title: Temperature below absolute zero?

text: I saw this Nature article today, which cites e.g. arXiv:1211.0545. And it makes no sense to me. The temperature of a collection of particles is the average kinetic energy of those particles. Kinetic energy cannot be less than zero (as far as I'm aware), so I don't understand what this article is trying to say, unless they're playing around with the conventional definition of "temperature". The only thing I can thing of is if you have something like: $$\frac{1}{kT} ~=~ \left(\frac{\partial S}{\partial E}\right)_{N,V}$$ And they've created a situation where entropy decreases with increasing energy.


title: Thermodynamics, temperature below 0 Kelvin

text: I read a news article about how they were able to create a negative temperature, below absolute zero, and my question is how does this work? I know that there are different definitions of temperature, and for the negative temperature, it's referring to entropy, the disorder. Can someone explain to me how is the negative temperature created and why is it in some situations when we add energy to a system, entropy does not increase?


title: Is It Possible To Have Temperature Below Absolute Scale?

text: Guys I have a doubt Is it possible to maintain the temperature of any Substance Below the Absolute Scale?


title: Less than absolute zero possible?

text: > **Possible Duplicate:**   >  Temperature below absolute zero? According to this article http://www.sciencemag.org/content/339/6115/52 (preprint: http://arxiv.org/abs/1211.0545) it is. What do you think. Is this team of scientists a faker team or you could actually go under absolute zero? Could this be considered a way to dark energy something which is less than absolute zero?


title: Quantum phyics project for a high schooler

text: > **Possible Duplicate:**   >  Study Quantum Physics I am a high schooler who is interested in physics and mathematics, and I have a kind of 'high-school thesis' coming up in a year and a half or so. I want mine to be about Quantum Physics, and I have already prepared by self-studying Linear Algebra (and I'm planning on starting with Differential Equations too). I just have a couple of obstacles:   * If I know Linear Algebra and (Ordinary & Partial) Differential Equations, what parts of Quantum Physics will be in my reach?   * What other fields of mathematics would you recommend me studying to gain a grasp of rudimentary quantum physics and to make me able to make a thesis on quantum physics?   * What would you guys consider interesting for a thesis on quantum physics? What are fascinating experiments, concepts, results, etc. of quantum physics that would be worth making a thesis about?


title: Why is pressure in a liquid the same in all directions?

text: I'd like answers both in the more intuitive side an on the more precise side. Thinking of water as "cubes" of water, for example, would allow pressure in the z axis to be independent of the y or x axis. Choosing other "shapes", like a triangular prism, yield different results. Will this fact, then, be dependent on the format of the molecules of the liquid ? I've heard of the rotational symmetry of liquids. What is its precise statement ? Why is it true ?


title: use of Mobile phone on Petrol pump

text: I was trying to find out the cause behind the Explosion at petrol pump due to the Mobile phones but eventually it turns out it is not Radio Frequency which may cause the explosion rather it is totally the static electricity which may cause serious explosion and ignition of vapors, The other reason I found is, due to the spark by the mobile phone battery in case the mobile has fallen out of your pocket and it seems more legit. The second scenario, Its been said that Electromagnetic Waves absorb in liquid across a wide wavelength , is there any possibility that the absorption of radio Frequency in Petrol may cause any explosion ? My question is, anything can produce Static electricity then why mobile phones are prohibited ? Do they produce more Static Electricity than other equipments ?


title: Physical reason for annihilation?

text: What is the fundamental reason as to why matter and antimatter annihilate? Is it because both particles and antiparticles are excitations of quantum fields, and the annihilation process corresponds to a transition to the _ground state_?


title: Question on "pocket" universes

text: I recently heard about the possibility of the creation of a "baby" universe or "pocket" universe from our space time,the theory says that concentrating enough mass into a tiny volume of space (like in black holes) a pocket of false vacuum would form e perhaps it'll continue to expand following it's own inflation period and go on forever completly disconnected from our space time without influencing it any way. Now my question is : supposing that we live in a metastable vacuum we know that true vacuum bubble could form via quantum tunnelling and begin to expand into our space time where these false vacuum bubble (or pocket/baby universe) are supposed to go and expand ?


title: Are bubbles in ice at a lower or higher pressure than atmospheric pressure?

text: When water phase-transitions into ice it expands. The water usually contains dissolved air. Freezing forces the air out of the solution into bubbles. Are these bubbles at a lower or higher pressure than atmospheric pressure? I can see arguments both ways: when a washer expands, both the outer and inner radius increase. This would imply bubbles are at a lower pressure, but perhaps the ice expands differently and squeezes the bubble and it is at a higher pressure. However, if they were squeezed I would expect them to try to expand, and propagate along weak fraction lines in the ice, and I do not see this - bubbles are usually round, suggesting low pressure environments. **Edit** : Two papers on the subject, but neither sheds direct light on the answer:   * "Bubbles and bubble pressures in Antarctic glacier ice" (A. J. Gow, _J. Glaciol._ **7** no. 50 (1968), pp. 167-182) shows that the bubbles shrink with increasing pressure. These are therefore presumably high-pressure bubbles. However, the internal bubble pressure is due to ice compressing as more ice forms on top, not just the freezing process.   * "Air Bubbles in Ice" (A. E. Carte, _Proc. Phys. Soc._ **77** no. 3 (1961), p. 757) talks about bubble formation, but not internal bubble pressure.


title: Casimir force between plate and sphere

text: In our quantum optics lecture we derived the formula for the Casimir force between two plates (or at least the force per unit surface, since for two infinite plates we would have an infinite force). It was then said, that in experiments one often uses a sphere and a plate to measure the force, where the small distance between the sphere and the plate is varied by a piezo. I'm not particularly interested in how exactly the force is measured but rather how to theoretically calculate it in this case. Our professor said, the measured forces agree with the theory up to 1% so I guess, there is a way to calculate it. Let's say I have a sphere of radius $R$ in a distance $d$ from a plate where $R\gg d$. How can I theoretically calculate the Casimir force between a plate and a sphere?


title: Hydrostatic pressure?

text: What what I understand, hydrostatic pressure is the "weight" of the water pushing against objects. But if this is true, why is hydrostatic pressure perpendicular to the surface it acts on instead of always going down? For example, if you placed a book on a desk, the book's weight would push against the desk, but gravity is pulling it "down". But if you put another book beside this book the first book wouldn't apply any force on it. The weight exerted from the book always _goes down_. Similarly, why doesn't hydrostatic pressure always go down? I understand that if you put a plate on the seabed, the weight of the water would push down on it, but if you just had a vertical plate standing on the bed, why would force push it from the sides? To sum up, why does hydrostatic pressure act perpendicular to the object instead of always down?


title: Fluorescence lifetime imaging

text: Actually I am not a physics student but I have to give a lecture about Fluorescence lifetime imaging(FLIM). I am thinking of comparing this techniques to another imaging techniques. Does anyone know what is the best imaging techniques that is comparable to FLIM? Thank you Soha


title: How do we know we've unified two interactions?

text: 1. What is the precise definition of unification of fields (in classical and quantum mechanics)?   2. In general, does unification of a field mean that we can write both of them at both sides of an equation (like Maxwell's laws)? Or does it mean that one of them can produce the other (like $E$ and $B$)?   3. Is there any intuitive explanation of how electroweak unification works? Like an electric charge will feel a weak field or that a flavoured particle will produce a weak field?


title: How to understand holography and hologram

text: I've spent some time reading wiki etc. What I get now is that apart from the normal light amplitude information, holograms also record the phase information of light. But this is so difficult for me to understand. In my understanding to display the 3D objects you have to reconstruct the 3D object field; the light source and exact light field at that moment, what does the phase mean here? why phase alone makes all these happen? Apart from the principles, what is restraining holography display from being popular at the moment?


title: Physics Standard Deviation

text: I am a physics enthusiast and I have a question: Why is it meaningless to express the '$\pm$' (standard deviation) value as a percentage?


title: Density Matrix Characterization

text: I am working a two dimensional Hilbert space with basis {$|0\rangle, |1\rangle$} and I am trying to show that the density matrix is characterized by 3 real numbers and show that these three numbers are the expected values of the Pauli matrices. I understand that a density matrix is given by $M_{ij} = \langle i|\rho|j\rangle$ where $|i\rangle$ is an orthonormal basis of $H$ and $\rho$ is the density operator. How do I go about showing this?


title: Tennis serving machine--- How does a spinning ball bounce?

text: I have an idea of making a tennis serving machine. I will briefly describe what it is:   * The machine is configured to serve the ball at a fixed speed to the center of the left (or right) service court   * Based on the rotation in 3 directions (X,Y,Z) of the ball, the machine would let the ball land in various positions in the left service court. How can I determine the relationship between the angular velocity of the ball and the location where it lands?


title: Tensor of inertia

text: The tensor of inertia of a solid sphere is $I_{ii}=\frac{2}{5}MR^2$ about an axis passing through its CM. Why would the tensor of inertia of each hemisphere about that axis be $I_{ii}=\frac{2}{5}mR^2$, where $m=\frac{M}{2}$ is the mass of the hemisphere? Is it because the tensor of inertia of each hemisphere would be expected to be $I_{ii}=\frac{1}{5}MR^2$, but then $M = 2m$, hence $I_{ii}=\frac{2}{5}mR^2$?


title: why F is to be defined as ma?

text: First, I would like to apology for my lousy English used here, as being a non- native english speaker, I am currently using Classical Mechanics as a tool to learn how to construct a theory as an end. Therefore, I have a few questions, and a few reasonings behind them (I only consider reality, not imagined world), and I would like your suggestions or corrections :D To make things simple, I first consider when mass is a constant, so F=dp/dt can be written as F=ma. Why F=ma? Instead of, say, I define new constants or variables, and claim 1.) $I=N \frac{d^3x}{dt^3} $ ? Or using the same F both literally and physically, and claim 2.) $F=N(x,\dot x,\ddot x,...x^{(n)};t) \frac{d^3x}{dt^3} $ or 3.) $F=Q \frac{dx}{dt} $? I mean, force is not something that you can touch nor see, either is mass; so, we pretty much define them via indirect observations, recording the data, and then comparing with other objects, in order to give operational definitions on mass and force (in case of "mass", we will have to define a unit mass first, and then use F= ma as operational definition for mass.). Under this line of reasoning, So why can't we work things in a simple way - $I=Q \dot x $? Yes, we can indeed. for question 1.) given reality: A falling object a=g: $I=Q \frac{dx}{dt}$ $\dot x=gt$ $I=t$ $Q=1/g$ So if I define such way $I=Q \dot x $, it is just the law of falling. So my answer to my 1st question is: Yes, we can define things such way, but it has nothing to do with the subject we are dealing right now - inertia. Lesson learned: Know what you want to deal with. For question 2.) $F=N(x,\dot x,\ddot x,...x^{(n)};t) \frac{d^3x}{dt^3} $ Since I only consider our reality, that means F=ma is truth. So I will comparing F=Na' with F=ma with a few cases (constant a, and SHM) And it turns out that F=Na' is far more complicated (often a variable) and N is going to be ill defined given constant a. Reality: A falling object a=g: $F=N\dot a=0 =>N\dot a=mg =>N=mg/\dot a$. as $\dot a=0$, N(x,t) is ill defined. Reality: SHM: $ m\ddot x=-Ax =>\dddot x= -A\dot x/m $. Given $ Na'=ma => N= ma/a'=-m^2\ddot x/\dot x A=(-m^2/A)(d\dot x/dx) $ Too complicated. (comparing with the fact that m is just a constant.) Lesson learned: 1.) best work for most of the cases. 2.)When a constant will do, don't use a function. 3.)Theory is about productivity. For question 3.) $F=Q \frac{dx}{dt} $ or any other orders, actually, I am just asking: for $ F= \frac {d^nR(W,\dot W,...,W^{(n)},x,\dot x,...,x^{(n)};t)}{dt^n}= \frac {dP(m,\dot m,x,\dot x;t)}{dt} $ 4.) is there exist a n, such that there exist a R makes things simpler than $F=\dot P$ ? Define $R(W,x;t)=W(x,\dot x,\ddot x,...x^{(n)};t) x$, (R, W new variables, x is position function of time) if there doesn't exist such an n, then 5.) why F= ma turns out to be the simplest way to work things out? And thanks for your reading. Please give me some criticisms to my reasonings. p.s. I think I am making the reinvent wheel mistake. I hope these fancy mathematical philosophy will end up some educational purpose on physics.


title: Is it possible to escape from within event horizon?

text: I always think that it is not possible to escape from within event horizon. However, some one recently told me with deep conviction that it _is_ possible with sustained energy output. I countered with calculations in Schwarzschild metric showing that any objects below event horizon in the simplistic black hole will hit the singularity within a finite proper time, and was met with the response that Schwarzschild metric was a bad choice for coordinate system. Now I understand that the singularity at $r=\frac{2GM}{c^2}$ in Schwarzschild metric is an artifact of the coordinate choice, and thus it seems that he could have a point. So my question is, can one prove that it is (or not) possible to escape from within the event horizon of a simple static black hole using a better metric?


title: What is the difference between QFT and elementary particle physics?

text: I'm a little unclear as to how QFT differs from Elementary particle physics. They both use pictorials of Feynman graphs, is it that Elementary particle physics assumes the point particle perspective, while QFT treats them as fields?


title: Material implementations of the holographic principle

text: I'm afraid this question is a little too open-ended, but bear with me while I find a better formulation. carbon allotropes (like fullerenes and graphene) are regular patterned. Conduction bands of electrons in graphene in particular, have energy-momentum relationships that make them behave as massless carriers (sticking to what is called a Dirac cone). More interestingly the propagation dynamics seem to be at first order, direction independent. Given the above facts, is possible that the interactions of carriers on such 2D sheet is describable by the limit of some Conformal Field Theory. If that were to be the case, and I have a big closed surface of graphene material, what does the holographic principle will say about the fields in the volume bounded by the surface? What is the equivalent of the Gravitational Theory in that volume?


title: What is a complete book for quantum field theory?

text: I am searching for a complete and comprehensive book for QFT. What is, in your opinion, a good one?


title: Books dealing with Quantum field theory

text: > **Possible Duplicate:**   >  What is a complete book for quantum field theory? I am looking for good books that deal with Quantum Field theory starting from basics. Please suggest something that deals with the physical interpretation of the subject, and not just the mathematical formalism. Books dealing with particle sources and particle detectors will be much appreciated.


title: Jauch, Piron, Ludwig -> QFT?

text: > **Possible Duplicate:**   >  What is a complete book for quantum field theory? At the moment I am studying Piron: Foundations of Quantum Physics, Jauch: Foundations of Quantum Mechanics, and Ludwig: Foundations of Quantum Mechanics All of them discuss nonrelativistic quantum mechanics. Now my question is, if there are corresponding approaches to quantum field theory (in particular aiming to the standard model of particle physics). **Edit** (in response to noldorin's comment): I think if one knows the books mentioned above, the question is not very vague. Take for example Ludwigs approch. For instance it is written with the aim in mind to provide clear foundations of nonrelativistic quantum mechanics which match Ludwigs epistemological theory about physics ("A new foundation of Physical Theories"). It is axiomatic, mathematical sound, emphasises the idea of preperation and registration procedures, uses the mathematical language of lattice theory etc. However since I am studying this book at the moment I might miss some important points in Ludwigs approach. So I ask a bit vague if there is an approach to QFT which corresponts **in your view** to Ludwigs approach to nonrelativistic quantum mechanics in **the most essential points (from your point of view)** (in style, epistemological background, mathematical language etc.). Perhaps there are students of Ludwig which transfered and developed his ideas for QFT as well, I don't know.


title: Recommendation: Advanced topics in quantum field theory

text: I have read Srednicki's Quantum Field Theory book. I want to learn more about advanced topics in field theory, such as geometry and topology in field theory, topology defect, anomaly, soliton, instanton, renormalization，advanced application in condensed matter physics and so on. Does anyone have anything recommended? Thanks


title: QFT Literature recommendation

text: > Before answering, please see **our policy** on resource recommendation > questions. Please try to give substantial answers that detail the **style, > content, and prerequisites** of the book or paper (or other resource). > Explain what the resource is like as much as you can; that way the reader > can decide which one is most suited for them rather than blindly relying on > the suggestions of others. Answers which just suggest a book or paper will > be deleted. > > And please, note that any answers to this are **community-owned** , which > means that they are subject to heavy editing, especially to make them comply > with the book policy, to avoid deletion. I am really getting confused. This term, I am sitting for a QFT 1 lecture and it is what I consider really hard for a physics student! But this is not my problem. My problem is that other than Weinberg, there is absolutely no QFT literature for a modern course. Our literature recommendation for our course is first of all, Peskin and Schroeder. That is good, but not very detailed. The the other is Itzykson Zuber. Another problem is that while our course is very excellent which also means that the script is really good (you would have to have listened to very good lecture series on Quantum Mechanics (for the math).), to understand read "between the lines", like representation theory, what is spin, etc., one needs to read Weinberg. But the style of writing is very strenuous for me. And thanks to a bad theoretical quantum mechanics lecture, I still really don't know what a spin is! I only know so far that this has something to do with a representation of the Lorentz algebra. So, do you know some good resources or books which find a good way between the physics and the mathematics?


title: Top spun up with string under tension problem

text: > **Possible Duplicate:**   >  Homework about spinning top I have a top with an unknown mass. It has a moment of inertia of 4.00 * 10^-7 kgm^2 a string is wrapped around the top and pulls it so that its tension is kept at 5.57 N for a distance of .8 m. Could somebody help me derive some equations to help with this? Or to get me in the right direction? I have been trying to derive some sort of equations from KEr = 1/2 * I * w^2 but I cant get anywhere without ending up at radius = radius or mass = mass.


title: Cosmic background radiation

text: Where/what is emitting cosmic background radiation, and when did it come into existance, was immediately after the big bang? I know that the universe isn't 3D in the traditional sense, and I don't pretend to understand it. However if the big bang originated from a single point, and the cosmic radiation was emitted from that point at the very start, then unless he explosion/expansion/inflation (whatever happened) wasn't faster the speed of light, then the CMB would have reached the edge of the universe and gotten out of it.


title: What are the most important papers in physics?

text: Recently I got the book "On the Shoulders of Giant" from Stephen Hawkings. It consists of more than 1000 pages of classical publications in physics. However 900 pages are given to the work of Copernicus, Galilei and Newton and 100 pages are given to the papers of Einstein. Therefore I think this books gives a wrong impression about what is really important for our knowledge in physics today. If you could fill a book with the most important papers relevant for our knowledge in physics today, which ones would you select?


title: Conduction– conservation of energy

text: So I know that conduction is transfer of thermal energy by direct contact via molecular collisions. For conduction, $Q/t$ is the rate of heat flow and is heat current ($I$). My textbook says that the rate of heat flow is constant across any number of slabs between 2 heat reservoirs, and I understand why this is so because if I use electric current as an example, the current in constant through resistors of different ohms in a series circuit too. But then, the textbook goes on to show how conservation of energy also explains why. It says that "If the rate of energy transfer were not steady across all slabs, the slab that conducted heat the fastest would become cold. A cold slab could not conduct heat to warmer slabs because heat moves from hot to cold." I don't understand their explanation that they offered from a conservation of energy standpoint. Could you please clarify this for me? Why would the slab that conducted heat the fastest become cold if the rate of energy transfer were not steady? For your reference, the equation for conduction is: $$\frac{Q}{t} = kA\frac{T[\text{hot}] – T[\text{cold}]}{L}$$


title: Lagrangian and grassmann numbers

text: Why sometimes we remember that "classical" lagrangians of fermions are constructed from grassmann numbers, while sometimes don't? For example, for Majorana's field in terms of 2-component spinors (you can read about this representation here in details), $$ L = \bar{\psi}(\alpha^{\mu}\partial_{\nu} - m)\psi , \quad \alpha^{\mu} = (\sigma_{2}K, \sigma_{1}, \sigma_{2}, \sigma_{3}), \quad K\psi = \psi^{*}, \quad \bar{\psi} = -\psi^{\dagger}\alpha_{0}, $$ and the mass term become $$ \bar{\psi} \psi = -\psi^{\dagger}\sigma_{2}\psi^{*} = -\begin{pmatrix}\psi_{1}^{*} & \psi_{2}^{*}\end{pmatrix}\sigma_{2}\begin{pmatrix} \psi_{1}^{*} \\\ \psi_{2}^{*}\end{pmatrix} = 0 $$ if the fields are "classical". So we must assume their grassmanian nature before making all calculations (there this expression won't be equal to zero). But when we assume the mass matrix for see-saw type 1 model, we introduce the term $$ -M_{ij}(\psi_{R}^{T})^{i}\hat{C}\psi_{R}^{j} + h.c., \quad M_{ij} = M_{ji}, \quad \hat{C} = i\gamma_{2}\gamma_{0}. $$ If the fields are grassmann numbers, this expression is equal to zero (don't forget about symmetric mass matrix). So we forget about grassmannian nature of the fields even in "classical" level. Here "classical" means that we set all of anticommutators to zero. If we set them not to zero, we will have new problems, but it's the other story.


title: Mainstream view on the measurement problem

text: What is the modern mainstream way to deal with the measurement problem? For this question, the measurement problem is the problem of formalizing wave function collapse when the observer is part of the model (say I want to simulate an observer+observee quantum system on a computer). I am aware of proposed solutions: Everett, Bohm, and decoherence (though decoherence apparently does not explain collapse). How does mainstream physics deal with this issue? For example: Given two electrons initially isolated from each other and the universe which then interact , can each electron be considered an observer? With no-collapse theories each electron sees the other election's wave function collapse relatively. If the mainstream view holds that a single electron cannot be an observer what is the criterion for an observer?


title: Optical absorption in a semiconductor for $E<E_g$

text: Quoting from Solid State Electronic Devices (by Ben G. Streetman and Sanjay Banerjee): > A photon with energy less than $E_g$ is unable to excite an electron from > the valence band to the conduction band. Thus in a pure semiconductor, there > is negligible absorption of photons with $hv<E_g.$ This explains why some > materials are transparent in certain wavelength regions. Does this mean that electrons don't absorb photons with $E < E_g$? If so, then how is it possible that an electron 'knows' the energy of the photon without absorbing it?


title: Bondi-Metzner-Sachs (BMS) symmetry of asymptotically flat space-times

text: I started studying the BMS symmetry in connection with the paper: http://arxiv.org/abs/1312.2229 and there are a few strange things I noticed. First of all, from reading the original papers by Bondi, Metzner and Sachs, I know that the "BMS symmetry" is just an allowed subset of coordinate diffeomorphisms which leaves the asymptotic flatness of the space-time intact. However, when I read the paper above, the BMS symmetry is stated in form of a vector field eq. (2.10) and (2.14). Therefore, my first question: > How do I obtain from a given subset of coordinate diffeomorphisms (which is > being considered a symmetry) a corresponding vector field a'la Killing > vector? Furthermore, later in the paper they go from the BMS vector fields to generators of BMS symmetry in eq. (3.3). They do not mention how to do it, so I'd like to know: > How do I go from vector fields characterizing a symmetry to an actual > generator of the symmetry? I realize that this is some pretty advanced stuff. I am grateful for any help or suggestion!


title: Diffusion in the standard map

text: Consider the standard map (also known as Chirikov map): $$ p_{n+1} = p_n + K \sin(\theta_n) \\\ \theta_{n+1} = \theta_n + p_{n+1} $$ I know that the diffusion coefficient according to the Einstein relation is defined through the relation: $$ \sigma^2 = 2Dt $$ being $\sigma$ the variance of the position in the case of the $1$-D random walker. Can you figure out why the diffusion coefficient for the standard map is given by: $$ D = \lim_{n \rightarrow \infty} \frac{<(p_n - p_0)^2>}{n} \quad ? $$ I mean, the $n$ makes sense, but what about the variance, why this is given just in terms of $p$, what happens with the $x$.


title: What is the optimal diameter for the exhaust hole in a pressurized vessel to deliver highest acceleration ?

text: Imagine you have a sealed cylindrical vessel with a given radius with a compressed gas inside. Let's give some numbers, 5 cm radius and 100 atm pressure. You poke a hole in the vessel and the gas will flow out, propelling the vessel in the opposite direction with a given force. Intuition tells me that this force depends on the hole size. Suppose the pressure does not change as time goes by, what's the optimal hole size to deliver the best acceleration ?


title: Question about gravity felt at various radii within a massive sphere

text: Imagine there's a sphere of radius $R$ which has constant density $\rho$ and you can stand anywhere inside of the sphere. Wherever you stand within the sphere, you'll only feel gravity coming from the mass at smaller radii than the point at which you're standing. That is, you could be standing at some point with radius $r$ and if somebody come along and added more mass to the sphere so that the radius was extended to $R'>R$ you would feel no effect because you're insensitive to it. Puzzle: What happens if somebody comes by and adds mass to the sphere until it extends out to infinity? Now the mass distribution is uniform everywhere in space so you'd expect no gravitational pull in any direction, by symmetry. What happens? Do you suddenly stop feeling a force? If you have an issue with extending the mass to infinity then imagine that we've added a point to $\mathbb R^4$ so that we're on $S^4$. What's the intuitive way to understand what's happening in this limit? What's the technical explanation? Is it a boundary value issue?


title: semiclassical quantization failure

text: could exist 2 different Hamiltonians $ H1 $ and $ H2$ so a) The SEMICLASSICAL eigenvalue staircase is equal i mean the expression $$ \oint _{C} p.dq = n+\alpha $$ but the real eigenvalues $ E_{n} (1,n) $ and $ E_{n}(2,n) $ are completely different


title: Understanding the Selection Rules of a Spin-Forbidden, Magnetic Dipole Transition in Molecular Oxygen

text: I am studying the transition from the second excited electronic state of molecular oxygen, $b^1\Sigma_g^+$ , to the ground state, $X^3\Sigma_g^-$. I know that the ground state has total angular momentum $J=1$, total spin $S=1$, and three spin sublevels $(m_s=-1,0,+1)$. The upper state has $J=0$, $S=0$, and one sublevel with $m_s=0$. I am specifically interested in the transition from the upper state to the ground state $m=\pm1$ level and I will refer to it as the $b-X,1$ transition. I would like to understand the the selection rules that govern this transition and the language that is used to describe this transition. Here is what I know so far about this transition.   * Brecha, Pedrotti, Krause, "Magnetic rotation spectroscopy of molecular oxygen with a diode laser" JOSA 1998: They describe this transition as "doubly weak" and I quote, "First, it is a magnetic dipole transition and thus is roughly 5 orders of magnitude weaker than a normal electric dipole transition. Second, the transition is a singlet-triplet intercombination band, making it 3 orders of magnitude weaker still."    * Minaev, Agren, 1997: They describe this transition as "magnetic dipolar" and "spin-forbidden." They also say that it is spin-orbit coupling that accounts for this "doubly weak" transition being as large as it is.    * Wikipedia-Selection Rules: Provides the rules for Magnetic Dipole (M1) transitions and discusses Spin-Obrit (LS) Coupling but I do not understand where this came from.    * Sannigrahi, "Derivation of Selection Rules for Magnetic Dipole Transitions," 1982: This is pretty clear but just seems to say that $\Delta m_s=\pm1$. This is true for the $b-X,1$ transition but how do I tie in $L$ and $J$ into the selection rules? Most of what I have found on the internet and in textbooks relating to selection rules is directed to electric dipole transitions. Where I have found something discussing magnetic dipole transitions, it is either a summary of the rules or I do not have the required background to understand it.   * What does it mean for a transition to be spin-forbidden?   * What is a singlet-triplet intercombination band? Does this change the selection rules? How is this related to being spin-forbidden?   * What does spin-orbit coupling have with this? I look forward to any feedback on this topic. # Follow up Questions ## Meaning of Spin-Forbidden Does 'spin-forbidden' just mean that the transition from a $J=1$ to $J=0$ state is not allowed because the selection rules for magnetic dipole transitions say that $J$ cannot change? I expected 'spin-forbidden' to imply something about the change of the spin between the initial and final states. For example, suppose I have a time-dependent perturbation like $V_{md}(t) = \frac{e}{m} \vec{S}\cdot \vec{B}(t)$ and I am interested in the transition rate between the initial state $\left| s m_s \right\rangle$ and the final state $\left| s' m_s' \right\rangle$ with the quantization axis in the z direction. As you pointed out, the transition rate will be proportional to $\left\langle s' m_s'\right| \vec{S} \cdot \vec{B}(t) \left| s m_s \right \rangle$. Now if $\vec{B}(t)$ is circularly polarized, the rules for an allowed transition will be $s'-s=0$ and $m_s'-m_s = \pm1$. For $\vec{B}(t)$ polarized in the z direction, $s'-s=0$ and $m_s'-m_s = 0$ so there is not a transition to other states. I would think that if $s'-s=0$ and $m_s'-m_s = \pm1$ are **not** true (like if $s'=1$ and $s=0$), then the magnetic dipole transition between $\left| s m_s \right\rangle$ and $\left| s' m_s' \right\rangle$ would be called 'spin-forbidden.' The same arguments could be made for $\vec{L}$ or $\vec{J}$ as I did with $\vec{S}$. Would you also call a magnetic dipole transition between $\left| L=0 \right\rangle$ and $\left| L=1 \right\rangle$ 'spin-forbidden?' ## Spin-Orbit Coupling Now the difference in the energies of the singlet $(b)$ and triplet, ground state $(X)$ is $1.63\text{ eV}$. That seems too large to be due to spin-orbit coupling breaking a degeneracy. If I was pretending this was a hydrogen atom, I would say this was like a transition where the principle quantum number $n$ changed. I am not sure how to talk about this in a molecule. This isn't key to the rest of your explanation but I did want to clarify to make sure we were on the page. ## Singlet-Triplet Intercombination Band Do you know what the expression 'singlet-triplet intercombination band' is referring to? After your explanation, it seems to refer to the mixing of the singlet and triplet states due to spin-orbit coupling (SOC). Is this true? ## Mixing of unperturbed states How did you know that the perturbed upper state could be written as a combination of the unperturbed upper state and the $M_s=0$ ground state? It makes sense that the states would get mixed up by SOC but I don't know how. If this is a messy explanation, don't worry about it. **I would like to reiterate that your (George G's) explanation has been incredibly helpful. Thank you.**


title: Comparing two infinite sets

text: 1. All the linearly independent eigenfunctions of the parity operator $\mathcal{P}$ form an infinite set and all the linearly independent eigenfunctions of the unit operator $\bf 1$ also form an infinite set. Is the cardinality of these two sets same or different?    2. Every function is an eigenfunction of the unit operator $\bf 1$. Right? Can any arbitrary function be written as the linear combination of the eigenfunctions of $\mathcal{P}$? If not, doesn't this imply that the set linearly independent eigenfunctions of unit operator $\bf 1$ forms a much bigger set than that of $\mathcal{P}$? **EDIT:** I just want to know since $[\mathcal{P},\bf{1}]=0$, whether the total number of linearly independent eigenvectors of $\mathcal{P}$ and $\bf{1}$, forms a basis i.e., do they have same number of linearly independent eigenvectors and and whether both spans the space?


title: Determinate the right force intensity given a certain amount of energy to spend

text: I am using Chipmunk's physics engine to simulate 2D classical mechanics of solid polygons (excellent by the way!). The only way you can alter the spontaneous behaviour of the the simulated objects is to apply forces and impulses to them. I am given a certain amount $E$ of energy to spend (= to input into the system) over a certain period of time $\Delta T$, that I need to use to alter the motion of a given body. By the time I need to do so, it is rather difficult to access its mass, instantaneous speed, kinetic moment, energy, or other forces applied to it.. (I can still track its shape, position and rotation.) However, I feel like I don't need to access all these to actually spend this energy and alter its motion (with the only tools I am given) provided I decide I will do it by applying a single force $F$ to it over the period $\Delta T$, and provided I give myself an application point $P$ and a direction $d$ (both relative to the body's actual position and rotation). * * * To be more precise about "my feeling": no matter the actual situation of the body, I will apply this force and spend this energy over $\Delta T$:   * if it turns out to be a very heavy object, then it won't move a lot   * if it turns out to be in a super-fast race against $d$, then its speed won't change a lot   * if it turns out to be blocked under a stone, then it won't move a lot   * if it turns out to be free, light, resting.. then the change will be visible.. (I am aware that the concept of _force_ might be the wrong one to consider here, so I just want to make sure we'll agree about what I am talking about, whatever it might be. It is exactly like this body had a certain amount of energy to spend and used it to _move itself_ without actually knowing anything about the world around.) * * * If I was right about this feeling that I do have everything I need to perform this alteration without spending more than $E$, then how would I compute the actual intensity of $F$ to be applied? If I was wrong, then what information am I missing? * * * [EDIT]: It turns out from the discussion that I actually need to give myself a certain _cost of applying a force_ $\beta\ (m.s^{-1})$, inspired from biophysics. Then the actual energy spent is simply $$E = \beta \int_{\Delta t} F(t) \mathrm{d}t,$$ and I just have to take care that it will never be lower than the actual work $W$ done by $F$ over $\Delta t$. To achieve this, one solution seems to keep constantly track of the actual useful energy spent, which is difficult but.. not undoable.


title: what to use for mass in a 2D FEM simulation

text: I am trying to find the energy of a wave travelling through a solid material in a 2D Finite Element Method (FEM) - Simulation. As a general approach I would try to use $E_{kin}=\frac{1}{2}mv^2$ at the highest particle velocity $v$, as this would be the point, where all the energy in the wave is kinetic and potential energy should be zero. So I would know the energy of the wave But what would I use for mass? Since it is a 2D simulation, I image that, as the third dimension is infinitely small, there is no mass. Now I remember seeing in a paper once (sorry, I don't know the source anymore), that people used density $\rho$ instead of mass in their 2D FEM simulations. My questions have 2 parts:   * Is using density instead of mass justifiable? If not, what other approach could someone point me to?   * The units would not match up anymore. So if it was ok to use density how would I use it.


title: Alignment of Fermi Levels between Metal and Insulator

text: If I stick a metal and insulator together, will the Fermi level of the metal align with the insulator? When people draw a band diagram for a metal-oxide-semiconductor heterostructure, I never see them label the Fermi level of the oxide. What is the Fermi level of the oxide relative to the metal and insulator's? I'm particularly interested in graphene/SiO2/Si field effect transistors. When we change the gate voltage between the graphene and Si, what happens to the SiO2 conduction band, valence band, and Fermi level? If the SiO2 Fermi level aligns with the graphene Fermi level, what is the mechanism that keeps graphene and SiO2 in thermal equilibrium so quickly?


title: Particle/Pole correspondence in QFT Green's functions

text: The standard lore in relativistic QFT is that poles appearing on the real-axis in momentum-space Green's functions correspond to particles, with the position of the pole yielding the invariant mass of that particle. _(Here, I disregard complications tied to gauge-fixing and unphysical ghosts)_ Schematically, I take this to mean: $$\text{one particle state} \Leftrightarrow \text{pole on real axis}$$ It is easy to show the correspondence going $\text{particle} \Rightarrow \text{pole} $. This is done in many textbooks, _e.g._ Peskin and Schroeder's text, where a complete set of states diagonalizing the field theoretic Hamiltonian is inserted into a $n$-point correlator, and a pole is shown to emerge. However, how does one make the argument going the other way $\text{particle} \Leftarrow \text{pole} $? That is, the appearance of a pole in a Green's function indicates an eigenstate of the Hamiltonian, with the eigenvalue corresponding to the particle energy?


title: Global vs. local gauge group in mathematical sense - physics examples?

text: Upon reading about the principal bundle picture of (quantum) field theory I encountered two different definitions of the gauge group:   * **Local gauge group $G$**. Corresponds to the fibers of the $G$-bundle. Local gauge transformations correspond to the change of coordinates in which fields and form are written in trivializations. In other words, as far as I understand, local gauge transformations = transition functions for trivializations of the principal bundle and the associated bundles.   * **Global gauge group $\mathfrak G=Aut(P)$**. Is the group of principal bundle diffeomorphisms, and is claimed to be much bigger then $G$. I would like to understand which notions in physics correspond to these two, apparently very distinct, notions. First, in physics, there is the notion of _global symmetry_ , as far as I understand this has nothing to do with gauge/bundles. But on the other hand, the limit of a gauge symmetry where the parameter becomes constant is a global symmetry. Typical $U(1)$ example: $\exp(i\alpha(x))$ = $U(1)$ gauge transformation, but if $\alpha(x)=\alpha=const.$, then it is a global symmetry. Is there any deeper connection between global symmetry and gauge "symmetry"/bundles? Second, I would appreciate some examples which clarify the distinction between local and global gauge groups mentioned above. If, for example we take a prototype $SU(2)$ gauge theory of the form $$ \mathcal L = -\frac{1}{4}(F_{\mu\nu}^a)^2+|(D_\mu\phi)^a|^2-V(\phi) $$ then we can put it in the bundle picture as a $SU(2)$ principal bundle, the fields $A_\mu^a$ living in the associated bundle $$(P\times\mathfrak su(2))/\\{(p,A)\sim(pg,Ad_{g^{-1}}A)\\}$$ and the doublet field $\phi$ in fundamental representation of $SU(2)$ living in the associated bundle $$(P\times\mathbb C^2)/\\{(p,v)\sim(pg,\rho(g^{-1})\phi)\\}.$$ So, the $SU(2)$ group which we are referring to here is what was the local gauge group $G$ above, is this correct? The gauge transformations apply to coordinate versions of the fields in the trivializations of the associated bundles to change between different coordinates. Now if, this is the case, what is the global gauge group $\mathfrak G$ then? In particular I am seeking the answer for the following questions:   * Do global gauge transformations apply to the model $SU(2)$ theory above?   * In what sense is the global gauge group much bigger then the local gauge group?   * Is there a model example where one can clearly write down the local gauge group and the global gauge group?   * Are there cases where the two groups are the same?   * How does this discussion relate to quantization of classical fields (if it does at all)? * * * # Edit I'll post here answers to various questions in the comments/answers. About the definitions I used in my text. I took them from my lecture notes from a course on mathematical gauge theories taught by a mathematician. Although there is no on-line version of this course, I can provide more detailed definitions if needed. Specifically, concerning the distinction global/local gauge, I have found this piece of information on nLab, which seems to be agreement with my definition: http://ncatlab.org/nlab/show/gauge+group. Although this page provides some explanations, it uses some mathematical terminology which I am not entirely familiar with. The problem both with the nLab page and with the lecture I attended is to make the connection to physics (I am a physicist). Ok, I'll just write down the definitions from my lecture notes. ## Local gauge Let $P\to M$ a principal $G$-bundle, $\omega$ a connection 1-form on $P$. $U_i, U_j\subset M$ such that $U_i\cap U_j\neq\emptyset$ over which P is trivial: $$ \psi_i:\pi^{-1}(U_i)\to U_i\times G\\\ \psi_j:\pi^{-1}(U_j)\to U_j\times G $$ with transition functions $$ \psi_i\circ\psi_j^{-1}:(U_i\cap U_j)\times G\to (U_i\cap U_j)\times G\\\ \phantom{\psi_i\circ\psi_j^{-1}:XXX}(m,g)\mapsto (m,\psi_{ij}(m)g) $$ where $\psi_{ij}:U_i\cap U_j\to G$. Using these trivializations we can construct preferred sections as follows: $$ \sigma_i:U_i\to\pi^{-1}(U_i)\\\ \phantom{\sigma_i:x}m\mapsto \psi_i^{-1}(m,e)\,. $$ Define $$ \omega_i:=\sigma_i^*\omega\,, $$ which is a $\mathfrak g$-valued 1-form on $U_i$. Changing between coordinates on $U_i$ and $U_j$ is what is called choice of local gauge. We would like to write down a transformation prescription for $\omega_{i/j}$. On G we have the canonical 1-form $\theta$ with values in $\mathfrak g$ defined as follows: $$ \theta_g(X_g)=A\in\mathfrak g\quad\text{if }(A^*)_g=X_g $$ where by $A^*$ we mean the fundamental vector field on $G$ corresponding to $A$. One can show that the 1-forms transform in the following way: $$ \omega_j=Ad_{\psi_{ij}}^{-1}\omega_i+\psi_{ij}^*\theta\,. $$ This looks to me like a gauge transformation of guage fields $A=A_\mu dx^\mu$ in physics where we write $$ A'=gAg^{-1}+gdg^{-1} $$ and I even believe that the above transformation for $\omega_j$ can be rewritten as something like this $$ \omega_j=Ad_{\psi_{ij}}^{-1}\omega_i+\psi_{ij}^{-1}d\psi_{ij}\,, $$ Then it looks like the transformation for $A$ with $g=\psi_{ij}^{-1}$. So much about the local gauge transformation. ## Global gauge group The group of global gauge transformation $\mathfrak G$ is the set of automorphisms of the principal bundle P: $$ \mathfrak G = Aut(P)\,. $$ This must not be confused with the structure group $G$, which is sometimes called gauge group in leterature, but is much smaller. The (global) gauge group can be described in three ways:   1. $\mathfrak G=\\{\phi:P\to P|\phi\text{ is a diffeo., }\pi\circ\phi=\pi, \phi(pg)=\phi(p)g\,\forall g\in G\\}$   2. $\mathfrak G=\\{u:P\to G|u\text{ smooth s.t. }u(pg)=g^{-1}u(p)g\,\forall g\in G\\}$. Note that $\phi(p)=p\cdot u(p)$.   3. $\mathfrak G=\\{\text{sections }s:M\to F\\}$, where $F=(P\times G)/\sim$ with $(p,h)\sim(pg,g^{-1}pg)\forall g\in G$. If $\omega$ is a connection 1-form which corresponds to a choice to the horizontal tangent space $H$ on $P$, then $\phi^*\omega$ is also a connection 1-form, defining the pull-back connection $$ (\phi^*H)_p:=(D_p\phi)^{-1}H_{\phi(p)}\quad\Leftrightarrow\quad D_p\phi((\phi^*H)_p):=H_{\phi(p)} $$ We conclude that $\mathfrak G$ acts on connections, but the explicit form of this action if fairly complicated. In contrast, the action of $\mathfrak G$ on the curvature is easy to understand. From the last equation above we conclude that $\phi$ maps $\phi^*H$ to $H$, and so it maps $\tilde\Omega$, the curvature of $\phi^*H$, to $\Omega$, the curvature of $H$: $$ \phi(\tilde\Omega(X,Y))=\Omega(X,Y)\quad\forall X,Y\in T_mM\,. $$ One can show that the following transformation relation holds: $$ \tilde\Omega=Ad_{u^{-1}}\Omega\,. $$ where $u:P\to G$ is the map from the second definition of $\mathfrak G$ above corresponding to $\phi$. Two connections $H_1$ and $H_2$ on P are called gauge equivalent if there is a $\phi\in\mathfrak G$ with $\phi^*H_2=H_1$. * * * One of the things which I am curious about is that from the local gauge transformations the transformation law for the 1-form $\omega$ seems to be what we call the gauge transformation of gauge fields $A$ (see above), but from the global gauge group transformation, the transformation of the curvature is what we see in physics, namely if $F$ is the field strength corresponding to $A$, and $g$ an element of the gauge group then $$ F\to F'=gFg^{-1}\,. $$ Also, as I said at the very beginning, I would like to pin down what these mathematical definitions correspond to in what we learn in physics. In physics, when we discuss a certain gauge theory, there is THE gauge group, such as $U(1)$, $SU(2)$, etc. under which fields transform in certain representations, as well as the gauge fields themselves, which transform in the adjoint representation. Now in mathematics I see the distinction local/global gauge group. The essence of my question is do understand this distinction and to relate it to phyics.


title: Explosive energy in a container

text: What happens when the explosive energy of a grenade or a bomb is contained in a container, if you will, and no energy can be released through the container in the form of heat or sound etc...? Just theoretically asking? I just want to know what happens when we take a huge explosion or any explosion and shrink it to a finite point. Theoretically, what happens?


title: Is there some quantum potential producing exponential eigenvalues?

text: Usual central potentials produce quantum spectra with energy levels going as $n$, $n^2$, $n^3$ and so on, being $n$ the quantum number of the orbit. In the other extreme we have "dirac-delta" potentials which have only a single discrete eigenvalue. I was wondering, what kind of potential do we need for producing an exponential $e^n$ set of discrete eigenvalues?


title: Missing step while (classically) deriving Friedmann equation

text: I'm trying to understand the classical derivation of Friedmann equation but I'm missing one step. So, I start with accelerations, where $a$ is a scale factor $\ddot{a}=-\frac{GM}{a^{2}}$ $\ddot{a}=-\frac{4\pi G}{3}\rho a$ Now multiply both sides by $\dot{a}$, and here is the question: Why LHS is $\dot{a}\ddot{a}=\frac{1}{2}\frac{d}{dt}\left(\dot{a}^{2}\right)$


title: random particles on a grid: Effect of increasing density on distance between them

text: ![enter image description here](http://i.stack.imgur.com/O9S2S.png) Say I have two boxes which both contain, say, 25 red particles (as shown in picture). These particles are randomly placed in a 2d grid, and in one the total area $A_{1}=20000$ and the other has area $A_{2}=10000$. If the average distance between any two particles before was $d_{1}$, what is the average (expected) distance between any two particles after, $d_{2}$? I have worked it out as: Since we have vertically compressed the box by factor 0.5: ![enter image description here](http://i.stack.imgur.com/kdKaW.png) Then we get a factor of $\frac{\sqrt{1.25}}{\sqrt{2}}$ = 0.7905.... of the distance $d_{1}$. However, I am not sure if this is correct because surely the particles would be closer together in the x-direction as well.


title: Degrees of freedom of the graviton versus classical degrees of freedom

text: I have a puzzle I can not even understand. A graviton is generally understood in $D$ dimensions as a field with some independent components or degrees of freedom (DOF), from a traceless symmetric tensor minus constraints, we get:   1. A massless graviton has $D(D-3)/2$ d.o.f. in $D$-dimensional spacetime.   2. A massive graviton has $D(D-1)/2-1$ d.o.f. in $D$-dimensional spacetime.  Issue: In classical gravity, given by General Relativity, we have a metric (a symmetric tensor) and the Einstein Field Equations(EFE) provide its dynamics. The metric has 10 independent components, and EFE provide 10 equations. Bianchi identities reduce the number of independent components by 4. Hence, we have 6 independent components. However, for $D=4$, we get   1. 2 independent components.   2. 5 independent components.  Is the mismatch between "independent" components of gravitational degrees of freedom (graviton components) one of the reasons why General Relativity can not be understood as a quantum theory for the graviton? Of course, a massive graviton is a different thing that GR but even a naive counting of graviton d.o.f. is not compatible with GR and it should, should't it? At least from the perturbative approach. Where did I make the mistake?


title: What is the preferred basis objection to the many-worlds interpretation of quantum mechanics?

text: I've seen the preferred basis problem referred to in many places, but have not seen a clear explanation of what the problem is. For example, this question asks whether the problem has been solved, but I wasn't able to glean anything from the discussion on what the problem was. Zurek 2001 gives the following: > the original MWI does not address the "preferred basis question" posed by > Einstein with a footnote quoting Einstein as saying this: > When the system is a macrosystem and when ψ1 and ψ2 are 'narrow' with > respect to the macrocoordinates, then in by far the greater number of cases > this is no longer true for ψ = ψ1 + ψ2. Narrowness with respect to > macrocoordinates is not only independent of the principles of quantum > mechanics, but, moreover, incompatible with them. The first sentence seems obvious to me. I'm baffled by the second sentence. Some states are narrow and some are not. What's the problem, and what does this have to do with a basis or a preferred basis? Zurek, "Decoherence, einselection, and the quantum origins of the classical," http://arxiv.org/abs/quant-ph/0105127


title: Beads sliding on a hoop

text: ![enter image description here](http://i.stack.imgur.com/PxLaK.png)Two particles, $P_1$ and $P_2$, of equal masses $m$ are linked by a spring of stiffness $k$ and natural length $a$. They are sliding freely without friction along a horizontal fixed hoop of radius $R$. I need some help in writing Lagrangian for generalized coordinates $\theta_1$ and $\theta_2$ given as $\theta_1=\frac{\phi_1+\phi_2}{2}$, $\theta_2=\frac{\phi_1-\phi_2}{2}$. I wrote that position vectors for $P_1$ and $P_2$ are $\vec r_1$ and $vec r_2$, respectively.   $\vec r_1=-R\cos\phi_1 \vec i+R\sin\phi_1\vec j$   $\vec r_2=R\cos\phi_2\vec i+R\sin\phi_2\vec j$   From this   $\dot{\vec r_1}=R\dot\phi_1\sin\phi_1\vec i+R\dot\phi_1\cos\phi_1\vec j$   $\dot{\vec r_2}=-R\dot\phi_2\sin\phi_2\vec i+R\dot\phi_2\cos\phi_2\vec j$ Kinetic energy is $T=\frac{1}{2}m(\dot{\vec r_1})^2+\frac{1}{2}m(\dot{\vec r_2})^2=\frac{1}{2}mR^2(\dot\phi_1^2+\dot\phi_2^2)=mR^2(\dot\theta_1^2+\dot\theta_2^2)$ I need help about forces that are doing work.   I know that there is a force in the string, $F=ke$ where $e$ is extension, but I don't know how to determine its direction. I wrote that   $\vec {P_1P_2}=\vec r_2-\vec r_1$   $|\vec{P_1P_2}|=R\sqrt{2(1+cos(\phi_1+\phi_2))}$ and $|\vec {P_1P_2}|=a+e$ This is all I did. I'm not sure if normal reaction does any work. Any help would be appreciated. Left particle is $P_1$, right one is $P_2$, I forgot to write it on the picture.


title: OPE and 4-point correlation function in CFT_d

text: I'm reading this paper where to determine the coefficient $C^{\phi\phi O}(x_{12},\partial_2)$ of the OPE (p.10) $$\phi^\alpha (x_1)\phi^\beta (x_2)=C_\phi \frac{\delta^{\alpha\beta}}{x_{12}^{2\eta}}+C^{\phi\phi O}(x_{12},\partial_2)O(x_2)\delta^{\alpha\beta}$$ they require consistency of the OPE with the 3-point correlation function, i.e. comparing the fixed form of the 3-point correlation function to what happens when we apply the OPE to two channels in the 3-point function and then use the form of the 2-point function. Anyway, to solve the form of the coefficient that satisfies eq. 2.29 they use the integral representation $$\frac{1}{(ab)^\rho}=\frac{1}{B(\rho,\rho)}\int_0^1 dt \frac{[t(1-t)]^{\rho-1}}{[at+(1-t)b]^{2\rho}}$$ which comes from the book Table of Integrals, Series and Products by I.S. Gradshteyn and I.M. Ryzhik. However, looking at that book (I only have access to the 7th ed.) I only found the relation $$B(\mu,\nu)\frac{1}{(a+c)^\mu(b+c)^\nu}=\int_0^1 dx x^{\mu-1}(1-x)^{\nu-1}\frac{1}{[ax+(1-x)b+c]^{\mu+\nu}}$$ which reduces to the desired relation when $c=0$ however, in the book it says that the relation is valid for $c>0$, so how can they use that relation?


title: Does the electromagnetic field "spin"?

text: Due to electron "spin", a small magnetic field is produced. Maxwell's equations imply that magnetic fields are due to changes in electric fields. Is the magnetic field produced then because the electric field is "spinning" with the "spinning" electron, in the quantum sense of "spinning" and this change in electric field is generating the magnetic field? Can one generalize to say that the magnetic field thus would spin when a magnet is spun?


title: Why infrared absorption is a nonlinear technique?

text: I am looking for a good explanation and/or reference quotation explaining why infrared absorption technique is essentially nonlinear (eg. for carbon monoxide quantification). When using UV/visible/near-IR absorption technique, Beer-Lambert Law often is valid and then you have quite straight way to convert transmittance into absorbance which linearly respond to concentration (provided BLL limitations are not met). Above technique relies on electronic transitions, when IR absorption technique relies on vibrational and rotational transitions for molecules which has permanent or transient dipole moments. I would like to figure out why such technique is essentially nonlinear.


title: How fast do molecules move in objects?

text: I guess it depends on the heat or the type of the material but can you give some examples or formulas to calculate it ? The best example would be the average speed of the air molecules (all types in the air) at room temperature or water molecules at human body temperature.


title: Are the $10^{500}$ different string theories being whittled down?

text: An example of a test: Ask each variant whether its estimate of the electron mass lies within $\pm\,x\%$ of the known value. This surely can't take long per theory. Although $10^{500}$ is huge, whittling them down could be essentially a background task for a few thousand computers. But is it being done, and if not, why not?


title: Finding the stabilizer group given a state

text: Consider general pure state $|\psi\rangle$ in some hilbert space $\mathcal{H}$ (which could be a tensor product of other Hilbert spaces) **I would like to know whether there is a way to systematically find the group of all operators on $\mathcal{H}$ under which $|\psi\rangle$ is invariant.** A little more precisely: the group I'm looking for is $$ G=\\{A \in H(\mathcal{H}): A|\psi\rangle = |\psi\rangle\\} $$ Where $ H(\mathcal{H})$ is the set of Hermitian operators acting on the Hilbert space. This group I defined here is a stabilizer group in the usual sense in mathematics, and a slight generalization of the ones popularized by Gottesman in quantum information as we're considering a general state.


title: Flow rate is calculated only using the parallel component of the velocity vector

text: Flow rate is calculated using only the parallel component of the velocity vector to the area vector. Why is this? How can I mathematically prove this? Namely, how do I prove any perpendicular component of the velocity vector is not contributing to any volume output? I know this is the result of the dot product; but I want to know why the dot product is valid reasoning for this scenario.


title: Why don't the black holes appear black in color in images of galaxies taken from HST?

text: According to NASA > a black hole is anything but empty space. Rather, it is a great amount of > matter packed into a very small area According to the documentary Space Unraveling The Cosmos about Black Holes > Gravity here is so strong even light cannot escape Which leads me to believe that Black Holes basically are masses so compact but great which gives them very strong gravitational influence (Newton's Gravitational Law) that light is not allowed to go out of, hence a "Black" hole. If my above understanding of Black Holes is correct, then why is those Spiral Galaxies pictured using Hubble Telescope is showing a big shiny ball instead of a "Black" hole ? What is the obvious clue that I am missing here ? Here is the image of Andromeda Galaxy image taken using Hubble. ![Andromeda Image taken using HST](http://i.stack.imgur.com/YkdZ2.jpg)


title: Does the strong (nuclear) force ever contribute to decay?

text: Does the strong (nuclear) force ever contribute to decay ? Or is the weak nuclear force the only decaying force ?


title: Length of solid part of object shadow

text: Little background first:   I take part in preparing environmental reports for wind turbines/farms and now I'm exploring shadow flickering effect of moving blades. From some sources (in Polish) I got to know that the shadow casted by turbine blades is solid, black only to some distance and above this distance the shadow is blurred. This distance was described as approx. $108\ * \ width$ of blade. As an explanation of this arguments the umbra-penumbra-antumbra diagram was presented. I wanted to find where from this $108$ value comes from.   I found that for spherical bodies, when the occulting object is smaller than the star, the length (L) of the umbra's cone-shaped shadow is given by:   $L=\frac{rR_o}{R_s-R_o}$   where $R_s$ is the radius of the star, $R_o$ is the occulting object's radius, and $r$ is the distance from the star to the occulting object.   I've also found that this equation comes from simple triangles similarity in Sun-Earth arrangement. When I use the radius of the Sun and the distance from the sun to the Earth, the equation actually simplifies ($R_s >> R_o=3.5m$) to: $L=215R_o \approxeq108D_o$ ($D_o$ - object diameter) My question is:   Can this equation be used to approximate the length of solid part (umbra part) of a shadow of objects on the Earth?


title: Area under the graph of squared wave function

text: I was given a graph of square of the wave function of a hydrogen atom, against the distance of the electron from the nucleus (denoted by r). What I know is that the square of the wave function gives the probability of finding an electron at a particular position, but could anyone explain to me that why does the area under this graph has to do with the probablity of locating an electron? Because I thought that on the graph where the highest value of square of wave function is, the corresponding value of r will be the answer. Isn't that so??


title: Sound localization using cross-correlation

text: I am using cross correlation for sound localization. I have a four microphone array (arranged linearly). My problem is, sometimes the time diff found by cross correlation between a given pair of mics jumps to some eratic value. For example, if for the first three times it gives a value of 14 then the next time it suddenly jumps to some totally different value like 153 for the sound generated from the same point. Is this normal or am I doing anything wrong? My data collection procedure is as follows:   1. Start listening   2. Keep listening till trigger event occurs (which is a peak in the sound amplitude)   3. As soon as the trigger event is noticed, collect sound sample for the next 300ms from all mics   4. Calculate cross-correlation to find time diff In order to detect the trigger event, I am listening for 10ms and checking the mic data if there is a peak in it. If yes, I record for the next 300ms and process it. The ambient noise in my setup is very minimal so I dont think that is effecting the calculations. If anyone has come across a similar situation, could you please shed some light?


title: Proton as superposition of hadrons: $\vert p\rangle = c_0\vert p_0\rangle+c_1\vert h\rangle+\cdots$

text: I have a question regarding hadron fluctuations. For instance on page 85 in Feynman's "Photon-Hadron Interactions" equation 15.2 reads: $$\tag1\vert \omega\rangle = \vert \omega_0\rangle+\frac{\delta}{m_\omega- m_\rho-\frac{\mathrm{i}\Gamma_\omega-\mathrm{i}\Gamma_\rho}{2}}\vert \rho_0\rangle$$ where $\vert\delta\vert\approx 3.7$ MeV (see p. 86 same book). My question is where I can find an analogous expansion for the proton (say), i.e. something like $$\tag2\vert p\rangle = c_0\vert p_0\rangle+c_1\vert h\rangle+\cdots$$ where $h=$ some hadron.


title: How does a 2D MEMS based optical cross-connect control its mirrors?

text: Concerning fiber optics, I'd like to know how a 2D MEMS based optical cross- connect is able to switch optical signals. I've read that these mirrors are controlled with electrical signals to be either "on" or "off" to redirect light to output ports. As far as I know, all-optical based switches don't use buffers on the incoming data. However, an electrical signal must be generated to control the MEMS mirrors in order to redirect the signal to the output port. How can this electrical signal be fast enough to switch the optical signal? I assume this electrical signal comes from the header of the packet, but I'm not sure how an optical switch obtains that information and generates an electrical signal.


title: Ward Identity in CFT

text: This is about Polchinski's eq(2.3.11). It says that $$Res_{z\rightarrow z_0}j(z)\mathcal A(z_0,\bar z_0)+\bar{Res}_{z\rightarrow z_0}\tilde j(\bar z)\mathcal A(z_0,\bar z_0)=\frac1{i\epsilon}\delta \mathcal A(z_0,\bar z_0)$$ from (2.3.10), $$\oint_{\partial R}(jdz-\tilde jd\bar z)\mathcal A(\sigma_0)=\frac{2\pi}{i\epsilon}\delta\mathcal A(\sigma_0).$$ ``Res'' is the coefficient of $\frac1{z-z_0}$. But how did we know that $j\sim\frac{1}{z-z_0}$, i.e., it has a pole at $z=z_0$? I know nothing about $j$ except that it is a function of $z$.


title: How does escape velocity relate to energy and speed?

text: I have several confusions regarding escape velocity. I am sure I am missing something(s) obvious or maybe I am taught wrong.   1. Lets say we throw an object of any mass at exactly escape velocity of earth calculated from $v^2=\frac{2GM}{r}$ which is almost $11 \text{km s}^{-1} $ but I am talking about exact escape speed. That ball initially has $KE=\frac{1}{2}mv^2$ and $PE=mgh$. Wikipedia says and I quote  > In physics, escape velocity is the speed at which the kinetic energy plus > the gravitational potential energy of an object is zero. How is that possible?   2. As $F=\frac{GmM}{r^2}$ no matter how much the particle travels away from earth's surface it will always be accelerated towards earth. With increasing $r$ the $F$ will decrease but it will never reach $0$. That means that there will be no point where the particle will stop and will continue to move with slower and slower _speed will never reaches zero_. Am I right?   3. A request: Please explain exactly what happens to particle's $KE$ and $PE$ at different points such as at $r=0$ and at $r=\infty$.


title: Finding interaction Hamiltonian for nano particles

text: I am studying Quantum electrodynamics (QED) and the core part of this theory is the light-matter interaction (quantized light and matter). I referred many papers and books (Molecular quantum electrodynamics, Interaction Hamiltonian of quantum optics etc), and in all the places the interaction Hamiltonian is derived based on the dipole approximation or multipole. However if we are to find the interaction Hamiltonian for more realistic situation like quantumdots/nano wire to light interaction, what should be considered in the first place? Thank you very much.


title: Why would a fat skydiver fall first in free fall?

text: I was having one of those obnoxious conversations with a friend where he was arguing that a fat skydiver would reach the ground faster than a skinny skydiver. To me it seemed as obvious that the world is round (that is: not very obvious unless you have some education). We weren't getting anywhere so I punched in the data in Wolfram Alpha and I was surprised that indeed a fat person falls faster. I don't quite grasp why a heavier person with the same surface area has less `slowdown factor`. Can anybody explain this to me? ![enter image description here](http://i.stack.imgur.com/LjDM9.png)


title: What equation of state is needed for liquid states?

text: I'm familiar with the ideal gas law $$PV=nRT$$ but I don't think it applies to liquids like water. If I'm wrong, please correct me! If I'm right, then what equation of state applies to liquids such as water?


title: Can a wavefunction be solved to any arbitrary precision, given enough computer time?

text: I learned that the wavefunction for the hydrogen atom can be solved analytically (we did the derivation in class), but that for more complicated atoms it is "impossible" to solve and that only approximations can be obtained. Well, I'm not sure I like the vague wording with "impossible". Basically, a hydrogen atom can be solved in terms of elementary functions. Elementary functions, as far as I know, are just arbitrarily defined (Is $\sqrt{}$ one? What about $\text{erfc}$?). For instance, I could say that $\sin{x}$ cannot be solved exactly except in a few special cases (x = some rational multiples of $\pi$). To analogize, a general quintic equation cannot be "solved" exactly using elementary functions -- that is, unless you define a Bring radical as a new elementary function. To me, it's really arbitrary and all of these solutions come down to numerical, iterative refinement. Solved exactly vs solved approximately seems to be an ill-defined concept. So what I am asking -- given enough computer time -- can I solve the Schrodinger equation of, say, Lithium, to whatever precision I like? Much as how I can solve $\sin{x}$ to however many decimal places I want? Or is there some _other_ limit to such a calculation?


title: Rotation of Spacetime => Change in orbit/path

text: Along the idea of frame-dragging; Will the rotation of a black hole, which has some velocity _v_ and angular momentum, influence its path in 3D space? I've seen the fact that depending on the direction of the rotational velocity, the Kerr metric can give different revolution periods for orbiting masses. (e.g. as in our solar system) So why (or why not) is this so? How does the metric show that the direction of the velocity can influence orbital periods?


title: Tachyonic field

text: i'm working on a paper about symmetron cosmology. symmetron is a scalar field that by its symmetry breaking can explain the dark energy. the action is: ![enter image description here](http://i.stack.imgur.com/5WGnl.jpg) ans A , V are assumed to be: ![enter image description here](http://i.stack.imgur.com/THXhv.jpg) ![enter image description here](http://i.stack.imgur.com/BJqVP.jpg) where M and ![enter image description here](http://i.stack.imgur.com/Saayr.jpg) are mass scales and lambda in a coupling. somewhere in this paper (http://arxiv.org/abs/1107.2112) "we get interested in the case where the field becomes tachyonic around current cosmic density, ![enter image description here](http://i.stack.imgur.com/qFKaH.jpg) my questions are: >   1. what is the meaning of a field becomes tachyonic? >   2. where the above approximation come from? > thank you so mach


title: Two electron beams exert different forces on each other depending on frame of reference?

text: I am sure there is a simple explanation for my confusion, but I am a little puzzled: We are dealing with two parallel electron cannons that each produces a straight beam of electrons. They are placed at a distance d next to each other, which means there will be a repulsive electrostatic force between the electrons of the two beams. Also, according to the Maxwell equations, every moving electron will produce a magnetic field which then exerts another force on the other beam's moving electrons (similar to the effect that two currents in wires will cause an attractive force between the wires). However, if we change the reference frame such that we are moving at the same speed alongside the electrons, there will be no magnetic field and no Lorentz Force. We are only left with the electrostatic force. Do you guys know what might be the fallacy?


title: 1D Ising Model with different boundary conditions

text: The Hamiltonian for one-dimensional Ising model is given by, \begin{equation} \mathcal{H} = -J\sum_{<ij>} S_iS_j; \quad i,j=1,2,...,N+1 \end{equation} where $<ij>$ denotes that there is nearest neighbor approximation. The partition function is given by, \begin{equation} \mathcal{Z}=\sum_{\\{S_i\\}} e^{-\beta \mathcal{H}(S_i)} \end{equation} with $\beta = \frac{1}{k_BT}$, where $k_B$ is the Boltzmann constant and $T$ is the temperature. Now my questions are: **1\. How to compute the partition function when $S_{N+1}=+1$ while other spins ($S_i$ for $i=1,2,...,N$) may take value $+1$ or $-1$?** **2\. How to compute the partition function when $S_{N+1}=-1$ while other spins ($S_i$ for $i=1,2,...,N$) may take value $+1$ or $-1$?**


title: Can I levitate soon?

text: Whereas I have seen frogs fly in high magnetic fields and Sumo ringers stand on high-temperature superconductors it seems that these are not the only forms of levitation. Recently a group of physicicsts managed to levitate coffee without a magnetic field (PNAS link, only abstract is free though). So while I am still trying to grasp how this is even possible my question is a bit more far fetched: Can I levitate using this approach?


title: Does speed of writing data on an optical disk affect the speed of reading data?

text: Does speed of writing data on an optical disk ( like DVD ) affect the speed of reading data from that disk ? even a little?


title: Force exerted by rotating bar

text: Say we have a bar immersed in liquid, and that the bar in being rotated at a given constant angular speed. What force is exerted on the liquid by the faces of the bar moving through the liquid (so the force is tangent to the circular motion of the bar). My guess is that the expression involves the distance from the centre of the bar, $r$, and the angular velocity $\omega$. But I have no clue what exactly the relationship should be; can anybody help?


title: The role of metric in the Wave Equation

text: The wave equation is often written in the form $$(\partial^2_t-\Delta)u=0,$$ involving the Laplace-Beltrami operator $\Delta$. However, the Laplace- Beltrami operator $\Delta$ is defined only in the presence of Riemannian metric $g$. Indeed, one can think of $\Delta u$ as of difference between $u$ and its mean value in the balls, and the notion of ball only makes sense in the presence of a metric. What is the physical meaning of metric $g$ here? My guess is it's connected with the property of isotropy of space: in the vacuum, the waves should propagate uniformly in all directions, and this situation corresponds to Euclidean metric. The case of general Riemannian (= ‘perturbed Euclidean’) metric should then correspond to the propagation of waves in a medium of some sort, so that the property of ambient space isotropy is not satisfied, hence balls in it do not look like usual ‘Euclidean’ balls, and wave equation takes its general form $(\partial^2_t-\Delta)u=0$. Is my guess true? If not, what is the correct interpretation of metric in the general wave equation?


title: Potentiometer voltage change

text: I'm trying to understand simple things about electricity reading allaboutcircuits.com web. This chapter includes image which´s principle I don't understand. Here it is: ![enter image description here](http://i.stack.imgur.com/gizaD.png) When supplying constant voltage to circuit where resistance and current change, how is it possible that change of resistance affects voltage on resistor? Please help.


title: Sunflowers and radioactivity

text: I read yesterday that sunflowers were used and to cleanup radioactivity at Chernobyl and the Atomic Bomb sites in Japan and may be used as part of a campaign to clean up the Fukushima area. But my questions are:   1. How does a sunflower absorb radioactivity?   2. Is there a way to measure how well the sunflowers are working?    3. Can sunflowers planted in not particularly radioactive places concentrate otherwise harmless radiation? _I was tempted to ask this on Skeptics.SE, but there's not much to be skeptical about, except whether or notit's a good idea_


title: Can it require different amounts of energy to generate the same impulse?

text: According to impulse principle the impulse is the same as the change in the object's momentum: $\bar I = \delta p$ Because the momentum can be calculated like this: $\bar p = m\bar v$. If we solve speed from that: $\bar v = \frac{\bar p}m$ We can conclude that a certain impulse causes certain change in speed as long as the mass is a constant: $\delta \bar v = \bar v_2 - \bar v_1 = \frac{\bar p_2}m - \frac{\bar p_1}m = \frac{\bar p_2 - \bar p_1}m = \frac{\delta \bar p}m = \frac{\bar I}m$ However, if we solve how much energy it takes to cause a certain change in speed it depends on the original speed: $W = \delta E = E_2 - E_1 = \frac 1 2mv_2^2 - \frac 1 2mv_1^2 = \frac 1 2m(v_1 + \delta v)^2 - \frac 1 2mv_1^2 = mv_1\delta v + \frac 1 2m(\delta v)^2$ So my question is: Does the energy needed to generate an impulse of certain magnitude depend on the original speed of the object?


title: String Theory and Standard Model in CERN

text: I don't know how to say it, but in the TV dominatrices and the popular science books we see the string theory as "the best theory to explain everything", and as "the only game in town"... etc. And yet, in CERN -The largest physics laboratory-, they are just using the Standard Model (SM) as the only way to find the theory of everything (ToE), and sometime they call it: " _theory of almost everything_ ". Also, they didn't even mention anything about string theory. (as if it doesn't exist!) So my question is: **Why?** Isn't it easier for them to take the string theory as a Theory of Everything? Also, the Standard Model is not able to explain the gravitational force from the SM equation! So, the SM is essentially _incomplete_! Why do they keep working on the SM? Am I missing something here? -P.S.: I'm an engineer, not a physicist! So, I don't know much about the details.


title: Examples of incorrect physics education

text: Can you give me examples where the standard physics curriculum in high school or college is inaccurate? When did you realize you had been fed lies?


title: Question about Calabi-Yau manifolds and quantum fluctuations at the Planck scale

text: Super string theory establishes that at each point of our 4 dimensional spacetime, there is attached a 6-dimensional Calabi-Yau (CY) manifold which encodes the geometry of the additional compactified spatial dimensions needed to formulate the theory. It is postulated that the size of these compactified dimensions is of the order of the Planck scale. On the other hand, I've always understood that in quantum gravity, we are confronted with the big problem of wild quantum fluctuations -due to the Heisenberg principle- at the Planck scale.These fluctuations should lead to wild shifts and fluctuations in the metric coefficients and hence, they would make hard to properly talk in the terms of differential geometry (manifold, distance, metric, curvature) that are the terms in which we talk about CY manifolds. I'd like to know how string theory solves this contradiction that has arisen due to my lack of knowledge of the full technical details of the theory. Is it possible that, the conformal invariance required for the string worldsheet during its propagation, prevents the pathological effects of these quantum fluctuations allowing us to talk in terms of geometry even at the Planck scale?


title: Difference between Residual Function and Excess Property

text: Sengers describes the deviation function as the difference of the actual mixture property and as determined from a model. If this model happens to be the ideal gas model, then it is also called residual function. Lemmon describes excess property as the amount that a mixture property changes solely due to the mixing of the components. Does this mean that a deviation function is dependent on the type of model that you specify whereas excess property is independent of that? What are the differences between these two terms?


title: Does the photoelectric effect obey Ohm's law?

text: So, I've been reading about the photoelectric effect for my modern physics class, and I was confused about how Ohm's law works in relation to it. Say we have a photoelectric apparatus that simply generates some current and then applies a stopping potential V to bring the current to zero. If we increase the intensity, this increases the current. However, then V = IR, and resistance remains the same, so shouldn't voltage increase? This is against the fact that intensity and stopping potential are independent, so does this mean that Ohm's law just fails in this case? Thanks so much.


title: How does autorefractor work?

text: Autorefractors are being used by eye opticians for eye diagnosis. I searched internet for articles and wiki page as well but I wasn't satisfied. I am interested to know how do they actually work. How the machine is able to focus sharp on retina automatically. How do they figure out spherical/cylindrical aberration for human eye. I even asked optician as well but he tricked out of this position saying it does all happen automatically. LOL! A detailed answer would be much appreciated. Please don't hesitate to write any technical details or mathematical derivations. I would love to know how these machines actually work. Thanks in advance!


title: Supersymmetry definition

text: Can someone outline the difference between supersymmetry and supersymmetric quantum mechanics? I often hear the two used interchangeably but I'm almost certain they are not the same.


title: How do ice spikes form?

text: I recently saw this picture posted on Twitter which shows a so-called ice spike rising from an ice cube tray. ![Ice Spike from Ice Cube Tray](http://i.stack.imgur.com/DwBoZ.jpg) I have read the Wikipedia page, but it doesn't mean much to me. My instinct was that it is caused by vibrations from the freezer setting up a resonant frequency on the surface of the water, and the spike has gradually "built up" as the water freezes. But I'm not a physicist, so I don't know if this makes sense. I'm ideally looking for a layman's explanation of why this happens.


title: Is the conjectured noncommutative heavy scalar "brother" of the already detected Higgs boson is a pseudo scalar?

text: This is a technical (may be trivial?) question about this sigma scalar field advertised by Chamseddine and Connes to improve the electroweak vacuum stability involved by the weak mass of the already found Higgs-like particle. **Is it a scalar or a pseudo scalar field?** (motivation: I was looking for a possible connection with Majoron models)


title: About the stability of the ground state of the bosonic string

text: In Polchinski's string theory vol 1, p. 23, it is said "It is a complicated question whether the bosonic string has any stable vacuum, and the answer is not known." The book was published on 1998. What is the current answer of this question? Is bosonic string a realistic theory nowadays? In addition, the ground state here is $$|0;k \rangle $$, not $$| \mathrm{vacuum} \rangle$$. Is that because the latter one is absolute nothing so we don't study?


title: What does a point in space mean? What we humans think a point in space is in scientific term? (a broader picture)

text: I am a first year and general student. I Am just curious about the vacuum or space. I just don't get what actually space is. Well, I know our scientists are still working on it but still I just want to know what is the accepted and best definition of space or a point in space that is the basic building block of this medium that allows matter to exist and its properties. Is space accepted as absolute zero or we believe that space itself lies within another medium having higher dimension or something like this. One more question when we say universe is expanding, in reference to what we are saying it is expanding? I mean is space itself stretching (which means space in between basic building blocks of matter is increasing, and if it is like this then how can we measure it as even our standard of measurement will be effected and hence net change will be zero) **OR** just heavenly bodies are going away from each other but then what is the reference ? Let us suppose it is Milkyway, but is Milkyway itself not moving and gaining velocity in the direction of the objective (the heavenly body which we say is going far from us)? Hence net change in velocity here should be zero?


title: The connection between classical and quantum spins

text: I have two questions, which are connected with each other. **The first question**. In a classical relativistic (SRT) case for one particle can be defined (in a reason of "antisymmetric" nature of angular momentum) 4-tensor $$ L_{\alpha \beta} = x_{\alpha }p_{\beta} - x_{\beta }p_{\alpha } = (\mathbf L, \mathbf G), $$ where $\mathbf L = [\mathbf r \times \mathbf p ], \quad \mathbf G = \frac{E}{c}\mathbf r - ct\mathbf p $ are angular momentum and center of energy vectors. It is called angular momentum tensor. In a case of many particles it's convenient to introduce Pauli-Lubanski 4-vector of spin: $$ S_{\nu} = \frac{1}{2}\varepsilon_{\nu \alpha \beta \gamma }L^{\alpha \beta}U^{\gamma}, $$ where $U^{\gamma}$ is summary 4-speed. The physical meaning of this vector is representation of own angular momentum of particles (without summand connected with motion of system as one body): $$ S_{0} = (\mathbf L \cdot \mathbf U), \quad \mathbf S = \mathbf L U_{0} - [\mathbf G \times \mathbf U] = \frac{E}{M}\left( \mathbf L - [\mathbf R_{E} \times \mathbf P]\right). $$ By the way, for one particle in classical case $$ \mathbf S = \mathbf L - [\mathbf R_{E}\times \mathbf P ] = [\mathbf R_{E}\times \mathbf P ] - [\mathbf R_{E}\times \mathbf P ] = 0 . $$ For many particles in general case $\mathbf L $ include orbital angular momentum, so $\mathbf L - [\mathbf R_{E}\times \mathbf P ] \neq 0$. There, if we introduce an operator representation, it will be easy to show, that $$ [\hat {L}_{x}, \hat {L}_{y}] = [\hat {S}_{x}, \hat {S}_{y}] = i\hat {L}_{z}, $$ where equality of first and second commutators is possible in the rest frame (add: i.e., for a case $\hat {\mathbf P}| \Psi_{\mathbf P = 0} \rangle = 0|\Psi_{\mathbf P = 0} \rangle $). That, after simple transformations, leads us to the discrete spectra of eigenvalues of $\hat {L}_{z}$ (or $\hat {s}_{z}$ in a case of the rest frame): $$ m = s, s - 1, ..., -s, $$ where s is the "full momentum" (or spin in the rest frame). So, the question: can we talk about classical analogue of spin in a case of quantum mechanics, when we analyze the quantum system (not one particle)? In a classical case the analogue of spin is an own angular momentum determined above. **The second question**. In a field theory it is well known that angular-spin tensor, which is a current of a Lorentz-symmetry of Lagrangian (Noether formalism). Does it connect with a spin of the quantum field in terms of first question? If the answer is positive, there is another question: has Belifante procedure an analogue of a quantum case?


title: Gauss law in classical U(1) gauge theory

text: I can see that $a_{0}$ is not an independent field and Gauss law is a constraint on the theory arising from field equations. But, I don't get the geometrical picture. Let $A$ be the space of all field configurations and let $G$ be the group of gauge transformations. Then, the actual configuration space $C=A/G$ where we consider $A$ modulo the action of gauge transformations. So, first of all, why don't we do our variational calculus to obtain field equations on configuration space? It has always been taught to me that I should consider variation of "paths" in configuration space only while applying Hamilton's principle. Instead, we first obtain field equations regarding $A$ as our configuration space and then "project" them perpendicular to gauge orbits using Gauss law. Are these two methods equivalent? Moreover, I don't understand how Gauss law really projects my fields orthogonal to gauge orbits. According to Manton's book (see the relevant pages scanned below) on topological solitons, he starts with a variation of the fields $\phi$ and $a$ in a small interval of time. Then, he claims that a "naive contribution to the kinetic energy" would be some expression involving these variations (see below) which is non-zero for infinitesimal gauge transformations also. But, when I look at the expression of kinetic energy (which is $T=\int(\frac{1}{2}e_{i}e_{i}+\frac{1}{2}\bar{D_{0}\phi}D_{0}\phi)d^{d}x$) in terms of fields, then I find that the expression is gauge invariant. I don't see why the particular expression $\frac{1}{2}\int\frac{1}{(\delta t)^{2}}(\delta\vec{a}.\delta\vec{a}+\bar{\delta\phi}\delta\phi)d^{d}x$ be the change in kinetic energy. In short, I want to understand this geometric insight into Gauss law whose details are attached below. Also, I wonder if this Gauss law has a relation to the Gauss law I learned while doing high school electrodynamics. Any information or resource on this would be very helpful. ![enter image description here](http://i.stack.imgur.com/3rP4Y.jpg) ![enter image description here](http://i.stack.imgur.com/nRlWL.jpg)


title: Refraction, reflection, and what is total reflection?

text: So if light travels from one media to another with a different refraction index, what may happen happen? Refraction, reflection or total reflection? I am quite confused as to the differences between these three concepts. Does 1 of these always happen in this situation? What exactly is total reflection and how/why does it occur?


title: From where do the permanent magnets get energy from?

text: I have a doubt about permanent magnets. If a magnet is permanent it can attract some materials permanently. Attracting something involves energy. If a permanent magnet can do this forever, from where does this energy come from? How can it not run out of energy? Doesn't it contradict with the laws of energy?


title: Why can one equate the the zeroth order coefficient with the initial state in time-dependent perturbation theory in quantum mechanics?

text: # Setup In the typical treatment of time-dependent perturbation theory in quantum mechanics, one arrives at the set of equations $$ i \dot{a}^{(r + 1)}_m(t) = \sum_n \langle m |H_1(t)|n \rangle e^{i \omega_{mn} t} a^{(r)}_m(t) $$ for integers $r > 1$ and $\dot{a}^{(0)}_m(t) = 0$. Here, the Hamilton is $H(t) = H_0 + \lambda H_1(t)$, the states $\\{|n\rangle\\}$ satisfy $H_0 |n\rangle = \varepsilon_n |n\rangle$ and $\omega_{mn} = \varepsilon_m - \varepsilon_n$. The expansion coefficients are defined by $$ a_m(t) = \sum_{r = 0}^{\infty} \lambda^r a^{(r)}_m(t) $$ where $a_m(t)$ are the coefficients in the expansion $$ |\psi(t)\rangle = \sum_n a_n(t) e^{-i \varepsilon_n t} |n\rangle $$ where $$ i \frac{d |\psi(t)\rangle}{dt} = H(t) |\psi(t)\rangle $$ Note then that $a^{(0)}_m(t) = a^{(0)}_m(0) = a^{(0)}_m$, say. # Question The literate then claims that if $|\psi(0)\rangle$ is known, then $a^{(0)}_m = \langle m |\psi(0)\rangle$. Does this really follow from the above? I feel an additional assumption (perhaps a boundary condition on $t$) is required to conclude the association of the initial state with the zeroth order coefficient. (Though obviously we have $a_m(0) = \langle m |\psi(0)\rangle).$


title: What is the typical orbital life of an artificial satellite?

text: The orbit of satellites around Earth eventually decays, or so I read. This is typically caused either by atmospheric drag, or by tides. I would assume most satellites have a limited service life in orbit. Hence the question - What is the typical orbital life of a satellite? How long before it's successor must be launched into orbit ? EDIT: For instance, a weather satellite


title: Determining axis of rotation from angular speeds about axes

text: I think my pure-math head is messing with me on the question below: my physics and CS friends both seemed to think it was a simple computational thing, and my program says the method works, but now I've confused myself about rotations vs. angular velocities. The problem: you have data from gyroscopes on a rigid body that give you the angular speeds of the object around three orthogonal axes through the body. These measured angular speeds are constant, call them $\omega_x$, $\omega_y$, and $\omega_z$. Sine the angular speeds are constant, the axis of rotation of the object is also constant, and it's not too much trouble to find it by considering the combined angular velocity vector $\Omega$ = ($\omega_x$,$\omega_y$,$\omega_z$). I am then confused when I think about this in terms of rotations. There's a unique axis of rotation because every composition of rotations is equivalent to a single rotation about some axis. It seems, then that you should be able to determine this axis by determining the eigenvectors of the combined matrix ABC, where A, B, C each represent the rotation about one of the axes. But of course, rotations in general don't commute. I'm forced to conclude that what makes the unique determination of an axis work is that infinitesimal rotations commute. So, if I'm right about this, my question is -- how can I see that infinitesimal rotations commute? My intuition fails me here. If they don't - what's the link between the rotation and angular velocity vector way of looking at this that removes the usual problems dealing with rotations that don't commute?


title: diesel steam engine

text: Let's imagine a diesel engine's cylinder head would be heated by external means, to some hundred degree celsius, and the fuel would be replaced by water. We've made modifications so that water would'nt destroy the fuel system and the cylinders would sustain the high temperature. Now the injector would squirt some water into compressed hot air. It would instantly vaporize and the steam expands with the moving piston. Would this give a working engine? What degree of efficiency can be expected compared to internal combustion?


title: Why does a wall act as a low-pass filter?

text: Learning about the fourier transform and its connection to filtering/convolution got me curious about naturally occurring filters. Why/how is it that brick walls naturally act as a low-pass filter (which requires something as seemingly complicated as convolution with the `sinc` function) to sound waves?


title: Is total angular momentum conserved in particle interaction?

text: Imagine that two electrons interact by exchanging a virtual photon. I know that the total energy and linear momentum of the two electrons is conserved by the interaction. Is the total (orbital) angular momentum of the electrons conserved as well? If the photon transfers energy and momentum between the electrons does it also transfer angular momentum between them?


title: One more time about LSZ-theorem

text: This question is the continuation of this one. For simplicity, let's use $(1)$ from the linked question (it is called n-point Green function and in particle case coincides with internal diagram), $$ G(p_{1},...,p_{n}) = \int d^{4}x_{1}...d^{4}x_{n}e^{-i\prod_{i =1}^{n}p_{j}x_{j}} \langle |\hat {T}\left( \hat {O}_{l}(x_{1})\hat {A}_{2}(x_{2})...\hat A_{n}(x_{n})\right) |\rangle , \quad (1) $$ (in accordance with the linked question $\hat {O}_{l}(x)$ transforms as some free field $\hat {\Psi}_{l}(x)$ under the irreducible rep of the Lorentz group, and the other operators are local functions of the free fields) for the case when $\hat {A}_{2}(x) = \hat {O}_{l'}^{\dagger}(x)$ and the other operators $\hat {A}_{i}(x_{i})$ is equal to the unity. Then it's not hard to show that $$ G (p_{1}, p_{2}) \to \frac{-2i|N|^{2}\sqrt{\mathbf p_{1}^{2} + m^{2}} }{p_{1}^{2} - m^{2} - i\varepsilon}\sum_{\sigma}u^{\sigma}_{l}(\mathbf p_{1})(u_{l'}^{\sigma}(\mathbf p_{1}))^{*}\delta (p_{1} + p_{2})(2 \pi )^{4}, \qquad (2) $$ which coincide with propagator of the free field $\langle |\hat {T}\left(\hat {\Psi}_{l}(x) \hat {\Psi}_{l'}(y)\right)|\rangle$ in the momentum representation excepting $|N|^{2}$ factor. For the simplicity lets assume that $\hat {O}_{l}(x) = \hat {O}(x)$, i.e. we working with the scalar operator. Then it's not hard to show that $|N|^{2} = (2 \pi )^{3}\left|\langle \mathbf p| \hat {O}(0) |0\rangle \right|^{2} = \frac{Z}{2\sqrt{\mathbf p^{2} + m^{2}}}$ (this result may be easily compared with the paragraph about renormalization of the field strength of Peskin and Schroeder QFT; the result above coincides with the result from Peskin QFT up to the normalization condition). So, the question: why in the LSZ formalism $|N|^{2} \neq \frac{1}{2\sqrt{\mathbf p^{2} + m^{2}}}$ as for case of free theory? I.e., why $Z \neq 1$?


title: How do I calculate the position on the Bloch sphere of a quantum gate with a given diagonal matrix?

text: In quantum computation there are several principal quantum gates that have corresponding matrix representations. One of these is the Z gate, whose matrix is $\left[\begin{smallmatrix} 1 & 0 \\\ 0 & -1 \end{smallmatrix}\right]$. ... anyway, I've found the eigenvalues (equal to +1, -1) using the characteristic equation, and used them to derive the corresponding eigenvectors, which come together quite nicely in a 2×2 matrix $\left[\begin{smallmatrix} 1 & 0 \\\ 0 & 1\end{smallmatrix}\right]$, equal to the identity. So, in diagonalizing this matrix, I find that the diagonal matrix $D$ is the same matrix as the one for gate $Z$. ... the next step and where I'm stuck is to find the corresponding point on the Bloch sphere for this gate. In order to do that, I need to compute how to take the diagonalized matrix call it $D_z$ and derive two things: **(a)**  its diagonal representation $| 0 \rangle \langle 0 | - | 1 \rangle \langle 1 |$, and **(b)**  the normalized eigenvalues $a, b$ for $Z$, where $Z = a|0\rangle + b|1\rangle$ and must be orthonormal i.e. $a^2 + b^2 = 1$. The $a$ and $b$ terms correspond to the probabilities of measuring 0 or 1 for the state, respectively (I think). After I have the values for $a$ and $b$, I'll be able to locate the gate on the Bloch sphere because the calculation of its coordinates on the sphere is straightforward: $a = \cos(\theta / 2)$, and $b = e^{i\phi}\sin(\theta/2)$.