interaction with matter. The electromagnetic field will have gravitational effect by the general relativity, but have not other effects at all. Now, the electromagnetic field just like dark matter. It is just a inappropriate analogy. Maybe the dark matter or dark energy are just a free field but its charge do not exist in our universe, or its charge is too big or heavy to be created by natural process. Perhaps, someone has came up a similar idea and published. It is just a naive idea, and I just want some discussions or comments.


title: Charge distribution on plates

text: ![enter image description here](http://i.stack.imgur.com/8y7dY.png) $A,B,C$ are $3$ identical metallic plates. Initially, charges $Q$, $4Q$ and $2Q$ were given to $A$,$B$ and $C$ respectively. Find final charge distributions when $B$ was earthed and $A$ and $C$ were connected with a conducting wire. My attempt : Using charge conservation on $A$,$C$ $$q_3-q_1-q_2+q_4=3Q $$ Electric field is zero inside each of these(give same equation) : $$q_3=q_4$$ Also.$$V_{AB}=V_{CB}$$ $$\frac{q_1d}{A\epsilon_0}=\frac{2q_2d}{A\epsilon_0}$$ $$ q_1=2q_2$$ I know now I have to use the fact that potential of $B $ is earthed and at $0V$, but I can't seem to frame that into an equation. Any help will be appreciated. Sorry for the dirty diagram. **Edit :** Please tell if my way is correct : We can use the fact that potential is zero at infinity as well as $B$. Let a point $P$ be $x$ distance left from $A$. Then, $$V_{BP}=V_{BA}+V_{AP}$$ $$\frac{2q_1d}{A\epsilon_0}+\frac{q_3+q_4}{A\epsilon_0}x=0$$ $$q_3+q_4=\frac{-2q_1d}{A\epsilon_0 x}$$ As $q_1=0$ will violate charge conservation, we can say that as $x\to\infty$, $q_3+q_4\to 0$. Hence, $q_3+q_4=0$ in an additional constraint which enables us to solve all the four variables. Am I right?


title: Is the giant Newton's cradle in the Kit-Kat ad feasible?

text: Apologies in advance if this is too basic a question for Phys.SE. I don't want to dumb down this venerable institution. :) My wife and I just watched this TV ad for Kit-Kat where a crew of crane operators line their wrecking balls up to form an enormous Newton's Cradle. It left us wondering if the basic premise is feasible. If the wrecking balls were perfectly aligned would it work? I imagine the lack of lateral stability inherent in having only one anchor point per "ball" would be a problem, but what else (if anything) stops this from working for real?


title: Partition function of a hydrogen gas

text: Hi I have a doubt (I'm not very expert in statistical mechanics, so sorry for this question). We consider a gas of hydrogen atoms with no interactions between them. The partition function is: $$ Z=\frac{Z_s^N}{N!} $$ where $Z_s$ is the partition function of one atom. So we write $$ Z_s =Tr\\{e^{\beta \hat H}\\} $$ and we must consider sum to discrete and continous spectrum. So I'd write for the contribute of the discrete spectrum: $$ Z_{s_{disc}}=\sum_{n=1}^{+\infty}n^2 e^{\beta \frac{E_0}{n^2}} $$ but this serie doesn't converge. For continuum spectrum, I'm not be able to write the contribute to the sum, because I have infinite degeneration, so where's my mistake? I have thought that the spectrum tha I considered for energy values is for free atom in the space, and partition function maybe is defined for systems with finite volume. So, this doubt however would me ask a problem. Can whe prove that operator $e^{-\beta \hat H}$ always has got finite trace for halll phsical systems?


title: asymptotic curvature of the universe and correlation with local curvature

text: There is not-so-rough evidence that at very large scale the universe is flat. However we see everywhere that there are local lumps of matter with positive curvature. So i have several questions regarding this: 1) Does the fact that a manifold with a) asymptotic (space) curvature zero and b) local inhomogeneities with positive (space) curvature imply that there will be regions with negative (space) curvature? 2) a Region of negative (space) curvature implies dark energy in that region? 3) assuming answer to both 1) and 2) are true: does this represent an independent confirmation of dark energy? or there is somehow an geometric relationship relating asymptotic flatness to accelerated expansion (the traditional reason to introduce dark energy in the first place)? EDITED: to reflect distinction between space and space-time curvatures.


title: Time-Energy Uncertainty Principle and Operators

text: In most of examples, I notice that uncertainty principle for time & energy is given between mass & lifetime. The UP for time and energy is $$ \Delta t\,\Delta E\geq\frac h{4π} $$ where $$Δt =σ_A \frac{dt}{d\langle \hat A\rangle}$$ with $σ_A$ being the standard deviation of operator $\hat{A}$. But what operator should one define so that $Δt$ becomes the lifetime of a particle?


title: Origins of the principle of least time in classical mechanics

text: Is it possible to derive the principle of least time from the principle of least action in lagrangian or hamiltonian mechanics? Or is Fermat's principle more fundamental than the principle of least action?


title: Predicting the ratio of translational diffusion coefficients for a sphere and a sheet

text: Imagine that I have a spherical particle of molecular weight $M$, volume $V$, and some experimentally observed center-of-mobility translational diffusion coefficient $D_{sphere}$ in water. I take this sphere and compress it into a sheet with thickness $L$, keeping the $M$ and $V$ constant, and experimentally measure a new translational diffusion coefficient $D_{sheet}$. Can I ab initio predict the ratio $\frac{D_{sheet}}{D_{sphere}}$? Does the rigidity of the sheet matter?


title: Distance formula in kinematics?

text: I've always seen it $d = v_i t + \frac{1}{2} at^2$ However, I also saw it as $d= d_i + v_i t + \frac{1}{2} at^2$ where $d_i$ = initial position. This was applied to 2D kinematics. Can anyone explain this?


title: Are there any versions of LQG that claim to not violate Lorentz symmetry?

text: LQG formulations have a minimum length/area. Since say, a Planck area can always be boosted, any minimum area in space can be shrunk. Do LQG proponents worry about local Lorentz invariance violation, and if not, why not? In LQG, does considering length to be a quantum operator really get rid of the boost problem?


title: Drying up a wet object

text: I am wondering whether an object which has been wet with hot water always dries up more easily and more quickly than one wet with cold water. How much do the shape, roughness, material, structure of the object matter? Practically: if for example I needed to wash a can and a sweater, would in both cases hot water allow me to dry them up more quickly, just by means of natural ventilation?


title: Renormalization and Infinites

text: Measuring a qubit and ending up with a bit feels a little like tossing out infinities in renormalization. Does neglecting the part of the wave function with a vanishing Hilbert space norm amount to renormalizing of Hilbert space?


title: LSZ theorem for photons

text: Is there any discussion on what the LSZ theorem for photons looks like? I would rather like a discussion of this in axial gauge $A_0 = 0$ (instead of the usual Lorenz gauge), but anything would do.


title: Physical Interpretation of Relationship Between Hall Conductivity and Berry Curvature?

text: Why is the Hall conductivity in a 2D material $$\tag{1} \sigma_{xy}=\frac{e^2}{2\pi h} \int dk_x dk_y F_{xy}(k)$$ where the integral is taken over the Brillouin Zone and $F_{xy}(k)$ is the Berry curvature of the filled bands? What is the physical interpretation of this equation? Also, can we re-parametrize all of the filled states by another pair of variables $A$ and $B$ and conclude that $$\tag{2} \sigma_{xy}=\frac{e^2}{2\pi h} \int F(A,B)dAdB$$ where $F(A,B)$ is the Berry curvature with respect to the $A$ and $B$ parameter space?


title: How do you derive the Lagrangian for the Standard Model?

text: Is there a way to derive the Lagrangian for the Standard Model just like one would for Einstein's field equations for instance? Also, how do you derive the Laganrigans for QCD and QED? Is it possible to do so from first principles? In addition, what are these Lagrangians for these quantum field theories describing? An action? Thanks,


title: Beginners questions concerning Conformal Field Theory

text: I started reading about Conformal Field Theory a few weeks ago. I'm from a more mathematical background. I do know Quantum Mechanics/Classical Mechanics, but I'm not really an expert when it comes to Quantum Field Theory. So while reading through several papers, some questions appeared and I would be more than happy if someone could answer them:   1. First I'm refering to http://www.phys.ethz.ch/~mrg/CFT.pdf (which is a german document) Chapter 3.2.1 (starting at page 29). The main topic here is the Free Boson as an example of a CFT. In this chapter he tries to proof the conformal invariance of the quantized theory. He's doing this by constructing the generators of conformal symmetry. Now here comes the first question: Why does this show conformal invariance? In the classical case you show conformal invariance by showing that the integrand of the action functional doesn't change under a conformal group action (which seems reasonable). **But why does constructing generators of conformal symmetry imply conformal invariance in the quantum case?** Next I'm refering to equation (3.2.26) in the same chapter. Here he states that the equation (3.2.26) $[L_{m},\phi]=z^{m+1}\partial_{z}\phi(z,\bar{z})$ proofs that the operators L_{m} actually do implement conformal transformations of the type $L_{n}=-z^{n+1} \partial_{z}$. Why is that so? Why does this proof that the L_{m} actually implement these conformal transformations $L_{n}=-z^{n+1} \partial_{z}$? Or better: **What does he mean by "implementing a transformation"?** (What's the defintion if you want). The equation $[L_{m},\phi]=z^{m+1}\partial_{z}\phi(z,\bar{z})$ kind of looks like an eigenvalue-equation (which of course it isnt). But if you think that there's no Lie-Bracket. It looks like $z^{m+1}\partial_{z}$ being an "eigenvalue" of $L_{m}$ of the eigenvector $\phi$.   1. I'm now refering to the paper of David Tong: http://www.damtp.cam.ac.uk/user/tong/string/four.pdf (which is english). In Chapter 4.5.1 (on page 92) he adresses radial quantization. So he has a Theory living on a cylinder and maps this theory with the exponential map to the complex plane. So first of all, this seems a bit restrictive to me. I mean **why should we only consider theories living on a cylinder as the general case?** What I've heard is that that since you have a finite spatial direction on the cylinder you avoid singularities. But since I don't know Quantum Field Theory well enough I'm not so sure about this. N* _ext it's not so obvious where the actual quantization appears in this process._ * For me quantization means imposing certain commutation relations on operators/fields. And I don't see where this is happening here? Maybe it happens implicitly when you transform the energy momentum tensor from the cylinder to the plane by using the Schwartzian derivative??  I'd really be more than happy if someone can provide me with some detailed explanations. (because I'm really not an expert yet in this subject) Thanks in advance!!


title: What is the difference between the Balmer series of hydrogen and deuterium?

text: In my quantum mechanics textbook, it claims that the Balmer series between hydrogen and deuterium is different. However, I was under the impression that the Balmer series $$H_\alpha, H_\beta, H_\gamma$$ is related by the equation $$\lambda=C\frac{n^2}{n^2-4}$$ where $$ C=3646 \mathring{\text{A}} $$ $$n=3,4,5$$ Is there a equation relating the mass and the Balmer series? Any hint would be appreciated


title: Forces acting on a double pulley

text: In a double pulley system, if the mass of the object is 10 kg and the object is accelerating upwards at 2 m/s$^2$, what is the tension in the ropes? I figured the weight of the box is 100N and the upward force is 20 N. Since these forces are in opposite direction I would think to subtract them and get 80N and divide it by the two ropes. 40N However my textbook says $$2T =m(a+g)\\\ 2T = 10kg (2m/s^2 + 10m/s^2)\\\ 2T = 10kg \cdot 12m/s^2\\\ 2T= 120N \\\ T = 60N $$ Basically its saying that the 2 forces are added together. This does not make intuitive sense to me, even when looking at a diagram of a double pulley. Can someone explain this to me.


title: If humans were able to catch all sun energy reaching the earth for their use, will the climate change?

text: I guess that energy will be used up and, at the end, will contribute to heat the earth, so I see no big differences... please explain your point of view.


title: Can heat be extracted and concentrated?

text: If one could devise an answer to this question, electricity would be free for everyones disposal. (seems like a good dream but in reality free energy will be this worlds downfall)


title: conformal anomaly of free scalar in 2D

text: I'm trying to calculate the conformal anomaly $c$ of a free scalar on a 2-sphere. I've seen other, indirect ways to do this, but since this is a free theory I feel like it should be possible to see it directly as the anomalous depenedence of the partition function on the radius of the sphere. However, I'm having a difficult time actually doing this. In principle, one starts with the naive expression for the partition function, the inverse square root of the determinant of the Laplacian. This is obtained as a product of its eigenvalues, $l(l+1)$, raised to their multiplicity $2l+1$. We get: $$Z = \prod_l (l(l+1)/\pi)^{-l-1/2}. $$ This diverges, so needs to be regulated. For example, we might impose a cutoff above momenta $\Lambda=l_{max}/R$. By dimensional analysis, the regulated partition function must be a function of $\Lambda R$, although ideally it would be cutoff independent. The point of the conformal anomaly is then that there is some dependence like: $$\log Z \sim c \log (\Lambda R), $$ where, for the free scalar, $x=1$. This cannot be removed by a local counterterm, and so the partition function depends on the radius (it also depends on the cutoff, which, as far as I understand, is why we say the partition function is ill-defined). This is how it should work, anyway. But inspecting the expression above, it seems like we'll actually get something like: $$ \log Z \sim (\Lambda R)^2 \log (\Lambda R) .$$ Can anyone explain this discrepancy? Maybe someone could give a sketch of the correct way to do this calculation. Moreover, why exactly do we think of this as a dependence on the radius? Can't we just talk about the cutoff in terms of the dimensionless quantitiy $l_{max}$, without mentioning $\Lambda$ or $R$ seperately? For example, $l_{max}$ is related to the number of points in a lattice regularization, with no reference to the size of the sphere.


title: Find electric field given the magnetic field

text: Given the magnetic field as a vector, how do I find the electric field?


title: On the relationship between entropy and chaotic noise

text: I have few conceptual questions related to application of chaos in communications. Kolmogorov-Sinai Entropy1 , Kolmogorov-Sinai Entropy2 and Kolmogorov-Sinai Entropy3 KS is an entropy metric for dynamical system. Also, given as the sum of the positive Lyapunov Exponents of the chaotic system. If KS entropy is greater than 1, then it means the system is chaotic and if it is infinity then it means the system is stochastic or random. Now, in real world scenarios most of the noise is chaotic in nature. If we don't assume that the measurement noise is Gaussian or colored, then if the signal is impaired with chaotic noise then the entropy used is KS entropy. Considering an Autoregressive model of order 2 driven by chaotic noise, $\eta(t)$ generated by a Logistic Map (a discrete system) $y(t) = ay(t-1)+by(t-2)+ \eta(t)$ where $a,b$ are the coefficients. I need to show a relationship that if chaotic noise decreases, then KS entropy also decreases. Any ideas on how I can proceed? Thank you


Can we see refraction of light in the air with our eyes? I am asking that because every night I see that the light in the street moving little is that because of change in the air? I do not mean Street light interference.


title: A question about particle scattering

text: For massless spin-1/2 fermions say $N$ I am using the spinors as given here say - http://theory.fnal.gov/people/ellis/Calctools/spinor.pdf \- the $u_{+}(k)$ and $u_{-}(k)$ on page 2. So the $u_{+}(k)$ are the eigenspinors of the right chirality projector ($P_R = diag (I,0)$) and the $u_{-}(k)$ are the eigenspinors of the left chirality projector, ($P_L = diag(0, I )$). Now say I have the Lagrangian which interacts between a positively charged scalar $S^{+}$, a massles spin-1/2 fermion $N$ and another massive fermion $B$ as, $L = (\partial_\mu S^{+} ) \bar{N} \gamma^\mu P_L B + (\partial_\mu S^{-} )\bar{B}\gamma^\mu P_L N$ The two terms are Hermitian conjugates of each other.   * Now I want to know as to from knowing this Lagrangian how (or does it?) follow that this reaction is possible of ($\bar{B} \rightarrow S^{+} + \bar{N} $) ?    * Why I am not sure of the above is because of this - lets say that the $N$ are the known left-handed neutrinos (so we need to represent them as the $u_{-}(k)$) - then it should follow that the anti-neutrinos are right-handed and hence they need to be represented as the $u_{+}(k)$ spinors - right? Now a matrix element of the above reaction, $\bar{B} (p) \rightarrow S^{+}(k) + \bar{N}(q)$ will look like, $v_B(p)\gamma^\mu k_\mu P_L u_{+}(q)$ -But isn't this term starightaway $0$ since $P_L$ annihilates $u_{+}$?


title: Kolmogorov entropy and noise

text: There are various ways of calculating the Kolmogorov entropy (KE)of dynamical system. According to Pesin's theorum, it is the summation of the lyapunov exponents; frominformation theoretic concet, it is the supremum of the Shannon entropy defines for all the partitions. I am using the formula given here. > (Jaap C. Schouten, Floris Takens, and Cor M. van den Bleek , " Maximum- > likelihood estimation of the entropy of an attractor" Phys. Rev. E 49, 126) The desired signal $x(t)$ whose entropy is to be measured is corrupted with additive white gaussian noise. Say: $$x(t)= ax(t-1) + bx(t-1) + \text{Chaos_Signal}+\text{White noise}$$ I want to calculate the KE of $x$. Assuming, the chaotic signal to be logistic map, my question are (Q1) for different values of the parameters $(a_1,b_1),(a_2,b_2),\ldots$ will the KE measure be same for a particular signal-to-noise ratio (SNR)? (Q2) Will KE decrease with increase of SNR? Intuitively, entropy is the measure of randomness. So, higher the level of noise (low SNR) then greater is the entropy. Again, noise also changes the value of the Lyapunov exponents of the system. I guess, Lyapunov exponent increases. Is there any proof that shows relationship between KE of time series increasing or decreasing with noise?


title: What does a MEMS gyro measure during off-axis rotations?

text: I've been working with MEMS gyroscopes in hobbyist projects for years--but it's recently come to my attention that I don't have any principled understanding of what rotation rate measurements mean, physically speaking, and now it's bugging me. My immediate motivation for asking is that I want to understand the design decisions in Paul Riseborough's Kalman filter for UAVs (see my earlier attempt at this question). Now that I'm aware of my confusion, I have the same issue with interpreting output from the Portland State Aerospace Society's inertial measurement unit, which has a similar package of three-axis accelerometers/gyroscopes/etc. Paul states (at the above link) that it's standard in the literature to interpret the three-axis rate-gyro measurements as a kind of axis-angle representation of rotation, where the direction of the 3-vector is the axis of rotation, and the magnitude of the vector is the angular rotation rate around that axis. I just don't understand why that would be true in general, although it's clearly true in the special case where rotation is exactly around one gyro axis. I think that if someone can show me the derivation of what each MEMS gyro axis should read, given an arbitrary 3-D rotation rate, that may lead to a deeper understanding of how I should interpret these sensor readings.


title: Particles Associated With Gravitational Waves

text: I've been reading about linearized GR and the study of gravitational waves, and an odd thought popped into my head. According to wave-particle duality (admittedly, usually used in quantum mechanics!), particles can act like waves, and waves can act like particles. My logic was that, therefore, there should be a particle associated with a gravitational wave. There is one thing, though, that might interfere with this: gravitational waves are plane waves, not longitudinal waves - which is how we describe, for example, photons. But my original question still stands: Is there a particle associated with a gravitational wave? If not, why not? If so, is there a separate mathematical structure associated with it (aside from the metric for the wave)? FYI: Nope, I'm not talking about gravitons. I know those are completely different and unrelated (and, at the moment, hypothetical).


title: Why is the range for a particle thrown at 90 degree same as that of a particle thrown at 0 degree?

text: There is a derivation given in my book which concludes to the fact that the range is same for two angles of projection, provided they are complementary. I fail to understand this fact. For example, what if we put the angle as 90 and 0 which are complimentary. Practically how is the particle projected at zero degree have 0 range? Isn't rolling a ball on a floor have a range? Or maybe hitting the ball during billiards. Basically at zero degree I feel the motion would become rectilinear.whereas at 90 degree the range is zero as the object goes up and comes down. I have understood it mathematically by putting the angles in the range formula but not practically. Please help. (This question is not the same as a question I posted earlier. Since there were too many mistakes in that question I decided to post the edited question again although the I edited the question there also but no one seems to be responding.)


title: Understanding the time dilation calculation in Brian Cox's Doctor Who lecture

text: ## Background My son has just watched Brian Cox's fascinating "The Science of Doctor Who" lecture on space-time and was particularly intrigued by the part (at 22m:22s) where he said: > Let us say that we catapult Jim off at 99.94% of the speed of light for five > years, according to his watch. Then, we tell Jim to turn around and come > back. It takes another five years to get back to the Earth, so for him the > journey would take ten years. But for us, with our watches ticking faster > than Jim's, 29 years would have passed. He wanted me to show him how to calculate this, but I need help! ## Question How do you calculate that a trip for 10 years at 99.94% of the speed of light results in 29 years being passed at Earth? ## What I've tried First we tried applying a relativistic distortion based on the rocket instantly moving to the speed of light, but this resulted in Earth dwellers having aged by 288 years. Second we tried assuming the rocket had a constant acceleration (in its frame of reference) and first accelerated to 99.94%, then decelerated to come to a stop, then accelerated backwards to 99.94%, and then decelerated to stop at the Earth. This results (according to our sums based on the wikipedia page on time dilation) in the people on the rocket aging 4 years, not 10. Perhaps we have made a mistake in the calculations (shown below)? Or perhaps there is another common model of acceleration that we should use? (Perhaps one that assumes we have more fuel at the start so our acceleration is slower initially?) ## Method Python code:               from math import *     c=299792458.     v=c*99.94/100.     year=365.25*24*60*60.          print 'First assume we instantly accelerate to our final speed'     rocket=10*year     earth=rocket/sqrt(1-v**2/c**2)     print 'On Earth people have aged by',earth/year,'years'          print     t=29*year/4     g = sqrt(v**2/(1-v**2/c**2))/t     print 'Now assume a constant acceleration of',g,'metres per second per second'     print 'After 29/4 years we accelerate to a speed of',g*t/sqrt(1+(g*t)**2/c**2)     pt=(c/g)*asinh(g*t/c)     print 'This takes',pt/year,'years of proper time'     print 'so the complete journey will last',4*pt/year,'years for the spacemen'


title: A few simple questions about Grassmann numbers: commutation relations and derivatives

text: I'm trying to learn about Grassmann numbers from the book "Condensed Matter Field Theory" by Altland and Simons, but I am currently encountering some difficulties. I have several smaller questions which are connected.   * Does a Grassmann number $\eta_i$ and a creation operator $a^\dagger_j$ commute or anti-commute? Altland and Simons only defined $[\eta_i, a_j]_+ =0$. I would suggest to take the conjugate of this and arrive at an equation for the creation operator, but the authors refused to define a conjugation for Grassmann numbers as it can lead to some difficulties.   * The authors defined functions of Grassmann numbers by its Taylor expansion: $f(\eta) = f(0)+f'(0)\eta$. But how do I get the equation $e^{-\eta a^\dagger} = 1-\eta a^\dagger$ which I encountered in several notes? The derivative of $e^{-x a^\dagger}$ is $-a^\dagger e^{-x a^\dagger}$. So I would guess that $e^{-\eta a^\dagger} = 1 - a^\dagger \eta$. One solution would be that a creation operator commutes with a Grassmann number, but I am not sure if this holds.   * Can anyone recommend a good introduction to Grassmann numbers and their use in field theory which does not delve too deep into the mathematics but concentrates on performing calculations?


title: Anti-matter as matter going backwards in time? (requesting further clarification upon a previous post)

text: I understand this question has already been asked here, however, I don't have enough reputation points to place a comment (I suppose that's the reason) on a specific answer to request a reference. A user has answered (I'm not sure I can cite his name here, so I won't): > There is one technical inaccuracy in saying that antimatter moves back in > time (whatever it might mean). In quantum field theory we get positive > energy solutions (usual particles) and negative energy solutions. Negative > energy solutions behave in time as if they were propagating backward in > time. But they are not the antiparticles, they are just the "negative-energy > particles". Antiparticles are positive energy solutions, and they are > obtained by acting with charge conjugation operator on the negative-energy > solutions. So, antiparticles move forward in time, as usual particles. and, even though someone else comments > This is interpreting "going back in time" differently than anyone else does. I'd like a reference for (or further clarification on) that, since I have already heard that before (also informally). I understand this is a repost but if someone could at least give me some reference before closing it, I'd be thankful. PS: My first intention was to ask how to proceed on this case in "meta", but I don't have reputation points to place a question there. **EDIT:** To clarify further my question: I had originally the same question as the OP from the post I quoted. At first, I thought that an explanation of what are anti-particles and the interpretation as particles going back in time could be found in quantum mechanics alone, without more advanced quantum field theory. The argument I knew is that anti-particles are really related to the originally negative-energy solutions, that can be reinterpreted as positive- energy solutions, as long as one invert the sign of time in the unitary time- evolution operator. This would hence lead to the reinterpretation of such states as positive-energy anti-particles _going back in time_ or, at least, with _anomalous time dependency_. The particular answer I quoted, seems to contradict this interpretation, saying that it is a mistake. I'd like a further clarification here. Is this interpretation really wrong? From what is said on Wikipedia, looks like the _Feynman–Stueckelberg interpretation_ relies on the fact that anti-particles are to be understood as particles going backwards in time and then related to the latter by charge conjugation. Where does this interpretation primordially comes from? Where is it primordially needed if their association with the negative energy solutions of Dirac's equation (say) is a mistake? I understand that in quantum field theory one uses complex fields to describe both particles and anti-particles, relating them by charge conjugation. But isn't this done this way to accommodate Feynman–Stueckelberg's interpretation?


title: How much time has passed for Voyager I since it left the Earth, 34 years ago?

text: 34 years have passed since Voyager I took off and it's just crossing the solar system, being approximately at 16.4 light-hours away. How much time have passed for itself, though?


title: Understanding the concept of period of motion in simple harmonic motion formula

text: I have a spring system, whose position equation is $$x(t) = c_1cos(8 \sqrt{2}t) + c_2sin(8 \sqrt{2}t)$$ The textbook says it will have a period of motion of $\frac{2 \pi}{(8 \sqrt{2}t)}$. I understand roughly what a period of motion is (it's the time it takes for the system to make a complete cycle) but how is the figure $\frac{2 \pi}{(8 \sqrt{2}t)}$ determined?


title: Practical meaning of making a measurement/observation in QM?

text: When an argument like 'measure the spin along the $x$ axis', 'observe the position of a particle' and so on is made, what is the implied experimental procedure? Since laboratory equipment is macroscopic, I have a hard time relating this to measuring an observable. For example, if I have a QM harmonic oscillator, how do I actually determine which eigenmode it is in?


title: Two masses with interacting forces and an external force

text: Two masses in 3d space attract each other with a potential relative to the distance between them. There is also an external force on each particle based on the distance from a origin. I want to find the equations of motion. I thought about reducing it to the 1 body problem or use Lagrangian formalism to find the equation of motion, however, the external force kind of ruins that approach. Any ideas on how to approach this?


title: How is it that the cross product of two vectors is always perpendicular to the given vectors?

text: Vector addition, subtraction and dot product seem logical enough, but I don't understand how two vectors acting on the same plane maybe, can give a perpendicular resultant.


title: "Spooky action at a distance" or particle correlation at the time of their creation?

text: In Bell test experiments, two particles that are a singlet are measured and are found to be correlated. The correlation leads us to believe that they are somehow connected non- locally: the measurement of one particle results in affecting the other particle instantaneously. My question is this: have we ruled out the possibility that there is no transmission of state and particles are actually created in such a way that their state seems to be transmitted to each other? In other words, have we ruled out the possibility that the particles are created with all the properties required so as that when we measure them a correlation shows up? Perhaps what we think of as state transmission is nothing more than particles created with such properties that when measured after their creation, they seem to affect each other, but they actually do not, they are simply created with specific properties. I apologise for my poor language, I am neither a native English speaker nor a Physics student, just a curious mind.


title: Inside a huge sphere, which is being converted to a black hole

text: Suppose the following scenario: You reside inside a huge stable spherical star with non-lethal environment at its core. The object is so huge and massive, that its radius is only slightly above its Schwarzschild radius. Then the sphere is bombarded with high-density energy (sth. like neutron soup) from all sides, increasing the Schwarzschild radius to the point where an event horizon forms. All this would happen at the perimeter of your habitat, without you noticing it. Are you now trapped inside an event horizon (I think, yes)? Would your local metric change as soon as you "see" the horizon forming events? Or would it remain the same until the gravitational collapse "gets" you? Would it in theory be possible to generate enough outward pressure to stop the gravitational collapse, i.e. is the pressure of gravitational collapse finite? * * * The motivation of this question is the description of spherical objects by an outer Schwarzschild metric and an inner part, which is "stapled", so that it fits and solves Einstein's equations. I always asked myself, whether there was a solution which stapled a non-vacuum metric to the Schwarzschild metric _inside_ the event horizon.


title: Questions on the $N=2$ superconformal algebra

text: In my understanding, mirror symmetry in physics originates from representation of the $N=2$ superconformal algebra. Why do we need precisely 2 supersymmetries (why not 1 or 4)? Moreover, a chiral (anti-chiral) field is defined as a state that is annihilated by $G^+_{-1/2}$ ($G^-_{-1/2}$), where $G^+_{-1/2}$ and $G^-_{-1/2}$ are coefficients of the Fourier mode expansion of some anti- commuting current $G^+(z)$ and $G^-(z)$ of conformal weight $3/2$. How should I understand this chiral (anti-chiral) field? In $N=(2,2)$ superconformal algebra, there are four rings: $(c,c),(a,a),(a,c),(c,a)$. It is known that the first two are charge conjugate, but what does theism mean? Right and left-moving, and chiral and anti-chiral rings... these all confuse me.


title: How does heat pass through glass?

text: So infra-red and ultra-violet waves from the sun heat up the glass by conduction then it then radiates this heat to heat up a room or a car? Is that right? How would one calculate how much heat a type of glass keeps out?? Also how does glass coatings like tinting 'reflect' heat?


title: How does a one-sided glass work?

text: How does a one-sided glass work? If seen from the outer side, we can see through thinking it is transparent, while if seen from the inside, the glass acts as a mirror. How?


title: Variation of electric field and current flowing

text: Theoretically, a change in either electric or magnetic field will cause a current to flow , I am already familiar to Faraday's law of electromagnetic induction, so I tried to search about producing a current using a varying electric field, didn't find anything, I found an interesting apparatus though: Tesla's electrostatic induction apparatus. My question here is about Lenz's law, will this law be applicable here? I know the law is in case of magnets , but I tried to imagine what will happen if we tried the coil magnet situation where approaching coil to the coil a resisting force will develop , but if we replaced the magnet with an electret (electrostatic equivalent of permanent magnet), as it approaches the loop the side near the electret will obtain a partial charge opposing to charge of the electret causing it to approach faster (the opposite of what happens with a magnet). In the end of the article Tesla wrote > The output of such an apparatus is very small, but some of the effects > peculiar to alternating currents of short periods may be observed. The > effects, however, cannot be compared with those obtainable with an induction > coil which is operated by an alternate current machine of high frequency, > some of which were described by me a short while ago. Why was the output of such an apparatus small? Does an electric field have a smaller influence on electrons than a magnetic field?


title: Earth-Moon system

text: While learning about motion of centre of mass, I came across this statement online, > _It is also more accurate to say that the Earth and Moon together revolve > about their common center of mass._ I am not able to understand this. Can anyone please explain this statement?


title: clear picture about charge and flux

text: What is the difference between charge and electric flux?? According to my knowledge electric flux is nothing but charge enclosed by the closed surface.


title: 1-particle momentum eigenfunction in terms of field operator for real Klein-Gordon field

text: Suppose $\phi(x)$ is a real Klein-Gordon field, then the single-particle wave function $\psi(x)$ corresponding to a momentum $p$ is given by (QFT, Ryder) $$\psi_p(x)=\langle0|\phi(x)|p\rangle.$$ The calculated result is just as expected ($e^{-ipx}$). But is this by definition or does it come from somewhere else?


title: A particle has $\overrightarrow{r}(0)=4m(\hspace{2pt}\hat{j}\hspace{2pt})$ and $\overrightarrow{v}(0)=(2m/s^2)\hat{i}$

text: I am having trouble with these problems, and I want to gain a understanding of how to solve these. I'll put what I have tried at the end, even though I don't think it'll be of help. A particle has $\overrightarrow{r}(0)=4m(\hspace{2pt}\hat{j}\hspace{2pt})$ and $\overrightarrow{v}(0)=(2m/s)\hat{i}$. If it's acceleration is constant and given by$\overrightarrow{a}-(2m/s^2)(\hspace{2pt}\hat{i}\hspace{2pt}+\hspace{2pt}\hat{j}\hspace{2pt})$, at what time $t$ does the particle first cross the x-axis? $$$$And a related question: At what time $t$ is the particle moving parallel to the y-axis, that is in the $\hat{j}$ direction?$$$$So what I did is draw a vector going up $4m$ because of the $4m\hat{j}$ and wrote down $a_y=-2m/s^2$ and $a_x=-2m/s^2$ and that is pretty much it. A complete explanation of how to solve this problem would be greatly appreciated, and sorry for not having much insight, i'm just really lost with this problem. Thanks for all the help in advance. EDIT: My work for question 1 $$$$So for the first question I used $$\vec{x}_f=\vec{x}_i+\vec{v}_it+\frac{1}{2}\vec{a}t^2$$ and plugged in $$0=4m+2m/s(t)+\frac{1}{2}(-2m/s^2)t^2,$$ and ended with the quadratic function of $4+2t-t^2=0$ or $t^2-2t-4$ and found the roots at $t=1\pm \sqrt{5}$ but the answer was wrong. What did I do wrong? (I inputed both values) $$$$EDIT #2: I can't input any more times, so I cannot check my answer but I think I the answer is $2s$. In my first attempt I put $2m/s$ as the initial velocity, when it was actually 0. So I think the correct formula inputs are $0=4-t^2$ and therefore $t=2$, would that be right?


title: Mathematical Prerequisites for QFT

text: I am curious about which areas of _mathematics_ one should be comfortable with before learning QFT. I am familiar with the "learn-it-as-you-go" approach often advocated in physics, but would like to know how to avoid that in learning QFT. Naming of specific textbooks is appreciated. (For the sake of this discussion, "learning QFT" can be taken to mean "learning at the level of/from Peskin and Schroeder's text.") In particular, I would like to know what, in addition to the material in Rudin's _Real and Complex Analysis,_ one should know. Would this be sufficient? How about Folland's _Real Analysis_?


title: Oscillation and wave

text: A box of mass $0.2 \,\text{kg}$ is attached to one end of a spring whose other end is fixed to a rigid support. When a mass of $0.8 \, \text{kg}$ is placed inside the box, the system performs four oscillations per second and the amplitude falls from $2 \, \text{cm}$ to $1\, \text{cm}$ in $30 \, \text{s}$ . Calculate the force constant, the relaxation time, and the Q-factor.


title: Remote EEG Detection / Reading

text: I recently noted that Phillips, Accenture, and Emotive grouped together to create a proof of concept that would assist ALS patients to gain greater control of their lives through brain, voice, and eye commands. See this link here. While a breakthrough for ALS patients, it remains limited as such that the patient must wear a headpiece, or use close to surface biomedical implants for the EEG to be read. This made me wonder how one might 'read' EEG signals using remote means? This could be used for many purposes: those suffering with ALS, remote monitoring of elderly or special needs patients, hospitals and emergency care units, combat air pilots and other military purposes, and etc... I am curious as to how one might be able to remotely detect biological EMF signals. In my review thus far I am to understand that noise would be an issue in obtaining a clear EMF reading, as most devices require close, physical contact, or for the subject to be secured in a EMF 'clean room'. I have found a lot of patents regarding how this could be done, but always seem to be able to find a way of defeating them (some flaw in their design/theory). I am to understand that lower frequencies penetrate the body best, and the depth depends on the frequency used. I also have read that the human body works as a dipole antenna. However, I am to understand that biological matter is nonlinear, and as such signals are not modulated when passing through tissue.   * Could this be done, perhaps, by using externally applied rf signals?   * Is there any point/part in the body to which the biologically generated signals are naturally modulated, or combined (eSum) - perhaps for processing by the brain? Thanks!


title: Do wide-angle videos make the first-person view seem slower than perceived in real life?

text: I considered posting this on other SE sites such as _Audio-Video Production_ and _Photography_ , but I didn't feel I'd get the definitive, fact-based (rather than experience-based) answer I'm seeking. **Background** I used a GoPro HD Hero (which has a 170-degree view) to record a track day (driving a car) and a snowboarding trip. Playing back the videos after each event, I sensed immediately that they seemed _slow_. However, since clocks and speech seemed to progress at a normal rate, I dismissed my senses, figuring adrenaline and wind must've made me perceive _real life_ faster than it actually was. And there's probably some truth to this adrenaline-and-wind explanation, but watching the videos again today (more than a year later), I sensed the slowness as I always did. I came up with a potential, partial explanation, but still lacking; I was hoping someone might be able to shed some light as to whether this "slow video" phenomena is merely psychological, or an expected effect of optics. **Partial Explanation** I think there _may_ be an effect, due to _depth perception_ being altered by wide-angle lenses. Suppose we were to record a skateboarder from the side, _i.e._ so the skateboarder would skate from left to right in our camera's field of view. In this case, because the axis of movement (the skateboarder's "line") would be perpendicular† to the axis of distortion (the camera's "line", or depth), there would be no distortion of perceived speed: the skateboarder moves relative to an undistorted frame of reference (the ground). However, if we were to mount the camera on the skateboarder's helmet, _i.e._ record in a first-person view, then the axes of movement and distortion would be one and the same. From here though, I'm not sure how to explain why this translates to slower-seeming playback. **Question** In layman's terms, my problem is, "The ground and the trees pass by me slower," when playing back a wide-angle video.   1. Is this psychological, or is there a physical explanation?   2. Is there a mathematical formula that can tell by how much one must speed up playback in order for "the ground and the trees" to pass by as they seemed to in real life? **Footnotes** † Technically, I guess "perpendicular" would be a curve, _i.e._ staying equidistant from the camera, but for simplicity, suppose the skateboarder is far enough from the lens that we may consider the curve a "line".


title: Do viruses suffer from quantum de-localization?

text: Consider some microscopic life form. It should obviously be localized in space, in the quantum-mechanical sense, if it is treated as a single particle (though it is composite). If its characteristic length is $l$, then its wavefunction would de-localize in the typical timescale $$ \tau = \frac{ml^2}{\hbar}$$ If we estimate this for a virus (~typical diameter 100 nm), and assume roughly water mass density, we obtain $$ \tau \approx \frac{1 \frac{g}{cm^3}\cdot(100\ nm)^5}{\hbar} \approx100\ seconds$$ So technically speaking, after about two minutes, the virus had doubled its uncertainty in position. Even if the mass assumption is off by two orders of magnitude, we still obtain everyday timescales. How should this estimate be interpreted? Was is observed in biology?


title: How does velocity cause time dilation?

text: I've been trying to learn about the speed of light and time dilation, but I'm at an impasse. The presented facts say that if I travel around the solar system at 50% the speed of light and then come back to earth I will have experienced less local passage of time than them. I will effectively have traveled to their future. I've also read that gravity causes time dilation too. I understand that space-time is a sort of unified thing and that affecting space affects time. It makes sense then that gravitational forces bending space will also bend time. But velocity? I can't wrap my head around it and I can't find a good explanation for it. People cite orbiting craft and planes as proof of time dilation, since their clocks will go slower than those in the surface (and vice-versa). This certainly explains gravitational time-dilation, but not necessarily velocity. Can't the change in local time passage be caused solely by the gravitational bending of space-time? Trying to find an answer, I came to a very recurring and frustrating example in texts that seek to explain time dilation. A man on a moving train throws a ball forward. Since he's moving with the train and the train is his point of reference, the ball to him only moves at the speed he threw it. But to a woman on the station the ball is moving at the speed of the train plus the speed it was thrown with. To some authors, this seems to open the mind to time-dilation understanding. To me it only explains the relative nature of motion. It says nothing of time. Another example I've found: if person A speeds away from person B very quickly, A's clock will seem to advance slower from B's point of view. How is this time dilation, though? The difference can be explained by the longer time it takes light to get to B, can't it? I realize I can't be right against the fine physicists out there, so I was hoping someone here could enlighten me. Where does the notion that velocity causes time-dilation come from?


title: Quantum fluctuations and expanding universe

text: As far as I understand, Hawking radiation is formed at the edge of a black hole, when a particle/anti particle pair is formed and one of the particles falls into the black hole before the particles has time to annihilate. This made me think of other situations where this could occur which led me to think about the expansion of the universe. Since rate of expansion of the universe is increasing all the time due to dark energy, and at some point in the future the expansion will be so fast as to tear matter apart, wouldn't there be a point where particle/anti particle pairs that spontaneously form from quantum fluctuations be torn apart faster than they can annihilate, thus forming infinite amounts of new matter in the ever expanding universe? Can anyone with a better understanding of the maths behind this explain what would actually happen?


title: LHC Big Bang Temperatures

text: It's been claimed that the LHC's 14 TeV energy produces temperatures comparable to that which occurred very soon after the Big Bang. The well-known $E=1.5kT$ formula from classical statistical mechanics predicts LHC produced temperatures in the $2.44\times 10^{17}$ K range. This temperature is much higher than I expected. Is there an agreed upon method of calculating temperature equivalents of 14 TeV p-p interactions, and if so, what is the result?


title: I need help with finding distance traveled

text: How do I find the distance traveled of an object if the speed is not constant?


Are many-worlds and the multiverse really the same thing? Not too long ago, Susskind and Bousso uploaded the article "The Multiverse Interpretation of Quantum Mechanics" with the thesis that the many-worlds interpretation and the multiverse of eternal inflation are one and the same thing. The parallel worlds of one are exactly the same thing as the parallel worlds of the other. First, they claim decoherence can't happen over a complete description of the future light-cone of the measurement. Then, they apply that principle to eternal inflation. Without decoherence, superpositions of nucleating bubbles and metastable vacua can't decohere. According to the anthropic principle, most bubbles have no conscious observers, but an exponentially small minority do. Apply black hole complementarity to causal horizons. Then, somehow, in a way I can't follow, they combine causal diamond worlds into a global multiverse. Then they claim decoherence is reversible. My head is spinning. What are your opinions on this paper?


title: A question on smooth 1-manifolds

text: Consider two people living on two different smooth 1-manifolds $S$ and $T$ as shown in figure 1. The manifold $S$ is a bump function joining the points $A$ and $B$ and the manifold $T$ is formed by joining two identical bump functions at the points $P$ and $Q$. Let the points marked as $H$ denote the homes of the two guys $X$ and $Y$ living on the manifolds $S$ and $T$ respectively with distances (along the curve) from $H$ and $B$ (and $H$ and $Q$) being 40 km. Let $X$ and $Y$ are equipped with cars which do not have a reverse gear. Every time $X$ starts on a drive from his home $H$ it is certain that he reaches back home after a 80Km drive. But same is not the case with $Y$. It is not sure how much long a drive does he need to return back to his home or will he even ever return back to his home. My question is what kind of a manifolds are $S$ and $T$ and how are they related. Is there any paradox here which needs to be resolved. Life of $Y$ can be made more complicated by joining the points $P$ and $Q$ with some more bump functions each lying on different planes. ![figure 1](http://i.stack.imgur.com/jnuOv.jpg)


title: Hexagonal Warping

text: Hexagonal warping had observed in $Bi_2Te_3$. Is it related somehow with the topological insulator type? Is it a characteristic of weak topological insulator or are there other reasons for this phenomenon?


title: How to get the $i\epsilon$ prescription for a Faddeev-Popov ghost propagator?

text: In path integral formalism, for a physical field there will be an $i\epsilon$ term in the action, which comes from identifying the in and out vacuum, and in turn this $i\epsilon$ will naturally appear in the denominator of the corresponding propagator. However for FP ghost, it is only introduced to rewrite the functional determinant in an exponential form, and the issue of identifying an in and out ghost vacuum never enters the picture, thus no $i\epsilon$ term in the ghost part of the action. Yet all ghost propagators I've seen do have an $i\epsilon$ in the denominator, so where does it come from?


title: Friction due to air drag at high speeds

text: I am trying to set up this problem, but I am not sure how to go about doing so. (From University Physics, Young & Freedman): > You throw a baseball straight up. The drag force is proportional to $v^2$. > In terms of $g$, what is the y-component of the ball’s acceleration when its > speed is half its terminal speed and (a) it is moving up? (b) It is moving > back down? I am not sure if I am on the right track, but when they state that the drag force is proportional to $v^2$, it suggests to me that I need to use this relationship: $v^2 = \frac{mg}{D}$. Accordingly friction due to air drag is: $$f = Dv^2 = mg \rightarrow f = w$$ I am not sure how to find the y-component of the acceleration vector upwards or downwards at half the terminal speed - is the acceleration vector distinct from $g$ in this case?


title: EM force, blocking force carrier photons in a static electric field

text: I am doing some personal research in this specific area and wanted to ask something related to photons and EM force. are involved. Here is a thought experiment that doesn't add up to observed results, what's the flaw? Initial assumption: Virtual photons are the carriers for EM force even in static situations. Initial setup: Two oppositely charged parallel metallic plates are set up like a simple capacitor, inside the plates are an uniform static electric field, fringe fields are ignored. Based on the force carrier model, exchange of virtual photons are responsible for this uniform electric field. A plate of non-conductive, non-polar material should be capable of blocking any photons. In this case, the force carriers would be unable to reach their destination and there would be no electric field. But in reality the electric field passes straight through the non-conductive plate. Where have I gone wrong? Cheers Zed.


title: Is the total energy of the universe constant?

text: If total energy is conserved just transformed and never newly created, is there a sum of all energies that is constant? Why is it probably not that easy?


title: What does it mean (how is it visualized) for a particle to act as a wave?

text: I have no background in physics. This isn't for homework, just for interest. In quantum physics, it's described that a particle can act as both a particle and a wave. Quoted from HowStuffWorks "Wave-Particle Duality" > Today, physicists accept the dual nature of light. In this modern view, they > define light as a collection of one or more photons propagating through > space as electromagnetic waves. This definition, which combines light's wave > and particle nature, makes it possible to rethink Thomas Young's double-slit > experiment in this way: Light travels away from a source as an > electromagnetic wave. When it encounters the slits, it passes through and > divides into two wave fronts. These wave fronts overlap and approach the > screen. At the moment of impact, however, the entire wave field disappears > and a photon appears. Quantum physicists often describe this by saying the > spread-out wave "collapses" into a small point. I have trouble visualizing a _particle_ transforming into a _wave_ and vice- versa. The quote says that light travels away from a source as an electromagnetic wave. What does that even look like? How can I visualize "a wave"? Is that supposed to look like some thin wall of advancing light? And then, the quote says, at the moment of impact, the wave disappears and a photon appears. So, a ball of light appears? Something that resembles a sphere? How does a sphere become something like an ocean wave? What does that look like? My (completely uneducated) guess is, by a particle becoming a wave, does that mean that this expansive wave is filled with tons of ghost copies of itself, like the one electron exists everywhere in this expansive area of the wave, and then when it hits the wall, that property suddenly disappears and you're left with just one particle. So, this "wave", is really tons of identical copies of the same photon in the shape and form and with the same properties of, a wave? My guess comes from reading about how shooting just one photon _still_ passes through _two_ slits in the double-slit experiment. So the photon actually _duplicated_ itself?


title: What is the expectation value of the number operator when the vacuum has a VEV?

text: The number operator N applied to a field whose vacuum has zero VEV gives $N|0>=0$. What if we apply it to the Higgs field? The background of this question is that in popular scientific accounts, the Higgs field is sometimes described as a 'sea' of particles. I would like to clarify the meaning of this, and what is the physical interpretation of a non-zero VEV.


title: Why is it desirable to have a symmetry to make cosmological constant zero?

text: 1. It is sometimes stated that absence of a symmetry to make cosmological constant zero is a problem. But observed value of dark energy is very small and non-zero. So why is it desirable to have a symmetry to make cosmological constant zero?    2. Second, why is the energy density due to vacuum fluctuations expected to be Lorentz invariant? What happens if it is not so? (In other words, why should the vacuum expectation value of T_{ij} be of the form \Lambda g_{ij}?)


title: Relating the variance of the current operator to measurements

text: ( **EDIT** : Thanks to Nathaniel's comments, I have altered the question to reflect the bits that I am still confused about.) This is a general conceptual question, but for definiteness' sake, imagine a quantum dot sandwiched between two macroscopic metal leads with different chemical potentials. The chemical potential difference drives a current of electrons that flow through the quantum dot from one lead to another. Conservation of charge leads to the continuity equation for the charge density operator $\hat{\rho}$: $$ \frac{ \mathrm{d} \hat{\rho}}{\mathrm{d}t} = \mathrm{i}[\hat{H},\hat{\rho}] = - \nabla \hat{j}. $$ Given a Hamiltonian $\hat{H}$, one can in principle use the above formula to calculate the form of the current operator $\hat{j}$, whose expectation value gives the number of electrons that pass from one reservoir to the other per unit time. The expectation value of the current is independent of time in the steady state. The operational procedure to measure this expectation value is clear: sit there and count the number of electrons $n_i$ that pass through the quantum dot in time $t$, then repeat this procedure $M$ times, giving $$ \langle \hat{j} \rangle \approx \frac{1}{M}\sum\limits_i^M \frac{n_i}{t}. $$ The angle brackets on the left mean quantum mechanical average: $\langle \hat{j} \rangle = \mathrm{Tr}(\hat{\chi} \hat{j})$, where $\hat{\chi}$ is the density operator describing the quantum dot. (I am assuming that conceptual issues relating to quantum measurement are unrelated to this problem -- although please tell me if I'm wrong -- because a similar question can be posed for a classical stochastic system.) Now, each measurement $n_i$ will not not be exactly $\langle \hat{j} \rangle t$ due to fluctuations of the current. One can write down the variance of the current $$(\Delta j)^2 = \langle \hat{j}^2\rangle - \langle \hat{j} \rangle^2. $$ (As Nathaniel pointed out, the calculated variance in the current depends on the choice of time units.) However, the quantity you actually measure is the following: $$ (\Delta n)^2 = \overline{n^2} - \overline{n}^2, $$ where the overline means the average over the $M$ realisations of a measurement of $n_i$ electrons hopping between the reservoirs in time $t$, i.e. $\overline{n} = \frac{1}{M}\sum_i n_i$. My confusion relates to the fact that the quantity $\Delta n(t)$ must depend on the measuring time $t$. You can see this easily by considering the limit $t\to\infty$: if you watch and wait for long enough then the fluctuations will average to zero and every measurement $n_i$ that you make will be exactly the expected value. On the other hand, (I think) $\Delta j$ is the expected RMS fluctuation over a single unit of time, i.e. $\Delta j = \Delta n(1)$. Is there a simple relationship between $\Delta j$ and $\Delta n(t)$ measured over arbitrary times?


title: Would a gauss rifle based on generated magnetic fields have any kickback?

text: In the case of currently developing Gauss rifles, in which a slug is pulled down a line of electromagnets, facilitated by a micro-controller to achieve great speed in managing the switching of the magnets, does the weapon firing produce any recoil? If so, how would you go about calculating that recoil?


title: How can some-one independently do research in particle physics?

text: I'm not affiliated with a physics department and I want to do independent research. I'm working my way through Peskin et. al. QFT now. Let's say that I've finished Peskin et. al. and Weinberg QFT books and want to do some independent research on phenomenology although I don't have a specific topic in mind. Would I be prepared for doing research at this level? Where should I start? Should I just begin reading papers until I figure out current unanswered problems in the field? What journal would accept my paper provided that It doesn't have scientific errors and well-written? Can some-one knowledgeable in these topics give me an advice? I didn't go to graduate school so I don't know how one usually do research in particle physics. EDIT: Can I just visit a physics department and seek advice from some-one working on the field who can also give me an endorsement so that I can publish my papers on arXiv?


title: What is the maximam distance of light that is released by a flashlight?

text: Recently I looked at the cloudy night sky and noticed a pale green color light moving around the clouds for about 20 min.After observation I came a conclusion that it is probably because of a light house near by or any powerful light emitter.I could even barely see the light is being flashed from ground to clouds.Then I thought,If I focus my flashlight in to the night sky what would be the farthest distance of the light can reach?


title: Bound states and continua in the spectral function

text: Okay, let me try hard to pose this question as clear as I can. Let's take a quantum system where a single charge carrier interacts with a bosonic mode. Examples would be the Holstein model where a tight-binding carrier interacts with dispersionless phonons, or maybe a single carrier in a magnetic lattice where it can excite magnons. The typical single particle spectral function then has a bunch of generic features: There might be a bound state, appearing as a sharp peak in the spectrum, and there'll also be a continuum of states that are associated with the "free" particle plus a bosonic excitation. For example, in the Holstein model there'll be a sharp peak for the Holstein polaron, at some energy $E_0$, and then a continuum of "polaron + a free phonon" states starting at energy $E_0 + \Omega$ where $\Omega$ is the energy of the bosonic mode (with $\hbar = 1$). So far, so good, I've worked with the Holstein model and some more complicated variatons of it for quite some time. Now I'm looking at something more complicated, and I'm looking at it in the two-particle sector, and here's a simple question: When I see a continuum in the spectrum, should there always be a simple interpretation for that continuum? As in, I look at where the continuum starts and then I can tell immediately: "Ah yes, this is the continuum of two independent single-carrier bound objects" or "Yes, this is the continuum of ground-state-plus-one-bosonic-excitation states"? Or could there be continua that are more complicated in nature and aren't amenable to a straightforward interpretation? I'm asking because I'm trying to decide whether something my simulation spits out is a bug or a feature... I see a continuum of states well below where the "two independent single-particle-bound-states" continuum would lie, but below this continuum I'd only expect a bound state...


title: Applications of physics beyond QFT

text: Are there any applied results (inventions) based on physics beyond quantum field theory? By beyond QFT I mean any physical theory that is (maybe slightly) different from classical physics, relativity, quantum physics and quantum field theories. By applied results I mean not-only-on-paper results.


title: Moment of Inertia

text: Let $f(x) = \frac{1}{L}$ be a probability function, where $L$ is constant. Find the mean and variance. Discuss your results by making a connection to the moment of inertia definition.


title: How far are constellations recognizable?

text: A novel of the 'Foundation' series mentioned that the universe being 3-D the shape of a constellation is recognizable only in the near vicinity of a given planet. In other words move out from the planet's surface some distance, and the shape of the constellation changes dramatically. It feels true, but is this observation true? How far from the planet may one travel, and still recognize the constellation/s that are typically visible on that planet's surface ?


title: Why isn't absolute $0 K$ temperature possible?

text: So $T$ is defined as $$T = \left(\frac{\partial E}{\partial S}\right)$$ and $S$ is defined as $$S = k_B \ln \Omega$$ where $\Omega$ is the number of accessible states of the system for a given $E$. I suppose you could write it as $\Omega(E)$. Hence, for $T$ to be zero, it must be possible to prepare a system that has an equivalent number of possible $|\psi\rangle_E$ for different $E$. It seems to me like you could find some weird system for which this applies. Why can't this be true?