title: How are anyons possible? (another version)

text: I know that this question has been submitted several times (especially see How are anyons possible?), even as a byproduct of other questions, since I did not find any completely satisfactory answers, here I submit another version of the question, stated into a very precise form using only _very elementary general assumptions of quantum physics_. In particular I will not use any operator (indicated by $P$ in other versions) representing the swap of particles. Assume to deal with a system of a couple of identical particles, each moving in $R^2$. Neglecting for the moment the fact that the particles are indistinguishable, we start form the Hilbert space $L^2(R^2)\otimes L^2(R^2)$, that is isomorphic to $L^2(R^2\times R^2)$. Now I divide the rest of my issue into several elementary steps. **(1)** Every element $\psi \in L^2(R^2\times R^2)$ with $||\psi||=1$ defines a state of the system, where $|| \cdot||$ is the $L^2$ norm. **(2)** Each element of the class $\\{e^{i\alpha}\psi\:|\; \psi\\}$ for $\psi \in L^2(R^2\times R^2)$ with $||\psi||=1$ defines the same state, and a state _is_ such a set of vectors. **(3)** Each $\psi$ as above can be seen as a complex valued function defined, up to zero (Lebesgue) measure sets, on $R^2\times R^2$. **(4)** Now consider the "swapped state" defined (due to (1)) by $\psi' \in L^2(R^2\times R^2)$ by the function (up to a zero measure set): $$\psi'(x,y) := \psi(y,x)\:,\quad (x,y) \in R^2\times R^2$$ **(5)** The physical meaning of the state represented by $\psi'$ is that of a state obtained form $\psi$ with the role of the two particles interchanged. **(6)** As the particles are identical, the state represented by $\psi'$ must be the same as that represented by $\psi$. **(7)** In view of (1) and (2) it must be: $$\psi' = e^{i a} \psi\quad \mbox{for some constant $a\in R$.}$$ **Here physics stops.** I will use only mathematics henceforth. **(8)** In view of (3) one can equivalently re-write the identity above as $$\psi(y,x) = e^{ia}\psi(x,y) \quad \mbox{almost everywhere for $(x,y)\in R^2\times R^2$}\quad [1]\:.$$ **(9)** Since $(x,y)$ in [1] is _every_ pair of points up to a zero-measure set, I am allowed to change their names obtaining $$\psi(x,y) = e^{ia}\psi(y,x) \quad \mbox{almost everywhere for $(x,y)\in R^2\times R^2$}\quad [2]$$   (Notice the zero measure set where the identity fails remains a zero measure set under the reflexion $(x,y) \mapsto (y,x)$, since it is an isometry of $R^4$ and Lebesgues' measure is invariant under isometries.) **(10)** Since, again, [2] holds almost everywhere for every pair $(x,y)$, I am allowed to use again [1] in the right-hand side of [2] obtaining: $$\psi(x,y) = e^{ia}e^{ia}\psi(x,y) \quad \mbox{almost everywhere for $(x,y)\in R^2\times R^2$}\:.$$   (This certainly holds true outside the union of the zero measure set $A$ where [1] fails and that obtained by reflexion $(x,y) \mapsto (y,x)$ of $A$ itself.) **(11)** Conclusion: $$[e^{2ia} -1] \psi(x,y)=0 \qquad\mbox{almost everywhere for $(x,y)\in R^2\times R^2$}\quad [3]$$ Since $||\psi|| \neq 0$, $\psi$ cannot vanish everywhere on $R^2\times R^2$. If $\psi(x_0,y_0) \neq 0$, $[e^{2ia} -1] \psi(x_0,y_0)=0$ implies $e^{2ia} =1 $ and so: $$e^{ia} = \pm 1\:.$$ And thus, apparently, anyons are not permitted. Where is the mistake? **ADDED REMARK.** (10) is a completely mathematical result. Here is another way to obtain it. (8) can be written down as $\psi(a,b) = e^{ic} \psi(b,a)$ for some **fixed** $c \in R$ and **all** $(a,b) \in R^2 \times R^2$ (I disregard the issue of negligible sets). Choosing first $(a,b)=(x,y)$ and then $(a,b)=(y,x)$ we obtain resp. $\psi(x,y) = e^{ic} \psi(y,x)$ and $\psi(y,x) = e^{ic} \psi(x,y)$. They immediately produce [3] $\psi(x,y) = e^{i2c} \psi(x,y)$. So the _physical_ argument (4)-(7) that we have permuted again the particles and thus a further new phase may appear does not apply here. **2nd ADDED REMARK.** It is clear that as soon as one is allowed to write $\psi(x,y) = \lambda \psi(y,x)$ for a **constant** $\lambda\in U(1)$ and **all** $(x,y) \in R^2\times R^2$ the game is over: **$\lambda$ turns out to be $\pm 1$ and anyons are forbidden.** This is just mathematics however. My guess for a way out is that the true configuration space is not $R^2\times R^2$ but some other space whose $R^2 \times R^2$ is the universal covering. An idea (quite rough) could be the following. One should assume that particles are indistinguishable from scratch already defining the configuration space, that is something like $Q := R^2\times R^2/\sim$ where $(x',y')\sim (x,y)$ iff $x'=y$ and $y'=x$. Or perhaps subtracting the set $\\{(z,z)\:|\: z \in R^2\\}$ to $R^2\times R^2$ before taking the quotient to say that particles cannot stay at the same place. Assume the former case for the sake of simplicity. There is a (double?) covering map $\pi : R^2 \times R^2 \to Q$. My guess is the following. If one defines wavefunctions $\Psi$ on $R^2 \times R^2$, he automatically defines many-valued wavefunctions on $Q$. I mean $\psi:= \Psi \circ \pi^{-1}$. The problem of many values physically does not matter if the difference of the two values (assuming the covering is a double one) is just a phase and this could be written, in view of the identification $\sim$ used to construct $Q$ out of $R^2 \times R^2$: $$\psi(x,y)= e^{ia}\psi(y,x)\:.$$ Notice that the identity cannot be interpreted literally because $(x,y)$ and $(y,x)$ are the same point in $Q$, so my trick for proving $e^{ia}=\pm 1$ cannot be implemented. The situation is similar to that of $QM$ on $S^1$ inducing many-valued wavefunctions form its universal covering $R$. In that case one writes $\psi(\theta)= e^{ia}\psi(\theta + 2\pi)$. **3rd ADDED REMARK** I think I solved the problem I posted focusing on the model of a couple of anyons discussed on p.225 of this paper matwbn.icm.edu.pl/ksiazki/bcp/bcp42/bcp42116.pdf suggested by Trimok. The model is simply this one: $$\psi(x,y):= e^{i\alpha \theta(x,y)} \varphi(x,y)$$   where $\alpha \in R$ is a constant, $\varphi(x,y)= \varphi(y,x)$, $(x,y) \in R^2 \times R^2$ and $\theta(x,y)$ is the angle with respect to some fixed axis of the segment $xy$. One can pass to coordinates $(X,r)$, where $X$ describes the center of mass and $r:= y-x$. Swapping the particles means $r\to -r$. Without paying attention to mathematical details, one sees that, in fact: $$\psi(X,-r)= e^{i \alpha \pi} \psi(X,r)\quad \mbox{i.e.,}\quad \psi(x,y)= e^{i \alpha \pi} \psi(y,x)\quad (A)$$ for an anti clock wise rotation. (For clock wise rotations a sign $-$ appears in the phase, describing the other element of the braid group $Z_2$. Also notice that, for $\alpha \pi \neq 0, 2\pi$ the function vanishes for $r=0$, namely $x=y$, and this corresponds to the fact that we removed the set $C$ of coincidence points $x=y$ from the space of configurations.) However a closer scrutiny shows that the situation is more complicated: The angle $\theta(r)$ is not well defined without fixing a reference axis where $\theta =0$. Afterwards one may assume, for instance, $\theta \in (0,2\pi)$, **otherwise $\psi$ must be considered multi-valued**. With the choice $\theta(r) \in (0,2\pi)$, (A) does not hold everywhere. Consider an anti clockwise rotation of $r$. If $\theta(r) \in (0,\pi)$ then (A) holds in the form $$\psi(X,-r)= e^{+ i \alpha \pi} \psi(X,r)\quad \mbox{i.e.,}\quad \psi(x,y)= e^{+ i \alpha \pi} \psi(y,x)\quad (A1)$$ but for $\theta(r) \in (\pi, 2\pi)$, and **always for a anti clockwise rotation** one finds $$\psi(X,-r)= e^{-i \alpha \pi} \psi(X,r)\quad \mbox{i.e.,}\quad \psi(x,y)= e^{- i \alpha \pi} \psi(y,x)\quad (A2)\:.$$ Different results arise with different conventions. In any cases it is evident that the phase due to the swap process is **a function of $(x,y)$** (even if locally constant) and not a constant. This invalidate my "no-go proof", but also proves that the notion of anyon statistics is deeply different from the standard one based on the groups of permutations, where the phases due to the swap of particles is **constant** in $(x,y)$. As a consequence the **swapped state is different from the initial one** , differently form what happens for bosons or fermions and **against the idea that anyons are indistinguishable particles.** [Notice also that, in the considered model, swapping the _initial pair_ of bosons means $\varphi(x,y) \to \varphi(y,x)= \varphi(x,y)$ that is $\psi(x,y)\to \psi(x,y)$. That is, **swapping anyons does not mean swapping the associated bosons** , and it is correct, as it is another physical operation on different physical subjects.] Alternatively one may think of the anyon wavefunction $\psi(x,y)$ as a **multi-valued** one, again differently from what I assumed in my "no-go proof" and differently from the standard assumptions in QM. This produces a truly constant phase in (A). However, it is not clear to me if, with this interpretation the swapped state of anyons is the same as the initial one, since I never seriously considered things like (if any) Hilbert spaces of multi-valued functions and I do not understand what happens to the ray- representation of states. **This picture is physically convenient, however, since it leads to a tenable interpretation of (A) and the action of the braid group turns out to be explicit and natural.** Actually a last possibility appears. One could deal with (standard complex valued) wavefunctions defined on $(R^2 \times R^2 - C)/\sim$ as we know (see above, $C$ is the set of pairs $(x,y)$ with $x=y$) and we define the swap operation in terms of phases only (so that my "no-go proof" cannot be applied and the transformations do not change the states): $$\psi([(x,y)]) \to e^{g i\alpha \pi}\psi([(x,y)])$$ where $g \in Z_2$. This can be extended to many particles passing to the braid group of many particles. Maybe it is convenient mathematically but is not very physically expressive. In the model discussed in the paper I mentioned, it is however evident that, up to an unitary transformation, the Hilbert space of the theory is nothing but a standard bosonic Hilbert space, since the considered wavefunctions are obtained from those of that space by means of a unitary map associated with a singular gauge transformation, and just that singularity gives rise to all the interesting structure! However, in the initial bosonic system the singularity was pre-existent: the magnetic field was a sum of Dirac's delta. I do not know if it makes sense to think of anyons independently from their dynamics. And I do not know if this result is general. I guess that moving the singularity form the statistics to the interaction and _vice versa_ is just what happens in path integral formulation when moving the external phase to the internal action, see Tengen's answer.


title: How does Newton's first law asserts the existence of inertial frames?

text: Recently I've seem here one answer telling that Newton's first law really assures the existence of inertial reference frames. But how is that? I really can't see it. As I know, Newton's first law says: > _Every body continues in its state of rest, or of uniform motion in a right > line, unless it is compelled to change that state by forces impressed upon > it._ Now, what I've seem done is to use this to define that whenever this law holds in some reference frame we call it inertial. But how this law states that such a frame does exist?


title: Is Newton's first law something real or a mathematical formalism?

text: Why do objects always 'tend' to move in straight lines? How come, everytime I see a curved path that an object takes, I can always say that the object tends to move in a straight line over 'small' distances, but as you take into account the curvature of the path, a force acting on the particle appears. I mean, I can always take a small enough portion of the curve, zoom in enough, and conclude that the object is moving in a straight line, but then as I zoom out I find out that a force is acting on the particle. The force of gravity is everywhere and, no matter how weak it is, it will make the particle take a path which is different from a straight line. This is my question: since particles are, in reality, never moving in straight lines, is Newton's first law a mathematical formalism or some true property of material objects?


title: Why does only one side of a neon lamp glow?

text: When applying DC to a neon lamp, only one electrode glows: ![http://commons.wikimedia.org/wiki/File:Neonlamp3.JPG](http://i.stack.imgur.com/3poGs.jpg) The voltages across the lamps are left: DC (left lead positive), middle: DC (right lead positive), and right: AC. But... why? The electrodes are the same shape, so the electric field around them should be same shape, and the gas should break down at the same electric field strength. It seems like the fields would be symmetrical. Is there a difference in threshold between positive and negative coronas? If so, do both sides light up at high enough voltage? Or maybe only one type of corona is possible in neon since it's a noble gas? If it contained air would it glow at both electrodes?


title: Can gravity be shielded, like electromagnetism?

text: If I remember well, they said that it can't, but I do not know why. Yes, I meant if gravity can be shielded using something like a Faraday cage (or something else?). Thank you.


title: Can a solar thermal collector designed for use in homes ever power a stirling engine designed for use in homes?

text: Can a solar thermal collector common on private roofs today designed for collecting heat for the household ever collect enough heat to power a stirling engine of say this model: http://www.whispergen.com/main/achomesspecs_info/ Normally a stirling engine of this model is powered by combustion of natural gas.


title: Temperature change effected by electric heater

text: A 40-gallon electric water heater has a 10kW heating element. What will the water temperature be after 15 min of heating if the start temp is 50F degrees. There must be an equation. I can't find it in notes or text.


title: Can one submit/publish a paper between postdocs?

text: I've almost finished one postdoc, and haven't found another yet. If I don't find one in time, can I keep publishing papers? I have 2 scenarios:   1. If I did most of the research during the postdoc, but submit it afterwards.   2. If I start the research while I'm unemployed. If the answer is no to either/both of these, is there some way around this? (Such as becoming affiliated somehow to a nearby group, without actually being employed by them).


title: Fairly Broad Spectrum Light Source Options

text: I tried this sight before to find filtering options and got fantastic results, so let's try again! I am setting up an experiment that requires light of two different frequencies (445nm and 350nm). The light ultimately needs to be focused on a small area. I can think of two good ways to make this happen:   1. Get a broad spectrum light source that emits over this range - maybe a halogen lamp? - and buy two filters: A high pass that cuts off at ~420nm and a low pass that cuts off at ~380nm. For the low pass, I may also include a high pass in order to make a band pass (or just flat out buy a band pass filter). I will then simply wrap the filters in a black paper cone to "focus" the light (for this experiment, light spread after hitting my target doesn't seem to important). We would use a power meter to tell us what flux is actually hitting our experiment. My questions for this approach: What kind of broad spectrum source would you recommend? Can you recommend a retail outlet for this source? Our grant ain't so grand, so afford ability is important. Can you think of a better way to focus the light after it passes through the filters? Again, any retail sources would be appreciated.   1. Get two more highly focused sources. Given our small area focusing requirements, we are looking at lasers (Again, 340nm and 445nm). I have found a couple of retail sources that have such lasers, but they are fairly cost prohibitive. Do you know of any sources that sell such lasers - power sources and all - for <500? Thank you in advance for any help and recommendations. If a paper is produced from this - which I very much am counting on - I will thank the guy who provides the best answer in it (if that is an incentive).


title: How do we know the size of the universe?

text: Ok, from astronomical observations we can tell that the observable matter is separating - so rewind the clock about 13.7 billion years and it was all at a single point. However, how do we distinguish between the following two options:   1. Universe is expanding    2. Matter distribution is increasing into infinite void **Clarification 1:**   (My notion of) The traditional notion is that all time/space/matter was created at the instant of the big bang. I.e. BB was inital conditions of: $t=0$, $V=0$, $E=very big$ People say "the universe is expanding", rather than "observable matter is separating". Why is this? How do we know that the big bang event wasn't started by all matter condensed at a single point within a larger (otherwise empty) universe? How do we know that BB wasn't: $t=0$, $V(universe)>0$ but $V(matter)=0$, $E=very big$?


Why is a wine glass shaped the way it is? And why are there different shapes for different wines? Is this a tradition, or is there any scientific reason behind it?


title: Plasma Treatment of LaAlO3: Surface Roughening

text: I'm trying to understand something I've observed in exposing LaAlO3 substrates to an oxygen plasma (yielding atomic oxygen). In literature, these substrates are frequently "cleaned" in a oxygen rich environment by exposing the surface to atomic oxygen (via plasma source) at substrate temperatures greater than 700 C. This typically results in a clear diffraction pattern indicative of a flat surface. In my experiments, I'm noticing that initially, the surface improves upon plasma exposure (enhanced diffraction intensity), but after a certain point, the intensity of the diffraction pattern begins to fade, and (presumably) the surface becomes disordered. I'm looking for insight into why this might be happening, because typically the surface will be exposed to the atomic oxygen plasma during growth, but this seems to be damaging my film. My first guess would be that initially, atomic oxygen reacts with surface species to produce a clean surface, and, after a point, begins to somehow erode the surface, reducing crystallinity.


title: Graviton Emission from D-Branes

text: I'm working through Polchinski's book on string theory, and I ran into something that I don't think I understand. I'm hoping that someone who knows this stuff can help me out. Before calculating the Dp-brane tension in Chapter 8, Polchinski says that we could have obtained the same result by calculating the amplitude for graviton emission from the D-brane (instead of calculating closed string exchange between two D-branes). It seems like we would do this by placing a graviton vertex operator on the disk with Dirichlet boundary conditions on (25-p) coordinates and Neumann on the rest. But doesn't the amplitude with only one vertex operator vanish, since there aren't enough ghost insertions to get a nonzero result? I must be misunderstanding something, because it seems like if we only fix the real and imaginary parts of the position of the vertex operator, then we have to divide by the volume of the rest of the CKG of the disk, which is infinity. Any ideas or hints? Thanks.


title: Einstein Relative Motion and Time Order of Events

text: According to Einstein, do observers in relative motion agree on the time order of all events? I don't think they would agree on the timing of events, but I am having trouble figuring out why they wouldn't agree. Any thoughts?


title: De Broglie wavelength, frequency and velocity - interpretation

text: Two fundamental equations regarding wave-particle duality are: $$ \lambda = \frac{h}{p}, \\\ \nu = E/h .$$ We talk about de Broglie wavelength, is it meaningful to talk about de Broglie frequency ($\nu$ above) and de Broglie velocity ($\nu \lambda$)? Are these two equations independent or can one derive one from the other? Or mid-way, does one impose constraint on other? In case of light or photons we can relate frequency and wave, is there similar interpretation in case frequency and wavelengths in above equations? Comparing with that of light, if we multiple $\lambda$ and $\nu$ we get velocity, what does this velocity mean here? If we do the above calculation for an average human, what would be the meaning of $\nu$ and $\nu \lambda$? Are we jiggling with that $\nu$?


title: Why does the metastable form of Technetium-95 have a longer half-life than its most stable state?

text: I was looking at the isotopes of technetium page on Wikipedia recently, and it seems that the metastable ${^{95m}Tc}$ has a substantially longer half-life (61d) than its most stable state of ${^{95}Tc}$ (20h). Several alternative sources gave the same numbers. That seems quite peculiar to me. Is it just a general thing that I shouldn't count on the most stable state of an isotope having the longest half-life, or is there some quirkiness there (alternatively, an inaccuracy that got replicated in a bunch of places)?


title: If you could look at the LHC beam, what would you see?

text: Kind of a weird question compared to what I normally ask on here, but I've been wondering about this for a while. Does the LHC beam generate any photons within the visible light spectrum? Assuming the beam was traveling through air instead of a vacuum, would it interact with the nitrogen or oxygen to generate visible light?


title: How does the energy in a standing wave travel beyond a node?

text: In a standing wave, how does energy travel past a node? It should just get reflected. Assume the case of first overtone and you strike the string at a place. How will energy distribute itself? If it distributes over the whole string, then how is it the wave traveled past a node? Basically the problem is a standing wave has been set up in the string . Now you strike the string at a place, and this creates a new pulse. Now this pulse will travel past the next node in the string (not the boundary) or will it reflect off even that node and remain confined to a particular region of the string?


title: When will the Moon reach escape velocity?

text: From what I know, the Moon is accelerating away from the Earth. Do we know when it will reach escape velocity? How do we calculate this?


title: How can I take the Wigner transform of an operator with an absolute value?

text: I want to be able to find the Wigner transforms of operators of the form $\Theta(\hat{O})$, where $\Theta$ is the Heaviside function and $\hat{O}$ in general depends on both $x$ and $p$. For the operators of interest, $\tilde{O}$ is known. Since the Wigner transform is linear, I can use the expression $$ \Theta(x)=\frac{1}{2} \frac{\left( |x| - x \right)}{x}$$ Unfortunately, I am no better off than I was when I started, as I can't figure out how get the Wigner transform of the absolute value of an arbitrary operator in terms of the Wigner transform of that operator. **Is there an expression for the Wigner transform of the absolute value of an operator in terms of the Wigner transform of that same operator?** **If so, what is it?**


title: Magnetic force and terminal velocity of falling loop and falling disc?

text: According to my textbook, if a square loop with mass $M$ is allowed to free fall (with a magnetic field at the bottom side and as soon as the square loop is dropped then the area inside the loop begins to be filled with more and more magnetic field) through a magnetic field, it will reach a terminal velocity. This is due to the increase in magnetic flux through the loop which creates a current in the loop opposing the change in magnetic flux and then a force is then exerted on the bottom side which balances out the gravitational force. The force on the loop sides balance each other out and by the time the top side enters the magnetic field then the flux is not changing. The terminal velocity is given as: $$V=\frac{MgR}{B^2w^2}$$ (first picture below) where $M$ is loop mass, $g$ is gravity, $R$ is resistance of loop, $B$ is magnetic field and $w$ is width of the loop However, when a metal disc is dropped from just below an infinite wire carrying a current I, there is no terminal velocity even though the magnetic flux through the disc (falls so that magnetic field is perpendicular to the circular face where $A=\pi r^2$ is changing due to the change in magnetic field strength as the disc falls further from the wire. (Second picture below) Why is this? ![Square loop](http://i.stack.imgur.com/xr9G3.png) ![Washer](http://i.stack.imgur.com/yQUTO.png)


title: Rigid body problem

text: I have some doubts about the next excercise: > _A bar of length $2a$ and mass $m$ moves freely with both of its extremes on > a ring of radius $\sqrt2a$. The ring can rotate freely in a certain > diameter, remaining the center fix. Find the equations of motion._ I did it using the balance of angular momentum, but I want to try to do it with the Lagrangian. But I have a problem. The energy of the ring is easy to calculate (just a rotation), but the energy of the bar I am not sure how to do it. ¿Is a translation of the center of mass plus a rotation? ¿Or just translation? ¿Or just a complicated rotation? I did this picture to illustrate: ![enter image description here](http://i.stack.imgur.com/DhYke.png) Thanks!


title: Intuitive explaination for why higher engine compression ratio is more efficient?

text: Intuitively to me it seems likes increasing compression ratio would require more work to compress the gasses before ignition, so you'd just end up getting back what you put in - like a spring. What am I missing?


title: half Skyrmion vs Meron

text: Is there a difference between a half skyrmion and a meron? I'm asking this in regard to half skyrmion theories of High Tc Superconductors. It would be interresting to know if the proposed half skyrmions which emerge in the disturbed antiferromagnetic spin structure are actually merons.


title: Compatible Observables

text: My QM book says that when two observables are compatible, then the order in which we carry out measurements is irrelevant. When you carry out a measurement corresponding to an operator $A$, the probability that the system ends up in the eigenvector $\psi_n$ is $$P_n ~=~ \frac{|<\psi|\psi_n>|^2}{<\psi|\psi>} ~=~ \frac{|a_n|^2}{<\psi|\psi>},$$ where $a_n$ is the eigenvalue corresponding to $\psi_n$. (Assume degeneracy) But compatible operators are guaranteed only to have the same eigenvectors, not the same eigenvalues. So if I have observables with operators $A$ and $B$, then after the first measurement of $A$ or $B$, subsequent measurements of $A$ or $B$ will not change the state of the system. But whether the first measurement is $A$ or $B$ will definitely effect things. Is this correct or am I misunderstanding my textbook?


title: Sun reaching zenith at a particular latitude

text: I need to find when the sun reaches the Zenith at a given latitude. What I've done so far: $L=23.5 \cdot \sin(\frac{2\pi}{365.25}\cdot D) $ Here L is the latitude (<23.5) and D is number of days from March 21. This is based on my understanding that the sun executes SHM around the celestial equator. I then solve this equation and get two roots, which are the required days. My questions are:   1. Since the orbit of the earth is not exactly spherical, how much deviation actually occurs?   2. Am I right in understanding that since the declination is continuously shifting, the sun may actually never reach the Zenith? In such a scenario, I am only calculating the day when the little circle that the sun is instantaneously travelling on, meets the Zenith.


title: Results for the path integral formalism for a system with known start and end configuration?

text: The path integral provides a method for computing a time evolution by a weighted summing up all possible deviations. Is there such a method for a system, where one not only knows the initial condition, but also how the system end up? I.e. given is the beginning and the end configuration and what one is interested is the average field configuration in between this interval. Of course the path integral $$\langle \psi_{t_2},\psi_{t_1} \rangle=\int D\psi... $$ gives a value for both states at $\psi_{t_1}$ (start) and $\psi_{t_2}$ (end) fixed, but that's just interpreted as "what is the probability for this state later, if I start out like that". I want to consider a situation, where that is known to be equal to 1 and I'm interested in the evolution in between. One might think of a transition from one field equilibrium to another here. The point being that in such a situation, it is a priori clear that the summing will 100% end up in one specific field situation. I think knowing the end configuration must sure give new information/restriction to get more out of it. For example, a naive idea I come up with right now are the possibility that one can consider the time evolution from beginning _and_ end, i.e. one would approach a field configuration in between from both sites, and they would have to coincide.


title: Why doesn't the dark matter halo co-rotate with the luminous disk?

text: What keeps it from falling into the center if not angular momentum?


title: Does light have timbre?

text: Timbre is a property associated with the shape of a sound wave, that is, the coefficients of the discrete Fourier transform of the corresponding signal. This is why a violin and a piano can each play the same note at the same frequency yet sound completely different. Now for light: if you mix red and blue light, the resulting wave has on average the same frequency as green, but a different shape. Is this why we observe it as magenta instead of green? So can we say light has timbre? But other colours mixtures don't work! Red+green is yellow, but yellow is a pure colour, so apparently red+green has no unique timbre in this case; the shape of the red+green wave is identical to a pure yellow wave.


title: Tensor Introduction

text: I have recently started learning about tensors during my course on Special Relativity. I am struggling to gain an intuitive idea for invariant, contravariant and covariant quantities. In my book, invariant quantities are described as being those physical quantities associated with a point in space e.g. temperature at point P. Thus, when changing coordinate system, the value associated with the point is unchanged: you may represent the point by a different set of coordinates, but on locating the point in the new coordinates, the temperature value will still be the same. Here comes my struggle: the archetypical example of a covariant quantity is the gradient vector. Is it not the case that we require it to transform between coordinate systems in a way that preserves the vector associated with each point? Thus, in a way, the gradient vector is an invariant quantity? Why then are many derivations in my course driven by the fact that physical quantities like Action, Lagrangians etc should be "Lorentz invariant"?


title: Free charge movement in an electric field - including bremsstrahlung

text: Let us imagine a free, negatively charged object that is in rest and placed in an elecric field of a point positive charge. The positive charge has a huge mass and cannot move, so we consider only the movement of the negative charge. The negative charge starts falling the center at time t0. It is straight forward to find a velocity at given time when the charge simply falls toward the center if we consider only the Coulomb force. But I would like to find the velocity at a given time, when not only the Coulomb force applies, but we consider bremsstrahlung too. Namely - some of the energy of the accelerating charge is lost due to emision of the electromagnetic radiation. I look for classical electromagnetic solutions, not quantum mechanical. How could I do that? In case of the acceleration parallel to the velocity the formula for the bremsstrahlung is rather simple, but I still don't know how to start the calculation. Any hints?


title: Spin of a decay product

text: > _A particle A decays into particles B, C and D. The spin of A, B and C > particles is 1/2 each. What are the possible spins of particle D?_ My attempt is the following: Since B and C have spin 1/2 each, these can form singlet or a triplet. So the spin of D can only be 1/2 to get a spin of 1/2 for the decayed particle A. But the correct answer which is given is that D can have values like 1/2, 3/2, 5/2, 7/2.... Can any enlightened expert in Physics explain this please. his reaction looks like the decay of a neutron(A) into proton(B), electron(C) and anti-neutrino(D). So I believe my answer to be correct because neutrino has spin=1/2. I would still like confirmation of my reasoning from others.


title: Proof that the One-Dimensional Simple Harmonic Oscillator is Non-Degenerate?

text: The standard treatment of the one-dimensional quantum simple harmonic oscillator (SHO) using the raising and lowering operators arrives at the countable basis of eigenstates $\\{\vert n \rangle\\}_{n = 0}^{\infty}$ each with corresponding eigenvalue $E_n = \omega \left(n + \frac{1}{2}\right)$. Refer to this construction as the **abstract solution**. How does the abstract solution also prove uniqueness? Why is there only one unique sequence of countable eigenstates? In particular, **can one prove the state $\vert 0\rangle$ is the unique ground state without resorting to coordinate representation?** (It would then follow that the set $\\{\vert n \rangle\\}_{n = 0}^{\infty}$ is also unique.) The uniqueness condition is obvious if one solves the problem in coordinate representation since then one works in the realm of differential equations where uniqueness theorems abound. Most textbooks ignore this detail (especially since they often solve the problem both in coordinate representation and abstractly), however I have found two exceptions:   * Shankar appeals to a theorem which proves one-dimensional systems are non-degenerate, however this is unsatisfactory for two reasons:     1. Not every one-dimensional system is non-degenerate, however a general result can be proven for a large class of potentials (the SHO potential is in such a class).     2. The proof requires a departure from the abstract solution since it classifies the potentials according to their functional properties.   * Griffiths addresses this concern in a footnote stating that the equation $a \vert 0\rangle = 0$ uniquely determines the state $\vert 0\rangle$. Perhaps this follows from the abstract solution, however I do not see how.


title: Fringe Pattern Brightness for Young's Double-slit experiment

text: Perhaps I have missed something in my notes, but I have noticed when looking at different sources that some textbooks/sites state that the fringe brightness for the young's experiment is the same for all the bright fringes. Others, say that the brightness "falls off" with angle theta. Well, which is it? And how do you calculate it, not using derivatives if possible... I'm preparing for the MCAT exam and calculus is not tested. Thanks! Here are some pictures that I've come across either in different textbooks or websites: ![Knight college physics textbook](http://i.stack.imgur.com/peng0.png) ![http://cnx.org/content/m42508/latest/?collection=col11406/latest](http://i.stack.imgur.com/f3QES.png) ![http://www.physicsclassroom.com/class/light/u12l3d.cfm](http://i.stack.imgur.com/JwvGF.gif) ![Serway Physics textbook](http://i.stack.imgur.com/7IJPr.jpg) ![http://en.wikipedia.org/wiki/Double_slit](http://i.stack.imgur.com/lZpSO.jpg)


title: What happens when you apply the path integral to the Einstein-Hilbert action?

text: The Einstein Field Equations emerge when applying the principle of least action to the Einstein-Hilbert action, and from what I understand the path integral formulation generalizes the principle of least action. What happens when you apply the path integral instead of the action principle to the Einstein-Hilbert action?


title: In the αβγ cosmology paper, how do the authors assume the integral of density over time in the early universe?

text: In the famous Alpher-Bethe-Gamow paper, the authors say: "it is necessary to assume the integral of $\sigma_n dt$ during the building-up period is equal to $5 \times 10^4 \frac{\text{g sec}}{\text{cm}^3}$" How do they determine this integral of density over time of the early universe?


title: Does the amount of gravitational potential energy in the universe increase as it expands?

text: It seems to me that extra gravitational potential energy is created as the universe expands and the distance between massive objects such as galaxy clusters increases; this implies that energy is not conserved in the universe. Is that right?


title: Does there exist a mathematically rigorous formulation of curved Hilbert Space?

text: The Hamiltonian is traditionally defined as \begin{align} H_{\text{flat}} = U^{\dagger}DU \end{align} where $D$ is a diagonal matrix with real eigenvalues and $U^{\dagger}U=I$ are the unitary transformations to generalize $H_{\text{flat}}$ in any traditional base. Thus the hermitian relation $(H_{\text{flat}})^{\dagger}=H_{\text{flat}}$ holds. However I discovered that some operators can have real eigenvalues, but are not necessarily hermitian. For example the following matrix \begin{align} H_{\text{curved}} = \begin{pmatrix} 3 && 4 && 1 && 2 \\\ 2 && 4 && 2 && 4 \\\ 1 && 3 && 2 && 3 \\\ 2 && 4 && 4 && 1 \end{pmatrix} \end{align} has the real eigenvalues $(10.73814 , -2.34185 , 1.78222 , -0.17850)$, but is not hermitian. I further explored this generalization and discovered a way to understand how $H_{\text{curved}}$ differs from $H_{\text{flat}}$ by showing how the hilbert space is different between them. The diagonalization of $H_{\text{curved}}$ requires a left and right eigenvector matrix such that \begin{align} H_{\text{curved}} = RDR^{-1} \end{align} where $L = R^{-1}$ and $R^{\dagger}\neq R^{-1}$. The orthogonality of the eigenvectors is achieved from the following relation $LR=R^{-1}R=I$, however $R^{\dagger}R\neq I$ illustrates how the orthogonal hilbert space no longer makes sense in this non-traditional context. To make sense of this is to form the inner product of the eigenvectors as if it were in a curved base. To do this we define a metric of the form $G^{-1} = R^{-\dagger} R^{-1} = (R R^{\dagger})^{-1}$, such that the orthogonality relation $R^{\dagger}G^{-1}R=I$ holds. I dubbed $H_{\text{curved}}$ as the curved hamilotian, because its hilbert metric $G^{-1}$ mimics similar properties of the metric of curved space time $g^{ab}$ in general relativity. I present the following question using my own vocabulary: does there exist a mathematically rigorous formulation of curved Hilbert Space similar to how general relativity is treated?


title: How can a black hole produce sound?

text: I was reading this article from NASA -- it's _NASA_ \-- and literally found myself perplexed. The article describes the discovery that black holes emit a "note" that has physical ramifications on the detritus around it. > **Sept. 9, 2003:** Astronomers using NASA’s Chandra X-ray Observatory have > found, for the first time, sound waves from a supermassive black hole. The > “note” is the deepest ever detected from any object in our Universe. The > tremendous amounts of energy carried by these sound waves may solve a > longstanding problem in astrophysics. > > The black hole resides in the Perseus cluster of galaxies located 250 > million light years from Earth. In 2002, astronomers obtained a deep Chandra > observation that shows ripples in the gas filling the cluster. These ripples > are evidence for sound waves that have traveled hundreds of thousands of > light years away from the cluster’s central black hole. > > “The Perseus sound waves are much more than just an interesting form of > black hole acoustics,” says Steve Allen, of the Institute of Astronomy and a > co-investigator in the research. “These sound waves may be the key in > figuring out how galaxy clusters, the largest structures in the Universe, > grow.” Except:   * Black holes are so massive that light, which is faster than sound, can't escape.   * Sound can't travel in space (space has too much, well, space)   * It's a b-flat? So: **How can a black hole produce sound if _light_ can't escape it?**


title: Construction of Bound states in Lattice QCD

text: How the proton state is found and renormalized in lattice QCD? Is there any literature shows the method step by step? What about other bound states in QCD if we somehow know their quantum numbers?


title: Finding equation of state from thermal expansion coefficient and isothermal compressibility

text: I'm stuck on a problem that I found in a book (Modern Thermodynamic with Statistical Mechanics, Helrich S., problem 5.2). The text of the problem is that: > Consider a solid material for which: > > $$ \frac{1}{\kappa_T} = \frac{\varepsilon}{2V_0}\left[\frac{2\Gamma c_v > T}{\varepsilon}\,\frac{V_0}{V} - 3\left(\frac{V_0}{V}\right)^3\right] $$ $$ > \beta = \frac{1}{T}\left[1 + 3\,\frac{\varepsilon}{2\Gamma c_v > T}\left(\frac{V_0}{V}\right)^2\right] $$ > > Where $\varepsilon$ is a constant with the units of energy, $\Gamma$ is a > dimensionless constant and $V_0$ is a reference volume less than $V$. The > temperature range is such that we may assume that the specific heat at > constant volume $c_v$ is independent of temperature. Find the thermal > equation of state. In this book the convention defines $\kappa_T$ as the isothermal compressibility and $\beta$ as the thermal expansion coefficient. $$ \beta = \frac 1V \left(\frac{\partial V}{\partial T}\right)_P $$ $$ \kappa = - \frac 1V \left(\frac{\partial V}{\partial P}\right)_T $$ The answer key for this problem says: $$ P(V,T) = \frac{\Gamma c_vT}{V} + \frac{\varepsilon}{2V_0}\left[\left(\frac{V_0}{V}\right)^5 - \left(\frac{V_0}{V}\right)^3\right] $$ Here's the procedure I tried to apply. I wanted to do a contour integration of this equation because I can express the partial derivatives as a function of $\kappa _T$ and $\beta$: $$ dP(T,V) = \left(\frac{\partial P}{\partial V}\right)_TdV + \left(\frac{\partial P}{\partial T}\right)_VdT $$ given that: $$ \left(\frac{\partial P}{\partial T}\right)_V = \frac{\beta}{\kappa_T} $$ $$ \left(\frac{\partial P}{\partial V}\right)_T = - \frac{1}{V \kappa_T} $$ But by using this procedure I have a problem with the integration of $c_v$ with respect to the volume, and even if I assume this constant I still don't get the result. Which approach would you suggest?


title: Sign issue on electrostatic potential energy

text: I have been working on a problem about finding the electrostatic potential energy stored on a capacitor of concentric spheres with inner radius $a$ and outer radius $b$ and with charge $Q$. I've got to first calculate the energy using the capacitance and then integrating the energy density. I did the following: first of all I've used Gauss' Law to find the electric field a distance $r$ from the center of both spheres with $a < r < b$. As expected I've got the field of a point charge $Q$ at the center. Then I've integrated the field along a segment joining the two spheres and I've got the following difference of potential $$V=\frac{1}{4\pi\epsilon_0}\frac{Q(a-b)}{ab}$$ Then I've found the capacitcante using $Q=CV$ and finally I've found the energy using $U=Q^2/2C$. That's fine, I've got the value: $$U_1=\frac{1}{8\pi\epsilon_0}\frac{Q^2(a-b)}{ab}$$ Now the second part, I should find the same value integrating the energy density. So, the energy density is $\mathcal{u}=\epsilon_0E^2/2$ and hence it is: $$\mathcal{u}=\frac{1}{2}\epsilon_0 \frac{1}{16\pi^2\epsilon_0^2}\frac{Q^2}{r^4}$$ Since I must integrate on the region between the two spheres I've used spherical coordinates and computed the following integral getting the energy: $$U_2=\int_0^\pi\int_0^{2\pi}\int_a^b \frac{1}{2}\epsilon_0 \frac{1}{16\pi^2\epsilon_0^2}\frac{Q^2}{r^4}r^2\sin\phi dr d\theta d\phi$$ And this integral gave simply: $$U_2 =\frac{1}{8\pi\epsilon_0}\frac{Q^2(b-a)}{ab}$$ But wait a moment, I've got $U_2 = -U_1$ instead of $U_2 = U_1$ as expected. I've calculated it once again and once again and I've got the same problem. Can someone point out what's happening? Where's my mistake? Thanks in advance for your help!


title: Given a potential energy function, find expression of the force of a particle?

text: This comes from an AP review packet. I'm given a potential energy functon, $$U(r)=br^{-3/2} + c,$$ where $b$ and $c$ are constants, and need to find the expression for the force on the particle. There's a graph of $U(r)$ given with the problem, but I'm not sure if it's needed or not. I'm just looking for how to go about solving this problem.


title: Intro to Solid State Physics

text: I didn't see this listed on the books page so here it is. I'm currently in an introductory Solid State course, and we are using Kittel's book. I have been having a rough time with this book although I am starting to get used to it as we get farther in. What are good introductory solid state books?


title: Relativistic center of mass

text: Recently I realized the concept of center of mass makes sense in special relativity. Maybe it's explained in the textbooks, but I missed it. However, there's a puzzle regarding the zero mass case Consider any (classical) relativistic system e.g. a relativistic field theory. Its state can be characterized by the conserved charges associated with Poincare symmetry. Namely, we have a covector $P$ associated with spacetime translation symmetry (the 4-momentum) and a 2-form $M$ associated with Lorentz symmetry. To define the center of mass of this state we seek a state of the free spinning relativistic point particle with the same values of conserved charges. This translates into the equations $$x \wedge P + s = M$$ $${i_P}s = 0$$ Here $x$ is the spacetime coordinate of the particle and $s$ is a 2-form representing its spin (intrinsic angular momentum). I'm using the spacetime metric $\eta$ implicitely by identifying vectors and covectors The system is invariant under the transformation $$x'=x+{\tau}P$$ where $\tau$ is a real parameter For $P^2 > 0$ and any $M$ these equations yields a unique timelike line in $x$-space, which can be identified with the worldline of the center of mass of the system. However, for $P^2=0$ the rank of the system is lower since it is invariant under the more general transformation $$x'=x+y$$ $$s'=s-y \wedge P$$ where $y$ satisfies $y \cdot P = 0$ This has two consequences. First, if a solution exists it yields a null hyperplane rather than a line*. Second, a solution only exists if the following constaint holds: $$i_P (M \wedge P) = 0$$ > Are there natural situations in which this constaint is guaranteed to hold? > In particular, does it hold for zero mass solutions of common relativistic > field theories, for example Yang-Mills theory? I'm considering solutions > with finite $P$ and $M$, of course *For spacetime dimension $D = 3$ a canonical line can be chosen out of this hyperplane by imposing $s = 0$. For $D = 4$ this is in general impossible


title: Index of Refraction

text: The scenario: A ray of light strikes the center of the (a) flat surface and (b) curved surface of a semicircular glass medium with the angle of incidences in degrees of 10, 20, 30, 40 and 50. The angles of refraction for each angle of incidence and for each surface were: (a) flat surface               Angle of incidence (deg) | Angle of Refraction (deg) | Index of refraction     10                       | 6                         | 1.66     20                       | 12                        | 1.65     30                       | 19                        | 1.54     40                       | 25                        | 1.52     50                       | 30                        | 1.53      (b) curved surface               Angle of incidence (deg) | Angle of Refraction (deg)      10                       | 16       20                       | 31      30                       | 50      40                       | 73       50                       | N/A       I used the formula: `index of refraction` $=\sin(\theta_{incident}) / \sin(\theta_{refracted})$ Questions: a). Do I still use the same formula to solve for the index of refraction for the curved surface? Because if I use it, I'll get a different index of refraction. b). If ever they are really different, why are the calculated indexes of refraction for both media are different for each angle of incidence? Is the difference expected? Thank you!


title: What does Bell's Theorem really violate?

text: Well this is a fairly straightforward question. I know it states that either hidden variables is wrong or Quantum Mechanics. But indirectly hidden variables is a part of QM due to uncertainty. So what exactly does it violate. I am just a bit confused.


title: Is a preferred reference frame of the universe the old aether?

text: About two years ago I posted a question about a symmetrical twin paradox: Here. Recently a new answer was posted and an intense discussion ensued: Here. One of the points discussed concerns a preferred reference frame in this universe: > _The asymmetry comes from the fact that the universe itself has a reference > frame, and its size will lorentz contract. This is measurable by the people > themselves--all that needs to happen is to send out a light ray and wait for > the light ray to go around the world. The 'diameter of the universe' will be > (light orbit time)/c. This time will be observed to be smaller the faster > the observer is travelling. So all observers will agree that there is a > global, absolute notion of motion, and this will pick out who ages when._ **My questions**     * Which (mathematical) characteristics determine whether there is a preferred reference frame in a universe?   * Does our universe have a preferred reference frame?   * If a universe has a preferred reference frame is this comparable with the old aether?   * If a universe has a preferred reference frame don't we get all the problems back that seemed to be solved by RT (e.g. the 'speed limit' for light because if there were a preferred frame you should be allowed to classically add velocities and therefore also get speeds bigger than c?


title: Is GR vacuum equation unique?

text: The title question would be too long if I tried to specify it clearly. So let me be more clear. Consider the class of theories having the following properties:   1. The langrangian density is only dependent of scalars created from the curvature tensors (for example $R, R^{ab}R_{ab}, R^{abcd}R_{abcd}, C^{abcd}C_{abcd}, (R)^2, R^{ab}R_{bc}R^{cd}R_{da}$, etc.) + generic matter using the usual minimal coupling.   2. The field theory is generated from the lagrangian by considering variations in the metric (ie. we will not consider Palatini or torsion theories)   3. The field equation automatically provides a covariantly conserved stress energy tensor (this means like in GR, the covariant derivative of the Ricci terms are collectively zero just from geometric requirements and not an extra postulate. I've seen papers claim this follows automatically from 1, but I have not seen proof, so I'm specifying it just in case.)   4. Let's stay with 3+1 dimensions. My question is then: GR is the theory starting from using $R$ as the lagrangian density, and the vacuum equation in GR reduces to $R^{ab}=0$. Is there another function of curvature scalars in this class which would yield the same vacuum equations? (For clarity: Yes, for this question I don't care if the field equations differ in non-vacuum. I'm just curious if they can possibly match in vacuum.) One may be tempted to immediately say it is unique, but consider the theories from $(R)^2$, $R^{ab}R_{ab}$, and $R^{abcd}R_{abcd}$.(1) One might intuitively expect that $C^{abcd}C_{abcd}$ would then add yet another theory, but it turns out this theory can be written as a combination of the previous three. (2)(3) And it turns out that in 4-dimensions, there are actually only two linearly independent combinations of these four theories. (4) Playing with it to try to find if there is a theory equivalent to GR in vacuum, I was looking for a theory which in vacuum after I contract the indices, I can show the only possible result is R=0 (For an example if the contraction gave a polynomial like $R + R^3 + R^{abcd}R_{abcd}R = 0$, then the only solution with real curvature is $R=0$). And then hoping that applying this to the field equations reduces everything to $R^{ab}=0$. This hunt and peck method is failing me, as I haven't found anything yet and clearly won't let me prove GR is unique if that is the answer. Can someone give an example which yields the GR vacuum equation or prove mathematically no such theory exists and thus the GR vacuum equation is unique in this class?


title: Have we ever seen an atom? If not, how do we know they exists?

text: Question is what exactly title says that have we ever seen an atom through any means (microscopes/equipments.etc)? If not, how do we know they exists?


title: Introduction to string theory

text: I am in the last year of MSc. and would like to read string theory. I have the Zwiebach Book, but along with it what other advanced book can be followed, which can be a complimentary to Zwiebach. I would like a more mathematically rigorous book or lecture notes along with Zwiebach. Specifically, mention whether the book discusses string theory   * Rigorously?   * Intuitively? What's the scope of the book? Does it cover the advanced materials, e.g. Matrix string theory, F-theory, string field theory, etc. Maybe even String Phenomenology?


title: Prerequisites and introduction to string theory

text: Can someone please give me the prerequisites and mathematics required for string theory? Are there some good references to study it, both online and in a book? Please consider I am a newbie in string theory, and only about up to the level explained in Hawking's Brief History of Time and Penrose's Road to Reality. I would prefer something which does not omit rigor, but builds it up.Also, are there good online lectures on this? (I have covered most of GR, but I am not really comfortable with advanced Quantum mechanics. I have studied QM only from Griffith's book, and a few McGraw hill lecture notes.)


title: Wave Packet in Curved Spacetime

text: It is known that the classical equation of motion for a scalar field wave packet on a curved spacetime background gives the geodesic trajectory (the e.o.m. is $(\nabla_\mu \nabla^\mu + m^2) \Phi=0$). However, I couldn't see that. How can one derived the geodesic equation from the above e.o.m. ?


title: Coherence with an infinfinite number of waves

text: **If I had an infinite number of sine waves with frequencies between 0 and 2, and I know what amplitude each wave has, is there a way for me to predict how they interfere?** **for example if I have:** frequencies=[ 0 ......................... 2] amplitudes=sin(frequencies*(pi/1))+sin(frequencies*(pi/2)) wave=sum(amplitudes*sin((pi*2)*frequencies)) **what would the wave's phase and amplitude be at any given point?** **thanks.**


title: Gravitational potential energy

text: Consider two places next to each other: Place 1, where there is a gravitational field whereas Place 2 - there's no field. Now if we lifted a box in place 1, it gains potential energy. Then, we move this box horizontally to place 2. What happens to this energy?


title: Acceleration or Deacceleration. Can either be identified?

text: An accelerated clock will measure a smaller elapsed time between two events than that measured by a non-accelerated (inertial) clock between the same two events. . ![Valid XHTML](http://www.outersecrets.com/real/image/rsri.gif) **Experiment #1** Imagine that you have a space station. Docked at the space station you have two rockets. Each rocket is pointing away from the space station. One is on the left side(1) and one is on the right(2). At 12:00 noon both rockets depart and quickly accelerated in opposite directions to 260,000 kilometers per second. Once reaching this speed they both send a radio pulse signal back to the space station and do the same an hour later. At this speed their clocks are ticking at half the speed of which the clocks are ticking back at the space station. Thus at the space station the time measured between the two radio pulse signals sent from the rocket off to the left would be two hours rather than one. Also at the space station the time measured between the two radio pulse signals sent from the rocket off to the right would also be two hours rather than just one. Is this not true without question? Please answer this question prior to proceeding onward to the following paragraphs. **Experiment #2** However, what if during sleep hours, the space station along with the two rockets, were all accelerated off to the right to a velocity of 260,000 kilometers per second, and done so by let's say Aliens from far away. Those within the space station and in the rockets, due to being asleep, were therefore completely unaware of this "Acceleration". If the above two 12:00 noon rocket departures were repeated under these new conditions, will those at the space station obtain the same measurement results? After all, the rocket on the left will deaccelerate back to the original state, yet the rocket on the right will have been accelerated a second time. ( Please note insightful correction made by Godparticle down below. )


title: Non-commuting operators can't share any eigenvector

text: In an introductory Quantum Mechanics textbook, I found the following statement: > For two Hamiltonians $H$ and $H'$, non commuting with each other, but > commuting with the same group of translations ${\cal{T}} (\vec{R})$ an > eigenvector of $H$ can't be an eigenvector of $H'$. But I don't see how $[H,H']\neq 0$ implies that $[H,H']$ cannot vanish for a specific eigenvector $\alpha$ of $H$, making it a shared eigenvector with $H'$.


title: Does quantum reversibility require many worlds?

text: The source S sends a photon into the beam splitter below. There is a 50% chance that it will be detected at A and a 50% chance it will be detected at B.               S -----------\---------> A                  |                  |                  |                  v                  B      Now if we assume that physics is (generally) reversible we should be able to time-reverse this process. This would imply that if we send in a photon at A or B it should _always_ appear back at S. How can this happen? We know that if we send in a photon at A (or B) we would expect it to appear 50% of the time at S and 50% at C as below.                            C                                         C                  ^                                         ^                  |                                         |                  |                                         |                  |                                         |     S <----------\--------- A               S <------------\                                                            |                                                            |                                                            |                                                            B      In order that the photon always appears back at S and never at C we need a superposition of the two situations above such that the paths to S constructively interfere and the paths to C destructively interfere. Thus it seems that in order to retain time-reversibility we need to assume a many-worlds view. As a photon is sent in at A (or B) it must be assumed that another photon is simultaneously sent in at B (or A) and both states weighted with the correct amplitude such that a photon appears with certainty back at S. What do people think?


title: How light know which path is smallest?

text: We know from fermat's principle that light follows the smallest path. But how light know that which path is smallest?


title: What would happen if I take a glass of water in space?

text: What would happen if I take a glass of water in space i.e. outside the gravitational influence of earth? My teacher said that the water would vaporize but I am not completely satisfied by the answer. I feel that it will still be in the glass because of adhesion forces. Am I correct?


title: How Do Electrons "Resonate Against the Restoring Force Of Positive Nuclei?"

text: While reading "Surface Plasmon Resonance," I came across the following: > "The resonance condition is established when the frequency of light photons > matches the natural frequency of surface electrons oscillating against the > restoring force of positive nuclei." 1.) What is the "restoring force of positive nuclei"? 2.) How do electrons resonate against the restoring force of the positive nuclei? I realize that electrons and protons attract each other and that they both behave like waves, but the concept of a "restoring force" eludes me.


title: Chemical reaction as state transition?

text: When considering diffusion of chemicals, the reaction part is business of chemical kinetics, where the relevant characteristics of different substances come from collision theory together with some classical statistics. If one want to go deep down, one can try to compute the collision rates with quantum theoretical methods. For some time now I wonder if there is a reason that it might not be possible to just compute the whole chemical reaction process using a path integral. Can't one encode chemical substances (atoms to molecules) in a Hilbert space'ish manner and come up with some sort of Lagrangian, mirroring the change of species concentration from one equilibirum (with seperated chemicals) to another? $$``\ |\text{CH}_4,2 \text O_2 \rangle\ \ \overset{\text {burn}}\longrightarrow \ \ |\text{CO}_2, 2 \text H_2\text O\rangle\ "$$ Instead of iterating non-linear differential equations, which arise from classical statistical consideration, and which involve empircal or quantum chemically computed rate constants, can't one transfer all of this to computing Feynman diagrams? (Not implying that that would be easier.)


title: Proof of $Dq-qD=1$ where $D=\frac{\partial }{\partial q}$ is the differential operator

text: Can anyone provie me the proof of $Dq-qD=1$ where $D=\frac{\partial }{\partial q}$ refers to the differential operator? Or if it's something special to quantum mechanics, why is it? Is this following from $[\hat{q},\hat{p}] =i\hbar ~{\bf 1}$?


title: Pressure in waves on a string

text: We know that when we speak sound waves are created. The air particles compress and rarefy and pressure is more at the nodes and less at anti-nodes. But can we say the same thing about waves on a string,- that pressure is more at the nodes than the anti-nodes?


title: Finding V in a electric field?

text: How will we find $V_{ab}$ when $\vec E = (2i+3j+4k)$ **N/C** , $\vec R_a = (i-2j+k)$ **m** and $\vec R_b = (2i+j-2k)$ **m** ? I know $\vec E= -\frac{d V}{d\vec r}$, but I don't know what should be my initial approach toward the question.


title: What do you really see on a line of clocks as you pass by them at high speed?

text: According to my understanding of SR, if I travel at 0.8c relative to a line of clocks, I should see the clocks in front of me going 3 times faster than my own, and those behind me going 3 times slower than my own (Doppler effect). OK, so what happens at my exact location? I reckon that as I look nearer and nearer to my origin, I would see a discontinuity between the forwards and backwards directions. That's bad enough, but at my origin there is no light travel delay, so my local time is my proper time, and the speed of the clocks should then be in accordance with gamma (4/3) at 0.8c. So, my question is: how do I reconcile the 3 contradictory speeds that I should observe for a clock at my origin? I think it should be the value given by gamma, but I can't explain the discontinuity resulting from the Doppler effect both forward and back in close proximity to the origin.


title: Question on luminosity of a radiant source and its dependence on temperature, and involvement of Doppler Effect

text: A few days ago, I happened to go through the chapters on Radiation, and Photometry, studying them at quite an elementary level. I studied Wien's displacement law, and the dependence of luminous flux of a radiant source on the total radiant flux of the source, and the wavelength distribution. From studying them, I think that on changing the temperature of a radiant source, the relative-luminosities of the different component wavelengths of the radiation must change- for the following reason: on changing the temperature of the source, energy-density of the radiation redistributes itself over the entire range of the component wavelengths. Now, I ask another _(I cannot decide whether it is related with the previous one, or not)_ question. I assume a space of homogeneous R.I., and let, it has a remarkably high coefficient of viscosity. Let, a meteor passes through this space at a very, very high speed, away from an observer such that the viscous- drag is able to stop the meteor at a very very large distance from the observer, at rest, at origin (considering only two dimensional Cartesian coordinate system). Due to Doppler effect, the color-shift of the radiation (emitted due to action of viscous-drag on meteor-consequent heating-consequent temperature rise) should be towards red, but, due to Wien's displacement law, the dominance of the wavelength involved in the radiation (gradual increase in relative luminosity, in direction of violet-region---this is where I guess there might be the relation I spoke of earlier), gradually builds up in the direction of the violet-region. **My Question** Is it possible that during the course of the journey, the color of radiation emitted by the meteor, as noticed by the observer, is white, due to wavelength compensation by Wien's displacement law, and Doppler effect? Please, answer this question in a way as you seem most suitable, for a high- school student, with not much knowledge on Quantum-Mechanics (I guess, as I don't know what else I should know).


title: A question about electric field

text: I would like to understand why is it the charge density while dealing with currents is $\mathop{\mathrm{div}}(E)/4\pi$, while when dealing with insulators is $-\mathop{\mathrm{div}}(E)/4\pi$? Thank you very much, and sorry if it's too obvious.


title: Energon: is it possible?

text: I'm always quite curious about the "Energy cube" in Transformers, or namely Energon. Is it really possible to store energy, such as electricity, into such a compact form? safe to distribution, and seems nothing left after being consumed? ps. Wikipedia has a page for Spark, which is more for transformer's soul. I'm not asking for that yet.


title: Derivation of Higher-order correlation functions from definition

text: I'm trying to understand the definition of the n-th order correlation function. My aim is to translate the math into a numerical implementation in order to compute the correlation function $g^{(n)}$ for a distribution of spherical particles with $n=\lbrace3,4,5,\ldots\rbrace$. I'm going to present you what I have understood for now, in order for you to get an better insight of what I'm looking for. (To reduce the verbiage of the following, I over-abbreviated sometimes $\mathbf r_1, \mathbf r_2$ into $r$, sorry if it kills the readability) I'm used to the radial distribution (or pair distribution function) that we call $g(r)$ (which should be called rigorously $g^{(2)}(r)$). I want to derive the expression of $g^{(2)}(r)$ from the definition of $g^{(n)}$ (because I hope that if I'm able to do it for $n=2$, I will be able to do it for any $n$ !) So let's start with the definition of $g^{(n)}$ one can find in wikipedia or any stat mech book :   $$\rho^{(n)}(\mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{n}) = \rho^{n}g^{(n)}(\mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{n}) \, ,$$ with $\rho$ the particle number density and $\rho^{(n)}$ the probability of (all the permutations of) elementary configuration $(\mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{n})$ : $$ \rho^{(n)}(\mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{n}) = \frac{N!}{(N-n)!} \frac{1}{Z_N} \int \cdots \int \mathrm{e}^{-\beta U_N} \, \mathrm{d} \mathbf{r}_{n+1} \cdots \mathrm{d} \mathbf{r}_N \, .$$ with $U_N(\mathbf{r}_{1}\, \ldots, \, \mathbf{r}_{N})$ the potential energy of the configuration and $Z_N$ the configurational integral, taken over all possible combinations of particle positions. Rewriting the previous with $n=2$ leads to : $$ \rho^{(2)}(\mathbf{r}_{1}\, \mathbf{r}_{2}) = N(N-1) \frac{1}{Z_N} \int \cdots \int \mathrm{e}^{-\beta U_N} \, \mathrm{d} \mathbf{r}_{3} \cdots \mathrm{d} \mathbf{r}_N \, .$$ From that starting point, I should be able to derive the expression for $g^{(2)}(\mathbf{r}_{1}\, \mathbf{r}_{2})$ but I have no clue about the origin of the Dirac Delta function that appears in the definition of $g^{(2)}(\mathbf{r}_{1}\, \mathbf{r}_{2})$ or $\rho^{(2)}(\mathbf{r}_{1}\, \mathbf{r}_{2})$ $$ \rho^{(2)}(\mathbf{r}_{1}\, \mathbf{r}_{2}) = \left\langle \sum_i\sum_j' \delta(\mathbf r_1 - \mathbf r_i) \delta(\mathbf r_2 - \mathbf r_j) \right\rangle $$ I think there is something appearing from the potential energy that was defined before, but I'm not able to understand exactly the origin. Any help will be highly appreciated :) Thank you in advance ! ( _Note_ : related questions like "Use and understanding of higher-order correlation functions" is absolutely not helpful, and the reference on the wikipedia page leads to a paper that has nothing to deal with correlations functions IM(H)O...)


title: What really are exotic supersymmetric black holes?

text: I have just read (in the black holes chapter 14 on p244 of this book Ref.1) that in string theory, when one adds an (electric?) charge $Q$ to a static black hole, one can arrive at an exotic supersymmetric black hole. This sentence is not explained further and there are several (I think related enough) things I dont understand about it, which can be summerized under the question what really is an exotic suppersymmetric black hole? First, how exactly does the addition of a charge (if it is not outright a supercharge) lead to supersymmetry? Second, what is meant by an exotic black hole, conversely to for example an extremal black hole that has just the maximum charge allowed given its mass? Third, what does it mean for a black hole to be supersymmetric anyway? References:   1. D. McMohan, _String Theory Demystified_ , McGraw-Hill, 2009


title: Stochastic process corresponding to Schrödinger evolution

text: In probability theory, the Fokker-Planck equations governs the trajectory of a sample path of a stochastic process -- say the heat equation in the case of a Wiener process. Consider a Gaussian wave-function evolved with respect to the free-Hamiltonian -- for the sake of simplicity it may be a one dimensional solution. It is a classical result that in both forward and backwards time the wave disperses, and the modulus squared, i.e. the probability density, is a Gaussian with increasing variance. **For this PDE, does there exist a corresponding stochastic process? If so, is there a physical interpretation of it?** After all quantum probability is said to be different from classical probability, so perhaps this construction is too naive.


title: Would infinite time elapse relative to an outside observer if an object was completely at rest?

text: Here's my reasoning... time dilation due to velocity: t'=t√(1-v^2) v expressed as a % of the speed of light. If you are moving through distance at the speed of light, to an observer at rest relative to you, zero time has elapsed. So if you are at complete rest (no particle movement at all) is infinite time elapsing to you relative to an observer? Would infinite time, mean that you would witness the end of the universe?


title: What caused a fluctuating cloud to form in a jet engine intake on a humid day?

text: I was on a commercial flight in a 737 stationary at idle on a taxiway. It had recently stopped raining so the relative humidity was likely near 100% and the air temperature was about "light jacket" level. I was in a window seat forward of the engine intake so I could see the whole intake. Every 30 seconds or so a cloud would burst into existence across the entire jet intake and be sucked away in perhaps one quarter of a second. Was it really a tiny cloud that I was seeing and what bit of thermodynamics was causing it? Why was it periodic? (Yes, I know "Boeing 737" under-specifies the engine type but it should give an approximate size and configuration. Intuitively I expect that the engine design was not crucial for the effect, but I've not had any university-level physics, so my intuition is worth what I'd paid for it.)


title: Are gravitational waves longitudinal or transverse?

text: Waves are generally classified as either transverse or longitudinal depending on the they way the propagated quantity is oriented with respect to the direction of propagation. Then what is a gravitational wave? It doesn't make sense to me that a disturbance in the curvature of spacetime has a "direction", so I would say they're neither, much like a wave packet in quantum mechanics.


title: Can symmetry generators be used for quantization?

text: Take the Poincaré group for example. The conservation of rest-mass $m_0$ is generated by the invariance with respect to $p^2 = -\partial_\mu\partial^\mu$. Now if one simply claims > The state where the expectation value of a symmetry generator equals the > conserved quantity must be stationary one obtains $$\begin{array}{rl} 0 &\stackrel!=\delta\langle\psi|p^2-m_0^2|\psi\rangle \\\ \Rightarrow 0 &\stackrel!= (\square+m_0^2)\psi(x),\end{array}$$ that is, the Klein-Gordon equation. Now I wonder, is this generally a possible quantization? Does this e.g. yield the Dirac-equation for $s=\frac12$ when applied to the Pauli-Lubanski pseudo-vector $W_{\mu}=\frac{1}{2}\epsilon_{\mu \nu \rho \sigma} M^{\nu \rho} P^{\sigma}$ squared (which has the expectation value $-m_0^2 s(s+1)$)?


title: What Is The Difference Between The Maths That Physicists Use And The Maths On A Typical Mathematics Degree

text: Cross-posted to Math.SE here. Physicists are widely respected for using and sometimes even inventing mathematics yet physicists study Physics which is a subject in its own right. So surely someone studying physics spends less time studying maths than the mathematics student? If this is the case how does the Physicist achieve the required high level of mathematical maturity? Is it a case of just knowing what mathematical methods to use and what results to look for or does the physicist study things in pure mathematics like analysis, etc? I ask this because I am a computer science student who wants to learn mathematics in my spare time, but I think it might be a better idea to learn mathematics like a physicist as a large portion of my time will be taken with my current studies. Also for clarity I only speak of study up to BSc level.


title: Projectile motion - ball hit 180m/s at 45 degrees, what is the hang time?

text: Assuming no air resistance etc, how long is the ball in the air for? This is actually part of a longer question concerning the ball trajectory on different planets but I think there must be an easier way than what I am doing. This is what I have - $$Y -Direction$$ $$v_y=u_y+a_yt$$ $$0=u_y-9.8t$$ $$u_y=9.8t$$ $$u_y=sin45\cdot{}v_0$$ Where $v_o$ is the inital velocity of the ball $$X-Direction$$ $$x=x_0+u_xt+\frac{1}{2}a_xt^2$$ $$180=0+u_xt+\frac{1}{2}0_xt^2$$ $$u_x=\frac{180}{t}$$ $$u_x=cos45\cdot{}v_0$$ From this I substitute to find t. $$9.8t=sin45\cdot{}v_0$$ $$v_0=\frac{9.8t}{sin45}$$ $$\frac{180}{t}=cos45\cdot{}v_0$$ $$V_0=\frac{180}{cos45t}$$ $$\therefore\frac{9.8t}{sin45}=\frac{180}{cos45t}$$ $$180=\frac{9.8t\cdot{}cos45t}{sin45}$$ The $sin45$ and $cos45$ cancel out $$180=9.8t^2$$ $$t^2=\frac{180}{9.8}$$ $$t=\sqrt{\frac{180}{9.8}}$$ $$t\approx4.29s$$ However I can't help but think this is too long winded and I am going about it the wrong way, especially considering it is exam prep where it will only be worth 1 mark out of 40.


title: Change in the width of depletion layer with doping

text: As previously learnt that increasing the doping will decrease the width of the depletion layer and vice-versa. However, I am unable to understand this. Does it have some relation with force of repulsion?


title: Quantum Mechanics proved incorrect. Bohr - Einstein debates concluded?

text: "Photons act like they go through two paths, even when we know which they took". Please refer the above link and its conclusion. I am an Engineer. What I infer from this is :-   1. This proves ERP.   2. Einstein Wins.   3. This basically proves that quantum mechanics is incomplete/incorrect.   4. There is a requirement for an extension for QM. What this does is :   1. "Declaration of completeness of quantum mechanics" by Heisenberg needs to be pulled down officially   2. Other theories needs to be thought about, like Bohm's. > Can somebody confirm my understanding ? Any help is appreciated.


title: Can one show that ${\gamma^5}^\dagger = \gamma^5$ directly from the anticommutation relations?

text: Is it possible to show that ${\gamma^5}^\dagger = \gamma^5$, where $$ \gamma^5 := i\gamma^0 \gamma^1 \gamma^2 \gamma^3,$$ using only the anticommutation relations between the $\gamma$ matrices, $$ \left\\{\gamma^\mu,\,\gamma^\nu\right\\}=2\eta^{\mu\nu}\,\mathbb{1},$$ and _without_ using any specific representation of this algebra and a unitary invariance argument, as is usually done?


title: Difference between torque and moment

text: What is the difference between torque and moment? I would like to see mathematical definitions for both quantities. I also do not prefer definitions like "It is the tendancy..../It is a measure of ...." To make my question clearer: Let $D\subseteq\mathbb{R}^3$ be the volume occupied by a certain rigid body. If there are forces $F_1,F_2,....,F_n$ acting at position vectors $r_1,r_2,...,r_n$. Can you use these to define torque and moment ?


title: Photons interact with themselves

text: We know that photons are the antiparticles of themselves and if they interact with each other through higher order process do they annihilate and again produce photons? Here is the Phys.SE question that made me ask this question.


title: Does humidity make winter air feel colder?

text: I sometimes hear stories where people compare their feelings in winter in different places in the world. It goes like > in city X the temperature was the same as in city Y, but the humidity made > me feel much colder... or > oh well, -20°C would be cold, but the humidity was low, so it felt OK so it implies that humidity somehow makes it feel colder. I am talking about temperatures below freezing (-30...0°C). Does this have any physical explanation, or is it some sort of psychological phenomenon?


title: Modelling noise with distance

text: I was wondering about the relation between noise with distance, assuming a point source, using sound as the method for communication and air as the medium of communication. Obviously as the distance from the sound source increases, noise should increase- but what is the nature of this relationship? Is it linear or non-linear? Any ideas? Is there any other way noise can be modeled (in the context of communication between individuals)? Thanks!


title: Symmetries of the Standard Model: exact, anomalous, spontaneously broken

text: There are a number of possible symmetries in fundamental physics, such as:   * Lorentz invariance (or actually, Poincaré invariance, which can itself be broken down into translation invariance and Lorentz invariance proper),   * conformal invariance (i.e., scale invariance, invariance by homotheties),   * global and local gauge invariance, for the various gauge groups involved in the Standard Model ($SU_2 \times U_1$ and $SU_3$),   * flavor invariance for leptons and quarks, which can be chirally divided into a left-handed and a right-handed part ($(SU_3)_L \times (SU_3)_R \times (U_1)_L \times (U_1)_L$),   * discrete C, P and T symmetries. Each of these symmetries can be   * an exact symmetry,   * anomalous, i.e., classically valid but broken by renormalization at the quantum level (or equivalently, if I understand correctly(?), classically valid only perturbatively but spoiled by a nonperturbative effect like an instanton),   * spontaneously broken, i.e., valid for the theory but not for the vacuum state,   * explicitly broken. Also, the answer can depend on the sector under consideration (QCD, electroweak, or if it makes sense, simply QED), and can depend on a particular limit (e.g., quark masses tending to zero) or vacuum phase. Finally, each continuous symmetry should give rise to a conserved current (or an anomaly in the would-be-conserved current if the symmetry is anomalous). This makes a lot of combinations. So here is my question: **is there somewhere a systematic summary of the status of each of these symmetries for each sector of the standard model?** (i.e., a systematic table indicating, for every combination of symmetry and subtheory, whether the symmetry holds exactly, is spoiled by anomaly or is spontaneously broken, with a short discussion). The answer to each particular question can be tracked down in the literature, but I think having a common document summarizing everything in a systematic way would be _tremendously_ useful.


title: Pendulum Confusion

text: This text in my book is pretty confusing:With my emphasis > A simple pendulum is a _heavy point mass_ (bob) suspended from a rigid > support by a _massless and inextensible string_. This is an ideal case > because **we cannot have** a _heavy mass having the size of a point_ and a > _string which has no mass_. At first they say that the string is massless, and bob a heavy point mass, and call it an ideal case. And then, they say that they cannot have a heavy point mass, with a massless string in an ideal case. I suppose this is a mistake. Or maybe is it something else ? Please clarfiy my doubts. Thank you!


title: Looking for the actual reason of refraction explained precisely without analogies

text: I'm a high school teacher trying to teach my students (15year olds) about refraction. I've seen a lot of good analogies to explain why the light changes direction, like the marching band analogy, that the light "choose" the fastest way etcetc, and for most of my students these are satisfying ways to explain the phenomenon. Some students, however, are able to understand a more precise and physically correct answer, but I can't seem to find a good explanation of why the lightwaves actually changes direction. So what I'm looking for is an actual explanation, without analogies, of how an increase/decrease in the speed of a lightwave cause it to change direction. Thanks a lot


If you put a latex balloon in a vacuum, how much would it expand? And would it pop? Assume it doesn't leak. EDIT: Some numbers: Ambient pressure is 100 KPa, balloon is perfectly spherical with a diameter of 300 mm, deflated it has a diameter of 25mm, temperature is always at equilibrium. Question components stated more formally:   * What pressure does the balloon need to have been inflated to to reach the 300 mm radius?   * What is the relationship between the (gauge) pressure and volume? _I think this simply comes down to PV=nRT_   * If there is some sort of spring constant involved, what is this value for a typical latex balloon?   * What is the relationship between the pressure (or volume) and the tension on the material the balloon is made from?      * What is the limit the tension can reach before popping? Semi-related: how much would a balloon expand if you sealed it deflated and put it in a vacuum?


title: How would we tell antimatter galaxies apart?

text: Given that antimatter galaxies are theoretically possible, how would they be distinguishable from regular matter galaxies? That is, antimatter is equal in atomic weight and all properties, except for the opposite reverse charge of the particles, identical to regular matter. Hence a star composed of antimatter hydrogen would fuse to anti-helium in an analogous way to our own Sun, and it would emit light and radiation at the same wavelengths as any regular matter star and would cause the same gravitational forces for planetary systems to form as in any other star system. Hence, what would be a telltale sign if you were observing a galaxy made up entirely of antimatter? Also, is there any evidence for that half of all galaxies are not made of antimatter -- while general theories currently assume that there is an imbalance of matter over antimatter in the universe, then what is the rationale for not assuming that there is in fact an even balance between the two?


title: Experimental observation of matter/antimatter in the universe

text: Ordinary matter and antimatter have the same physical properties when it comes to, for example, spectroscopy. Hydrogen and antihydrogen atoms produce the same spectroscopy when excited, and adsorb the same frequencies. The charge does not make a difference in the potential (regardless if it's generated by a proton or an antiproton) nor in how the positron behaves in this potential (being its mass equal to the mass of an electron) How can astronomy evaluate if a far galaxy is made of matter or antimatter, given that from the spectroscopy point of view, they behave in the same way? In other words, how do we know that an asymmetry exists between matter and antimatter in the universe?


title: Are there entire anti-matter galaxies?

text: Would we be able to tell if an entire neighboring galaxy was made up of anti- matter? What would we see in its signals? It seems like the anti-matter could be stable if it was completely surrounded by other anti-matter.


title: Is it possible that portions of the universe are made of antimatter?

text: > **Possible Duplicate:**   >  Experimental observation of matter/antimatter in the universe I've heard a bit about the antimatter, matter inbalance. But I don't understand how it has been decided highly unlikely/impossible that some areas of the universe are made of matter, and others from antimatter. not necessarily specific galaxies, but maybe whole clusters/super-clusters? Also I've also read that if this were the case then when opposite galaxies collide, it would create huge amounts of energy, which we would see, and as we don't see everything, all galaxies must be made of matter. However when normal galaxies collide, the stars themselves do not collide with each other, so why when matter stars collide with antimatter stars would it be any different? Am I missing something like, antimatter is repelled by gravity or something?


Many years ago, a discrepancy was found between the experimentally measured value of the muon magnetic moment, and the theoretically calculated value. Shockingly, most physicists were blase about it. It was no big deal to them. They dismissed it as either an experimental error, or some mistake in the QCD calculations, even though error bars have been painstakingly computed for both of them, and the discrepency still survived up to a few sigmas. What is the current status of the anomalous muon magnetic moment?


title: How is an inflatable parabolic antenna created?

text: I'm intrigued by this, and how it would work:- 3 sub-questions if I may:   1. Construction: As I understand it's a flexible sphere constrained by a rigid edge. a. Do we simply glue 2 flat circular pieces of flexible material together at the edges and inflate? (e.g. the inflatable dish on PE1RAH) Or, b. do you have to cut out sections to make it parabolic, and the inflation simply gives it the rigidity? (e.g. page 4 of this document)   2. Spherical Antenna, as manufactured by e.g. GATR. Would the design for this inflated dome with internal parabola be in effect 2 inflatable structures - 1 sphere (to give rigid circumference, and an internal parabola as in question 1, or could you do something like this with only 1 inflatable structure?   3. Design. I'm assuming the parabola generated using question 1.a would be a factor of gas pressure + stretch of fabric given there is no slack in the material? Or are other factors involved? Any suggestions for design or theory papers would be most appreciated. I'm going to have a try at building one of these for portable wifi :) Thanks EDIT: I just threw together a prototype from 2 plastic bags/drinking straw/tape - it's a bit rough to see a definitive answer, but looks parabolic. EDIT 2: on further digging, I confirmed that the inner parabola in a spherical antenna holds its shape by continually topping up the pressure in the 'top' of the sphere to keep the parabolic shape bowed the right way.


title: Casimir forces due to scalar field using Path integrals

text: I have just started learning QFT. I have just completed scalar fields, which I learnt in using Canonical Quantisation and Path integrals. I did calculation of Casimir force between two metal plates using just free scalar field theory (using the vacuum energy). However, I am not able to find a way to do this thing using Path integrals and propagators. The partition function for the case of free scalar field (i.e KG field) turns out to be, $$ Z[J] = \text{exp}\bigg(i\int \mathrm d^4x \;\mathrm d^4x'J(x')\Delta_F(x-x') J(x) \bigg) \qquad \qquad (1) $$ which after setting the $Z[J=0] =1$. I wish to know, how to approach my problem from here. PS : I have not learnt vector or spinor fields yet. Most of the references or notes that I checked either assumed a prior knowledge of that or did not say how to quantise scalar fields. EDIT : This is the integral to begin with right $$ Z[J] = \frac{1}{Z_0} \int [d \phi] \text{exp}\bigg(-i\int d^4x \bigg[ \frac{1}{2}\phi (\Box + m^2 - i\epsilon)\phi - \phi J\bigg]\bigg) $$ All I did was to introduce $\phi \rightarrow \phi + \phi_0 $ and demand that $$ (\Box + m^2 - i\epsilon)\phi_0 = J(x) $$ and $\Delta_F(x-x') $ is the Green's function involved in solving this equation. Then I obtain the equation (1).


title: Velocity and curved spacetime

text: It is possible that the universe has naturally a slight curve of spacetime, even in a place where there is not a massive object, and that this curve is responsible for the perception of the relative velocity between two bodies? For example: The body A and body B, has as its starting point the C point and arrival point D. The speed of A relative to C is 200 m / s and B to C, 100 m / s. It may be that the body A is in a less curved position of spacetime than that B and therefore the time interval and the spatial displacement experienced by A to arrive at D is less than that experienced by the B also to get D? I think of the possibility that at least 4-dimensional (3 space and 1 time) are curved, so that the universe is somewhat like a multidimensional bubble. From this point of view, then the body A would be traveling for the inner path (inside the bubble, so to speak) than the body B. The path from A to reach D, is shorter in time and space in the direction of its movement, than the path from B to get to D. That would not explain the space contraction and time dilation? I believe that only one spatial dimension is involved in the event, that is, the dimension in which the body moves. Therefore in the remaining two spatial dimensions, bodies with different velocities could be extremely near.


title: Sold-State Band Structure - connection between Fermi Energy, Fermi Level and Work Fuction

text: I've been struggling with the concepts of these three terms - Fermi Energy, Fermi Level and work fuction. Now, I was given these definitions (in the context of semi-conductors):   * Fermi Level is the energy where at $0\rm K$, all energy levels below it are occupied and all energy levels above it are empty.   * Fermi Energy depends on the temperature; it consists of the Fermi Level + thermal energy.   * Workfunction - the energy required to release an electron from the outer most shell. I'm not sure how to grasp the relationship between these definitions. In one of the classes, our TA gave us this "formula"; $WF=\left|FL+FE\right|$ Meaning, the workfunction of a metal equals the absolute value of the sum of its Fermi Level and Fermi Energy. I tried to visualize this through the Band Model, and this is the result: ![Band theory visualization... maybe](http://i.imgur.com/NtKZ0lO.png) Is this visualization correct? Are these definitions (again, in the context of semiconductors) correct? If not, what are the correct definitions and is there a connection between the three of them?


title: Definition of scattered particle?

text: Compare the number of scattered particles:   $N_s=Fa\int\sigma(\theta)d\Omega$   With the total number of incident particles:   $N_{in}=Fa$ Here, F is the flux of incoming beam, a the area. sigma the crossection and omega the solid angle. Why isnt $N_s=N_{in}$? How does one define which particles are scattered and which are not, arent they all interacting with the target to some degree? Isnt particles conserved normally? Do almost all the particles either pass right through almost undetected or are scattered signficiantly, so what we are really integrating over is a sphere surrounding the target except a spot of area $a$ where the beam exits?


title: Do burning red coals and red-hot iron have the same temperature?

text: It would seem that Planck's law implies that objects of similar radiation spectra have the same temperature if the objects are "similarly close" to being black bodies. Am I right to infer that burning red coals and red-hot iron have approximately the same temperature (which?) because they are both emitting mostly red light?


title: Why is the field inside a hollow sphere zero?

text: Say you have a hollow sphere with a uniformly distributed charge on the surface. Why is the electric field everywhere inside the sphere zero? For the centre, its easy to add the vectors from the surface charges and show they sum to zero because of symmetry. But can you show me how the field cancels out to zero for points other than the centre **using vector addition?** Thanks!


title: Is entropy in quantum mechanics emergent or fundamental?

text: Since a quantum mechanical system, even an isolated system containing one particle, can be described by a density matrix, with entropy for the system given by $\langle S\rangle=-k \rho\ln(\rho)$, is not entropy therefore a property of the system like mass or energy?


title: A theory about a travelling wave due to a vibrating rope seems to be contradictory, please spot what is wrong?

text: Consider a travelling wave produced by vibrating one end of a rope while the other end is made to freely move along a vertical line. Mathematically, the equation of the traveling wave that also represents the equation of motion of each point on the rope is given as follows. $$y(x,t)=A\sin(kx-\omega t)$$ In the book I read, a certain particle on the rope moves up and down perpendicular to the propagation vector as wave does not transfer material but energy. I don't agree with the statement that "a certain particle on the rope moves up and down perpendicular to the propagation vector". It is because I think to maintain the same point on the rope to move up and down, the arc length of $y(x,t)$ must be equal to that of $y(x,t+\Delta t)$ for $0\leq x \leq a$ and any $\Delta t$. However, according to the Notes below, their arc length are not generally equal. ![enter image description here](http://i.stack.imgur.com/8B30K.png) From the figure above, how can $P$ and $Q$ be the same point of the rope while the red and blue curves have different length? So my question are: **Does each point of the rope move up and down?** ## Notes From $x=0$ to $x=a$, generally the arc length of $y=\sin x$ is not equal to that of $y=\sin(x-x_0)$. Let $S(x_0)$ be the arc length of $y=\sin(x-x_0)$ for the given interval. \begin{align} S(x_0) &= \int_0^a\sqrt{1+\left(\frac{dy}{dx}\right)^2}\, \textrm{d}x\\\ &= \int_0^a\sqrt{1+\cos^2(x-x_0)}\, \textrm{d}x\\\ \end{align} If $S(0)=S(x_0)=\text{const}$ for any $x_0$ then $\frac{\textrm{d}S(x_0)}{\textrm{d}x_0}=0$. Thus I have to check whether or not $\frac{\textrm{d}S(x_0)}{\textrm{d}x_0}=0$. \begin{align} \frac{\textrm{d}S(x_0)}{\textrm{d}x_0} &= \frac{\textrm{d}}{\textrm{d}x_0}\int_0^a\sqrt{1+\cos^2(x-x_0)}\, \textrm{d}x\\\ &= \int_0^a\frac{\textrm{d}\left(\sqrt{1+\cos^2(x-x_0)}\right)}{\textrm{d}x_0}\, \textrm{d}x\\\ &= \int_0^a\frac{\cos(x-x_0)\sin(x-x_0)}{\sqrt{1+\cos^2(x-x_0)}}\, \textrm{d}x\\\ &= \int_{-x_0}^{a-x_0}\frac{\cos y\sin y}{\sqrt{1+\cos^2y}}\, \textrm{d}y\\\ &= \sqrt{1+\cos^2 x_0}-\sqrt{1+\cos^2(a-x_0)}\\\ &\not = 0 \end{align} It implies that generally they are not equal.


title: Decomposition of Wave function in infinite well

text: I am given that a wave function at time 0 is given as $$\psi(x,t=0)=\sin^6(\frac{\pi x}{2L})\cos(\frac{\pi x}{2L})$$ I am asked to find this wave function as a function of time. In order to do this, I feel like I need to find the combinations of eigenfunctions that make this function and then I can find how they each evolvewith time, so that $$\psi(x,t)=\sum_{n=1}^{\infty}c_n\psi_n(x)e^{-iE_nt/\hbar}$$ Am I correct in saying this? I feel that this makes sense, and then I can find $c_n$ as $$c_n=\int_0^{2L}\psi(x,t=0)\sin(\frac{n\pi x}{2L})dx$$


title: When heat is rejected by a system what will be $q$ according to sign convention?

text: When a system rejects heat it can be assumed that work is done by the system. So as per law the sign convention must be negative. But my physics textbook says the reverse. So is the question.