title: The principle behind door peepholes

text: Hotels usually install peepholes in their doors so that a person inside a suite can see who is at the door without having to open it. I understand that there should be a convex lens within the peephole so that the person inside can easily see the person outside, but what I'm wondering about is the trick such that the person outside can only see blurred shapes if he tries to look though the peephole. Surely, there is more to it than specially-arranged convex lenses!


title: does the current's direction on a loop with a magnetic field affect the torque?

text: I have a question in my book ![http://i.imgur.com/P5R55sR.png](http://i.stack.imgur.com/7yZhT.png) The question is very easy, but then my solutions manual gave me unexpected answers ![http://i.imgur.com/tSMmJEY.png](http://i.stack.imgur.com/cam6h.png) I don't get how in d) they conclude the vectors go in the opposite direction, but in b) they go in the same direction. The only thing I could guess is that there's some assumed current direction but I can't find anything in my book about that. Thanks for any help


title: Question on Uncertainty Principle

text: I have read about the uncertainty principle. And it applies to electrons. Then how is it that we can get exact tracks of electrons in cloud chambers?? That is to say that how is it that the position is fixed? I think I am missing a crucial point here. Please help.


title: errata for Morse & Feshbach - Methods of Theoretical Physics

text: Anyone knows where I can find an errata (or any related material, such as solution sheets, etc) for this book? Thanks. Note: This is not a physics question, but this book is so popular among physicists that I can't think of anyone else better to answer it.


title: How did Halley calculate the distance to the Sun by measuring the transit of Venus?

text: What numbers did Halley, Cook, et al. have? What was the strategy by which they calculated the AU?


title: Curved space to flat space calculation

text: When changing the curved space co-ordinate into a flat space co-ordinate if a cone. I got the result transformation that i cannot get a transformation at the vertex(apex) why?


title: If the temperature of 2 materials are the same does that mean the molecules are vibrating at the same speed?

text: Pretty much what the title says. My base question is this. Assuming I take a piece of steel, and a piece of PVC plastic and I measure both their temperatures and find they are the same. I then take a look at the vibration speeds of the individual molecules would they be the same as well? Here's a rough example: I measure both the steel and PVC and find them both at 100F, and then I measure the vibrations of a molecule in the steel and find it to be moving at 10 miles an hour. Would the PVC molecules also be moving at 10 miles an hour? I'm sure I'm not using the correct units of measure to measure the vibration, but I didn't know what the correct unit of measure is for something like that. Hopefully it gets the point across.


title: Work as an integral of mass over velocity?

text: As I've understood it, the area under $F$-$s$-graph is the work done, so then :$$W(s)=\int{F(s)ds}$$ I am also given this equation ($W_k$ is kinetic energy, which is equal the work done to set the object in motion): $$W_k=\frac{1}{2}mv^2$$ Does this mean that work is also the area under a $m$-$v$-graph, like so: $$W(v)=\int{m(v)dv}$$ Could anyone explain why this is, or isn't true?


title: Why is there a phase factor when the two composite angular momentum is exchanged in Clebsch–Gordan coefficients

text: An identity exists for CG coefficients: $$\langle j_1 m_1 j_2 m_2 |J M \rangle = (-1)^{j_1+j_2-J} \langle j_2 m_2 j_1 m_1|J M\rangle,$$ But why is there a phase factor $(-1)^{j_1+j_2-J}$? It seems to me that $$|JM\rangle =\sum_{m_{1},m_{2}}|j_{1}m_{1}\rangle\otimes|j_{2}m_{2}\rangle\langle j_{1}m_{1}j_{2}m_{2}|JM\rangle =\sum_{m_{1},m_{2}}|j_{2}m_{1}\rangle\otimes|j_{1}m_{2}\rangle\langle j_{2}m_{2}j_{1}m_{1}|JM\rangle $$ And since $|j_{1}m_{1}\rangle\otimes|j_{2}m_{2}\rangle$ and $|j_{2}m_{1}\rangle\otimes|j_{1}m_{2}\rangle$ are the same physical state, there should be no difference between $\langle j_1 m_1 j_2 m_2 |J M \rangle$ and $\langle j_2 m_2 j_1 m_1|J M\rangle$. What do I get wrong?


title: Why does the inertial mass of an object rise when a force is exerted on it in a particle accelerator?

text: A particle in a cyclotron requires more and more force to maintain the same acceleration as it accelerates.


title: Why do we hear a higher pitched sound out of water when we hit two stones against each other in the water?

text: The observer is outside the water; the stones are underwater (say, 1 m below surface, if that matters). This produces a higher pitched sound for the observer than when both the observer and the stones are in air. Is this because it takes more energy for sound waves to travel through water than through air, and so the ones that we hear from outside the water are the ones that had higher frequencies immediately after the collision to begin with? Does the density of the medium which is disturbed by a rigid body collision have any effect on the frequency distribution of sound waves that are generated by the collision? For example, does the higher "stiffness" of the cage of water molecules surrounding the vibrating stones post-collision mean higher frequency normal modes for water (than for air)? Finally, does refraction at the water-air interface play any role?


title: Physics for mathematicians

text: How and from where does a mathematician learn physics from a mathematical stand point? I am reading the book by Spivak Elementary Mechanics from a mathematicians view point. The first couple of pages of Lecture 1 of the book summarizes what I intend by physics from a mathematical stand point. I wanted to find out what are the other good sources for other branches of physics. Thanks


title: Prereqs for The Geometry of Physics by Frankel

text: I'm interested in giving The Geometry of Physics a read, and I was wondering what the mathematical and (more importantly) physical prerequisites are. My background is a bit stronger on the mathematical side, and I'm worried that if heavy (or even moderate) exposure to physics is required that I'll miss out on a lot. If someone has read it and has a good sense of this sort of thing, I'd very much appreciate the input! A link to the book.


title: What is a phonon?

text: I am trying to understand intuitively what a phonon is, but for the moment I find it quite difficult (having a limited background in quantum mechanics, an undergraduate course in non-relativistic QM). In fact, I find it hard to formulate good questions, so I hope my questions below make some sense. I read that phonons are (the quantum mechanical analog of) normal modes of vibration in a crystalline system of atoms or molecules, so I guess a superposition, i.e. a general vibration should also be a phonon. Is that so? Why would they then be described as normal modes? Could we say that a phonon is a particle whose position wave function extends over the whole crystal? Are the quantum mechanical frequency and wave vector the same as the frequency and wave vector of the corresponding classical oscillation (vibration in the crystal)? In what sense is it (like) a particle? In that it is always observed or it always interacts at a specific location?


title: What is "charge discreteness"?

text: I assume it is some kind of quantity. Google only made things more confusing. I get that it has something to do with circuits. I also get what a discrete charge is. In fact, I thought charges were, by definition, discrete - because each individual proton/electron/hole contributed one unit of charge.


title: Curved lines in a picture (Photography)

text: My problem is when I take a picture (a close one) the straight edge looks a little curved. In a standard camera, like a CyberShot. I would like to know if there is some relationship between the curvature of the glass and that effect or if you know more information about that.


title: Does the logistic map have an attractor for a particular value of the parameter?

text: Background: Currently I am studying a course on non-linear dynamics. We have been studying about attractors only intuitively, so I do not have a definition for an attractor. Let me give you a couple of examples. Consider a problem where a ball is slipping in a parabolic valley. It is clear intuitively and quantitatively that the lowermost point of the parabola is the attractor for it. Say we are talking about a Lorenz system. We may not guess intuitively that there is an attractor, a priori, but using simulations for different values of initial conditions, we may qualitatively (and maybe quantitatively too) compare the trajectories for different initial conditions and convince ourselves that we have an attractor. Problem: Consider the logistic map given as $$x_n = ax_{n-1}{(1-x_{n-1})}.$$ Let us fix a value of the parameter where we "know" that the logistic map exhibits chaos. Say $a=3.9$. Let us take the definition for the term attractor from Wikipedia: An attractor is a subset $A$ of the phase space characterized by the following three conditions:   * $A$ is forward invariant under $f$: if $a$ is an element of $A$ then so is $f(t,a)$, for all $t > 0$.   * There exists a neighborhood of $A$, called the basin of attraction for $A$ and denoted $B(A)$, which consists of all points $b$ that "enter $A$ in the limit $t → ∞$". More formally, $B(A)$ is the set of all points $b$ in the phase space with the following property: For any open neighborhood $N$ of $A$, there is a positive constant $T$ such that $f(t,b) ∈ N$ for all real $t > T$.   * There is no proper subset of $A$ having the first two properties. How might we demonstrate/convince ourselves of/disprove the statement that the logistic map does have an attractor for a particular choice of the parameter at which the map exhibits chaos?


title: Definition of Static Electricity

text: > The result of an imbalance of electrons between objects is called static > electricity. It is called "static" because the displaced electrons tend to > remain _stationary_ after being moved from one insulating material to > another. Please can any one explain to me what does it mean by the word _stationary_ in the definition? Does it mean that the displaced electrons do not spin around the nucleus in another material's atom?


title: Speed astronauts measure moving at the speed of light

text: Lets suppose a spaceship travels with v = 0.9c relative to the Earth. The time inside the spaceship would pass slower than on Earth. Would the astronauts measure a different speed (that means, a different one that the observer on Earth does) in relation to the same reference frame (Earth)?


title: Young's Modulus and Vibrating String Harmonics

text: I was wondering how Young's Modulus effects the resonant harmonics of a vibrating (string instrument) string. I know that the string's fundamental frequency is $$\frac{1}2 \times \text{Length} \times \frac{\text{Tension}}{\text{linear density}^{1/2}}$$ that Young's Modulus for a material is - $$\frac{\text{Force}\times \text{original length}}{\text{original cross section} \times \text{change in length}}$$ and that resonant harmonics of a string are even multiples of the string's fundamental frequency. Does the fundamental frequency of the string material itself (which I can calculate by figuring out the speed of sound in whatever material the string is made from and how thick the string is) effect the frequencies it vibrates at under tension?


title: More Heisenberg Uncertainty Principle (HUP) Clarification

text: If you look at the commutation relation of the position and momentum operators (in 1D position space), you get: $$[\hat{x}, \hat{p}_x] = [x,-i \hbar \frac{\partial}{\partial x}] = i \hbar$$ All this says to me is that if you prepare a system in state (A) and measure the position, the system is now in state (B), which is an eigenstate of the position operator. You then measure the momentum of state (B) and now you're in state (C), which is an eigenstate of the momentum operator. Note that these must be consecutive measurements. Alternatively, let's assume you reversed the order of the measurements. Beginning with state (A) again, you measure momentum first and put the system in state (D), an eigenstate of the momentum operator, but not necessarily state (C) [right?]. You then measure position, and put the system in state (E). $$(B) \neq (E)$$ $$(C) \neq (D)$$ That's all HUP says to me. It says nothing about simultaneous measurements of position and momentum — is that even possible? (What operator would that be?) It only says how much the final wavefunctions of two systems that started out the same differ if you perform two measurements in a different order. Where is the uncertainty? You know the position and momentum exactly — right when you measure them. You get some random value weighted by the coefficients of the eigenfunctions in the linear combination that forms $\psi$. So I don't think it's accurate to say you can't "simultaneously know position and momentum to arbitrary precision", because as far as I can tell, you can't even measure the two at the same time.


title: Is the speed of light related to the mass of the universe?

text: If the mass of the universe were cut in half, would it affect the speed of light? Would it be twice as fast? Would it stay the same? Do we have instruments that are sensitive enough to measure the speed of light at different positions relative to high-mass objects to empirically answer this question? The speed of light is (something of) a universal constant, but is it really dependent on the universe or on something intrinsic to photons? **EDIT:** Related question: Since gravity is a relationship between one atom and every other atom in the entire universe, and it takes all the energy in the universe to travel at the speed of light, is there something about the energy/gravity/mass of the universe that "slows" light from going a faster speed?


title: Examples of systems with energy as an intensive variable

text: I need to consider a couple of examples of systems which have energies that are intensive variables - not extensive. I'be been thinking about this and I am not coming up with anything. My understanding is that extensive variables (at least wrt usual energies) scales with mass or length (system size). It also seems that some 'energies' depend upon the model used, such as how strong the interactions are in neighbors of atoms or dipoles, etc., or whether one is considering chemical potential or not, etc. Any good suggestions?


title: Difference between vector and pseudo-scalar

text: In physics, a pseudo-scalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion such as improper rotations while a true scalar does not. Can someone show me the graphical picture or illustration that how a scalar changes under rotations? **Why we can't say it as vector, if just the sign of the quantity changes? What is the fundamental difference between a vector and pseudo-scalar?** **Why is the Klein-Gordon equation an equation of motion for a pseudo-scalar field?**


title: How does ice become thinner inside a no-frost fridge?

text: How does ice become thinner in a no-frost fridge? I thought ice can't evaporate at all since, well, it's below freezing temperature and it's solid, too. In my other yes-frost fridge, the ice never becomes thinner, but thickens over time, so I have to de-frost it. I don't believe the no-frost fridges have some patented magical mechanisms that make ice evaporate, but it has to somehow occur naturally, and they just speed up the process by drying the air. Still, I don't understand how the ice evaporates.


title: What is a quasicontinuum?

text: I'm currently going through statistical physics, especially on Fermi energy when I came across a term called "quasi-continuum", what exactly is it?


title: How to calculate displacement of an object

text: I am having a little bit of trouble with this following problem - Suppose a particle has initial velocity and is moving with a constant acceleration. After t seconds its velocity is v. What will its displacement be in 2t seconds? I have tried to solve this on my own with the help of these equations -   1. v = u + at ( a is acceleration, u is initial velocity)   2. s = ut + 1/2 at^2 ( s is the displacement)   3. v^2 = u^2 + 2as   4. s = (u+v)*t/2 I tried to replace all the acceleration parameter using equation 1(a = (v-u)/t) in all other equations. I've also tried some other ways but they didn't help. but I couldn't figure out how to do it. I wish I could have tagged it as `high-school physics`, but couldn't find such tag. Any help is appreciated!!!!!


title: What will happen when Russia shuts down the GPS stations?

text: Russian officials are seriously talking about shutting down US GPS ground stations within their borders and Deputy Prime Minister of Russia tweeted on the subject. What will happen to the GPS? My assumption would be that its accuracy will get worse, at least in some parts of the globe. If it is correct, then where, and by how much exactly?


If I put a ping pong ball in a vacuum, would it pop? If so, at what point would it happen? Any standard table ping pong ball is acceptable.


title: Particle in a 1D Box with Symmetric potential: How find solutions?

text: I am working on a problem in which I shall find the normalised solution to the 1D particle in a box. Solving for the particle in an asymmetric potential is quite straight forward, but I run into trouble when the potential is symmetric: $$ V(x) = \begin{cases} \infty & x < -\tfrac{L}{2} \\\ 0 & \- \frac{L}{2} \leq x \leq \frac{L}{2} \\\ \infty & x > \frac{L}{2} \end{cases} $$ The problems arise with the boundary conditions. We have $$ \frac{d^2\Psi (x)}{dx^2} = -k^2 \Psi (x) $$ where $k^2 = \frac{2mE}{\hbar ^2}$. The general solution is $$ \Psi (x) = Ae^{ikx} + Be^{-ikx} $$ Due to continuity and the nature of the potential, we must have $$ \psi (-\tfrac{L}{2}) = \Psi (\tfrac{L}{2}) = 0 $$ Plugging in: $$ \psi (-\tfrac{L}{2}) = Ae^{-ikL/2} + Be^{ikL/2} = 0 \\\ \psi (\tfrac{L}{2}) = Ae^{ikL/2} + Be^{-ikL/2} = 0 $$ I know that when the potential is symmetric, we will find even ($A=B$) and odd ($A=-B$) wave functions. We will see that for even functions, $n$ has to be odd whole numbers, and for odd functions $n$ has to be even whole numbers. This leads to a sequence of sine and cosine curves as $n$ increases by 1. I am trouble getting there, however, from those boundary conditions, and I would really appreciate pointers and help.


title: Electrical properties of molten gallium arsenide

text: I'm looking for the resistivity and magnetic permeability of molten gallium arsenide, but can only seem to find the values for the solid material at room temperature (e.g., Wikpedia). Not even temperature-dependency of these parameters seems to be available. Is this because the value just doesn't change with temperature? If it does, what's a good resource for such parameters?


title: Could a coronal mass ejection be responsible for this?

text: Around 9pm (GMT) on June 9th, we noticed that the two security cameras had gone dark. Around the same time, a photo voltaic sensor responsible for keeping the yard lights on at night also failed. In the morning the cameras were back to normal, which means that only the LEDs, responsible for night vision on the cameras, had failed. The cameras are powered by a 12V/2A power supply, which makes it even less likely that a power surge is responsible for the LEDs (and only the LEDs) burning out. Checking upon the space weather, I noticed that there was a solar flare (M5.9) emitted right around this time. I don't know much about solar flares, but from what I gather M5.9 is on the very low end of things. **Could this be the reason 2 cameras and one photo voltaic sensor failed at the same time?** Or what else should I look into? _Mods, sorry if the question is somewhat open ended. Feel free to close the question if not appropriate._


title: Gravity proofs/can it be measured?

text: Is it possible to measure gravity other than cause and effect. Gravity is a principle which can only be measured by cause and effect. Ie since an object falls we assume there must be something pushing it down and since this principle holds true for a variety of scenarios, we hold it as true. Is that all we have to go on with gravity? To put it into perspective electricity can be measured by other methods than turning on a light switch and observing the filament glowing. Infrared waves can be measured by heat. For gravity other than seeing an object fall, what other proofs do we have? (Info sourced by RMMS).


title: Nothing vs. empty space

text: This question quotes Hawking saying: > [...] _you enter a world where conjuring something out of nothing **is** > possible (at least, for a short while). That's because at this scale > particles, such as protons, behave according to the laws of nature we call > "quantum mechanics", and they really can appear at random, stick around for > a while, and then vanish again to reappear somewhere else._ Nowever, is empty space really nothing? is there a distinction between non- existence and the "nothingness" of space? Perhaps space is something, we just cannot grasp exactly what it is. Anyone can shed light on whether space is something and what exactly that "something" is.


title: Why Shock wave propagation is faster

text: From The Blast Wave > _A fraction of a second after a nuclear explosion, the heat from the > fireball causes a high-pressure wave to develop and move outward producing > the blast effect. The front of the blast wave, i.e., the shock front, > travels rapidly away from the fireball, a moving wall of highly compressed > air._ From Wikipedia > _A shock wave is a type of propagating disturbance. Like an ordinary wave, > it carries energy and can propagate through a medium (solid, liquid, gas or > plasma) or in some cases in the absence of a material medium, through a > field such as the electromagnetic field. Shock waves are characterized by an > abrupt, nearly discontinuous change in the characteristics of the medium. > Across a shock there is always an extremely rapid rise in pressure, > temperature and density of the flow. A shock wave travels through most media > at a higher speed than an ordinary wave._ How shock wave differs from ordinary wave, and how it can travel faster then ordinary wave in same medium. In first ex, In nuclear explosion why shock wave is traveling faster then fireball, when both are in same medium. Can someone please explain the phenomenon in detail.


title: Is quantum uncertainty principle related to thermodynamics?

text: Would like to ask a question, but first i would like to say **Hello Everybody** in a way that plays the system, since some geniouses decided that one should not be able to say hello in a question. The uncertainty principle in quantum mechanics is well known and considered one of most basic properties of natural reality. The 2nd Law of thermodynamics is also well known and also considered one of the most basic processes of natural reality. The uncertainty principle uses and is related to Planck's constant. Planck's constant has the dimensions of _action_ and in a statistical mechanics approach, also relates nicely with the partitioning of the phase-space providing the basic measure for the entropy functional (this answer provides a nice outline of this). Apart from that, there are relatively recent papers which relate the Heisenberg Uncertainty Principle in quantum mechanics directly and intuitively to the 2nd Law of Thermodynamics. Is this relation correct? And if so can we derive one from the other? Thank you PS. One can also check this question, which although not the same, is related in an interesting way. UPDATE: anna's answer is accepted since by mentioning the derivation of (part of) the 2nd law from unitary dynamics, answers the question at least in one way. Please consider this as still open so you can add another answer. There are more alternatives (and one of which is my stance, ie thermodynamics -> uncertainty)


title: am i finding the force correctly?

text: I am trying to see if I am doing a problem correctly. Problem > Suppose you carry a 50kg sack of potatoes up two flights of stairs, a total > height of 10m. How much work did you do? If it took you 20seconds, what was > your power outlet? (I am NOT looking for the answer, I am just trying to see if I am doing this correctly). Based on what I have learned to find Work you must do this equation: $W = Fd$ (force x displacement) This is where I am a little confused. Not sure what the force is, but going off of my notes to find the force you must do this equation: $F = ma$ (mass x acceleration). I already know the mass is 50kg, but to find acceleration I must find it in m/s^2. (Where I think I am getting this wrong). So I know that our meters is 10, and our seconds is 20 (10 m high, and 20 seconds going up stairs), but to find it squared I would do 10/20 / 20 m/s^2 which equals 0.025 m/s^2 for my acceleration. Going back to finding force I can now plug in my acceleration of 0.025 as follows: $F = (50kg) \times (0.025m/s^2) J$ $F = 1.25 N$ $W = (1.25) \times (10)$ $W = 12.5 J$ Did I do this correctly? If not, instead of answers, let me know what I did wrong, and maybe like a hint at what I should do next. I appreciate any help given.


title: Does the complex 3-sphere have a complex structure modulus?

text: This question has a flavor which is more mathematical than physical, however it is about a mathematical physics article and I suspect my misunderstanding occurs because the precise mathematical definition of the concepts used is different than what I think. But then again, it might be some foolish mistake on my part. I'm reading "Topological strings and their physical interpretation" by Cumrun Vafa and Andrew Neitzke. On p. 14 they introduce the "deformed conifold" (really complex 3-sphere) which is given by the equation xy - zt = μ (equation 2.27 in the article) in C^4 with coordinates (x, y, z, t) and μ a constant. According to the article, μ plays the role of a complex structure modulus, that is, varying μ we get diffeomorphic manifolds with different complex structure. On p. 14 below equation (2.27) they write > This gives a Calabi-Yau 3-fold for any value μ ∈ C, so μ spans the > 1-dimensional moduli space of complex structures Also on p. 16 they write > In summary, we have two different non-compact Calabi-Yau geometries, as > depicted in Figure 5: the deformed conifold, which has one complex modulus r > and no Kahler moduli, and the resolved conifold, which has no complex moduli > but one Kahler modulus t Here μ was replaced by r since they change coordinates to rewrite (2.27) in the form x_1^2 + x_2^2 + x_3^2 + x_4^2 = r (equation 2.30 in the article) However, for any lambda non-zero, multiplication of the coordinates by lambda yields a biholomorphic mapping between the complex manifolds corresponding to μ and lambda^4 μ. Thus they are all isomorphic. What am I missing here?


title: Counting complete sets of mutually unbiased bases composed of stabilizer states

text: Consider $N$ qubits. There are many complete sets of $2^N+1$ mutually unbiased bases formed exclusively of stabilizer states. How many? Each complete set can be constructed as follows: partition the set of $4^N-1$ Pauli operators (excluding the identity) into $(2^N+1)$ sets of $(2^N-1)$ mutually commuting operators. Each set of commuting Paulis forms a group (if you also include the identity and "copies" of the Paulis with added phases $\pm 1$, $\pm i$). The common eigenstates of the operators in each such group form a basis for the Hilbert space, and the bases are mutually unbiased. So the question is how many different such partitions there exist for $N$ qubits. For $N=2$ there are six partitions, for $N=3$ there are 960 (as I found computationally). The construction above (due to Lawrence et al., see below) may be an example of a structure common in other discrete groups - a partition of the group elements into (almost) disjoint abelian subgroups having only the identity in common. Does anyone know about this? Reference: Mutually unbiased binary observable sets on N qubits - Jay Lawrence, Caslav Brukner, Anton Zeilinger, http://arxiv.org/abs/quant-ph/0104012


title: What is the current state of research into $v$-representability?

text: In their proof, Hohenberg and Kohn (Phys Rev 136 (1964) B864) established that the ground state density, $\rho_\text{gs}$, uniquely determines the Hamiltonian. This had the effect of establishing an implicit relationship between $\rho_\text{gs}$ and the external potential (e.g. external magnetic field, crystal field, etc.), $V$, as the form of the kinetic energy and particle-particle interaction energy functionals are universal since they are only functions of the density. This implicit relationship defines a set of densities which are called $v$-representable. What is surprising is that there are "a number of 'reasonable' looking densities that have been shown to be impossible to be the ground state density for any $V$." (Martin, p. 130) On the surface, this restriction looks like it would reduce the usefulness of density functional theory, but, in practice, that is not the case. (See the proof by Levy - PNAS 76 (1979) 6062, in particular.) However, research continues into the properties of the $v$-representable densities, and I was wondering if someone could provide a summary of that work.


title: Characters of $\widehat{\mathfrak{su}}(2)_k$ and WZW coset construction

text: I am currently studying affine Lie algebras and the WZW coset construction. I have a minor technical problem in calculating the (specialized) character of $\widehat{\mathfrak{su}}(2)_k$ for an affine weight $\hat{\lambda} = [k-\lambda_1,\lambda_1]$. Given the generalized theta function $$\Theta_{\lambda_1}^{(k)}(z,\tau) = \sum_{n\in\mathbb Z}e^{-2\pi i\left[knz+\frac 12\lambda_1 z-kn^2\tau-n\lambda_1\tau- \lambda_1^2\tau/4k\right]}$$ I want to evaluate $$\chi^{(k)}_{\lambda_1} = \frac{\Theta^{(k+2)}_{\lambda_1+1} - \Theta^{(k+2)}_{-\lambda_1-1}}{\Theta^{(2)}_1 - \Theta^{(2)}_{-1}}$$ at $z=0$. Putting $z=0$ directly, both the numerator and denomerator vanish (since there is no difference between $\lambda_1$ and $-\lambda_1$ due to the sum). So my question is; what is the appropriate way to take the limit $z\rightarrow 0$? [This is from Di Francesco et al, section 14.4.2, page 585]. The result should be $$\chi^{(k)}_{\lambda_1} = q^{(\lambda_1+1)^2/4(k+2)-\frac 18}\frac{\sum_{n\in\mathbb Z}\left[\lambda_1 + 1 + 2n(k+2)\right]q^{n[\lambda_1+1+2(k+2)n]}}{\sum_{n\in\mathbb Z}\left[1+4n\right]q^{n[1+2n]}}$$ where $q=e^{2\pi i\tau}$. Since I fear the solution to my question is rather trivial, I have a bonus question. Do you know any paper which works out the details for the coset $$\frac{\widehat{\mathfrak{su}}(N)_k\oplus \widehat{\mathfrak{su}}(N)_1}{\widehat{\mathfrak{su}}(N)_{k+1}}$$ for arbitrary $N$? I am thinking about something like what Di Francesco et al. does in section 18.3 for $N=2$. It would be nice if the reference relates this to $\mathcal W$-algebras.


title: Optics for projecting OLED screen DLP style

text: I just bought a small OLED screen and was wondering is it possible to make a screen surface a little bit bigger ( 2x ) by projecting it through some optics on translucent surface. It reminds me of viewfinders in some old cameras or SLR lens adapters or DIY LCD projectors. Does it even make any sense? I'm positive that some fresnel lenses will be needed but can't get a grasp on their configuration for this to work. I've found pretty simple lens simulator and it makes it look like it can be done with only 2 lenses. But it don't seem right. I know that with LCD projectors you need a fresnel lens and a triplet. Can it be simplified at the expense of image distortion?


title: Why are particles in harmonic motion in normal modes?

text: Why do we assume that in normal modes, particles oscillate in form cos (wt) ? How do we know that the general motion of particles can be expressed as a superposition of normal modes? In both French and Crawford, the assumption of harmonic motion is made without any proof, please help.


title: Can we write the electromagnetic potential covariantly in terms of the four-current?

text: In the Lorenz gauge, we have a beautiful relation between the four-current and the four-potential: $$\Box A^{\alpha} = \mu_0 J^{\alpha}$$ To get $A$ in terms of $J$, however, we have to use a considerably uglier formula; or, at least, this is the formula presented in textbooks: $$A^{\alpha}(t, \mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{J^{\alpha}\left(t - \frac{1}{c}\|\mathbf{r} - \mathbf{r}'\|, \mathbf{r}' \right)}{\|\mathbf{r} - \mathbf{r}'\|} d^3\mathbf{r}'$$ The first equation is evidently Lorentz covariant. The second one, on the other hand, doesn't look covariant at all. The integrand has some messy dependence on $(t, \mathbf{r})$ and the integration goes over only the spatial dimensions. Can we rewrite the second equation in a covariant form? If not, then why not?


title: Weather Radar Interpretation - Radial Rays

text: I was browsing the NEXRAD radar feeds (I'm not an expert, just figuring them out) and I came across the following signature (visit the link to view the radar image) http://cl.ly/3n0y0p0g2M0K2B313g3U The radar in question was operating in clear air mode and the "ray" varies from about 0-16 dBZ. There were several of these all covering the Midwest, and I picked up one as far west as Kansas, at about the same return signal strength. Further west they weren't visible. Could this be caused by a cosmic phenomenon? IMMEDIATE FOLLOW-UP: I'm tracking it across the United States right now. I'll be following it below with updates.   * 9:16 PM: appears on KIND Indianapolis composite reflectivity   * 9:30 PM: appears on KSGF Springfield base tilt 1 and 2, KICT Wichita base tilt 2   * 9:31 PM: updated KSGF now shows only on base tilt 1, not 2   * 9:34 PM: now appears only on KICT Wichita base tilt 3? This doesn't make sense if it's a "stationary" cosmic object, that would mean it should be shifting to lower tilts. Might just be an outlier   * 9:37 PM: visible now on KDDC Dodge City base tilt 4, KICT tilts 2 and 3. Also now appears on KAMA Amarillo base tilts 4 and 3 (weaker on 3)   * 9:43 PM: visible on KICT base tilt 2 only


title: Is all kind of light same speed?

text: Is there any speed different between blue or red color? Is there speed different? or there are same speed?


title: What kind of light is needed to light up Venus?

text: Let's say I would want to light up Venus, such that we can see Venus all day long and not have to wait for a Venus Transit. What kind of light would I need for it? How powerful would it need to be?


title: How can one feel cold without touching it?

text: Let me explain. My shower is inconsistant in how hot it gets. Whenever I turn on the shower, I always put my hand really close to the water, but not touching it, so I can feel the heat radiate from the shower. That way, if I can feel it from a distance, I know it is too hot. I also notice that if the water is very cold, I can feel it as well. Do I feel the absence of radiating heat? Am I actually touching the water and not noticing it? Sorry if the questions isn't technical enough; it's more of an everyday physics question.


title: Why does dark energy produce positive space-time curvature?

text: My understanding is that dark energy, or equivalently a positive cosmological constant, is accelerating the expansion of the universe and I have read that this gives empty space-time positive curvature, ie de Sitter geometry. I also understand that parallel geodesics converge when curvature is positive and diverge when it is negative. I would expect accelerating expansion of space to make parallel space-time geodesics diverge and thus make curvature negative. Is there a nice visual explanation why dark energy actually produces positive curvature?


title: What is a fluid flux?

text: I need a concise definition of a fluid flux and an accompanying example. I've never taken a single physics course before, but I'm required to understand this concept so I can do the calculations for a Complex Analysis class.


title: Neutrino Oscillation and their gravitational implications

text: As I understand neutrinos, there are three different flavors, all with different masses. Although the masses of these neutrinos have not been directly measured, their mass differences have been. Current experiments, KATRIN and Project8 are going to measure neutrino masses and we shall know soon enough. Regardless, their mass states change as they travel through space. This leads to my question... Since an object's gravitational field is related to its mass and neutrinos have different mass states while they are traveling, it must mean that every point in space must be constantly altering in gravity intensity! Although every object alters the gravitational field intensity as it travels and passes a given point, neutrinos would do it differently because they keep changing mass states! Let's assume a constant stream of neutrinos pass by a point in space versus neutrinos that don't oscillate (This is hypothetical) doing the same thing. Wouldn't these extremely weak gravitational waves be different given oscillations than not?


title: Power of a runner

text: How can someone calculate the power in watt that a runner produces, when he runs uphill and downhill? Is there any algorithm? It is important to take under consideration the uphill and downhill elements of the run. * * * Thank you for your answers but i am confused since i don't have a background in physics. To make thinks simple, i have some values, and using these values and maybe some constants like gravity, i want to create a calculator for real time watt production. These values are: INSTANT VALUES: speed, distance, hr, kcal, ascent, descent, duration. OTHER VALUES: body mass, vertical speed (average), ascent time, descent time, max hr, rest hr, vo2max.


title: Relation between flow and pressure

text: if you have water flowing through a 6 or 4 inch pipe what would the pressure be if tapped off through a 1/2 inch BSP tap. What would the equation be to find out the pressure at different flows?


title: When recombination in PN junction occurs, which atom becomes an ion?

text: In an unbiased PN junction, when the carriers recombine to form a depletion layer , it is said that immobile ions are formed. We know that the conduction band electrons in N type are not associated with any particular atom. So when the conduction band electron diffuses to the P type region, which atom becomes an ion?


title: Is this hypo-theoretical model of future prediction feasible?

text: First let me state that I am not, nor ever have I been, a physics student. I am working on an idea for a book I'm writing. This is a thought experiment that posits the existence of a computer system that, to cut right to the point, has within its memory banks the thoughts, memories, and experiences of every human being who has ever died on Earth, and this data is continuously updated in real time. Also, this computer system has a constant stream of data fed to it from a network of planetary sensors that record every physical event that occurs on the planet, from the shifting of tectonic plates beneath the Earth's crust to the ol' hackneyed flapping of a butterfly's wings. This computer system is able to process this data through a realizable causal filter that allows it to create an unimaginably detailed model of, well, everything. This computer system should therefore be asymptotically omniscient, in regard to past and ever-present events that occurred and are occurring. Its data would not be 100% reliable, as there is a distinct difference between observed past (which is subject to interpretation), and actual past (which is indelible), and keep in mind that the physical data fed to it by the planetary sensors would also be considered as observations. I feel that the sheer amount of raw data available would make it so that, when looked at as a whole, multitudinous patterns of causation would emerge. My question is this: In the example that I described, is there anything that would physically prevent the computer system from anticipating (within specific parameters) future events to a given degree of probability, the accuracy of which would be proportional to the magnitude of the event, and inversely proportional to the time interval between the predicted event and the present? Furthermore, would it be conceivable that the computer system, when analysing a given prediction, could indicate (within a degree of probability) specific critical points in time and space that, if interfered with by the computer system's operator, could drastically raise or lower the odds of the original predicted event actually occurring? Further still, the computer system should also be able to simultaneously calculate the potential effects of such an interference, and suggest a course of action that is least likely to create unwelcome consequences further down the road. Assumptions: The computer system possesses near-infinite processing power and internal storage. The computer system is programmed to constantly scan for future catastrophic events within a specific magnitude, and when such an event is detected, the computer system is programmed to analyse the various patterns of causation leading up to the event, and suggest optimal points of interference, with the goal of averting the catastrophe. The computer system operates within a stochastic universe, thus 100% accurate predictions are asymptotically impossible.


title: About the duality when embedding Gopakumar-Vafa into superstring theory

text: Vafa proposed a duality when embedding the Gopakumar-Vafa duality into superstring theory. Vafa's duality is about a correspondence N=1 supersymmetric gauge theory and superstring propagating on noncompact CY manifolds with flux turned on. I am puzzled about this relation. a) On superstring theory side, is the total dimension six or ten? In the other words, it is the duality about ordinary string theory or topological string theory? b) Where does the N=1 gauge theory come from? IIA superstring theory compactification on conifold internal space with $D_6$ branes wrapped around 3-cycles, its geometric transition counterpart, or $D_6$ branes world volume theory? Thanks in advance.


title: Charge in a layered ball

text: A ball (radius $R$) has three layers. For $0<r<a$ it is a conductor with free charge $+Q$. For $a<r<b$ it is a linear dielectric $\epsilon$ with free charge embedded in it with density $\rho_{free}(r) = \left(\frac{\rho_0}{a^2}\right)r^2$. From $b<r<R$ it is a conductor again with some amount of charge on it $q$ which makes the field vanish for $r\lt R$. I'm supposed to:   1. Find the polarization $\vec P$ in the dielectric   2. Find bound volume charge density in the dielectric   3. Find free and bound surface densities at each surface $r=a,b,R$ So I tried using Gauss's law here with a sphere of radius $a<r<b$ as my surface to get $\vec P$:$$\oint \vec D\cdot d\vec A =Q_{free}+\int_a^r \rho_{free} dr$$$$D = {Q+\frac{\rho_0}{3a^2}(r^3-a^3)\over4\pi r^2}$$ Using $\vec D = \frac{\vec E}{\varepsilon}$ and $\vec P = \varepsilon_0\chi_e\vec E$ and $\rho_b=-\nabla\cdot\vec P$ I calculated: $$\vec P = \frac{\varepsilon_0\chi_eQ}{4\pi\varepsilon r^2}+\frac{\varepsilon_0\chi_e\rho_0}{12\pi\varepsilon}({r\over a^2}-{a\over r^2})\hat r$$$$\rho_b = \frac{\varepsilon_0\chi_e}{2\pi\varepsilon}(\frac{Q}{2r^2}-\frac{\rho_0r}{3a^2}+\frac{\rho_0}{6r^2})$$ Two questions:   1. Is that right?   2. How do I use that to calculate the surface charge densities $\sigma_b$ and $\sigma_f$ for each surface?


title: G(2) lattice and the M-theory landscape

text: In a previous question (Calabi-Yau manifolds and compactification of extra dimensions in M-theory), I was told that the $G(2)$ lattice can be used to compactify the extra 7 dimensions of M-theory and preserve exactly $\mathcal N=1$ supersymmetry. However, since there is only 1 $G(2)$ lattice, there should be only 1 4-dimensional M-theory. Then, why is there such a huge fuss about the M-theory landscape? Thanks!


title: Explosion in space

text: I'm curious about what happens if an explosive substance detonates in space. On Earth, I guess a good chunk of the energy released is carried away by shock waves in the atmosphere. But in space, the medium that supported the propagation of the shock wave on Earth is much more rarefied, so how does that work?


title: Is it possible to determine the universality class of phase transitions by just analysing symmetry?

text: Since phase transition is closely connected with symmetry, I am wondering whether it is possible to determine the universality class of phase transitions just by symmetry? Actually, I found it is quite boring to calculate critical exponent numerically and I want to find a new method.


title: Neutrons in nucleus

text: When I asked here why neutrons in nucleus (with protons) don't decay I was told that it would require energy for the neutron to decay, it wouldn't give energy. And since that wasn't really what I wanted to hear, since I already knew that, I'm now asking a similar question. I know that strong force holds protons and neutrons together in nucleus, but how does that effect a neutron in a way that it doesn't decay?


title: Motion of block on wedge

text: there is some confusion to me in the case of " **motion of block on a frictionless wedge** " Below is a simple diagram! ![enter image description here](http://i.stack.imgur.com/ET6uv.jpg) Let us consider a situation as above in which there is a block of mass $m$ moving with velocity $v$ in positive $x$ direction on a frictionless wedge as given above. Now since block doesn't have any velocity in $y$ direction initially so it moves horizontally as long as it is on table But **when at the instant it encounters the wedge it it gains some some velocity in upward direction and starts to move upward (i.e in positive $y$ direction) with velocity decreasing with time**. it approaches some height and after that comes back down the wedge.   1. What gives the block that upward velocity (i.e which force(s) changes the velocity in $y$ direction.) because of which it goes to some height and after that comes back?   2. What is the velocity of Block (net velocity which is along wedge). At the Instant it just starts to slide on the wedge?


title: How to determine the (n,m) dimensions of a carbon nanotube?

text: I've been reading about nanotubes lately, and I keep seeing the $ (n,m) $ notation. How does this describe a nanotube's structure? How do I determine which is $n$ and which is $m$ ? I'm familiar with matrix notation referring to rows and columns, but I couldn't connect it with the nanotube structure which is, albeit predictable, not quite like a row-column grid.


title: Why don't Microwave ovens use more energetic EM radiation?

text: Because in theory, visible light contains more energy than the microwaves actually used to cook your food. 1) Does it have to do with resonance frequency of the $H_2O$ molecule?   2) How can microwaves be dangerous when they are about the same wavelength as Radio waves? (unless it has to do more with intensity)


title: nuclear physics- Energetics and Mechanics of Nuclear Reaction Homework

text: > A sample of $^{24}_{12}\mathrm{Mg}$ is bombarded by a monoenergetic proton. > If the resulting nucleus in a $^{24}_{12}\mathrm{Mg}(p,\gamma)$ reaction; > $^{24}_{12}\mathrm{Mg}(p,\gamma)$ has its first energy level at 2.5 MeV. > What is the minimum energy of the of the proton to excite this level? * * * As I understand this problem, I assume that this is non-relativistic. I transformed the reaction from: $^{24}_{12}\mathrm{Mg}(p,\gamma)$ into: $^{24}_{12}\mathrm{Mg} + ^{1}_{1}\mathrm{H} \to \gamma + ^{25}_{13}\mathrm{Al}$ I found out that the product is not existing (because there is no 25,13 aluminum in our given table) The formula for non-relativistic would be: $$Q= KE_b + KE_y - KE_a + E$$ as part of my understanding to this problem, the 2.5 MeV will be the $KE_y$ and the variable to be find will be the internal energy ($E$). Solving for $Q$               Q= (summation of excess mass of resultant) - (summation of excess mass of product)     Q= (-13.933 + 7.289) - (0)      i made the summation of excess mass of product into 0 because gamma has no excess mass(in our table) and the product is not existing (in our table)               Q= -6.644 MeV               KEb = Q / (1+ [mass of b / mass of Y])          = -6.644 / (1 + 0)         = -6.644 MeV      from the formula of non-relativistic:               E = -6.644 + 6.644 - 2.5     E = -2.5 MeV      for the velocity, as my professor did on her example,               momentum (p)= mV     myVy = mbVb     Vy = (mbVb) / my      but in this problem, the unknown is Va (velocity of a or velocity of proton) I am confused in this problem because of the terms "monoenergetic proton" and "first energy level" which is not that familiar to us. I am doubtful to how I understand the problem. Does my understanding correct? does my internal energy (E) correct? i don't know how to find the velocity. please help me to solve this problem.. this will be passed the same day we are going to take exam on this topic. our homework serve as our reviewer so please, I don't want to be confused in our examination day.


title: How does one show using QED that same/opposite electric charges repel/attract each other, respectively?

text: Why do same charges repel each other and opposite charges attract each other (please explain the phenomenon using real laws of nature (QED) not with the approximation model)?


title: Velocity of a spacecraft in a circular orbit

text: What is a formula for velocity of a spacecraft in a circular orbit? Actually, is there a formula or I can find it from equation of motion?


title: Deriving the speed of the propagation of a change in the Electromagnetic Field from Maxwell's Equations

text: I've been told that, from Maxwell's equations, one can find that the propagation of change in the Electromagnetic Field travels at a speed $\frac{1}{\sqrt{\mu_0 \epsilon_0}}$ (the values of which can be empirically found, and, when plugged into the expression, yield the empirically found speed of light) I'm really not sure how I would go about finding $v = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$ simply from Maxwell's equations in the following form, in SI units -- $\nabla \cdot \mathbf{E} = \frac {\rho} {\epsilon_0}$ $\nabla \cdot \mathbf{B} = 0$ $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$ $\nabla \times \mathbf{B} = \mu_0 \left( \mathbf{J} + \epsilon_0 \frac{\partial \mathbf{E}} {\partial t} \right)$ Is what I believe true? (that the speed of propagation is derivable from Maxwell's Equations) If not, what else is needed? If so, can you provide a clear and cogent derivation?


title: Are all elementary particles of the same type exactly the same?

text: Are all elementary particles of the same type EXACTLY the same? Is there some variation in what an electron is, for example, or are they all the same?


title: Geometric Structure Factor for Monatomic FCC lattice

text: I am trying to find the geometric structure factor and my work here is clearly wrong. I will put my wrong answer and then I will throw up the link to wikipedia for the correct answer, because I cannot tell the difference. My attempt: BCC has a four atom basis. If x,y,z vectors are taken to be along the edges of the conventional cube like in the picture below (all credit due to Aschroft and Mermin) : ![enter image description here](http://i.stack.imgur.com/Qycg9.png) But the basis are located at: ![enter image description here](http://i.stack.imgur.com/3vP1X.png) Therefore the geometric structure factor is: $$F_{hkl} = \sum_{j=i}^{N} f_j e^{i \Delta \vec{k} \bullet \vec{r_j} }$$ But because the structure is monatomic, $f_j = f$ for all j. Also, $r_j$ denotes the location of the jth atom in the cell (I.E. $r_1 = \vec{0}, r_2 = \vec{ a_1} ,r_3 = \vec{ a_2}, r_4 = \vec{ a_3}$ in the picture ) $\Delta \vec{k}$ is just some vector that is an element of Reciprocal vector space. That is, $$\Delta \vec{k} = h \vec{b_1} + k \vec{b_2} + l \vec{b_3}$$ but because $\vec{a_i} \bullet \vec{b_j} = 2 \pi \delta_{ij}$ , $F_{hkl}$ reduces to: $$F_{hkl} = f [ e^0 + e^{i(h \vec{b_1} + k \vec{b_2} + l \vec{b_3}) \bullet \vec{a_1}} + e^{i(h \vec{b_1} + k \vec{b_2} + l \vec{b_3}) \bullet \vec{a_2}} + e^{i(h \vec{b_1} + k \vec{b_2} + l \vec{b_3}) \bullet \vec{a_3}} ] = f [ 1 + e^{i2 \pi h} + e^{i2 \pi k} + e^{i2 \pi l} ] = 4f$$ This answer is incorrect. The correct answer can be seen at this link to Wikipedia, scrolling down to fcc. From what I can tell, they must have defined their reciprocal lattice vector differently from me but I cannot see why.


title: Why are there no asteroids or meteoroids with relativistic speeds?

text: Cosmic rays can have energies going into the $10^{20}$ eV domain. Asteroids and meteoroids originating in the solar system are probably limited in their speed because they all started out from the same lump of matter having more or less the same speed, but what about rocks from other galaxies ? Could they reach relativistic speeds in relation to the solar system? Are there astronomical events that could be interpreted as a collision between such an asteroid and a planet or star? Just a mole of hydrogen atoms with that kind of energy would have three times the energy of the Chicxulub impact event, so I hope there's a reason for them not existing!


title: Hamiltonian from Euclidean lagrangian?

text: Can somebody help me in deriving the Hamiltonian of system starting from Euclidean Lagrangian? Say we are given the Minkowski Lagrangian $$L_m = \frac{\dot{\phi}^2}{2} - V(\phi).$$ The Hamiltonian can then be found by Legendre transformation $$H = \dot{\phi}\frac{\partial L_m}{\partial\dot{\phi}} - L_m,$$ which equals $$H = \frac{1}{2}\dot{\phi}^2 + V(\phi)$$ which was not hard. Now consider the corresponding Euclidean Lagrangian $$L_e = -\frac{\dot{\phi}^2}{2} - V(\phi).$$ How do I calculate the Hamiltonian in this formalism? The above way applied naively will not give the correct result.


title: minimising the mass or the lifetime of intermediary fermions in gamma-gamma physics

text: In gamma-gamma physics, what could be some way of either minimising the mass of the intermediary fermions or minimising the time for which those intermediary fermions could exist?


title: The square of Pauli-Lubanski operator

text: Let's have Pauli-Lubanski operator: $$ \hat {W}^{\alpha} = \frac{1}{2}\varepsilon^{\alpha \beta \gamma \delta}\hat {J}_{\beta \gamma}\hat {P}_{\delta} = \frac{1}{2}\varepsilon^{\alpha \beta \gamma \delta}\hat {S}_{\beta \gamma}\hat {P}_{\delta}, $$ where $\hat {J}_{\beta \gamma} = \hat {L}_{\beta \gamma} + \hat {S}_{\beta \gamma}$ and $\hat {S}_{\beta \gamma}$ is the spin-tensor. In some books there is a following expression for square of this operator: $$ \hat {W}_{\lambda}\hat {W}^{\lambda } = \hat {P}_{\alpha}\hat {P}_{\beta }\hat {S}^{\alpha \mu}\hat {S}_{\beta \mu} -\frac{1}{2}\hat {P}_{\alpha}\hat {P}^{\alpha}\hat {S}_{\alpha \beta}\hat {S}^{\alpha \beta }. $$ But I don't understand how to get it, because I can't claim, that $\hat {S}_{\alpha \beta}$ is antisymmetrical. So, I only get, using expression $$ \varepsilon_{\alpha \beta \gamma \lambda}\varepsilon^{\lambda \mu \nu \sigma} = \delta^{\mu}_{\alpha}\varepsilon^{\nu \sigma}_{\beta \gamma} + \delta^{\mu}_{\gamma}\varepsilon^{\nu \sigma}_{\beta \alpha} + \delta^{\mu}_{\beta}\varepsilon^{\nu \sigma}_{\gamma \alpha}=\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}\delta^{\sigma}_{\gamma} - \delta^{\mu}_{\alpha}\delta^{\nu}_{\gamma}\delta^{\sigma}_{\beta} + \delta^{\nu}_{\alpha}\delta^{\sigma}_{\beta}\delta^{\mu}_{\gamma} - \delta^{\nu}_{\alpha}\delta^{\mu}_{\beta}\delta^{\sigma}_{\gamma} + \delta^{\sigma}_{\alpha}\delta^{\mu}_{\beta}\delta^{\nu}_{\gamma} - \delta^{\sigma}_{\alpha}\delta^{\nu}_{\beta}\delta^{\mu}_{\gamma}, $$ following: $$ \hat {W}_{\lambda}\hat {W}^{\lambda} = \frac{1}{4}\varepsilon_{\lambda \alpha \beta \gamma }\hat {J}_{\alpha \beta}\hat {P}_{\gamma }\varepsilon^{\lambda \mu \nu \sigma}\hat {J}_{\mu \nu}\hat {P}_{\sigma} = -\varepsilon_{\alpha \beta \gamma \lambda}\varepsilon^{\lambda \mu \nu \sigma}\hat {J}_{\alpha \beta}\hat {P}_{\gamma }\hat {J}_{\mu \nu}\hat {P}_{\sigma} = $$ $$ = \frac{1}{4}\left(\delta^{\mu}_{\alpha}\varepsilon^{\nu \sigma}_{\beta \gamma} + \delta^{\mu}_{\gamma}\varepsilon^{\nu \sigma}_{\beta \alpha} + \delta^{\mu}_{\beta}\varepsilon^{\nu \sigma}_{\gamma \alpha}\right)\hat {J}_{\alpha \beta}\hat {P}_{\gamma }\hat {J}_{\mu \nu}\hat {P}_{\sigma} = $$ $$ =-\frac{1}{4}\left( \hat {J}_{\alpha \beta}\hat {P}_{\gamma}\hat {J}^{\alpha \beta}\hat {P}^{\gamma} - \hat {J}_{\alpha \beta}\hat {P}_{\nu}\hat {J}^{\alpha \nu}\hat {P}^{\beta} + \hat {J}_{\alpha \beta}\hat {P}_{\gamma}\hat {J}^{\gamma \alpha}\hat {P}^{\beta} - \hat {J}_{\alpha \beta}\hat {P}_{\gamma}\hat {J}^{\beta \alpha}\hat {P}^{\gamma} + \hat {J}_{\alpha \beta}\hat {P}_{\gamma}\hat {J}^{\beta \gamma }\hat {P}^{\alpha} - \hat {J}_{\alpha \beta}\hat {P}_{\gamma}\hat {J}^{\gamma \beta}\hat {P}^{\alpha}\right). $$ So, if spin-tensor operator is antisymmetrical, I'll get the answer easily using Poincare algebra. But if not, I can't continue. Can you help me to get the answer? Maybe, I can get the answer, because there is the identity of spin-tensor operator and orbital momentum tensor operator?


title: Does the non-relativistic conservation law of particles have an underlying (approximate) symmetry?

text: In momentum and energy is low enough, we end up with the same number of neutrons, protons and electrons after a collision as before it. This can be considered an approximate conservation law. Shouldn't there be a corresponding (approximate) symmetry corresponding to each of these particle groups?


title: Questions about the Solar System

text: 1. Most images you see of the solar system are 2D and all planets orbit in the same plane. In a 3D view, are really all planets orbiting in similar planes? Is there a reason for this? I'd expect that the orbits distributed all around the sun, in 3D.   2. Has an object made by man (a probe) ever left the Solar System?


title: Can low temperature plasma exist?

text: Plasma is ionized gas which as far as I know only occurs at high temperatures. When plasma cools down it tends to recombine with the electrons present and turn back into gas. But what if the disassociated electrons in the plasma were removed and the plasma were allowed to cool down in a vacuum, while being held in place by a strong magnetic field. Would this substance still be plasma? Is this possible?


title: How to make a nongrounded conductor have equipotential?

text: I'm studying the Method of Images and I seemed to have come to a conundrum. Method of Images takes advantage of grounded objects, (I am currently studying spheres), to set boundary conditions. However, how would one use the idea of MoI to set the potential of a conducting sphere as constant? Since $E = \nabla V$, a constant potential would be the electric field inside would be constant? Therefore, a density of charge inside the sphere?


title: Deriving the reduced Green's functions in Polchinski's volume 1

text: In equation 6.2.7, Polchinski defines his reduced Green's functions $G'$ on the 2-manifold to satisfy the equation, $$ \frac{-1}{2\pi \alpha '}\nabla ^2 G'(\sigma_1, \sigma_2) = \frac{1}{\sqrt{g}}\delta ^2 (\sigma_1 - \sigma_2) - X_0^2 $$ (..where $\sigma_1$ and $\sigma_2$ are two points on the manifold and $X_0$ is the zero-eigenvalue of the Laplacian..Why is he assuming that there is only one zero mode?..) Now at various place he has written down the solutions to the equation like,   * on a $S^2$ it is given by 6.2.9, $$G' = - \frac{\alpha'}{2}ln \vert z_1 - z_2\vert ^2 + f(z_1,\bar{z_1}) + f(z_2,\bar{z_2})$$ where $f(z,\bar{z}) = \frac{\alpha'X_0^2}{4} \int d^2 z' exp(2\omega(z,\bar{z}))ln \vert z - z'\vert ^2 + k$   * For the disk it is given by 6.2.32, $$G' = -\frac{\alpha'}{2}ln \vert z_1 - z_2\vert ^2 + \frac{\alpha'}{2}ln \vert z_1 - \bar{z_2}\vert ^2 $$   * For $\mathbb{RP}^2$ it is given by, $$G ' = -\frac{\alpha'}{2}ln \vert z_1 - z_2\vert ^2 + \frac{\alpha'}{2}ln \vert 1 +z_1\bar{z_2}\vert ^2 $$ I would like to know how these functions are derived.   * Also how is it that the dependence on the $f$ for the first case drops out in equation 6.2.17? If I plug in the functions I see a remnant factor in the exponent of the form, $\sum_{i,j,i<j,=1}^n k_i k_j (f(\sigma_i) + f(\sigma_j))+ \sum_{i=1}^n k_i^2 f(\sigma_i)$


title: How does positive charge spread out in conductors?

text: I know that when there are excess positive charges in a conductor, for example, a metal sphere, the positive charges will spread out over its surface. However, I am confused about how this excess charge spreads out over the surface, if protons cannot move and only electrons can move.   Can someone please inform me on how the excess positive charge spreads out over the surfaces of conductors?


title: How can we know the time frames for events in the early universe?

text: I just finished watching Into the Universe with Stephen Hawking (2010). Specifically the third episode titled 'The Story of Everything.' In the episode Hawking is explaining the mainstream theories of the events just after the Big Bang. What confuses me is that he just said at the beginning that the laws of physics were different and time (among other things) did not exist as we know it during this early stage in the life of the universe, but then goes on to say certain events took certain amounts of time. I don't know if the numbers match, but the same issue exists on this Wikipedia article. How can we even guess the time frames of these events if time itself is largely dependent on the current state of the universe?


title: What kinds of materials contract the most in cold temperatures?

text: I know that water expands in the freezer, but I'm curious about which materials _contract_ in response to cold temperatures --- and most importantly, which ones undergo the most drastic changes?


title: What happened with Hilbert's sixth problem (the axiomatization of physics) after Gödel's work?

text: I'll write the question but I'm not fully confident of the premises I'm making here. I'm sorry if my proposal is too silly. Hilbert's sixth problem consisted roughly about finding axioms for physics (and it was proposed in $1900$). I guess that at the time, such thing was impossible due to the nature of physics which is mainly based on observations and models. But it seems that after Gödel's work on $1931$, the axioms which were seen as self-evident truths started to be seen as unprovable statements and the job of a mathematician is grossly about deriving theorems from these axioms. So if this shift of axiomatic conception really happened, couldn't we just accept anything (including the physical observations) as axioms and reason about their consequences? Thus somehow _solving_ Hilbert's sixth problem?


title: Magnetic monopoles

text: I am a non-expert in this field, just have a layman's interest in the subject. Has anyone ever considered the possibility of magnetic monopoles (one positive and one negative charge) being confined together like quarks, and hence that could be the reason monopoles have never been observed? At the macro level the total magnetic charge would always be zero due to the charges being tightly bound.


title: Help calculating field of view for ball lens I just bought

text: ![Diagram](http://imgur.com/KwER0d8) I have a 5 mm (diameter) solid ball lens that I intend to place in front of a 36mm X 24mm camera sensor (I have already made preparations for the image to be focused and for the sensor to not be flooded with light). I want to calculate the resulting horizontal and vertical field of view from the ball lens when I take a picture. Please see the link for a diagram of the setup if the image does not work. ![http://imgur.com/KwER0d8](http://i.imgur.com/KwER0d8.jpg) I have already calculated the back focal length of the ball lens to be 1.479 mm.


title: Effect of wavelength on photon detection

text: When some photon detector detects a photon, is it an instantaneous process (because a photon can be thought of as a point particle), or does the detection require a finite amount of time depending on the wavelength of the photon? EDIT: I guess what I am wondering is if a photon has a wavelength and travels at a finite speed, then if a photon had a wavelength of 300,000,000m, would its interaction with the detector last 1s? Or does the uncertainty principle say that a photon with wavelength 300,000,000m (and therefore energy E), it cannot be known exactly when it hit the detector with an accuracy better than 1s. Or is it more like this: suppose there is a stream of photons moving towards the detector with wavelengths of 300,000,000m and they reach the detector at a rate of 10 photons/second and the detector has a shutter speed such that the shutter is open for 1s at a time, then it would record 10 photon hits (records all the photons). But if the shutter speed is only 0.5s, then it would record 2.5 hits on average? EDIT2: I'm not interested in the practical functioning of the detector and amplification delays. I'm looking at and ideal case (suppose the photon is 'detected' the instant an electron is released from the first photomultiplier plate). It is a question regarding the theory of the measurement, not the practical implementation.


title: Conservation law of equation involving Hilbert transform

text: I am trying to confirm a conservation law I cam across in a paper (Janssen 1983 "On a fourth-order envelope equation for deep-water waves" Journal Fluid Mechanics), and am having difficulty. In particular, I'm trying to confirm the conservation of linear momentum $P$, where $P = \frac{i}{2} \int AA^*_x - A^*A_x \ dx$, where $^*$ denotes complex conjugate and spatial integrals are taken to be over all space in this question. Now, the governing equation takes the form $A_t = \mathcal{N}(A,A^*) - P.V. \ i\alpha\ A\int_{-\infty}^{\infty} \frac{\partial|A|^2}{\partial x'}\frac{1}{x-x'} \ dx'$, where $\mathcal{N}$ is a (nonlinear) operator describing the rest of the dynamics, which is not important for this question, P.V. denotes the principal value and $\alpha$ is some real constant. Note, this integral is proportional to the Hilbert transform. Now, I want to look at the time evolution of the momentum $P$. To that end, we have $\frac{d P}{dt} = \frac{i}{2} \int \dot{A}A^*_x +A\dot{A}_x^* -\dot{A}^*A_x -A^*\dot{A}_x \ dx$ Substituting in the relation for our governing equation, and using integration by parts (we assume the field A is compact) we find $\frac{dP}{dt}=\frac{\alpha}{2} \int (|A|^2)_x \mathcal{H} (|A|^2_x) \ dx$ where $\mathcal{H}(|A|^2_x) \equiv P.V. \int_{-\infty}^{\infty} \frac{\partial|A|^2}{\partial x'}\frac{1}{x-x'} \ dx'$. Now, this term is non-zero, where as the author claims this integral is conserved. I don't see why this integral should vanish, but perhaps I'm not exploiting a property of the Hilbert transform. Am I missing something obvious? Thanks, Nick


title: Book with solved examples and exercise problems for SUSY

text: I am learning supersymmetry right now. I am mostly following Bailin and Love. I try to connect all the steps from the book and complete the derivations in order to get comfortable with the calculations but I feel the lack of a problem book with a lot of solved and unsolved problems, something like the Schaum's series which I can use together with Bailin and Love. Do you happen to know any such book that you think would be helpful for me?


title: Derivation of a formula concerning overlap between spin states

text: I am reading through Cavagna's Spin glass theory for pedestrians, but I am stuck at equation (35). I'll try to provide a little context. Given two spin configurations $\sigma$ and $\tau$, we define their overlap as: $$q_{\sigma\tau} = \frac{1}{N}\sum_{i=1}^N \sigma_i \tau_i$$ The configuration space can be split into sets of configurations called _pure states_ , which are characterized by the _clustering property_ , which states that for the set of configurations in a pure state, correlations drop to zero as the distance increases: $$\langle\sigma_i \sigma_j\rangle \rightarrow \langle\sigma_i\rangle \langle\sigma_j\rangle \qquad \text{for}\quad |i-j|\rightarrow \infty$$ The overlap between pure states $\alpha$, $\beta$ is defined as: $$q_{\sigma\tau} = \frac{1}{N}\sum_{i=1}^N \langle\sigma_i\rangle_\alpha \langle\sigma_i\rangle_\beta$$ An overlap distribution is defined as: $$P(q) = \frac{1}{Z^2} \int D\sigma D\tau e^{-\beta H(\sigma)} e^{-\beta H(\tau)} \delta(q-q_{\sigma \tau})$$ In the text it is claimed that because of the clustering property, $$P(q) = \sum _{\alpha \beta} w_\alpha w_\beta \delta(q-q_{\alpha\beta})$$ where $w_\alpha = Z_\alpha/Z$ (where $Z_\alpha$ is the partition function of all spin configurations in the pure state $\alpha$), and similarly for $w_\beta$. This is the formula I don't understand. I don't know how to derive this formula from the previous definitions and equations.


title: Lorentz invariance?

text: What exactly is meant by _Lorentz invariance_? Is it just an experimental observation, or is there a theory that postulates it? What quantities do we expect to be Lorentz invariant? Charge? Charge densities? Forces? Lagrangians?


title: How do we know if a formulation of classical mechanics is correct?

text: For example, the Lagrangian formulation. I may be missing something, i.e. not having done it in enough detail, but here is my issue: from the definition of the lagrangian ($\mathcal{L}$) and from the principle of least action ($S(t)$), we get that $$ S(t) = \int \mathcal{L} \ dt$$ has to be minimised. From this, one can derive the Euler-Lagrange equation, $$\frac{d}{dt} \left(\frac{\partial L}{\partial \dot{x}}\right) = \frac{\partial L}{\partial x}$$ and using $\mathcal{L} = T - V = m\dot{x}^2 - V(x)$ we get $$ m\ddot{x} = -V'(x),$$ i.e. Newton's second law of motion. **Now:** Does the fact that we get Newton's second law mean that the Lagrangian formulation (and therefore the principle of least action) is correct? I mean, could it not be a coincidence that we get this equation? But then again, is there a way that this equation could _not_ hold? The only difference between systems is the potential $V(x)$ (right?), so as long as it is included in a general form the equation should hold. (?). _In light of this_ , is there any physical meaning as to why the Lagrangian is defined as the difference between kinetic and potential energy, $\mathcal{L} = T - V $?


title: How loud is the thermal motion of air molecules?

text: In other words, given a magical room with walls that produce no vibration and transmit zero vibration from the outside, and nothing on the inside except room temperature air, what would be the noise level in dB SPL (sound pressure level) from the thermal motion of the air itself? (Similar to the noise floor of electronics being determined by thermal noise in the conductors.) What is the quietest possible anechoic chamber? For reference: Sound pressure is defined as the root-mean-square value of the instantaneous pressure, measured in pascal = N/m². SPL is the same number, but expressed in decibels relative to 20 µPa. (I assume that it has a white spectrum, but I could be wrong. Thermal noise in electronics is white, but other types of electronic noise are pink, and blackbody thermal radiation has a bandpass spectrum.) Here's an explanation in the context of underwater acoustics. Not sure how this applies to air: > Mellen (1952) developed a theoretical model for thermal noise based on > classical statistical mechanics, reasoning that the average energy per > degree of freedom is kT (where k is Boltzmann’s constant and T is absolute > temperature). The number of degrees of freedom is equal to the number of > compressional modes, yielding an expression for the plane-wave pressure > owing to thermal noise in water. For non-directional hydrophones and typical > ocean temperatures, the background level due to thermal noise is given by: > > NL = −15 + 20 log _f_ (in dB re 1 µPa) > > where _f_ is given in kHz with _f_ >> 1, and NL is the noise level in a 1 Hz > band. Note that thermal noise increases at the rate of 20 dB decade−1. There > are few measurements in the high-frequency band to suggest deviations from > the predicted levels. Citation is R. H. Mellen, The Thermal-Noise Limit in the Detection of Underwater Acoustic Signals, J. Acoust. Soc. Am. 24, 478-480 (1952).


title: Are there free data available online from cosmology (or astrophysics) experiments that anyone can analyse?

text: One can understand a subject better in physics by trying to solve as many problems as one can from a textbook say. When it comes to experimental physics and data analysis, no book on experimental (high energy or astrophysics or cosmology) physics comes with free data that one might analyze to test his understanding. Are there sites online, for experiments in cosmology or astrophysics (I know that particle physics data would be overwhelming), where such data is available for free that one can play with (i.e. to be able to make cuts and reproduce simple histograms...etc)? may be to reproduce roughly the CMB curve?


title: Why does the n=0 Landau level in graphene have half the degeneracy of the other levels?

text: I've looked through several papers that talk about the anomalous integer quantum Hall effect of graphene (such as http://journals.aps.org/prl/pdf/10.1103/PhysRevLett.95.146801), and they all state that the Hall effect is anomalous for graphene because the n=0 Landau level has half the degeneracy of the n>1 levels, with little to no explanation and without citing any (helpful) sources. One of the papers (http://journals.aps.org/prb/pdf/10.1103/PhysRevB.75.165411) states that "the Dirac fermions have acquired an effective gap in a form of a “relativistic mass” due to the Coulomb interaction. Such a gap reduces the degeneracy of _only_ the zeroth LL." Could someone explain this in more depth, specifically _why_ the Coulomb interaction only breaks the degeneracy of the zeroth LL?


title: Why can't an excess of electrons or holes by themselves cause current flow?

text: I am a beginner in electrical engineering. Often times (most cases actually), the underlying physics aren't really explained to us and we are just left to assume that it works "because it works." This is never enough for me in classes etc and I always end up doing a follow up of physics side of the spectrum. My question is, you have a battery with excess electrons built up on the negative lead, and excess holes on the positive lead, why is it that in our universe then connecting a conductive compound (like.. copper) to just the negative lead does not produce current, or vice versa connecting the wire to just the positive lead. After all, if we move a magnet passed a conductor, there is a tiny induced current. How can the magnet do this, but an excess of electrons or electron holes repelling each other cannot?


title: Rotational Friction

text: **This is the question-** Consider a cylinder of mass $M$ resting on a rough horizontal rug that is pulled out from under it with acceleration $a$ perpendicular to the axis of the cylinder. What is $F$ (friction) at a point _P_? It is assumed that the cylinder does not slip. ![Diagram](http://i.stack.imgur.com/b05sS.png) Options- A) $F = M g$ B) $F = M a$ C) $F = \frac{M}{2} a $ D) $F = \frac{M}{3} a $ **My attempt-** Since rolling is nothing but rotation about the point of contact (P in this case), according to me P should be at relative rest, therefore $F=Ma$. This should have been a short and crisp question, help me out please answer given is $F=\frac{M}{3} a$, and if you can perhaps please point me out to articles for such rolling friction question, I get stumbled whenever they ask friction in rolling. Thanks in advance.. **Edit-** This idea also came to my mind- Moment of inertia for a solid cylinder= I = $\frac{MR^2}{2}$ Let the frictional force be F, then, $F*R=(torque)=I*\alpha$ $F*R=\frac{(M*R^2)*(a)}{2R}$ (as the cylinder is in pure rolling) therefore- $F=\frac{Ma}{2}$ which is again incorrect any pointers to where I went wrong here again would be appreciated.. thanks


title: name for 2D Electrostatics as Integrable System

text: I am trying to understand 2D electrostatics of $n$ point charges. Roughly, $$ H = \sum_{i=1}^N n_i \ln |z- z_i|$$ However, I keep bumping across the Gaudin model instead with this Hamiltonian $$ H_j = \sum_{i \neq j} \frac{n_i}{z_j- z_i}$$ and these are often put into context with other integrable systems, such as Calogero-Moser, or Ruisenaars-Schneider. They talk about how $[H_i, H_j] = 0$ as operators and a torus fibration of the phase space. Is there a larger family of integrable systems that contains both cases I mentioned above? This is not electromagnetism, since that Hamiltonian looks rather different (I will try to look up the source for that). What I have written down is just the electrostatic potential. * * * The other strange part is the second Hamiltonian is a real part of the derivative $\frac{d}{dz}$ of my first example.


title: Does spacetime in general relativity contain holes?

text: > Are there physical models of spacetimes, which have bounded (four > dimensional) holes in them? And do the Einstein equations give restrictions to such phenomena? Here by holes I mean constructions which are bounded in size in spacetime and which for example might be characterized by nontrivial higher fundamental groups. On a related note: > Could relatively localized and special configurations of (classical) > spacetime be interpreted as matter? I.e. can field-like behavior emerge in characteristic structures of spacetime itself, like for example holes in the above sense, or localized areas of wild curvature? Gravitational waves certainly go in this direction, although they act on such a huge scale, that in their case we probably wouldn't recognize an organized, maybe even life-like behavior as such.


title: Which bonds are the cross-links and which are secondary bonds (in elastomers)?

text: Elastomeres are "defined" as: "linear-chain polymers with widely spaced cross-links attaching each molecule to its neighbours" Now I found sentences (talking about glass transition): "This means that at room temperature the secondary bonds are melted and the molecules can slide relative to each other with ease. Were it not for the cross-links, the material would be a viscous liquid, but the cross-links give it a degree of mechanical stability." So, what exactly are these bonds that melt? and which ones do not? An example would be nice. Chemistry is not my strongest field so be rigorous. The quotations are from cellular solids by gibson and ashby.


title: Efficiencies of Coupling Light into a Fiber

text: I am in AMO Physics and work a lot with optics. I just wanted to get an idea of what coupling efficiencies one "should" get in a "reasonable time"* by coupling light into a fiber using different couplers, like collimators, mounting plates, mounted lens systems, etc. I understand that this dependents on a lot of factors, so I will narrow it down for my specific case but would appreciate if ppl report some numbers with a short info about their setups. We use a cage system system with 5 mm lens and APC fiber plate to couple 633 nm laser light into an APC single mode fiber. Pretty much everything of the optical equipment is from Thorlabs.com. The reason for my question: My PI told me I should get efficiencies up 80% percent with our coupling scheme but I can only get 30%. I hope to get a better feeling for coupling efficiencies with this answer. Thanks for your input, n3rd *With "should" and "reasonable time" I refer to coupling efficiencies one can achieve at time scales on the order of tens of minutes rather than hours, days, weeks.


title: Real limits on the maximum obtainable resolution of an optical system

text: The maximum obtainable angular resolution of an optical system with some given aperture is well known, but it seems to me that this isn't a real theoretical limit. The assumption is that you are going to take a picture using the system and no further processing will take place. However, given the known point spread function of the optical system, you could perform a deconvolution calculation. The final resolution of the processed image should be limited by the noise. So, what then seems to matter is the observation time (the longer you integrate the signal the better the signal/noise ratio will be). So, what is the correct theoretical limit of the maximum resolution in terms of brightness of the two sources to be resolved, the aperture and the observation time (assuming that the only noise comes from the fluctuations in the finite numbers of photons from the sources)?


title: Galaxies that are newer than our own one

text: Can we see any galaxies/stars that are newer than our own galaxy? As light takes (c) amount of times to reach us - so relatively speaking, light which would have left the newer galaxy (and far enough from us) still would not have reached us. Therefore, can we ever see a galaxy that is newer than our own one. If we can see a new galaxy that is newer than our own one, how can we determine its age as light would not have reached us yet. EG : Light travels from galaxy (x) to us (y) and it has taken 100 million light years. However if (x) was less than 100 million years younger than us, how could it have reached us and can we determine it and its age ?


title: Where will New Horizons go after sailing past Pluto?

text: The New Horizons Spacecraft is scheduled to whiz by Pluto around 2015, and my understanding is that it is going to do exactly that — whiz by it. Where will it go after that? Or what else can it do once that mission is completed?


title: How are exoplanets confirmed?

text: In reference to the Kepler 22b news: > The Kepler team had to wait for three passes of the planet before upping its > status from "candidate" to "confirmed". This is possible because the planet has a orbital period ('year') of 289 days, and so it has transitioned more than once (as above) since the Kepler mission started. However I've seem some confirmed exoplanets that have orbital periods in 10s and 100s of years, and therefore could only have been detected once by Kepler. As far as I'm aware all methods of discovering exoplanets require the planet to pass in from of their parent star (i.e. we can't use a telescope to 'zoom in' and see them directly)? So how are exoplanets confirmed, when we haven't been searching long enough to cross in front of their parent star? How long does the transition even last?


title: Computing Planetary Positions

text: Shortly, I will be beginning my third year at University in Computer Science, I am a software developer and I will be required to work on a final year project. My idea for my final year project is to write a desktop application which would allow the user to specify a date/time and it would return all of the positions of the planets at that point in time. I was hoping someone could point me in the right direction of how I would begin to do the math or calculate this, as I'm aware that position is relative and a bit open ended - any scope refinements to my project would be gladly accepted too as I'd like to incorporate other things. Just to reiterate: This is not a programming question, and I'm not asking for someone to 'do my project for me' but any helpful pointers would be appreciated.


title: Current map of galaxies' z's and distances?

text: So, here's a diagram with some galaxies. I realize there are 500 billion galaxies out there (likely many more), but is there are fairly up-to-date diagram of all the galaxies, or a representative sample, at all observable distances? ![Hubble Diagram](http://i.stack.imgur.com/uCejG.gif)


title: What is the "center of gravity of a band"?

text: It seems there is a term called "center of gravity of a band" in solid state physics or chemistry which I'm confused about. Could anyone give a formal definition of the term or point to some reference that have a clear explanation of it's meaning?


title: Has the Double-Slit Quantum Eraser Experiment ever been tried on a large scale?

text: I was just reading about quantum entanglement and the example was the Double- Slit Quantum Eraser Experiment. Then this was used as a basis for saying that particles might be half a universe apart and still be just as connected. So I was just wondering if entanglement experiments have been done to show that this is really the case or if it is considered "academic" to conclude that there's no difference when the distances are huge? As a side query, quantum mechanical interference seems to be based on the existence of probability - does this mean probability should be thought of as something as tangible as matter/energy?


title: If I take an XRD image of a single cubic unit cell, would the diffraction pattern simply be its reciprocal lattice?

text: I've seen the rings from powder diffraction images, and I read that each line is made up of a lot of dots, I was wondering if these dots are reciprocal lattice points of the structure. And if we weren't looking at a powder but just a 2D hexagonal lattice, then the xrd pattern would simply be its reciprocal lattice?


title: representation of conformal group in d>2

text: In P. Di Francesco, P. Mathieu, D. Snchal they fix the generators of the conformal group acting on a scalar field by somewhat arbitrarily defining $$\Phi'(x)=\Phi(x)-i\omega_a G_a\Phi(x)$$ and by arbitrary I mean the sign. The "full" transformation would then be given by the exponentiation $$\Phi'(x)=e^{-i\omega_a G_a}\Phi(x)$$ the generators are given in eq. 4.18 and the commutation relations are in eq. 4.19. But by looking at the action of the group in the coordinates, I think we could get the same representation 4.18 by requiring $$x'^\mu=e^{ i\omega_aT_a}x^\mu=\alpha^\mu+\omega^\mu_{~~\nu} x^\nu + \lambda x^\mu +2(\beta \cdot x)x^\mu-\beta^\mu x^2$$ but the problem is that, in order to reproduce 4.18 the sign of the exponential is not the same as the one for $G_a$. I would have expected that the field transform in a similar way: $$\Phi'(x)=e^{i\omega_a G_a} \Phi(x)$$ In fact it seems that there is not a "standard" definition. For example in this article by Mack and Salam, as well as in this book by Fradkin and Palchik, the generators as well as the commutation relations disagree among each other. Anyway I would like to know how the generators should be defined so that everything is consistent from the Lie theory/representation theory point of view.


title: Frequency of earthquakes

text: Are earthquakes getting less frequent across the centuries? I know that more seismic stations have register more earthquakes in the last century, but that doesn't imply there were more. I am interested on the geophysical side of it. The logic is that things settle down, so there is less stuff to shuttle. Add to it that the earth (at its core) get colder with the passage of time.


title: Tree level QFT and classical fields/particles

text: It is well known that scattering cross-sections computed at tree level correspond to cross-sections in the classical theory. For example the tree level cross-section for electron-electron scaterring in QED corresponds to scattering of classical point charges. The naive explanation for this is that the power of hbar in a term of the perturbative expansion is the number of loops in the diagram. However, it is not clear to me how to state this correspondence in general. In the above example the classical theory regards electrons as particles and photons as a field. This seems arbitrary. Moreover, if we consider for example phi^4 theory than the interaction of the phi-quanta is mediated by nothing except the phi-field itself. What is the corresponding classical theory? Does it contain both phi-particles and a phi-field? Also, does this correspondence extend to anything besides scattering theory? Summing up, my question is: > What is the precise statement of the correspondence between tree-level QFT > and behavior of classical fields and particles?


title: Are possible gauge fields in a Lagrangian theory always determined by the structure of the charged degrees of freedom?

text: An elementary example to explain what I mean. Consider introducing a classical point particle with a Lagrangian $L(\mathbf{q} ,\dot{\mathbf{q}}, t)$. The most general gauge transformation is $L \mapsto L + \frac{d}{dt} \Lambda(\mathbf{q},t)$ which implies the usual transformations of the canonical momentum $p \to p+ \nabla_q \Lambda$. Generalizing this derivative as an extended one gives the connection of electromagnetism. Once the particle motion is quantized, we recognize this as a local $U(1)$ "internal" symmetry of the quantum-mechanical phase. Is this a fundamental property of gauges symmetries - being implied by non- dynamical symmetries of the action? By "non-dynamical symmetries" I mean those coming from the structure of and the freedom of labeling for the degrees of freedom under consideration. **EDIT** : After reflecting on the comments below, I'd re-formulate the question as: > Do non-dynamical symmetries of a local Hamiltonian exhaust all possible > types of gauges fields that it can couple to in a gauge-invariant manner? **EDIT-2** : The reason I'm asking is that it appears that the very possibility for a particle to couple to electromagnetism and gravity come from the applicability of the action formalism and is already built-in as the symmetry under addition of a total time derivative (which as I understand to be one of the possible general definitions of gauge symmetry). Some comments suggest the answer is a trivial yes, presumably because non- dynamical symmetries are gauge symmetries by definition. A concise expert answer would be helpful to close the question.


title: Is the distinction between the Poincaré group and other internal symmetry groups artificial?

text: For instance, given a theory and a formulation thereof in terms of a principal bundle with a Lie group $G$ as its fiber and spacetime as its base manifold, would a principle bundle with the Poincaré group as its fiber and $\mathcal{M}$ as its base manifold, where $\mathcal{M}$ is a manifold the group of whose isometries is $G$, lead to an equivalent formulation? Why? Why not? On a related note, can any Lie group be realized as the group of isometries of some manifold?


title: Reduced density matrices for free fermions are thermal

text: Many recent papers study entanglement in eigenstates of fermionic free hamiltonians (normally on a lattice) using the basic assumption that the reduced density matrices are _thermal_ (e.g. Peschel 2003). As I understand it, the theorem that we need in order to proceed with the computation of entanglement is the following: Consider a fermionic state, pure or mixed, which fulfills Wick's theorem (i.e.: all N-point functions can be expressed from the 2-point correlation function). Then, the reduced density matrix for a block can always be expressed as $\rho_B \propto \exp(-\beta H)$, where $H=\sum_k c_k d_k^\dagger d_k$. The $d_k$ and $d^\dagger_k$ are suitable fermionic operators which can be expressed as linear combinations of the original ones. The reference given is to Gaudin 1960. But that article only contains the reverse theorem: _if_ a fermionic density matrix is thermal, then Wick's theorem follows. So, my main question: Does anybody know about a proof of this theorem? Moreover: I guess this theorem is related to the thermalization theorem of QFT on curved spacetimes. Is this true?


title: Running of gauge couplings in the Standard Model

text: I'm sure many of us are familiar with the following plot showing the running of the inverse of the fine-structure constants of the SM. ![running of SM couplings](http://i.stack.imgur.com/1PZin.png) (I got the picture from google) At one-loop, the expressions for the fine-structure constants of the hypercharge, weak, and color gauge groups are, $$\frac{1}{\alpha_1 (\mu)} \approx 59 - \frac{41}{20 \pi}\log[\frac{\mu}{M_Z}]$$ $$\frac{1}{\alpha_2 (\mu)} \approx 30 + \frac{19}{12 \pi}\log[\frac{\mu}{M_Z}],$$ $$\frac{1}{\alpha_3 (\mu)} \approx 8.5 + \frac{7}{2 \pi}\log[\frac{\mu}{M_Z}],$$ where the numerical values of $\alpha$'s in the first terms in RHS are at the mass of the Z boson $M_Z$. Now, if I plot these (from $\mu=100$ GeV to $\mu=2\times 10^16$ GeV) using Mathematica, I get this: ![enter image description here](http://i.stack.imgur.com/M3A9X.jpg) The slopes of the $log$'s in the previous equations do not produce the shape appearing in the first plot. What am I missing here? How is the first plot, which we usually see in the literature, produced? **Update** : What I was missing is skills with Mathematica :p, Now I get the correct plot: ![enter image description here](http://i.stack.imgur.com/KQcdc.jpg)


title: Morphisms between chiral CFTs

text: This is a question about terminology. Given two vertex algebras $V_1$ and $V_2$ (= chiral CFTs), there are two kinds of maps $V_1\to V_2$ that one might want to consider. 1) Morphisms of VOAs that send the conformal vector of $V_1$ to the conformal vector of $V_2$. In other words, morphisms that send the copy of the Virasoro algebra in $V_1$ to the copy of the Virasoro algebra in $V_2$. Examples of those come from e.g. from conformal embeddings. This type of morphism forces the central charges of $V_1$ and $V_2$ to be equal. 2) Morphisms of VOAs that are not required to send the conformal vector of $V_1$ to the conformal vector of $V_2$. An example of the above notion is the inclusion of the trivial one dimensional VOA into some other VOA. **Question:** is there some standard terminology used in physics to refer to those two kinds of morphisms between chiral CFTs?


title: Symmetry factor of a second order four point function term of the $\phi^4$ theory

text: I am reading Cheng and Li. On page 9, it is written that the coefficient $\frac{1}{2 \cdot (4!)^2}$ for the second order term of the four point function becomes just $\frac{1}{2}$ for the following diagram: ![enter image description here](http://i.stack.imgur.com/hzbgJ.png) I am not getting that answer. Call the vertices in the diagram $y_1$ and $y_2$. Imagine $y_1$ and $y_2$ as a point with four lines coming out. (As it is $\phi^4$) My reasoning is as follows: $x_1$ can be connected to any one of the four lines emanating, after that $x_4$ has three choices for joining to one of the vertices. Ways of joining $x_1$ and $x_4$ is $4 \times 3$. Similarly for $x_2$ and $x_3$. Now $y_1$ and $y_2$ have two lines left which can be joined in a loop in two ways So total number of ways is $4 \times 3 \times 4 \times 3 \times 2$ . So I am getting the coeffecient as $\frac{4 \times 3 \times 4 \times 3 \times 2}{2 \times (4!)^2}=\frac{1}{4}$. Where am I going wrong?


title: Work done by isothermal expansion from two different viewpoints

text: Consider an adiabatic system as follows. It consists of a gas in a container and a piston. Initially, the system is at equilibrium and the gas inside it occupies a volume $V_i$ at a pressure $p_i$ which is equal to the outside pressure. Suddenly, the outside pressure changes and reduces to $p_{atm}$. The piston moves to equalize the pressure and the gas expands isothermally to obtain equilibrium. The gas now occupies a volume $V_f$ at a pressure of $p_f$ which is equal to $p_{atm}$ Now, two textbooks I have define the work done from two different viewpoints. **1: From the viewpoints of the surroundings** : The work done on the system by the surroundings equals $-p_{atm}\Delta V$. Since $p_{atm}$ is pretty much constant for the whole of the whole of the process, we can say that the work done equals: $$W_1 = -p_{atm}(V_f - V_i)\tag1$$ **2: From the viewpoint of the system:** We can write the internal pressure of the system as a function of its volume: $p_{in}(V)$. As during the expansion, the internal pressure changes, the work done by the system equals $$W_2 = \int_{V_i}^{V_f}p_{in}(V)dV$$ Now, I don't know which definition to use. The work is done by the system (from definition 2) is done on the surroundings. But what about the negative work that the surroundings did on the system? Where did that energy go? Maybe, the two definitions express the same thing: the work done by the system. The negative sign in definition 1 signifies that the work is done by the system. But that would mean that $$W_1 = W_2$$ We can simplify $W_2$ as follows. Clearly, $$p_{in}(V) = \frac{p_iV_i}{V}$$ $$W_2 = \int_{V_i}^{V_f}\frac{p_iV_i}{V}dV = p_iV_i \ln{\frac{V_f}{V_i}} \tag2$$ The internal pressure when the volume is equal to $V_f$ is $p_{atm}$ $$\implies p_{atm} = p_{in}(V_f) = \frac{p_iV_i}{V_f}$$ $$\implies V_f = \frac{p_iV_i}{p_{atm}}$$ Putting this into $(1), (2)$ gives us, $$W_2 = p_iV_i\ln{\frac{p_{i}}{p_{atm}}}$$ $$W_1 = -p_{atm}\left(\frac{p_iV_i}{p_{atm}} - V_i\right) = -V_i(p_{atm} - p_i)$$ Consider some values. Let $p_i = 5 \ \rm{Pa}, V_i = 1 \ \rm{m^3}, p_{atm} = 1 \ \rm{Pa}$. $$W_1 = -1\cdot(1-5) = 4$$ $$W_2 = 1\cdot1\cdot\ln{\frac{5}{1}} = 1.609$$ Where am I making a mistake?


Does there exist a single plate capacitor(conductor)? _if yes_ How will you define the **capacitance** and potential(difference) of such conductor?


title: Probability and the propagator

text: Due to the Wiki article, "...In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum...". Let's have the expression for the propagator of some field $\hat {\Psi}_{l}$ in the free theory: $$ \tag 1 D_{lm}(x - y) = \langle |\hat{T}\left( \hat {\varphi}_{l}(x)\hat {\varphi}_{m}^{\dagger}(y)\right) |\rangle = -\frac{i}{(2 \pi )^{4}}\int \frac{F_{lm}(p)e^{-ip(x - y)}d^{4}p}{p^{2} - m^{2} - i\varepsilon}. $$ Let's use two examples: spinor and scalar fields: for them $(1)$ takes the form $$ \tag 2 D_{lm}(x - y) = D(x - y) = -\frac{i}{(2 \pi )^{4}}\int \frac{e^{-ip(x - y)}d^{4}p}{p^{2} - m^{2} - i\varepsilon}, $$ $$ \tag 3 D_{lm}(x - y) = -\frac{i}{(2 \pi )^{4}}\int \frac{(\gamma^{\mu}p_{\mu} + m)_{lm}e^{-ip(x - y)}d^{4}p}{p^{2} - m^{2} - i\varepsilon} $$ respectively. How to convert $(2), (3)$ into the probability (which lies at interval $[0, 1]$)? Particularly i don't understand what to do with spinor indices in $(3)$ (the probability must be Lorentz scalar, so I need to sum over the indices (?)). And also, why doesn't probability depends on momentum (the propagator doesn't contain info about momentum)?